Axioms

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 2742

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 2533

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 - 1 Nov 2019

Cited by 3 | Viewed by 2285

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 3236

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 4 | Viewed by 2851

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 2502

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 - 2 Jul 2019

Cited by 4 | Viewed by 4620

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 - 6 May 2019

Cited by 10 | Viewed by 3366

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 2769

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 4399

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 2812

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 - 5 Feb 2019

Cited by 2 | Viewed by 2913

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 - 1 Feb 2019

Cited by 58 | Viewed by 4804

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 8044

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 - 9 Jan 2019

Cited by 11 | Viewed by 3460

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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