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Axioms, Volume 9, Issue 1 (March 2020) – 24 articles

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Open AccessArticle
Stability of Equilibria of Rumor Spreading Model under Stochastic Perturbations
Axioms 2020, 9(1), 24; https://doi.org/10.3390/axioms9010024 (registering DOI) - 18 Feb 2020
Abstract
The known mathematical model of rumor spreading, which is described by a system of four nonlinear differential equations and is very popular in research, is considered. It is supposed that the considered model is influenced by stochastic perturbations that are of the type [...] Read more.
The known mathematical model of rumor spreading, which is described by a system of four nonlinear differential equations and is very popular in research, is considered. It is supposed that the considered model is influenced by stochastic perturbations that are of the type of white noise and are proportional to the deviation of the system state from its equilibrium point. Sufficient conditions of stability in probability for each from the five equilibria of the considered model are obtained by virtue of the Routh–Hurwitz criterion and the method of linear matrix inequalities (LMIs). The obtained results are illustrated by numerical analysis of appropriate LMIs and numerical simulations of solutions of the considered system of stochastic differential equations. The research method can also be used in other applications for similar nonlinear models with the order of nonlinearity higher than one. Full article
Open AccessArticle
Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids
Axioms 2020, 9(1), 23; https://doi.org/10.3390/axioms9010023 (registering DOI) - 18 Feb 2020
Abstract
We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = i I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous [...] Read more.
We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = i I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results. Full article
(This article belongs to the collection Topological Groups) Printed Edition available
Open AccessReview
Cantor Waves for Signorini Hyperelastic Materials with Cylindrical Symmetry
Axioms 2020, 9(1), 22; https://doi.org/10.3390/axioms9010022 - 13 Feb 2020
Viewed by 96
Abstract
In this paper, local fractional cylindrical wave solutions on Signorini hyperelastic materials are studied. In particular, we focus on the so-called Signorini potential. Cantor-type cylindrical coordinates are used to analyze, both from dynamical and geometrical point of view, wave solutions, so that the [...] Read more.
In this paper, local fractional cylindrical wave solutions on Signorini hyperelastic materials are studied. In particular, we focus on the so-called Signorini potential. Cantor-type cylindrical coordinates are used to analyze, both from dynamical and geometrical point of view, wave solutions, so that the nonlinear fundamental equations of the fractional model are explicitly given. In the special case of linear approximation we explicitly compute the fractional wave profile. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
Open AccessArticle
A Comparative Assessment of Graphic and 0–10 Rating Scales Used to Measure Entrepreneurial Competences
Axioms 2020, 9(1), 21; https://doi.org/10.3390/axioms9010021 - 13 Feb 2020
Viewed by 109
Abstract
This article presents an empirical comparative assessment of the measurement quality of two instruments commonly used to measure fuzzy characteristics in computer-assisted questionnaires: a graphic scale (a line production scale using a slider bar) and an endecanary scale (a 0–10 rating scale using [...] Read more.
This article presents an empirical comparative assessment of the measurement quality of two instruments commonly used to measure fuzzy characteristics in computer-assisted questionnaires: a graphic scale (a line production scale using a slider bar) and an endecanary scale (a 0–10 rating scale using radio buttons). Data are analyzed by means of multitrait–multimethod models estimated as structural equation models with a mean and covariance structure. For the first time in such research, the results include bias, valid variance, method variance, and random error variance. The data are taken from a program that assesses entrepreneurial competences in undergraduate Economics and Business students by means of questionnaires administered on desktop computers. Neither of the measurement instruments was found to be biased with respect to the other, meaning that their scores are comparable. While both instruments achieve valid and reliable measurements, the reliability and validity are higher for the endecanary scale. This study contributes to the still scarce literature on fuzzy measurement instruments and on the comparability and relative merits of graphic and discrete rating scales on computer-assisted questionnaires. Full article
(This article belongs to the Special Issue Soft Computing in Economics, Finance and Management)
Open AccessArticle
Correlations in Two-Qubit Systems under Non-Dissipative Decoherence
Axioms 2020, 9(1), 20; https://doi.org/10.3390/axioms9010020 - 12 Feb 2020
Viewed by 152
Abstract
We built a new set of suitable measures of correlations for bipartite quantum states based upon a recently introduced theoretical framework [Bussandri et al. in Quantum Inf. Proc. 18:57, 2019]. We applied these measures to examine the behavior of correlations in two-qubit states [...] Read more.
We built a new set of suitable measures of correlations for bipartite quantum states based upon a recently introduced theoretical framework [Bussandri et al. in Quantum Inf. Proc. 18:57, 2019]. We applied these measures to examine the behavior of correlations in two-qubit states with maximally mixed marginals independently interacting with non-dissipative decohering environments in different dynamical scenarios of physical relevance. In order to get further insight about the physical meaning of the behavior of these correlation measures we compared our results with those obtained by means of well-known correlation measures such as quantum mutual information and quantum discord. On one hand, we found that the behaviors of total and classical correlations, as assessed by means of the measures introduced in this work, are qualitatively in agreement with the behavior displayed by quantum mutual information and the measure of classical correlations typically used to calculate quantum discord. We also found that the optimization of all the measures of classical correlations depends upon a single parameter and the optimal value of this parameter turns out to be the same in all cases. On the other hand, regarding the measures of quantum correlations used in our studies, we found that in general their behavior does not follow the standard quantum discord D . As the quantification by means of standard quantum discord and the measures of quantum correlations introduced in this work depends upon the assumption that total correlations are additive, our results indicate that this property needs a deeper and systematic study in order to gain a further understanding regarding the possibility to obtain reliable quantifiers of quantum correlations within this additive scheme. Full article
(This article belongs to the Special Issue Foundations of Quantum Computing)
Open AccessArticle
A Discussion on the Existence of Best Proximity Points That Belong to the Zero Set
Axioms 2020, 9(1), 19; https://doi.org/10.3390/axioms9010019 - 11 Feb 2020
Viewed by 102
Abstract
In this paper, we investigate the existence of best proximity points that belong to the zero set for the α p -admissible weak ( F , φ ) -proximal contraction in the setting of M-metric spaces. For this purpose, we establish φ [...] Read more.
In this paper, we investigate the existence of best proximity points that belong to the zero set for the α p -admissible weak ( F , φ ) -proximal contraction in the setting of M-metric spaces. For this purpose, we establish φ -best proximity point results for such mappings in the setting of a complete M-metric space. Some examples are also presented to support the concepts and results proved herein. Our results extend, improve and generalize several comparable results on the topic in the related literature. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
Open AccessFeature PaperArticle
Term Logic
Axioms 2020, 9(1), 18; https://doi.org/10.3390/axioms9010018 - 10 Feb 2020
Viewed by 113
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)
Open AccessArticle
Optimal Saving by Expected Utility Operators
Axioms 2020, 9(1), 17; https://doi.org/10.3390/axioms9010017 - 09 Feb 2020
Viewed by 182
Abstract
This paper studies an optimal saving model in which risk is represented by a fuzzy number and the total utility function of the model is defined by an expected utility operator. This model generalizes some existing possibilistic saving models and from them, by [...] Read more.
This paper studies an optimal saving model in which risk is represented by a fuzzy number and the total utility function of the model is defined by an expected utility operator. This model generalizes some existing possibilistic saving models and from them, by a particularization, one can obtain new saving models. In the paper, sufficient conditions are set for the presence of potential risk to increase optimal saving levels and an approximation formula for optimal saving is demonstrated. Particular models for a few concrete expected utility operators are analyzed for triangular fuzzy numbers and CRRA-utility functions. Full article
(This article belongs to the Special Issue Soft Computing in Economics, Finance and Management)
Open AccessEditorial
Mathematical Analysis and Applications II
Axioms 2020, 9(1), 16; https://doi.org/10.3390/axioms9010016 - 06 Feb 2020
Viewed by 121
Abstract
The present volume contains the invited, accepted and published submissions (see [1–17]) to
a Special Issue of the MDPI’s journal, Axioms, on the subject-area of “Mathematical Analysis and
Applications II” [...] Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
Open AccessArticle
On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform
Axioms 2020, 9(1), 15; https://doi.org/10.3390/axioms9010015 - 05 Feb 2020
Viewed by 126
Abstract
An interesting property of the inverse F-transform f ^ of a continuous function f on a given interval [ a , b ] says that the integrals of f ^ and f on [ a , b ] coincide. Furthermore, the same property [...] Read more.
An interesting property of the inverse F-transform f ^ of a continuous function f on a given interval [ a , b ] says that the integrals of f ^ and f on [ a , b ] coincide. Furthermore, the same property can be established for the restrictions of the functions to all subintervals [ a , p k ] of the fuzzy partition of [ a , b ] used to define the F-transform. Based on this fact, we propose a new method for the numerical solution of ordinary differential equations (initial-value ordinary differential equation (ODE)) obtained by approximating the derivative x · ( t ) via F-transform, then computing (an approximation of) the solution x ( t ) by exact integration. For an ODE, a global second-order approximation is obtained. A similar construction is then applied to interval-valued and (level-wise) fuzzy differential equations in the setting of generalized differentiability (gH-derivative). Properties of the new method are analyzed and a computational section illustrates the performance of the obtained procedures, in comparison with well-known efficient algorithms. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
Open AccessArticle
Oscillation Results for Higher Order Differential Equations
Axioms 2020, 9(1), 14; https://doi.org/10.3390/axioms9010014 - 03 Feb 2020
Viewed by 140
Abstract
The objective of our research was to study asymptotic properties of the class of higher order differential equations with a p-Laplacian-like operator. Our results supplement and improve some known results obtained in the literature. An illustrative example is provided. Full article
Open AccessArticle
On One Interpolation Type Fractional Boundary-Value Problem
Axioms 2020, 9(1), 13; https://doi.org/10.3390/axioms9010013 - 28 Jan 2020
Viewed by 165
Abstract
We present some new results on the approximation of solutions of a special type of fractional boundary-value problem. The focus of our research is a system of three fractional differential equations of the mixed order, subjected to the so-called "interpolation" type boundary restrictions. [...] Read more.
We present some new results on the approximation of solutions of a special type of fractional boundary-value problem. The focus of our research is a system of three fractional differential equations of the mixed order, subjected to the so-called "interpolation" type boundary restrictions. Under certain conditions, the aforementioned problem is simplified via a proper parametrization technique, and with the help of the numerical-analytic method, the approximate solutions are constructed. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
Open AccessArticle
Quantum Trapezium-Type Inequalities Using Generalized ϕ-Convex Functions
Axioms 2020, 9(1), 12; https://doi.org/10.3390/axioms9010012 - 26 Jan 2020
Viewed by 237
Abstract
In this work, a study is conducted on the Hermite–Hadamard inequality using a class of generalized convex functions that involves a generalized and parametrized class of special functions within the framework of quantum calculation. Similar results can be obtained from the results found [...] Read more.
In this work, a study is conducted on the Hermite–Hadamard inequality using a class of generalized convex functions that involves a generalized and parametrized class of special functions within the framework of quantum calculation. Similar results can be obtained from the results found for functions such as the hypergeometric function and the classical Mittag–Leffler function. The method used to obtain the results is classic in the study of quantum integral inequalities. Full article
(This article belongs to the Special Issue Special Functions and Their Applications)
Open AccessArticle
The Tubby Torus as a Quotient Group
Axioms 2020, 9(1), 11; https://doi.org/10.3390/axioms9010011 - 20 Jan 2020
Viewed by 174
Abstract
Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a [...] Read more.
Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem. Full article
(This article belongs to the collection Topological Groups) Printed Edition available
Open AccessArticle
Viscosity Approximation Methods for * −Nonexpansive Multi-Valued Mappings in Convex Metric Spaces
Axioms 2020, 9(1), 10; https://doi.org/10.3390/axioms9010010 - 17 Jan 2020
Viewed by 159
Abstract
In this paper, we prove convergence theorems for viscosity approximation processes involving * −nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points [...] Read more.
In this paper, we prove convergence theorems for viscosity approximation processes involving * −nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points of them. Our results improve and extend some corresponding results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
Open AccessArticle
Axiomatic Approach in the Analytic Theory of Singular Perturbations
Axioms 2020, 9(1), 9; https://doi.org/10.3390/axioms9010009 - 16 Jan 2020
Viewed by 161
Abstract
Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type [...] Read more.
Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality. Full article
Open AccessEditorial
Acknowledgement to Reviewers of Axioms in 2019
Axioms 2020, 9(1), 8; https://doi.org/10.3390/axioms9010008 - 15 Jan 2020
Viewed by 185
Open AccessArticle
Observations on the Separable Quotient Problem for Banach Spaces
Axioms 2020, 9(1), 7; https://doi.org/10.3390/axioms9010007 - 13 Jan 2020
Cited by 1 | Viewed by 148
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) Printed Edition available
Open AccessArticle
The Zahl-Anzahl Distinction in Gottlob Frege: Arithmetic of Natural Numbers with Anzahl as a Primitive Term
Axioms 2020, 9(1), 6; https://doi.org/10.3390/axioms9010006 - 31 Dec 2019
Viewed by 200
Abstract
The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the [...] Read more.
The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the original natural number (N) term is replaced by the term Anzahl (A). The functor of the successor (S) is defined in it. Full article
(This article belongs to the Special Issue Deductive Systems)
Open AccessArticle
F-Transform Inspired Weak Solution to a Boundary Value Problem
Axioms 2020, 9(1), 5; https://doi.org/10.3390/axioms9010005 - 31 Dec 2019
Viewed by 224
Abstract
We propose and show efficiency of a new fuzzy-transform-based numerical method of solving ordinary differential equations with boundary conditions. The focus is on weak solutions and a special construction of a two-parameterized family of test functions. On theoretical and computational levels, we show [...] Read more.
We propose and show efficiency of a new fuzzy-transform-based numerical method of solving ordinary differential equations with boundary conditions. The focus is on weak solutions and a special construction of a two-parameterized family of test functions. On theoretical and computational levels, we show how the proposed technique relates to and outperforms the Ritz–Galerkin method. We emphasize the importance of the proposed technique by considering its application to a real-life problem—the option pricing policy. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
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Open AccessArticle
On a Common Jungck Type Fixed Point Result in Extended Rectangular b-Metric Spaces
Axioms 2020, 9(1), 4; https://doi.org/10.3390/axioms9010004 - 27 Dec 2019
Viewed by 298
Abstract
In this paper, we present a Jungck type common fixed point result in extended rectangular b-metric spaces. We also give some examples and a known common fixed point theorem in extended b-metric spaces. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
Open AccessArticle
Harmonic Starlike Functions with Respect to Symmetric Points
Axioms 2020, 9(1), 3; https://doi.org/10.3390/axioms9010003 - 22 Dec 2019
Viewed by 295
Abstract
In the paper we define classes of harmonic starlike functions with respect to symmetric points and obtain some analytic conditions for these classes of functions. Some results connected to subordination properties, coefficient estimates, integral representation, and distortion theorems are also obtained. Full article
Open AccessArticle
Admissible Hybrid Z-Contractions in b-Metric Spaces
Axioms 2020, 9(1), 2; https://doi.org/10.3390/axioms9010002 - 21 Dec 2019
Viewed by 278
Abstract
In this manuscript, we introduce a new notion, admissible hybrid Z -contraction that unifies several nonlinear and linear contractions in the set-up of a b-metric space. In our main theorem, we discuss the existence and uniqueness result of such mappings in the [...] Read more.
In this manuscript, we introduce a new notion, admissible hybrid Z -contraction that unifies several nonlinear and linear contractions in the set-up of a b-metric space. In our main theorem, we discuss the existence and uniqueness result of such mappings in the context of complete b-metric space. The given result not only unifies the several existing results in the literature, but also extends and improves them. We express some consequences of our main theorem by using variant examples of simulation functions. As applications, the well-posedness and the Ulam–Hyers stability of the fixed point problem are also studied. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
Open AccessArticle
Relational Variants of Lattice-Valued F-Transforms
Axioms 2020, 9(1), 1; https://doi.org/10.3390/axioms9010001 - 19 Dec 2019
Viewed by 254
Abstract
Two categories of lower and upper lattice-valued F-transforms with fuzzy relations as morphisms are introduced, as generalisations of standard categories of F-transforms with maps as morphisms. Although F-transforms are defined using special structures called spaces with fuzzy partitions, it is shown that these [...] Read more.
Two categories of lower and upper lattice-valued F-transforms with fuzzy relations as morphisms are introduced, as generalisations of standard categories of F-transforms with maps as morphisms. Although F-transforms are defined using special structures called spaces with fuzzy partitions, it is shown that these categories are identical to the relational variants of the two categories of semimodule homomorphisms where these fuzzy partitions do not occur. This a priori independence of the F-transform on spaces with fuzzy partitions makes it possible, for example, to use a simple matrix calculus to calculate F-transforms, or to determine the image of F-transforms in relational morphisms of the two categories. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
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