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# Axioms, Volume 9, Issue 1 (March 2020) – 34 articles

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Article
On Grothendieck Sets
Axioms 2020, 9(1), 34; https://doi.org/10.3390/axioms9010034 - 24 Mar 2020
Cited by 3 | Viewed by 1087
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 $μ n n = 1 ∞$ in $b a ( A )$ which is pointwise convergent on $M$ is weakly convergent in $b a ( A )$ , i.e., if there is $μ ∈ b a A$ such that $μ n A → μ A$ for every $A ∈ M$ then $μ n → μ$ 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 $μ n n = 1 ∞$ 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 $Σ n : n ∈ N$ of subsets of $Σ$ there exists $p ∈ N$ 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 $Σ n : n ∈ N$ of subsets, there is $p ∈ N$ 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 $ℓ ∞ Σ$ is a Grothendieck space. Some applications to classic Banach space theory are given. Full article
Article
Quasinormal Modes of Charged Black Holes in Higher-Dimensional Einstein-Power-Maxwell Theory
Axioms 2020, 9(1), 33; https://doi.org/10.3390/axioms9010033 - 24 Mar 2020
Cited by 7 | Viewed by 1029
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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Article
On a Harmonic Univalent Subclass of Functions Involving a Generalized Linear Operator
Axioms 2020, 9(1), 32; https://doi.org/10.3390/axioms9010032 - 24 Mar 2020
Cited by 5 | Viewed by 949
Abstract
In this paper, a subclass of complex-valued harmonic univalent functions defined by a generalized linear operator is introduced. Some interesting results such as coefficient bounds, compactness, and other properties of this class are obtained. Full article
Article
Fixed Point Results under Generalized c-Distance in Cone b-Metric Spaces Over Banach Algebras
Axioms 2020, 9(1), 31; https://doi.org/10.3390/axioms9010031 - 21 Mar 2020
Cited by 3 | Viewed by 964
Abstract
In this work, we define the concept of a generalized c-distance in cone b-metric spaces over a Banach algebra and introduce some its properties. Then, we prove the existence and uniqueness of fixed points for mappings satisfying weak contractive conditions such [...] Read more.
In this work, we define the concept of a generalized c-distance in cone b-metric spaces over a Banach algebra and introduce some its properties. Then, we prove the existence and uniqueness of fixed points for mappings satisfying weak contractive conditions such as Han–Xu-type contraction and Cho-type contraction with respect to this distance. Our assertions are useful, since we remove the continuity condition of the mapping and the normality condition for the cone. Several examples are given to support the main results. Full article
Article
Systolic Aspects of Black Hole Entropy
Axioms 2020, 9(1), 30; https://doi.org/10.3390/axioms9010030 - 16 Mar 2020
Viewed by 711
Abstract
We attempt to provide a mesoscopic treatment of the origin of black hole entropy in (3 + 1)-dimensional spacetimes. We ascribe this entropy to the non-trivial topology of the space-like sections $Σ$ of the horizon. This is not forbidden by topological censorship, since [...] Read more.
We attempt to provide a mesoscopic treatment of the origin of black hole entropy in (3 + 1)-dimensional spacetimes. We ascribe this entropy to the non-trivial topology of the space-like sections $Σ$ of the horizon. This is not forbidden by topological censorship, since all the known energy inequalities needed to prove the spherical topology of $Σ$ are violated in quantum theory. We choose the systoles of $Σ$ to encode its complexity, which gives rise to the black hole entropy. We present hand-waving reasons why the entropy of the black hole can be considered as a function of the volume entropy of $Σ$ . We focus on the limiting case of $Σ$ having a large genus. Full article
(This article belongs to the Special Issue Theory and Mathematical Aspects of Black Holes)
Article
Dynamics of HIV-TB Co-Infection Model
Axioms 2020, 9(1), 29; https://doi.org/10.3390/axioms9010029 - 11 Mar 2020
Cited by 3 | Viewed by 1099
Abstract
According to World Health Organization (WHO), the population suffering from human immunodeficiency virus (HIV) infection over a period of time may suffer from TB infection which increases the death rate. There is no cure for acquired immunodeficiency syndrome (AIDS) to date but antiretrovirals [...] Read more.
According to World Health Organization (WHO), the population suffering from human immunodeficiency virus (HIV) infection over a period of time may suffer from TB infection which increases the death rate. There is no cure for acquired immunodeficiency syndrome (AIDS) to date but antiretrovirals (ARVs) can slow down the progression of disease as well as prevent secondary infections or complications. This is considered as a medication in this paper. This scenario of HIV-TB co-infection is modeled using a system of non-linear differential equations. This model considers HIV-infected individual as the initial stage. Four equilibrium points are found. Reproduction number R0 is calculated. If R0 >1 disease persists uniformly, with reference to the reproduction number, backward bifurcation is computed for pre-AIDS (latent) stage. Global stability is established for the equilibrium points where there is no Pre-AIDS TB class, point without co-infection and for the endemic point. Numerical simulation is carried out to validate the data. Sensitivity analysis is carried out to determine the importance of model parameters in the disease dynamics. Full article
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Article
Impossibility of Quantum Bit Commitment, a Categorical Perspective
Axioms 2020, 9(1), 28; https://doi.org/10.3390/axioms9010028 - 09 Mar 2020
Cited by 3 | Viewed by 1069
Abstract
Bit commitment is a cryptographic task in which Alice commits a bit to Bob such that she cannot change the value of the bit after her commitment and Bob cannot learn the value of the bit before Alice opens her commitment. According to [...] Read more.
Bit commitment is a cryptographic task in which Alice commits a bit to Bob such that she cannot change the value of the bit after her commitment and Bob cannot learn the value of the bit before Alice opens her commitment. According to the Mayers–Lo–Chau (MLC) no-go theorem, ideal bit commitment is impossible within quantum theory. In the information theoretic-reconstruction of quantum theory, the impossibility of quantum bit commitment is one of the three information-theoretic constraints that characterize quantum theory. In this paper, we first provide a very simple proof of the MLC no-go theorem and its quantitative generalization. Then, we formalize bit commitment in the theory of dagger monoidal categories. We show that in the setting of dagger monoidal categories, the impossibility of bit commitment is equivalent to the unitary equivalence of purification. Full article
Article
Genetic Algorithm for Scheduling Optimization Considering Heterogeneous Containers: A Real-World Case Study
Axioms 2020, 9(1), 27; https://doi.org/10.3390/axioms9010027 - 04 Mar 2020
Cited by 16 | Viewed by 1857
Abstract
In this paper, we develop and apply a genetic algorithm to solve surgery scheduling cases in a Mexican Public Hospital. Here, one of the most challenging issues is to process containers with heterogeneous capacity. Many scheduling problems do not share this restriction; because [...] Read more.
In this paper, we develop and apply a genetic algorithm to solve surgery scheduling cases in a Mexican Public Hospital. Here, one of the most challenging issues is to process containers with heterogeneous capacity. Many scheduling problems do not share this restriction; because of this reason, we developed and implemented a strategy for the processing of heterogeneous containers in the genetic algorithm. The final product was named “genetic algorithm for scheduling optimization” (GAfSO). The results of GAfSO were tested with real data of a local hospital. Said hospital assigns different operational time to the operating rooms throughout the week. Also, the computational complexity of GAfSO is analyzed. Results show that GAfSO can assign the corresponding capacity to the operating rooms while optimizing their use. Full article
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Article
Complex Dynamics in a Minimal Model of Protection-Based Mutualism
Axioms 2020, 9(1), 26; https://doi.org/10.3390/axioms9010026 - 02 Mar 2020
Viewed by 914
Abstract
This paper presents the first five variable model of mutualism motivated by the interaction between ants and homopterans. In this mutualism, homopterans benefit both directly through increased feeding rates and indirectly through predator protection. Results of our analyses show oscillatory, complex, and chaotic [...] Read more.
This paper presents the first five variable model of mutualism motivated by the interaction between ants and homopterans. In this mutualism, homopterans benefit both directly through increased feeding rates and indirectly through predator protection. Results of our analyses show oscillatory, complex, and chaotic dynamic behavior. In addition, we show that intraspecies interactions are crucial for closing trophic levels and stabilizing the dynamic system from potential “chaotic” behavior. Full article
(This article belongs to the Special Issue Numerical Computation and Nonlinear Dynamical Systems)
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Article
A Sequential Approach to Mild Distributions
Axioms 2020, 9(1), 25; https://doi.org/10.3390/axioms9010025 - 24 Feb 2020
Cited by 3 | Viewed by 829
Abstract
The Banach Gelfand Triple $( S 0 , L 2 , S 0 ′ ) ( R d )$ consists of $S 0 ( R d ) , ∥ · ∥ S 0$ , a very specific Segal algebra as algebra of test [...] Read more.
The Banach Gelfand Triple $( S 0 , L 2 , S 0 ′ ) ( R d )$ consists of $S 0 ( R d ) , ∥ · ∥ S 0$ , a very specific Segal algebra as algebra of test functions, the Hilbert space $L 2 ( R d ) , ∥ · ∥ 2$ and the dual space $S 0 ′ ( R d )$ , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce $S 0 ( R d ) , ∥ · ∥ S 0$ and hence $( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ )$ , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra $S 0 ( R d ) , ∥ · ∥ S 0$ can be used to establish this natural identification. Full article
Article
Stability of Equilibria of Rumor Spreading Model under Stochastic Perturbations
Axioms 2020, 9(1), 24; https://doi.org/10.3390/axioms9010024 - 18 Feb 2020
Cited by 3 | Viewed by 1065
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
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Article
Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids
Axioms 2020, 9(1), 23; https://doi.org/10.3390/axioms9010023 - 18 Feb 2020
Cited by 2 | Viewed by 785
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
Review
Cantor Waves for Signorini Hyperelastic Materials with Cylindrical Symmetry
Axioms 2020, 9(1), 22; https://doi.org/10.3390/axioms9010022 - 13 Feb 2020
Cited by 1 | Viewed by 749
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)
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Article
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
Cited by 4 | Viewed by 1021
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)
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Article
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 850
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
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Article
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
Cited by 4 | Viewed by 1027
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 [...] 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)
Article
Term Logic
Axioms 2020, 9(1), 18; https://doi.org/10.3390/axioms9010018 - 10 Feb 2020
Cited by 2 | Viewed by 905
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
Article
Optimal Saving by Expected Utility Operators
Axioms 2020, 9(1), 17; https://doi.org/10.3390/axioms9010017 - 09 Feb 2020
Viewed by 892
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)
Editorial
Mathematical Analysis and Applications II
Axioms 2020, 9(1), 16; https://doi.org/10.3390/axioms9010016 - 06 Feb 2020
Viewed by 858
Abstract
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
Article
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
Cited by 3 | Viewed by 1134
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)
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Article
Oscillation Results for Higher Order Differential Equations
Axioms 2020, 9(1), 14; https://doi.org/10.3390/axioms9010014 - 03 Feb 2020
Cited by 29 | Viewed by 1070
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
Article
On One Interpolation Type Fractional Boundary-Value Problem
Axioms 2020, 9(1), 13; https://doi.org/10.3390/axioms9010013 - 28 Jan 2020
Cited by 2 | Viewed by 871
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)
Article
Quantum Trapezium-Type Inequalities Using Generalized ϕ-Convex Functions
Axioms 2020, 9(1), 12; https://doi.org/10.3390/axioms9010012 - 26 Jan 2020
Cited by 10 | Viewed by 1127
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)
Article
The Tubby Torus as a Quotient Group
Axioms 2020, 9(1), 11; https://doi.org/10.3390/axioms9010011 - 20 Jan 2020
Viewed by 848
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
Article
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 870
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)
Article
Axiomatic Approach in the Analytic Theory of Singular Perturbations
Axioms 2020, 9(1), 9; https://doi.org/10.3390/axioms9010009 - 16 Jan 2020
Cited by 1 | Viewed by 715
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
Editorial
Acknowledgement to Reviewers of Axioms in 2019
Axioms 2020, 9(1), 8; https://doi.org/10.3390/axioms9010008 - 15 Jan 2020
Viewed by 693
Abstract
The editorial team greatly appreciates the reviewers who have dedicated their considerable time and expertise to the journal’s rigorous editorial process over the past 12 months, regardless of whether the papers are finally published or not [...] Full article
Article
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 917
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
Article
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
Cited by 1 | Viewed by 972
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
Article
F-Transform Inspired Weak Solution to a Boundary Value Problem
Axioms 2020, 9(1), 5; https://doi.org/10.3390/axioms9010005 - 31 Dec 2019
Cited by 1 | Viewed by 1147
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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