10th Anniversary of Axioms: Mathematical Physics

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 July 2022) | Viewed by 24205

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Special Issue Editor

1. Office for Outer Space Affairs, United Nations, Vienna International Centre, A-1400 Vienna, Austria
2. Centre for Mathematical and Statistical Sciences, Peechi Campus, KFRI, Peechi, Kerala 680653, India
Interests: special functions; fractional calculus; entropic functional; mathematical physics; applied analysis; statistical distributions; geometrical probabilities; multivariate analysis
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Special Issue Information

Dear Colleagues, 

Over the decade that has passed since the first issue, the journal Axioms has achieved a remarkable increase in the number of papers published, as well as a significant growth in citations. In order to mark this significant milestone, I am glad to lead this Special Issue entitled “10th Anniversary of Axioms: Mathematical Physics”. 

Mathematical physics applies rigorous mathematical ideas to problems inspired by physics and to investigate the mathematical structure of physical theories. Traditionally mathematical physics has been quite closely associated with ideas in calculus, particularly those of differential equations. In recent years, however, in part due to the rise of quantum field theory, quantum gravity, and cosmology, many more branches of mathematics have become major contributors to physics. The Section Mathematical Physics should cover a wide field for research in the mathematical and physical sciences and their applications, even including applications in chemistry, biology, and social sciences. Depending on the inclination of the authors of research papers for Axioms, one may prefer mathematics from the point of view of physics or, vice versa, physics from the point of view of mathematics. 

In 2021, we will celebrate the tenth anniversary of the journal Axioms, and we would be happy if you join us.

Prof. Dr. Hans Haubold
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Published Papers (15 papers)

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Editorial

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5 pages, 220 KiB  
Editorial
Henri Poincaré’s Comment on Calculus and Albert Einstein’s Comment on Entropy: Mathematical Physics on the Tenth Anniversary of Axioms
by Hans J. Haubold
Axioms 2023, 12(1), 83; https://doi.org/10.3390/axioms12010083 - 12 Jan 2023
Viewed by 1041
Abstract
This Special Issue of the journal Axioms collates submissions in which the authors report their perceptions and results in the field of mathematical physics and/or physical mathematics without any preconditions of the specific research topic [...] Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)

Research

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13 pages, 649 KiB  
Article
Existence and Uniqueness of Nonmonotone Solutions in Porous Media Flow
by Rouven Steinle, Tillmann Kleiner, Pradeep Kumar and Rudolf Hilfer
Axioms 2022, 11(7), 327; https://doi.org/10.3390/axioms11070327 - 05 Jul 2022
Cited by 1 | Viewed by 1171
Abstract
Existence and uniqueness of solutions for a simplified model of immiscible two-phase flow in porous media are obtained in this paper. The mathematical model is a simplified physical model with hysteresis in the flux functions. The resulting semilinear hyperbolic-parabolic equation is expected from [...] Read more.
Existence and uniqueness of solutions for a simplified model of immiscible two-phase flow in porous media are obtained in this paper. The mathematical model is a simplified physical model with hysteresis in the flux functions. The resulting semilinear hyperbolic-parabolic equation is expected from numerical work to admit non-monotone imbibition-drainage fronts. We prove the local existence of imbibition-drainage fronts. The uniqueness, global existence, maximal regularity and boundedness of the solutions are also discussed. Methodically, the results are established by means of semigroup theory and fractional interpolation spaces. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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24 pages, 485 KiB  
Article
A Theoretical Dynamical Noninteracting Model for General Manipulation Systems Using Axiomatic Geometric Structures
by Paolo Mercorelli
Axioms 2022, 11(7), 309; https://doi.org/10.3390/axioms11070309 - 25 Jun 2022
Cited by 4 | Viewed by 1341
Abstract
This paper presents a new theoretical approach to the study of robotics manipulators dynamics. It is based on the well-known geometric approach to system dynamics, according to which some axiomatic definitions of geometric structures concerning invariant subspaces are used. In such a framework, [...] Read more.
This paper presents a new theoretical approach to the study of robotics manipulators dynamics. It is based on the well-known geometric approach to system dynamics, according to which some axiomatic definitions of geometric structures concerning invariant subspaces are used. In such a framework, certain typical problems in robotics are mathematically formalised and analysed in axiomatic form. The outcomes are sufficiently general that it is possible to discuss the structural properties of robotic manipulation. A generalized theoretical linear model is used, and a thorough analysis is made. The noninteracting nature of this model is also proven through a specific theorem. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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13 pages, 297 KiB  
Article
On Fractional Inequalities Using Generalized Proportional Hadamard Fractional Integral Operator
by Vaijanath L. Chinchane, Asha B. Nale, Satish K. Panchal, Christophe Chesneau and Amol D. Khandagale
Axioms 2022, 11(6), 266; https://doi.org/10.3390/axioms11060266 - 01 Jun 2022
Cited by 1 | Viewed by 1475
Abstract
The main objective of this paper is to use the generalized proportional Hadamard fractional integral operator to establish some new fractional integral inequalities for extended Chebyshev functionals. In addition, we investigate some fractional integral inequalities for positive continuous functions by employing a generalized [...] Read more.
The main objective of this paper is to use the generalized proportional Hadamard fractional integral operator to establish some new fractional integral inequalities for extended Chebyshev functionals. In addition, we investigate some fractional integral inequalities for positive continuous functions by employing a generalized proportional Hadamard fractional integral operator. The findings of this study are theoretical but have the potential to help solve additional practical problems in mathematical physics, statistics, and approximation theory. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
7 pages, 1232 KiB  
Article
A Symplectic Algorithm for Constrained Hamiltonian Systems
by Jingli Fu, Lijun Zhang, Shan Cao, Chun Xiang and Weijia Zao
Axioms 2022, 11(5), 217; https://doi.org/10.3390/axioms11050217 - 07 May 2022
Cited by 3 | Viewed by 1706
Abstract
In this paper, a symplectic algorithm is utilized to investigate constrained Hamiltonian systems. However, the symplectic method cannot be applied directly to the constrained Hamiltonian equations due to the non-canonicity. We firstly discuss the canonicalization method of the constrained Hamiltonian systems. The symplectic [...] Read more.
In this paper, a symplectic algorithm is utilized to investigate constrained Hamiltonian systems. However, the symplectic method cannot be applied directly to the constrained Hamiltonian equations due to the non-canonicity. We firstly discuss the canonicalization method of the constrained Hamiltonian systems. The symplectic method is used to constrain Hamiltonian systems on the basis of the canonicalization, and then the numerical simulation of the system is carried out. An example is presented to illustrate the application of the results. By using the symplectic method of constrained Hamiltonian systems, one can solve the singular dynamic problems of nonconservative constrained mechanical systems, nonholonomic constrained mechanical systems as well as physical problems in quantum dynamics, and also available in many electromechanical coupled systems. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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17 pages, 796 KiB  
Article
Waves in a Hyperbolic Predator–Prey System
by Andrey Morgulis
Axioms 2022, 11(5), 187; https://doi.org/10.3390/axioms11050187 - 20 Apr 2022
Cited by 3 | Viewed by 1545
Abstract
We address a hyperbolic predator–prey model, which we formulate with the use of the Cattaneo model for chemosensitive movement. We put a special focus on the case when the Cattaneo equation for the flux of species takes the form of conservation law—that is, [...] Read more.
We address a hyperbolic predator–prey model, which we formulate with the use of the Cattaneo model for chemosensitive movement. We put a special focus on the case when the Cattaneo equation for the flux of species takes the form of conservation law—that is, we assume a special relation between the diffusivity and sensitivity coefficients. Regarding this relation, there are pieces arguing for its relevance to some real-life populations, e.g., the copepods (Harpacticoida), in the biological literature (see the reference list). Thanks to the mentioned conservatism, we get exact solutions describing the travelling shock waves in some limited cases. Next, we employ the numerical analysis for continuing these waves to a wider parametric domain. As a result, we discover smooth solitary waves, which turn out to be quite sustainable with small and moderate initial perturbations. Nevertheless, the perturbations cause shedding of the predators from the main core of the wave, which can be treated as a settling mechanism. Besides, the localized perturbations make waves, colliding with the main core and demonstrating peculiar quasi-soliton phenomena sometimes resembling the leapfrog playing. An interesting side result is the onset of the migration waves due to the explosion of overpopulated cores. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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11 pages, 305 KiB  
Article
Positive Solutions for a System of Fractional Boundary Value Problems with r-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters
by Alexandru Tudorache and Rodica Luca
Axioms 2022, 11(4), 164; https://doi.org/10.3390/axioms11040164 - 06 Apr 2022
Cited by 4 | Viewed by 1599
Abstract
In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change [...] Read more.
In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator A, the solutions of the integral system are the fixed points of A. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator A has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
17 pages, 4381 KiB  
Article
Control of String Vibrations by Displacement of One End with the Other End Fixed, Given the Deflection Form at an Intermediate Moment of Time
by Vanya Barseghyan and Svetlana Solodusha
Axioms 2022, 11(4), 157; https://doi.org/10.3390/axioms11040157 - 28 Mar 2022
Cited by 3 | Viewed by 1548
Abstract
We consider a boundary control problem for the equation of string vibration with given initial and final conditions, given the deflection form at an intermediate moment of time. The control is carried out by displacement of one end with the other end fixed. [...] Read more.
We consider a boundary control problem for the equation of string vibration with given initial and final conditions, given the deflection form at an intermediate moment of time. The control is carried out by displacement of one end with the other end fixed. The problem is reduced to the problem of a distributed action control with zero boundary conditions. We propose a constructive approach to constructing a boundary control action by the separation of variables and methods of the theory of control of finite-dimensional systems. The approach is applied to given functions. A computational experiment was carried out with the construction of the corresponding graphs and their comparative analysis. They confirm the obtained results. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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25 pages, 550 KiB  
Article
Deformed Mathematical Objects Stemming from the q-Logarithm Function
by Ernesto P. Borges and Bruno G. da Costa
Axioms 2022, 11(3), 138; https://doi.org/10.3390/axioms11030138 - 16 Mar 2022
Cited by 5 | Viewed by 2131
Abstract
Generalized numbers, arithmetic operators, and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previously described in the literature, while others are [...] Read more.
Generalized numbers, arithmetic operators, and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined. Commutativity, associativity, and distributivity, and also a pair of linear/nonlinear derivatives, are observed within each class. Two entropic functionals emerge from the formalism, and one of them is the nonadditive Tsallis entropy. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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8 pages, 273 KiB  
Article
A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators
by Vladimir E. Fedorov, Mikhail M. Turov and Bui Trong Kien
Axioms 2022, 11(3), 96; https://doi.org/10.3390/axioms11030096 - 24 Feb 2022
Cited by 6 | Viewed by 1553
Abstract
The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous [...] Read more.
The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials of an elliptic self-adjoint differential operator with respect to spatial variables as linear operators at the time-fractional derivatives. The nonlinear operator in the considered partial differential equations is assumed to be smooth with respect to phase variables. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
19 pages, 304 KiB  
Article
A New q-Hypergeometric Symbolic Calculus in the Spirit of Horn, Borngässer, Debiard and Gaveau
by Thomas Ernst
Axioms 2022, 11(2), 64; https://doi.org/10.3390/axioms11020064 - 04 Feb 2022
Cited by 2 | Viewed by 1805
Abstract
The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads to q-Euler integrals and a very similar canonical system of q-difference equations for multiple q-hypergeometric functions. q-analogues of recurrence formulas in [...] Read more.
The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads to q-Euler integrals and a very similar canonical system of q-difference equations for multiple q-hypergeometric functions. q-analogues of recurrence formulas in Horns paper and Borngässers thesis lead to a more exact way to find these Frobenius solutions. To find the right formulas, the parameters in q-shifted factorials can be changed to negative integers, which give no extra q-factors. In proving these q-formulas, in the limit q1 we obtain versions of the paper by Debiard and Gaveau for the solution of differential or q-difference equations. The paper is also a correction of some of the statements in the paper by Debiard and Gaveau, e.g., the Euler integrals and other solutions to differential equations for Appell functions, also without references to page numbers in the standard work of Appell and Kampé de Fériet. Sometimes the q-binomial theorem is used to simplify q-integral formulas. By the Horn method, we find another solution to the Appell Φ1 function partial differential equation, which was not mentioned in the thesis by Le Vavasseur 1893. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
16 pages, 1083 KiB  
Article
Bäcklund Transformations for Liouville Equations with Exponential Nonlinearity
by Tatyana V. Redkina, Robert G. Zakinyan, Arthur R. Zakinyan and Olga V. Novikova
Axioms 2021, 10(4), 337; https://doi.org/10.3390/axioms10040337 - 09 Dec 2021
Cited by 2 | Viewed by 1648
Abstract
This work aims to obtain new transformations and auto-Bäcklund transformations for generalized Liouville equations with exponential nonlinearity having a factor depending on the first derivatives. This paper discusses the construction of Bäcklund transformations for nonlinear partial second-order derivatives of the soliton type with [...] Read more.
This work aims to obtain new transformations and auto-Bäcklund transformations for generalized Liouville equations with exponential nonlinearity having a factor depending on the first derivatives. This paper discusses the construction of Bäcklund transformations for nonlinear partial second-order derivatives of the soliton type with logarithmic nonlinearity and hyperbolic linear parts. The construction of transformations is based on the method proposed by Clairin for second-order equations of the Monge–Ampere type. For the equations studied in the article, using the Bäcklund transformations, new equations are found, which make it possible to find solutions to the original nonlinear equations and reveal the internal connections between various integrable equations. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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17 pages, 369 KiB  
Article
Hot Spots in the Weak Detonation Problem and Special Relativity
by Satyanad Kichenassamy
Axioms 2021, 10(4), 311; https://doi.org/10.3390/axioms10040311 - 19 Nov 2021
Cited by 2 | Viewed by 1130
Abstract
Problem statement: The initiation of a detonation in an explosive gaseous mixture in the high activation energy regime, in three space dimensions, typically leads to the formation of a singularity at one point, the “hot spot”. It would be suitable to have a [...] Read more.
Problem statement: The initiation of a detonation in an explosive gaseous mixture in the high activation energy regime, in three space dimensions, typically leads to the formation of a singularity at one point, the “hot spot”. It would be suitable to have a description of the physical quantities in a full neighborhood of the hot spot. Results of this paper: (1) To achieve this, it is necessary to replace the blow-up time, or time when the hot spot first occurs, by the blow-up surface in four dimensions, which is the set of all hot spots for a class of observers related to one another by a Lorentz transformation. (2) A local general solution of the nonlinear system of PDE modeling fluid flow and chemistry, with a given blow-up surface, is obtained by the method of Fuchsian reduction. Advantages of this solution: (i) Earlier approximate solutions are contained in it, but the domain of validity of the present solution is larger; (ii) it provides a signature for this type of ignition mechanism; (iii) quantities that remain bounded at the hot spot may be determined, so that, in principle, this model may be tested against measurements; (iv) solutions with any number of hot spots may be constructed. The impact on numerical computation is also discussed. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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Review

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10 pages, 434 KiB  
Review
Near-Field Seismic Motion: Waves, Deformations and Seismic Moment
by Bogdan Felix Apostol
Axioms 2022, 11(8), 409; https://doi.org/10.3390/axioms11080409 - 17 Aug 2022
Cited by 1 | Viewed by 1267
Abstract
The tensorial force acting in a localized seismic focus is introduced and the corresponding seismic waves are derived, as solutions of the elastic wave equation in a homogeneous and isotropic body. The deconvolution of the solution for a structured focal region is briefly [...] Read more.
The tensorial force acting in a localized seismic focus is introduced and the corresponding seismic waves are derived, as solutions of the elastic wave equation in a homogeneous and isotropic body. The deconvolution of the solution for a structured focal region is briefly discussed. The far-field waves are identified as P and S seismic waves. These are spherical-shell waves, with a scissor-like shape, and an amplitude decreasing with the inverse of the distance. The near-field seismic waves are spherical-shell waves, decreasing with the inverse of the squared distance. The amplitudes and the polarizations of the near-field seismic waves are given. The determination of the seismic-moment tensor and the earthquake parameters from measurements of the P and S seismic waves at Earth’s’ surface is briefly discussed. Similarly, the mainshock generated by secondary waves on Earth’s surface is reviewed. The near-field static deformations of a homogeneous and isotropic half-space are discussed and a method of determining the seismic-moment tensor from epicentral near-field (quasi-) static deformations in seismogenic regions is presented. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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Other

12 pages, 769 KiB  
Perspective
Some Multifaceted Aspects of Mathematical Physics, Our Common Denominator with Elliott Lieb
by Daniel Sternheimer
Axioms 2022, 11(10), 522; https://doi.org/10.3390/axioms11100522 - 01 Oct 2022
Cited by 1 | Viewed by 1296
Abstract
Mathematical physics has many facets, of which we shall briefly give a (very partial) description, centered around those of main interest for Elliott and us (Moshe Flato and I), and around the seminal scientific and personal interactions that developed between us since the [...] Read more.
Mathematical physics has many facets, of which we shall briefly give a (very partial) description, centered around those of main interest for Elliott and us (Moshe Flato and I), and around the seminal scientific and personal interactions that developed between us since the sixties until Moshe’s untimely death in 1998. These aspects still influence my scientific activity and my life. They also had as a corollary a variety of “parascientific activities”, in particular, the foundation of IAMP (the International Association of Mathematical Physics) and of the journal LMP (Letters in Mathematical Physics), both of which were strongly impacted by Elliott, and Elliott’s long insistence that publishers do not demand “copyright transfer” as a precondition for publication but are satisfied with a “consent to publish”, which is increasingly becoming standard. This article being mainly a testimony to the huge scientific impacts of Elliott and also of Moshe, their intertwined aspects constitute the core of the present contribution. The last part deals briefly with metaphysical and metamathematical considerations related to axioms. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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