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Axioms, Volume 8, Issue 4 (December 2019)

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Open AccessArticle
Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable—Part II: Extremal Points, Convexity, Periodicity
Axioms 2019, 8(4), 114; https://doi.org/10.3390/axioms8040114 - 14 Oct 2019
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Abstract
We continue the presentation of new results in the calculus for interval-valued functions of a single real variable. We start here with the results presented in part I of this paper, namely, a general setting of partial orders in the space of compact [...] Read more.
We continue the presentation of new results in the calculus for interval-valued functions of a single real variable. We start here with the results presented in part I of this paper, namely, a general setting of partial orders in the space of compact intervals (in midpoint-radius representation) and basic results on convergence and limits, continuity, gH-differentiability, and monotonicity. We define different types of (local) minimal and maximal points and develop the basic theory for their characterization. We then consider some interesting connections with applied geometry of curves and the convexity of interval-valued functions is introduced and analyzed in detail. Further, the periodicity of interval-valued functions is described and analyzed. Several examples and pictures accompany the presentation. Full article
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Open AccessArticle
Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable. Part I: Partial Orders, gH-Derivative, Monotonicity
Axioms 2019, 8(4), 113; https://doi.org/10.3390/axioms8040113 - 14 Oct 2019
Viewed by 85
Abstract
We present new results in interval analysis (IA) and in the calculus for interval-valued functions of a single real variable. Starting with a recently proposed comparison index, we develop a new general setting for partial order in the (semi linear) space of compact [...] Read more.
We present new results in interval analysis (IA) and in the calculus for interval-valued functions of a single real variable. Starting with a recently proposed comparison index, we develop a new general setting for partial order in the (semi linear) space of compact real intervals and we apply corresponding concepts for the analysis and calculus of interval-valued functions. We adopt extensively the midpoint-radius representation of intervals in the real half-plane and show its usefulness in calculus. Concepts related to convergence and limits, continuity, gH-differentiability and monotonicity of interval-valued functions are introduced and analyzed in detail. Graphical examples and pictures accompany the presentation. A companion Part II of the paper will present additional properties (max and min points, convexity and periodicity). Full article
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Open AccessArticle
Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function
Axioms 2019, 8(4), 112; https://doi.org/10.3390/axioms8040112 - 11 Oct 2019
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Abstract
The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new [...] Read more.
The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. Full article
Open AccessArticle
Approximation Properties of an Extended Family of the Szász–Mirakjan Beta-Type Operators
Axioms 2019, 8(4), 111; https://doi.org/10.3390/axioms8040111 - 10 Oct 2019
Viewed by 102
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)
Open AccessArticle
A New Approach to the Interpolative Contractions
Axioms 2019, 8(4), 110; https://doi.org/10.3390/axioms8040110 - 10 Oct 2019
Viewed by 119
Abstract
We propose a refinement in the interpolative approach in fixed-point theory. In particular, using this method, we prove the existence of fixed points and common fixed points for Kannan-type contractions and provide examples to support our results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
Open AccessArticle
Deontic Logics as Axiomatic Extensions of First-Order Predicate Logic: An Approach Inspired by Wolniewicz’s Formal Ontology of Situations
Axioms 2019, 8(4), 109; https://doi.org/10.3390/axioms8040109 - 06 Oct 2019
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Abstract
The aim of this article is to present a method of creating deontic logics as axiomatic theories built on first-order predicate logic with identity. In the article, these theories are constructed as theories of legal events or as theories of acts. Legal events [...] Read more.
The aim of this article is to present a method of creating deontic logics as axiomatic theories built on first-order predicate logic with identity. In the article, these theories are constructed as theories of legal events or as theories of acts. Legal events are understood as sequences (strings) of elementary situations in Wolniewicz′s sense. On the other hand, acts are understood as two-element legal events: the first element of a sequence is a choice situation (a situation that will be changed by an act), and the second element of this sequence is a chosen situation (a situation that arises as a result of that act). In this approach, legal rules (i.e., orders, bans, permits) are treated as sets of legal events. The article presents four deontic systems for legal events: AEP, AEPF, AEPOF, AEPOFI. In the first system, all legal events are permitted; in the second, they are permitted or forbidden; in the third, they are permitted, ordered or forbidden; and in the fourth, they are permitted, ordered, forbidden or irrelevant. Then, we present a deontic logic for acts (AAPOF), in which every act is permitted, ordered or forbidden. The theorems of this logic reflect deontic relations between acts as well as between acts and their parts. The direct inspiration to develop the approach presented in the article was the book Ontology of Situations by Boguslaw Wolniewicz, and indirectly, Wittgenstein’s Tractatus Logico-Philosophicus. Full article
(This article belongs to the Special Issue Deductive Systems)
Open AccessArticle
A Kotas-Style Characterisation of Minimal Discussive Logic
Axioms 2019, 8(4), 108; https://doi.org/10.3390/axioms8040108 - 01 Oct 2019
Viewed by 116
Abstract
In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the [...] Read more.
In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the smallest one, this logic can be treated as a basis which could be extended to richer discussive logics that are obtained by varying accessibility relation and resulting in a lattice of discussive logics. One has to remember that while formulating discussive logics there is no one-to-one determination of discussive logics by modal logics. For example, it is proved that Jaśkowski’s logic D 2 can be expressed by other than S 5 modal logics. In this paper we consider a deductive system for the sketchily described minimal logic. While formulating the deductive system, we apply a method of Kotas that was used to axiomatize D 2 . The obtained system determines a logic D 0 as a set of theses that is contained in D 2 . Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain D 0 , so it is the smallest discussive logic. Full article
(This article belongs to the Special Issue Deductive Systems)
Open AccessCorrection
Corrections: Kim, T.; et al. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. Axioms 2018, 7, 56
Axioms 2019, 8(4), 107; https://doi.org/10.3390/axioms8040107 - 01 Oct 2019
Viewed by 139
Abstract
The authors, Kim and Ryoo in [1], studied Euler polynomials and Bernoulli polynomials with
an extended variable to a complex variable, replacing real variable x by complex variable x + iy,
and achieved several useful identities and properties [...] Full article
Open AccessArticle
Gabor Frames and Deep Scattering Networks in Audio Processing
Axioms 2019, 8(4), 106; https://doi.org/10.3390/axioms8040106 - 26 Sep 2019
Viewed by 161
Abstract
This paper introduces Gabor scattering, a feature extractor based on Gabor frames and Mallat’s scattering transform. By using a simple signal model for audio signals, specific properties of Gabor scattering are studied. It is shown that, for each layer, specific invariances to certain [...] Read more.
This paper introduces Gabor scattering, a feature extractor based on Gabor frames and Mallat’s scattering transform. By using a simple signal model for audio signals, specific properties of Gabor scattering are studied. It is shown that, for each layer, specific invariances to certain signal characteristics occur. Furthermore, deformation stability of the coefficient vector generated by the feature extractor is derived by using a decoupling technique which exploits the contractivity of general scattering networks. Deformations are introduced as changes in spectral shape and frequency modulation. The theoretical results are illustrated by numerical examples and experiments. Numerical evidence is given by evaluation on a synthetic and a “real” dataset, that the invariances encoded by the Gabor scattering transform lead to higher performance in comparison with just using Gabor transform, especially when few training samples are available. Full article
(This article belongs to the Special Issue Harmonic Analysis and Applications)
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