Axioms

17 pages, 5490 KiB

Open AccessEditor’s ChoiceArticle

Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

by Wolf-Dieter Richter

Axioms 2021, 10(4), 340; https://doi.org/10.3390/axioms10040340 - 10 Dec 2021

Cited by 5 | Viewed by 2462

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal [...] Read more.

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities. Full article

(This article belongs to the Special Issue Complex Analysis)

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11 pages, 288 KiB

Open AccessEditor’s ChoiceArticle

An Extension of Beta Function by Using Wiman’s Function

by Rahul Goyal, Shaher Momani, Praveen Agarwal and Michael Th. Rassias

Axioms 2021, 10(3), 187; https://doi.org/10.3390/axioms10030187 - 16 Aug 2021

Cited by 12 | Viewed by 2347

Abstract

The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this [...] Read more.

The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this extended beta function. Full article

(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)

10 pages, 285 KiB

Open AccessEditor’s ChoiceArticle

Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

by Andriy Zagorodnyuk and Anna Hihliuk

Axioms 2021, 10(3), 150; https://doi.org/10.3390/axioms10030150 - 07 Jul 2021

Cited by 7 | Viewed by 2126

Abstract

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the [...] Read more.

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained. Full article

(This article belongs to the Special Issue Analytic Functions and Nonlinear Functional Analysis)

6 pages, 243 KiB

Open AccessEditor’s ChoiceArticle

On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r

by Pavel Trojovský

Axioms 2021, 10(3), 144; https://doi.org/10.3390/axioms10030144 - 01 Jul 2021

Cited by 4 | Viewed by 2473

Abstract

For

r \geq 2

and

a \geq 1

integers, let

{(t_{n}^{(r, a)})}_{n \geq 1}

be the sequence of the

(r, a)

-generalized Fibonacci numbers which is defined by the recurrence [...] Read more.

For

r \geq 2

and

a \geq 1

integers, let

{(t_{n}^{(r, a)})}_{n \geq 1}

be the sequence of the

(r, a)

-generalized Fibonacci numbers which is defined by the recurrence

t_{n}^{(r, a)} = t_{n - 1}^{(r, a)} + \dots + t_{n - r}^{(r, a)}

for

n > r

, with initial values

t_{i}^{(r, a)} = 1

, for all

i \in [1, r - 1]

and

t_{r}^{(r, a)} = a

. In this paper, we shall prove (in particular) that, for any given

r \geq 2

, there exists a positive proportion of positive integers which can not be written as

t_{n}^{(r, a)}

for any

(n, a) \in Z_{\geq r + 2} \times Z_{\geq 1}

. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)

13 pages, 711 KiB

Open AccessEditor’s ChoiceArticle

Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal

by Faïçal Ndaïrou and Delfim F. M. Torres

Axioms 2021, 10(3), 135; https://doi.org/10.3390/axioms10030135 - 27 Jun 2021

Cited by 17 | Viewed by 2989

Abstract

We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations [...] Read more.

We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations are performed to ensure stability and convergence of the disease free equilibrium point. Full article

(This article belongs to the Special Issue Mathematics of the COVID-19)

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9 pages, 298 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs

by Praveen Agarwal, Jochen Merker and Gregor Schuldt

Axioms 2021, 10(2), 74; https://doi.org/10.3390/axioms10020074 - 24 Apr 2021

Cited by 11 | Viewed by 2217

Abstract

In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of [...] Read more.

In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant. Full article

(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)

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24 pages, 1814 KiB

Open AccessEditor’s ChoiceArticle

Optimal Control Analysis of Cholera Dynamics in the Presence of Asymptotic Transmission

by Emmanuel A. Bakare and Sarka Hoskova-Mayerova

Axioms 2021, 10(2), 60; https://doi.org/10.3390/axioms10020060 - 12 Apr 2021

Cited by 13 | Viewed by 2747

Abstract

Many mathematical models have explored the dynamics of cholera but none have been used to predict the optimal strategies of the three control interventions (the use of hygiene promotion and social mobilization; the use of treatment by drug/oral re-hydration solution; and the use [...] Read more.

Many mathematical models have explored the dynamics of cholera but none have been used to predict the optimal strategies of the three control interventions (the use of hygiene promotion and social mobilization; the use of treatment by drug/oral re-hydration solution; and the use of safe water, hygiene, and sanitation). The goal here is to develop (deterministic and stochastic) mathematical models of cholera transmission and control dynamics, with the aim of investigating the effect of the three control interventions against cholera transmission in order to find optimal control strategies. The reproduction number

R_{p}

was obtained through the next generation matrix method and sensitivity and elasticity analysis were performed. The global stability of the equilibrium was obtained using the Lyapunov functional. Optimal control theory was applied to investigate the optimal control strategies for controlling the spread of cholera using the combination of control interventions. The Pontryagin’s maximum principle was used to characterize the optimal levels of combined control interventions. The models were validated using numerical experiments and sensitivity analysis was done. Optimal control theory showed that the combinations of the control intervention influenced disease progression. The characterisation of the optimal levels of the multiple control interventions showed the means for minimizing cholera transmission, mortality, and morbidity in finite time. The numerical experiments showed that there are fluctuations and noise due to its dependence on the corresponding population size and that the optimal control strategies to effectively control cholera transmission, mortality, and morbidity was through the combinations of all three control interventions. The developed models achieved the reduction, control, and/or elimination of cholera through incorporating multiple control interventions. Full article

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

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19 pages, 558 KiB

Open AccessEditor’s ChoiceArticle

Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method

by Osman Tunç, Özkan Atan, Cemil Tunç and Jen-Chih Yao

Axioms 2021, 10(2), 58; https://doi.org/10.3390/axioms10020058 - 09 Apr 2021

Cited by 38 | Viewed by 2430

Abstract

The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining [...] Read more.

The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided. Full article

(This article belongs to the Special Issue Special Issue in Honor of the 60th Birthday of Professor Hong-Kun Xu)

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9 pages, 266 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Pseudo-Lucas Functions of Fractional Degree and Applications

by Clemente Cesarano, Pierpaolo Natalini and Paolo Emilio Ricci

Axioms 2021, 10(2), 51; https://doi.org/10.3390/axioms10020051 - 02 Apr 2021

Cited by 5 | Viewed by 1747

Abstract

In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show [...] Read more.

In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a

r \times r

matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case. Full article

(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)

25 pages, 388 KiB

Open AccessEditor’s ChoiceArticle

Fractional Newton–Raphson Method Accelerated with Aitken’s Method

by A. Torres-Hernandez, F. Brambila-Paz, U. Iturrarán-Viveros and R. Caballero-Cruz

Axioms 2021, 10(2), 47; https://doi.org/10.3390/axioms10020047 - 31 Mar 2021

Cited by 13 | Viewed by 3868

Abstract

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order

α

of the derivative [...] Read more.

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order

α

of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R. Full article

14 pages, 432 KiB

Open AccessEditor’s ChoiceArticle

A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries

by Ehab M. Almetwally, Randa Alharbi, Dalia Alnagar and Eslam Hossam Hafez

Axioms 2021, 10(1), 25; https://doi.org/10.3390/axioms10010025 - 26 Feb 2021

Cited by 51 | Viewed by 3480

Abstract

This paper aims to find a statistical model for the COVID-19 spread in the United Kingdom and Canada. We used an efficient and superior model for fitting the COVID 19 mortality rates in these countries by specifying an optimal statistical model. A new [...] Read more.

This paper aims to find a statistical model for the COVID-19 spread in the United Kingdom and Canada. We used an efficient and superior model for fitting the COVID 19 mortality rates in these countries by specifying an optimal statistical model. A new lifetime distribution with two-parameter is introduced by a combination of inverted Topp-Leone distribution and modified Kies family to produce the modified Kies inverted Topp-Leone (MKITL) distribution, which covers a lot of application that both the traditional inverted Topp-Leone and the modified Kies provide poor fitting for them. This new distribution has many valuable properties as simple linear representation, hazard rate function, and moment function. We made several methods of estimations as maximum likelihood estimation, least squares estimators, weighted least-squares estimators, maximum product spacing, Cram

\overset{´}{e}

r-von Mises estimators, and Anderson-Darling estimators methods are applied to estimate the unknown parameters of MKITL distribution. A numerical result of the Monte Carlo simulation is obtained to assess the use of estimation methods. also, we applied different data sets to the new distribution to assess its performance in modeling data. Full article

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17 pages, 2825 KiB

Open AccessEditor’s ChoiceArticle

On Λ-Fractional Viscoelastic Models

by Anastassios K. Lazopoulos and Dimitrios Karaoulanis

Axioms 2021, 10(1), 22; https://doi.org/10.3390/axioms10010022 - 20 Feb 2021

Cited by 5 | Viewed by 2116

Abstract

Λ-Fractional Derivative (Λ-FD) is a new groundbreaking Fractional Derivative (FD) introduced recently in mechanics. This derivative, along with Λ-Transform (Λ-T), provides a reliable alternative to fractional differential equations’ current solving. To put it straightforwardly, Λ-Fractional Derivative might be the only authentic non-local derivative [...] Read more.

Λ-Fractional Derivative (Λ-FD) is a new groundbreaking Fractional Derivative (FD) introduced recently in mechanics. This derivative, along with Λ-Transform (Λ-T), provides a reliable alternative to fractional differential equations’ current solving. To put it straightforwardly, Λ-Fractional Derivative might be the only authentic non-local derivative that exists. In the present article, Λ-Fractional Derivative is used to describe the phenomenon of viscoelasticity, while the whole methodology is demonstrated meticulously. The fractional viscoelastic Zener model is studied, for relaxation as well as for creep. Interesting results are extracted and compared to other methodologies showing the value of the pre-mentioned method. Full article

(This article belongs to the Special Issue Axioms on Advanced Differential Equations for Mathematical Modeling)

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8 pages, 228 KiB

Open AccessEditor’s ChoiceArticle

Global Optimization and Common Best Proximity Points for Some Multivalued Contractive Pairs of Mappings

by Pradip Debnath and Hari Mohan Srivastava

Axioms 2020, 9(3), 102; https://doi.org/10.3390/axioms9030102 - 07 Sep 2020

Cited by 20 | Viewed by 2386

Abstract

In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. [...] Read more.

In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. Next, we put forward the concept of multivalued Kannan-type contractive pair and also the concept of weak

Δ

-property to determine the existence of common best proximity point for such a pair of maps. Full article

(This article belongs to the Collection Mathematical Analysis and Applications)

18 pages, 345 KiB

Open AccessEditor’s ChoiceArticle

An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications

by Nopparat Wairojjana, Habib ur Rehman, Ioannis K. Argyros and Nuttapol Pakkaranang

Axioms 2020, 9(3), 99; https://doi.org/10.3390/axioms9030099 - 17 Aug 2020

Cited by 8 | Viewed by 3032

Abstract

Several methods have been put forward to solve equilibrium problems, in which the two-step extragradient method is very useful and significant. In this article, we propose a new extragradient-like method to evaluate the numerical solution of the pseudomonotone equilibrium in real Hilbert space. [...] Read more.

Several methods have been put forward to solve equilibrium problems, in which the two-step extragradient method is very useful and significant. In this article, we propose a new extragradient-like method to evaluate the numerical solution of the pseudomonotone equilibrium in real Hilbert space. This method uses a non-monotonically stepsize technique based on local bifunction values and Lipschitz-type constants. Furthermore, we establish the weak convergence theorem for the suggested method and provide the applications of our results. Finally, several experimental results are reported to see the performance of the proposed method. Full article

(This article belongs to the Special Issue Iterative Processes for Nonlinear Problems with Applications)

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18 pages, 2600 KiB

Open AccessEditor’s ChoiceArticle

The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem

by Konstantinos Kalimeris and Athanassios S. Fokas

Axioms 2020, 9(3), 89; https://doi.org/10.3390/axioms9030089 - 28 Jul 2020

Cited by 1 | Viewed by 2715

Abstract

Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on [...] Read more.

Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on each side of the hexagon. We show that if this function is odd, then this problem can be solved in closed form; numerical verification is also provided. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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18 pages, 450 KiB

Open AccessEditor’s ChoiceArticle

Facility Location Selection for B-Schools in Indian Context: A Multi-Criteria Group Decision Based Analysis

by Sanjib Biswas and Dragan Pamucar

Axioms 2020, 9(3), 77; https://doi.org/10.3390/axioms9030077 - 08 Jul 2020

Cited by 26 | Viewed by 4016

Abstract

Facility location is one of the critical strategic decisions for any organization. It not only carries the organization’s identity but also connects the point of origin and point of consumption. In the case of higher educational institutions, specifically B-Schools, location is one of [...] Read more.

Facility location is one of the critical strategic decisions for any organization. It not only carries the organization’s identity but also connects the point of origin and point of consumption. In the case of higher educational institutions, specifically B-Schools, location is one of the primary concerns for potential students and their parents while selecting an institution for pursuing higher education. There has been a plethora of research conducted to investigate the factors influencing the B-School selection decision-making. However, location as a standalone factor has not been widely studied. This paper aims to explore various location selection criteria from the viewpoint of the candidates who aspire to enroll in B-Schools. We apply an integrated group decision-making framework of pivot pairwise relative criteria importance assessment (PIPRECIA), and level-based weight assessment LBWA is used wherein a group of student counselors, admission executives, and educators from India has participated. The factors which influence the location decision are identified through qualitative opinion analysis. The results show that connectivity and commutation are the dominant issues. Full article

(This article belongs to the Special Issue Softcomputing: Theories and Applications)

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17 pages, 256 KiB

Open AccessEditor’s ChoiceArticle

Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces

by Jiří Močkoř

Axioms 2020, 9(2), 63; https://doi.org/10.3390/axioms9020063 - 02 Jun 2020

Cited by 6 | Viewed by 1996

Abstract

Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform [...] Read more.

Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Čech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Applications)

13 pages, 346 KiB

Open AccessEditor’s ChoiceArticle

Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique

by Choukri Derbazi, Zidane Baitiche, Mouffak Benchohra and Alberto Cabada

Axioms 2020, 9(2), 57; https://doi.org/10.3390/axioms9020057 - 21 May 2020

Cited by 34 | Viewed by 3394

Abstract

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the

ψ

-Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More [...] Read more.

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the

ψ

-Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More precisely we apply the monotone iterative technique combined with the method of upper and lower solutions to establish sufficient conditions for existence as well as the uniqueness of extremal solutions to the initial value problem. An illustrative example is presented to point out the applicability of our main results. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

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16 pages, 291 KiB

Open AccessEditor’s ChoiceArticle

Aristotle’s Syllogistic as a Deductive System

by Piotr Kulicki

Axioms 2020, 9(2), 56; https://doi.org/10.3390/axioms9020056 - 19 May 2020

Cited by 2 | Viewed by 6245

Abstract

Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the [...] Read more.

Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the Aristotelian system of syllogistic and Łukasiewicz’s reconstruction of it based on the tools of modern formal logic. We pay special attention to the notion of completeness of a deductive system as discussed by both authors. We describe in detail how completeness can be defined and proved with the use of an axiomatic refutation system. Finally, we apply this methodology to different axiomatizations of syllogistic presented by Łukasiewicz, Lemmon and Shepherdson. Full article

(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)

33 pages, 429 KiB

Open AccessEditor’s ChoiceReview

The Generalized Hypergeometric Structure of the Ward Identities of CFT’s in Momentum Space in d > 2

by Claudio Corianò and Matteo Maria Maglio

Axioms 2020, 9(2), 54; https://doi.org/10.3390/axioms9020054 - 14 May 2020

Cited by 5 | Viewed by 2471

Abstract

We review the emergence of hypergeometric structures (of F₄ Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to [...] Read more.

We review the emergence of hypergeometric structures (of F₄ Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to hypergeometric systems with four independent solutions. For symmetric correlators, they can be expressed in terms of a single 3K integral—functions of quadratic ratios of momenta—which is a parametric integral of three modified Bessel K functions. In the case of scalar 4-point functions, by requiring the correlator to be conformal invariant in coordinate space as well as in some dual variables (i.e., dual conformal invariant), its explicit expression is also given by a 3K integral, or as a linear combination of Appell functions which are now quartic ratios of momenta. Similar expressions have been obtained in the past in the computation of an infinite class of planar ladder (Feynman) diagrams in perturbation theory, which, however, do not share the same (dual conformal/conformal) symmetry of our solutions. We then discuss some hypergeometric functions of 3 variables, which define 8 particular solutions of the CWIs and correspond to Lauricella functions. They can also be combined in terms of 4K integral and appear in an asymptotic description of the scalar 4-point function, in special kinematical limits. Full article

(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)

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13 pages, 272 KiB

Open AccessEditor’s ChoiceArticle

Integral Representation of Coherent Lower Previsions by Super-Additive Integrals

by Serena Doria, Radko Mesiar and Adam Šeliga

Axioms 2020, 9(2), 43; https://doi.org/10.3390/axioms9020043 - 23 Apr 2020

Cited by 8 | Viewed by 2296

Abstract

Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet [...] Read more.

Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Applications)

14 pages, 463 KiB

Open AccessEditor’s ChoiceArticle

Quasinormal Modes of Charged Black Holes in Higher-Dimensional Einstein-Power-Maxwell Theory

by Grigoris Panotopoulos

Axioms 2020, 9(1), 33; https://doi.org/10.3390/axioms9010033 - 24 Mar 2020

Cited by 14 | Viewed by 3150

Abstract

We compute the quasinormal frequencies for scalar perturbations of charged black holes in five-dimensional Einstein-power-Maxwell theory. The impact on the spectrum of the electric charge of the black holes, of the angular degree, of the overtone number, and of the mass of the [...] Read more.

We compute the quasinormal frequencies for scalar perturbations of charged black holes in five-dimensional Einstein-power-Maxwell theory. The impact on the spectrum of the electric charge of the black holes, of the angular degree, of the overtone number, and of the mass of the test scalar field is investigated in detail. The quasinormal spectra in the eikonal limit are computed as well for several different space-time dimensionalities. Full article

(This article belongs to the Special Issue Theory and Mathematical Aspects of Black Holes)

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7 pages, 265 KiB

Open AccessEditor’s ChoiceArticle

On Grothendieck Sets

by Juan Carlos Ferrando, Salvador López-Alfonso and Manuel López-Pellicer

Axioms 2020, 9(1), 34; https://doi.org/10.3390/axioms9010034 - 24 Mar 2020

Cited by 5 | Viewed by 2319

Abstract

We call a subset

M

of an algebra of sets

A

a Grothendieck set for the Banach space

b a (A)

of bounded finitely additive scalar-valued measures on

A

equipped with the variation norm if each sequence [...] Read more.

We call a subset

M

of an algebra of sets

A

a Grothendieck set for the Banach space

b a (A)

of bounded finitely additive scalar-valued measures on

A

equipped with the variation norm if each sequence

{\{μ_{n}\}}_{n = 1}^{\infty}

in

b a (A)

which is pointwise convergent on

M

is weakly convergent in

b a (A)

, i.e., if there is

μ \in b a (A)

such that

μ_{n} (A) \to μ (A)

for every

A \in M

then

μ_{n} \to μ

weakly in

b a (A)

. A subset

M

of an algebra of sets

A

is called a Nikodým set for

b a (A)

if each sequence

{\{μ_{n}\}}_{n = 1}^{\infty}

in

b a (A)

which is pointwise bounded on

M

is bounded in

b a (A)

. We prove that if

Σ

is a

σ

-algebra of subsets of a set

Ω

which is covered by an increasing sequence

\{Σ_{n} : n \in N\}

of subsets of

Σ

there exists

p \in N

such that

Σ_{p}

is a Grothendieck set for

b a (A)

. This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a

σ

-algebra

Σ

is covered by an increasing sequence

\{Σ_{n} : n \in N\}

of subsets, there is

p \in N

such that

Σ_{p}

is a Nikodým set for

b a (Σ)

. This also refines the Grothendieck result stating that for each

σ

-algebra

Σ

the Banach space

ℓ_{\infty} (Σ)

is a Grothendieck space. Some applications to classic Banach space theory are given. Full article

(This article belongs to the Collection Topological Groups)

10 pages, 211 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Term Logic

by Peter Simons

Axioms 2020, 9(1), 18; https://doi.org/10.3390/axioms9010018 - 10 Feb 2020

Cited by 4 | Viewed by 2564

Abstract

The predominant form of logic before Frege, the logic of terms has been largely neglected since. Terms may be singular, empty or plural in their denotation. This article, presupposing propositional logic, provides an axiomatization based on an identity predicate, a predicate of non-existence, [...] Read more.

The predominant form of logic before Frege, the logic of terms has been largely neglected since. Terms may be singular, empty or plural in their denotation. This article, presupposing propositional logic, provides an axiomatization based on an identity predicate, a predicate of non-existence, a constant empty term, and term conjunction and negation. The idea of basing term logic on existence or non-existence, outlined by Brentano, is here carried through in modern guise. It is shown how categorical syllogistic reduces to just two forms of inference. Tree and diagram methods of testing validity are described. An obvious translation into monadic predicate logic shows the system is decidable, and additional expressive power brought by adding quantifiers enables numerical predicates to be defined. The system’s advantages for pedagogy are indicated. Full article

(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)

5 pages, 213 KiB

Open AccessEditor’s ChoiceArticle

Observations on the Separable Quotient Problem for Banach Spaces

by Sidney A. Morris and David T. Yost

Axioms 2020, 9(1), 7; https://doi.org/10.3390/axioms9010007 - 13 Jan 2020

Cited by 2 | Viewed by 2329

Abstract

The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, [...] Read more.

The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space

E^{*}

onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in

E^{*}

. It is shown that every dual-like Banach space has an infinite-dimensional separable quotient. Full article

(This article belongs to the Collection Topological Groups)

9 pages, 252 KiB

Open AccessEditor’s ChoiceArticle

A Versatile Integral in Physics and Astronomy and Fox’s H-Function

by Arak M. Mathai and Hans J. Haubold

Axioms 2019, 8(4), 122; https://doi.org/10.3390/axioms8040122 - 01 Nov 2019

Cited by 3 | Viewed by 2165

Abstract

This paper deals with a general class of integrals, the particular cases of which are connected to outstanding problems in physics and astronomy. Nuclear reaction rate probability integrals in nuclear physics, Krätzel integrals in applied mathematical analysis, inverse Gaussian distributions, generalized type-1, type-2, [...] Read more.

This paper deals with a general class of integrals, the particular cases of which are connected to outstanding problems in physics and astronomy. Nuclear reaction rate probability integrals in nuclear physics, Krätzel integrals in applied mathematical analysis, inverse Gaussian distributions, generalized type-1, type-2, and gamma families of distributions in statistical distribution theory, Tsallis statistics and Beck–Cohen superstatistics in statistical mechanics, and Mathai’s pathway model are all shown to be connected to the integral under consideration. Representations of the integral in terms of Fox’s H-function are pointed out. Full article

(This article belongs to the Special Issue Special Functions and Their Applications)

13 pages, 266 KiB

Open AccessEditor’s ChoiceArticle

Approximation Properties of an Extended Family of the Szász–Mirakjan Beta-Type Operators

by Hari Mohan Srivastava, Gürhan İçöz and Bayram Çekim

Axioms 2019, 8(4), 111; https://doi.org/10.3390/axioms8040111 - 10 Oct 2019

Cited by 27 | Viewed by 3043

Abstract

Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation [...] Read more.

Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation and other associated properties of a class of the Szász-Mirakjan-type operators, which are introduced here by using an extension of the familiar Beta function. We propose to establish moments of these extended Szász-Mirakjan Beta-type operators and estimate various convergence results with the help of the second modulus of smoothness and the classical modulus of continuity. We also investigate convergence via functions which belong to the Lipschitz class. Finally, we prove a Voronovskaja-type approximation theorem for the extended Szász-Mirakjan Beta-type operators. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

11 pages, 278 KiB

Open AccessEditor’s ChoiceArticle

Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics

by Erasmo Caponio and Antonio Masiello

Axioms 2019, 8(3), 83; https://doi.org/10.3390/axioms8030083 - 23 Jul 2019

Cited by 3 | Viewed by 2708

Abstract

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers–Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the [...] Read more.

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers–Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald. Full article

(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)

19 pages, 281 KiB

Open AccessEditor’s ChoiceArticle

Conditions of Functional Null Controllability for Some Types of Singularly Perturbed Nonlinear Systems with Delays

by Valery Y. Glizer

Axioms 2019, 8(3), 80; https://doi.org/10.3390/axioms8030080 - 15 Jul 2019

Cited by 5 | Viewed by 2370

Abstract

Two types of singularly-perturbed nonlinear time delay controlled systems are considered. For these systems, sufficient conditions of the functional null controllability are derived. These conditions, being independent of the parameter of singular perturbation, provide the controllability of the systems for all sufficiently small [...] Read more.

Two types of singularly-perturbed nonlinear time delay controlled systems are considered. For these systems, sufficient conditions of the functional null controllability are derived. These conditions, being independent of the parameter of singular perturbation, provide the controllability of the systems for all sufficiently small values of the parameter. Illustrative examples are presented. Full article

11 pages, 268 KiB

Open AccessEditor’s ChoiceArticle

Dual Numbers and Operational Umbral Methods

by Nicolas Behr, Giuseppe Dattoli, Ambra Lattanzi and Silvia Licciardi

Axioms 2019, 8(3), 77; https://doi.org/10.3390/axioms8030077 - 02 Jul 2019

Cited by 4 | Viewed by 4380

Abstract

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a [...] Read more.

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus. Full article

(This article belongs to the Special Issue Non-associative Structures and Other Related Structures)

14 pages, 1334 KiB

Open AccessEditor’s ChoiceArticle

Generating Root-Finder Iterative Methods of Second Order: Convergence and Stability

by Francisco I. Chicharro, Alicia Cordero, Neus Garrido and Juan R. Torregrosa

Axioms 2019, 8(2), 55; https://doi.org/10.3390/axioms8020055 - 06 May 2019

Cited by 10 | Viewed by 3156

Abstract

In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are [...] Read more.

In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are members of this family for particular choices of the weight function. The dynamical behavior of one of these choices is presented, analyzing the stability of the fixed points and the critical points of the rational function obtained when the iterative expression is applied on low degree polynomials. Several numerical tests are given to compare different elements of the proposed family on non-polynomial problems. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)

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15 pages, 238 KiB

Open AccessEditor’s ChoiceReview

Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey

by Paolo Emilio Ricci

Axioms 2019, 8(2), 50; https://doi.org/10.3390/axioms8020050 - 25 Apr 2019

Cited by 2 | Viewed by 2622

Abstract

By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

30 pages, 1942 KiB

Open AccessEditor’s ChoiceArticle

A New gH-Difference for Multi-Dimensional Convex Sets and Convex Fuzzy Sets

by Luciano Stefanini and Barnabas Bede

Axioms 2019, 8(2), 48; https://doi.org/10.3390/axioms8020048 - 24 Apr 2019

Cited by 12 | Viewed by 4214

Abstract

In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space [...] Read more.

In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space of compact convex sets in terms of a pseudo-norm, i.e., the magnitude of the difference set. A computational procedure for two dimensional sets is outlined and some examples of the new difference are given. Full article

(This article belongs to the Special Issue Softcomputing: Theories and Applications)

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27 pages, 344 KiB

Open AccessEditor’s ChoiceArticle

Euclidean Space Controllability Conditions for Singularly Perturbed Linear Systems with Multiple State and Control Delays

by Valery Y. Glizer

Axioms 2019, 8(1), 36; https://doi.org/10.3390/axioms8010036 - 21 Mar 2019

Cited by 4 | Viewed by 2678

Abstract

A singularly perturbed linear time-dependent controlled system with multiple point-wise delays and distributed delays in the state and control variables is considered. The delays are small, of order of a small positive multiplier for a part of the derivatives in the system. This [...] Read more.

A singularly perturbed linear time-dependent controlled system with multiple point-wise delays and distributed delays in the state and control variables is considered. The delays are small, of order of a small positive multiplier for a part of the derivatives in the system. This multiplier is a parameter of the singular perturbation. Two types of the considered singularly perturbed system, standard and nonstandard, are analyzed. For each type, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original system. It is established in the paper that proper kinds of controllability of the slow and fast subsystems yield the complete Euclidean space controllability of the original system for all sufficiently small values of the parameter of singular perturbation. Illustrative examples are presented. Full article

(This article belongs to the Special Issue Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution)

10 pages, 327 KiB

Open AccessEditor’s ChoiceArticle

Note on Limit-Periodic Solutions of the Difference Equation x_{t + 1} − [h(x_t) + λ]x_t = r_t, λ > 1

by Jan Andres and Denis Pennequin

Axioms 2019, 8(1), 19; https://doi.org/10.3390/axioms8010019 - 05 Feb 2019

Cited by 2 | Viewed by 2761

Abstract

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar [...] Read more.

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

14 pages, 40736 KiB

Open AccessEditor’s ChoiceArticle

Complex Soliton Solutions to the Gilson–Pickering Model

by Haci Mehmet Baskonus

Axioms 2019, 8(1), 18; https://doi.org/10.3390/axioms8010018 - 01 Feb 2019

Cited by 57 | Viewed by 4661

Abstract

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D [...] Read more.

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)

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15 pages, 336 KiB

Open AccessEditor’s ChoiceArticle

Harrod–Domar Growth Model with Memory and Distributed Lag

by Vasily E. Tarasov and Valentina V. Tarasova

Axioms 2019, 8(1), 9; https://doi.org/10.3390/axioms8010009 - 15 Jan 2019

Cited by 10 | Viewed by 7752

Abstract

In this paper, we propose a macroeconomic growth model, in which we take into account memory with power-law fading and gamma distributed lag. This model is a generalization of the standard Harrod–Domar growth model. Fractional differential equations of this generalized model with memory [...] Read more.

In this paper, we propose a macroeconomic growth model, in which we take into account memory with power-law fading and gamma distributed lag. This model is a generalization of the standard Harrod–Domar growth model. Fractional differential equations of this generalized model with memory and lag are suggested. For these equations, we obtain solutions, which describe the macroeconomic growth of national income with fading memory and distributed time-delay. The asymptotic behavior of these solutions is described. Full article

(This article belongs to the Special Issue Fractional Differential Equations)

50 pages, 598 KiB

Open AccessEditor’s ChoiceReview

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

by Domenico Perrone

Axioms 2019, 8(1), 6; https://doi.org/10.3390/axioms8010006 - 09 Jan 2019

Cited by 10 | Viewed by 3339

Abstract

There is one-to-one correspondence between contact semi-Riemannian structures

(η, ξ, φ, g)

and non-degenerate almost CR structures

(H, ϑ, J)

. In general, a non-degenerate almost CR structure is not a CR structure, that [...] Read more.

There is one-to-one correspondence between contact semi-Riemannian structures

(η, ξ, φ, g)

and non-degenerate almost CR structures

(H, ϑ, J)

. In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for

H^{1, 0} : = \{X - i J X, X \in H\}

is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

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