Axioms doi: 10.3390/axioms8030104

Authors: Jean-Pierre Desclés Anca Christine Pascu

In this paper, we give a mathematical model of the logic of determination of objects (LDO) based on preordered sets, and a mathematical model of the logic of typical and atypical instances (LTA). We prove that LTA is an extension of LDO. It can manipulate several types of &ldquo;exceptions&rdquo;. Finally, we show that the structural part of LTA can be modeled by a quasi topology structure (QTS).

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Authors: Urszula Wybraniec-Skardowska

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano&rsquo;s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier.

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Authors: Artur Ishkhanyan Clemente Cesarano

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes four restrictions imposed on the parameters: two of the singularities should have non-zero integer characteristic exponents and the accessory parameters should obey polynomial equations.

]]>Axioms doi: 10.3390/axioms8030101

Authors: Spiridonov Karchevskii Nosich

This study considers the mathematical analysis framework aimed at the adequate description of the modes of lasers on the threshold of non-attenuated in time light emission. The lasers are viewed as open dielectric resonators equipped with active regions, filled in with gain material. We introduce a generalized complex-frequency eigenvalue problem for such cavities and prove important properties of the spectrum of its eigensolutions. This involves reduction of the problem to the set of the Muller boundary integral equations and their discretization with the Nystrom technique. Embedded into this general framework is the application-oriented lasing eigenvalue problem, where the real emission frequencies and the threshold gain values together form two-component eigenvalues. As an example of on-threshold mode study, we present numerical results related to the two-dimensional laser shaped as an active equilateral triangle with a round piercing hole. It is demonstrated that the threshold of lasing and the directivity of light emission, for each mode, can be efficiently manipulated with the aid of the size and, especially, the placement of the piercing hole, while the frequency of emission remains largely intact.

]]>Axioms doi: 10.3390/axioms8030100

Authors: Alex Citkin

Using the defined notion of the inference with multiply-conclusion rules, we show that in the logics enjoying the disjunction property, any derivable rule can be inferred from the single-conclusion rules and a single multiple-conclusion rule, which represents the disjunction property. Also, the conversion algorithm of single- and multiple-conclusion deductive systems into each other is studied.

]]>Axioms doi: 10.3390/axioms8030099

Authors: Daripally Ram Prasad Gajula Naveen Venkata Kishore Hüseyin Işık Bagathi Srinuvasa Rao Gorantla Adi Lakshmi

In this paper, we establish some results on coincidence point and common fixed point theorems for a hybrid pair of single valued and multivalued mappings in complete C * -algebra valued fuzzy soft metric spaces. In addition, we provided some coupled fixed point theorems. Finally, we have given examples which support our main results.

]]>Axioms doi: 10.3390/axioms8030098

Authors: Modjtaba Ghorbani Matthias Dehmer Samaneh Zangi Abbe Mowshowitz Frank Emmert-Streib

This paper introduces a variant of entropy measures based on vertex eccentricity and applies it to all graphs representing the isomers of octane. Taking into account the vertex degree as well (degree-ecc-entropy), we find a good correlation with the acentric factor of octane isomers. In particular, we compute the degree-ecc-entropy for three classes of dendrimer graphs.

]]>Axioms doi: 10.3390/axioms8030097

Authors: Ramu Dubey Lakshmi Narayan Mishra Clemente Cesarano

In this paper, a new class of ( C , G f ) -invex functions introduce and give nontrivial numerical examples which justify exist such type of functions. Also, we construct generalized convexity definitions (such as, ( F , G f ) -invexity, C-convex etc.). We consider Mond&ndash;Weir type fractional symmetric dual programs and derive duality results under ( C , G f ) -invexity assumptions. Our results generalize several known results in the literature.

]]>Axioms doi: 10.3390/axioms8030096

Authors: Edraoui Mohamed Aamri Mohamed Lazaiz Samih

Our main goal of this research is to present the theory of points for relatively cyclic and relatively relatively noncyclic p-contractions in complete locally K -convex spaces by providing basic conditions to ensure the existence and uniqueness of fixed points and best proximity points of the relatively cyclic and relatively noncyclic p-contractions map in locally K -convex spaces. The result of this paper is the extension and generalization of the main results of Kirk and A. Abkar.

]]>Axioms doi: 10.3390/axioms8030095

Authors: Olga Kosheleva Vladik Kreinovich Thach Ngoc Nguyen

The main ideas of F-transform came from representing expert rules. It would be therefore reasonable to expect that the more accurately the membership functions describe human reasoning, the more successful will be the corresponding F-transform formulas. We know that an adequate description of our reasoning corresponds to complicated membership functions&mdash;however, somewhat surprisingly, many successful applications of F-transform use the simplest possible triangular membership functions. There exist some explanations for this phenomenon, which are based on local behavior of the signal. In this paper, we supplement these local explanations by a global one: namely, we prove that triangular membership functions are the only one that provide the exact reconstruction of the appropriate global characteristic of the signal.

]]>Axioms doi: 10.3390/axioms8030094

Authors: Vladik Kreinovich Olga Kosheleva Songsak Sriboonchitta

In many application problems, F-transform algorithms are very efficient. In F-transform techniques, we replace the original signal or image with a finite number of weighted averages. The use of a weighted average can be naturally explained, e.g., by the fact that this is what we get anyway when we measure the signal. However, most successful applications of F-transform have an additional not-so-easy-to-explain feature: the fuzzy partition requirement that the sum of all the related weighting functions is a constant. In this paper, we show that this seemingly difficult-to-explain requirement can also be naturally explained in signal-measurement terms: namely, this requirement can be derived from the natural desire to have all the signal values at different moments of time estimated with the same accuracy. This explanation is the main contribution of this paper.

]]>Axioms doi: 10.3390/axioms8030092

Authors: Gerardo L. Febres

This document introduces a method to solve linear optimization problems. The method&rsquo;s strategy is based on the bounding condition that each constraint exerts over the dimensions of the problem. The solution of a linear optimization problem is at the intersection of the constraints defining the extreme vertex. The method decomposes the n-dimensional linear problem into n-1 two-dimensional problems. After studying the role of constraints in these two-dimensional problems, we identify the constraints intersecting at the extreme vertex. We then formulate a linear equation system that directly leads to the solution of the optimization problem. The algorithm is remarkably different from previously existing linear programming algorithms in the sense that it does not iterate; it is deterministic. A fully c-sharp-coded algorithm is made available. We believe this algorithm and the methods applied for classifying constraints according to their role open up a useful framework for studying complex linear problems through feasible-space and constraint analysis.

]]>Axioms doi: 10.3390/axioms8030093

Authors: Wayne Lewis

A Structure Theorem for Protori is derived for the category of finite-dimensional protori (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a universal resolution for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces.

]]>Axioms doi: 10.3390/axioms8030091

Authors: Tibor K. Pogány

The sampling reconstruction theory is one of the great areas of the analysis in which Paul Leo Butzer earned longstanding and excellent theoretical results. Thus, we are forced either by earlier exhaustive presentations of his research activity and/or the highly voluminous material to restrict ourselves to a more narrow and precise sub-area in consideration; we discuss here, giving deeper insight, Paul Butzer&rsquo;s sampling theoretical work with special attention concerning sampling stochastic signals.

]]>Axioms doi: 10.3390/axioms8030090

Authors: Ged Corob Cook

In this paper, we establish a topological version of the notion of an Eilenberg&ndash;Mac Lane space. If X is a pointed topological space, &pi; 1 ( X ) has a natural topology coming from the compact-open topology on the space of maps S 1 &rarr; X . In general, the construction does not produce a topological group because it is possible to create examples where the group multiplication &pi; 1 ( X ) &times; &pi; 1 ( X ) &rarr; &pi; 1 ( X ) is discontinuous. This discontinuity has been noticed by others, for example Fabel. However, if we work in the category of compactly generated, weakly Hausdorff spaces, we may retopologise both the space of maps S 1 &rarr; X and the product &pi; 1 ( X ) &times; &pi; 1 ( X ) with compactly generated topologies to see that &pi; 1 ( X ) is a group object in this category. Such group objects are known as k-groups. Next we construct the Eilenberg&ndash;Mac Lane space K ( G , 1 ) for any totally path-disconnected k-group G. The main point of this paper is to show that, for such a G, &pi; 1 ( K ( G , 1 ) ) is isomorphic to G in the category of k-groups. All totally disconnected locally compact groups are k-groups and so our results apply in particular to profinite groups, answering a question of Sauer&rsquo;s. We also show that analogues of the Mayer&ndash;Vietoris sequence and Seifert&ndash;van Kampen theorem hold in this context. The theory requires a careful analysis using model structures and other homotopical structures on cartesian closed categories as we shall see that no theory can be comfortably developed in the classical world.

]]>Axioms doi: 10.3390/axioms8030089

Authors: Enrico Celeghini Manuel Gadella Mariano A. del Olmo

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space, &Phi; &times; , of a rigged Hilbert space, &Phi; &sub; H &sub; &Phi; &times; . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors &Phi; and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on &Phi; with its own topology, so that they admit continuous extensions to the dual &Phi; &times; and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider S O ( 2 ) and functions on the unit circle, S U ( 2 ) and associated Laguerre functions, Weyl&ndash;Heisenberg group and Hermite functions, S O ( 3 , 2 ) and spherical harmonics, s u ( 1 , 1 ) and Laguerre functions, s u ( 2 , 2 ) and algebraic Jacobi functions and, finally, s u ( 1 , 1 ) &oplus; s u ( 1 , 1 ) and Zernike functions on a circle.

]]>Axioms doi: 10.3390/axioms8030087

Authors: Wolfram Koepf Insuk Kim Arjun K. Rathie

In the theory of generalized hypergeometric functions, classical summation theorems for the series 2 F 1 , 3 F 2 , 4 F 3 , 5 F 4 and 7 F 6 play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions p F p for p = 2 , 3 , 4 , 5 and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings.

]]>Axioms doi: 10.3390/axioms8030088

Authors: Andriy Bandura Oleh Skaskiv

In this paper, for a given direction b &isin; C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t &isin; C } for any z 0 &isin; C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n &rarr; R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n &rarr; R + .

]]>Axioms doi: 10.3390/axioms8030086

Authors: Mikhail Tkachenko

We study factorization properties of continuous homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an &omega;-retractable submonoid of a product D = &prod; i &isin; I D i of topologized monoids and f : S &rarr; H is a continuous homomorphism to a topologized semigroup H with &psi; ( H ) &le; &omega; , then one can find a countable subset E of I and a continuous homomorphism g : p E ( S ) &rarr; H satisfying f = g ∘ p E ↾ S , where p E is the projection of D to &prod; i &isin; E D i . The same conclusion is valid if S contains the &Sigma; -product &Sigma; D &sub; D . Furthermore, we show that in both cases, there exists the smallest by inclusion subset E &sub; I with the aforementioned properties.

]]>Axioms doi: 10.3390/axioms8030085

Authors: Veronika Pitrová

Let A be an epireflective subcategory of the category of all semitopological groups that consists only of abelian groups. We describe maximal hereditary coreflective subcategories of A that are not bicoreflective in A in the case that the A -reflection of the discrete group of integers is a finite cyclic group, the group of integers with a topology that is not T 0 , or the group of integers with the topology generated by its subgroups of the form p n , where n &isin; N , p &isin; P and P is a given set of prime numbers.

]]>Axioms doi: 10.3390/axioms8030084

Authors: Vahid Parvaneh Nawab Hussain Aiman Mukheimer Hassen Aydi

In [Fixed Point Theory Appl., 2015 (2015):185], the authors introduced a new concept of modified contractive mappings, generalizing Ćirić, Chatterjea, Kannan, and Reich type contractions. They applied the condition ( &theta; 4 ) (see page 3, Section 2 of the above paper). Later, in [Fixed Point Theory Appl., 2016 (2016):62], Jiang et al. claimed that the results in [Fixed Point Theory Appl., 2015 (2015):185] are not real generalizations. In this paper, by restricting the conditions of the control functions, we obtain a real generalization of the Banach contraction principle (BCP). At the end, we introduce a weakly JS-contractive condition generalizing the JS-contractive condition.

]]>Axioms doi: 10.3390/axioms8030083

Authors: Erasmo Caponio Antonio Masiello

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers&ndash;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.

]]>Axioms doi: 10.3390/axioms8030082

Authors: Namhee Kwon

We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over sl ^ 2 . In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights.

]]>Axioms doi: 10.3390/axioms8030081

Authors: Hüseyin Işık Hassen Aydi Nabil Mlaiki Stojan Radenović

In this study, we establish the existence and uniqueness theorems of the best proximity points for Geraghty type Ƶ-proximal contractions defined on a complete metric space. The presented results improve and generalize some recent results in the literature. An example, as well as an application to a variational inequality problem are also given in order to illustrate the effectiveness of our generalizations.

]]>Axioms doi: 10.3390/axioms8030080

Authors: Valery Y. Glizer

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.

]]>Axioms doi: 10.3390/axioms8030079

Authors: Jingwei Too Abdul Rahim Abdullah Norhashimah Mohd Saad

To date, the usage of electromyography (EMG) signals in myoelectric prosthetics allows patients to recover functional rehabilitation of their upper limbs. However, the increment in the number of EMG features has been shown to have a great impact on performance degradation. Therefore, feature selection is an essential step to enhance classification performance and reduce the complexity of the classifier. In this paper, a hybrid method, namely, binary particle swarm optimization differential evolution (BPSODE) was proposed to tackle feature selection problems in EMG signals classification. The performance of BPSODE was validated using the EMG signals of 10 healthy subjects acquired from a publicly accessible EMG database. First, discrete wavelet transform was applied to decompose the signals into wavelet coefficients. The features were then extracted from each coefficient and formed into the feature vector. Afterward, BPSODE was used to evaluate the most informative feature subset. To examine the effectiveness of the proposed method, four state-of-the-art feature selection methods were used for comparison. The parameters, including accuracy, feature selection ratio, precision, F-measure, and computation time were used for performance measurement. Our results showed that BPSODE was superior, in not only offering a high classification performance, but also in having the smallest feature size. From the empirical results, it can be inferred that BPSODE-based feature selection is useful for EMG signals classification.

]]>Axioms doi: 10.3390/axioms8030078

Authors: Sergey V. Ludkowski

Nonassociative algebras with metagroup relations and their modules are studied. Their cohomology theory is scrutinized. Extensions and cleftings of these algebras are studied. Broad families of such algebras and their acyclic complexes are described. For this purpose, different types of products of metagroups are investigated. Necessary structural properties of metagroups are studied. Examples are given. It is shown that a class of nonassociative algebras with metagroup relations contains a subclass of generalized Cayley&ndash;Dickson algebras.

]]>Axioms doi: 10.3390/axioms8030077

Authors: Nicolas Behr Giuseppe Dattoli Ambra Lattanzi Silvia Licciardi

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.

]]>Axioms doi: 10.3390/axioms8020076

Authors: Yang-Hi Lee Gwang Kim

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x1 + y1, x2 + y2, …, xn + yn) = f(x1, x2, … xn) + f(y1, y2, …, yn). By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations.

]]>Axioms doi: 10.3390/axioms8020075

Authors: Maksim V. Kukushkin

In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann&ndash;Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann&ndash;Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces.

]]>Axioms doi: 10.3390/axioms8020074

Authors: Shin Min Kang Zain Iqbal Mustafa Habib Waqas Nazeer

Different results regarding different integro-differentials are usually not properly generalized, as they often do not satisfy some of the constraints. The field of fuzzy integro-differentials is very rich these days because of their different applications and functions in different physical phenomena. Solutions of linear fuzzy Volterra integro-differential equations (FVIDEs) are more generalized and have better applications. In this report, the Sumudu decomposition method (SDM) was used to find the solution to some linear and nonlinear fuzzy integro-differential equations (FIDEs). Some examples are given to show the validity of the presented method.

]]>Axioms doi: 10.3390/axioms8020073

Authors: Saida Mohamed Areeg Abdalla Robert John

In this paper, we propose a new approach to constructing similarity measures using the entropy measure for Interval-Valued Intuitionistic Fuzzy Sets. In addition, we provide several illustrative examples to demonstrate the practicality and effectiveness of the proposed formula. Finally, we use the new proposed similarity measure to develop a new approach for solving problems of pattern recognition and multi-criteria fuzzy decision-making.

]]>Axioms doi: 10.3390/axioms8020072

Authors: Erdal Karapınar

In this short survey, we aim to underline the importance of the non-unique fixed point results in various abstract spaces. We recall a brief background on the topic and we combine, collect and unify several existing non-unique fixed points in the literature. Some interesting examples are considered.

]]>Axioms doi: 10.3390/axioms8020071

Authors: Olga Tsekhan

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

]]>Axioms doi: 10.3390/axioms8020070

Authors: Nabil Mlaiki Katarina Kukić Milanka Gardašević-Filipović Hassen Aydi

In this manuscript, we introduce almost b-metric spaces and prove modifications of fixed point theorems for Reich and Hardy&ndash;Rogers type contractions. We present an approach generalizing some fixed point theorems to the case of almost b-metric spaces by reducing almost b-metrics to the corresponding b-metrics. Later, we show that this approach can not work for all kinds of contractions. To confirm this, we present a proof in which the contraction condition is such that it cannot be reduced to corresponding b-metrics.

]]>Axioms doi: 10.3390/axioms8020069

Authors: Badshah-e- Rome Muhammad Sarwar Poom Kumam

Some well known results from the existing literature are extended and generalized via new contractive type mappings in fuzzy metric spaces. A non trivial supporting example is also provided to demonstrate the validity of the obtained results.

]]>Axioms doi: 10.3390/axioms8020068

Authors: Julalak Prabseang Kamsing Nonlaopon Jessada Tariboon

The aim of this paper is to establish some new ( p , q ) -calculus of Hermite–Hadamard inequalities for the double integral and refinements of the Hermite–Hadamard inequality for ( p , q ) -differentiable convex functions.

]]>Axioms doi: 10.3390/axioms8020067

Authors: Taoufik Sabar Abdelhafid Bassou Mohamed Aamri

We first provide a best proximity point result for quasi-noncyclic relatively nonexpansive mappings in the setting of dualistic partial metric spaces. Then, those spaces will be endowed with convexity and a result for a cyclic mapping will be obtained. Afterwards, we prove a best proximity point result for tricyclic mappings in the framework of the newly introduced extended partial S b -metric spaces. In this way, we obtain extensions of some results in the literature.

]]>Axioms doi: 10.3390/axioms8020066

Authors: Alfonso Artigue Gonzalo Cousillas

In this article, we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property.

]]>Axioms doi: 10.3390/axioms8020065

Authors: Deepak Kumar Janak Raj Sharma Clemente Cesarano

Numerous higher-order methods with derivative evaluations are accessible in the literature for computing multiple zeros. However, higher-order methods without derivatives are very rare for multiple zeros. Encouraged by this fact, we present a family of third-order derivative-free iterative methods for multiple zeros that require only evaluations of three functions per iteration. Convergence of the proposed class is demonstrated by means of using a graphical tool, namely basins of attraction. Applicability of the methods is demonstrated through numerical experimentation on different functions that illustrates the efficient behavior. Comparison of numerical results shows that the presented iterative methods are good competitors to the existing techniques.

]]>Axioms doi: 10.3390/axioms8020064

Authors: Alfredo Roque Freire

In this review, I will discuss the historical importance of &ldquo;The Significance of the New Logic&rdquo; by Quine. This is a translation of the original &ldquo;O Sentido da Nova L&oacute;gica&rdquo; in Portuguese by Carnielli, Janssen-Lauret, and Pickering. The American philosopher wrote this book in the beginning of the 1940s, before a major shift in his philosophy. Thus, I will argue that the reader must see this book as an introduction to an important period in his thinking. I will provide a brief summary of the chapters, remarking on valuable features in each of them and positions Quine abandoned in his later work.

]]>Axioms doi: 10.3390/axioms8020063

Authors: Rekha Srivastava Humera Naaz Sabeena Kazi Asifa Tassaddiq

In this paper, we obtain a new series representation for the generalized Bose&ndash;Einstein and Fermi&ndash;Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from ( 0 &lt; &real; ( s ) &lt; 1 ) to ( 0 &lt; &real; ( s ) &lt; &mu; ) . This leads to fresh insights for a new generalization of the Riemann zeta function. The results are validated by obtaining the classical series representation of the polylogarithm and Hurwitz&ndash;Lerch zeta functions as special cases. Fractional derivatives and the relationship of the generalized Bose&ndash;Einstein and Fermi&ndash;Dirac functions with Apostol&ndash;Euler&ndash;N&ouml;rlund polynomials are established to prove new identities.

]]>Axioms doi: 10.3390/axioms8020062

Authors: Maxie D. Schmidt

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence&rsquo;s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way.

]]>Axioms doi: 10.3390/axioms8020061

Authors: Clemente Cesarano Omar Bazighifan

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the fourth order delay differential equations. Some new oscillatory criteria are obtained by using the generalized Riccati transformations and comparison technique with first order delay differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. The effectiveness of the obtained criteria is illustrated via examples.

]]>Axioms doi: 10.3390/axioms8020060

Authors: Florin F. Nichita

This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler&rsquo;s formula for hyperbolic functions is considered a consequence of a unifying point of view. Then, the unification of Jordan, Lie, and associative algebras is revisited. We also explain that derivations and co-derivations can be unified. Finally, we consider a &ldquo;modified&rdquo; Yang&ndash;Baxter type equation, which unifies several problems in mathematics.

]]>Axioms doi: 10.3390/axioms8020059

Authors: Francesca Mazzia Alessandra Sestini

The authors of the above mentioned paper specify that the considered class of one-step symmetric Hermite-Obreshkov methods satisfies the property of conjugate-symplecticity up to order p + r , where r = 2 and p is the order of the method. This generalization of conjugate-symplecticity states that the methods conserve quadratic first integrals and the Hamiltonian function over time intervals of length O ( h − r ) . Theorem 1 of the above mentioned paper is then replaced by a new one. All the other results in the paper do not change. Two new figures related to the already considered Kepler problem are also added.

]]>Axioms doi: 10.3390/axioms8020058

Authors: Mahmut Dirik Oscar Castillo Adnan Fatih Kocamaz

Mobile robot motion planning in an unstructured, static, and dynamic environment is faced with a large amount of uncertainties. In an uncertain working area, a method should be selected to address the existing uncertainties in order to plan a collision-free path between the desired two points. In this paper, we propose a mobile robot path planning method in the visualize plane using an overhead camera based on interval type-2 fuzzy logic (IT2FIS). We deal with a visual-servoing based technique for obstacle-free path planning. It is necessary to determine the location of a mobile robot in an environment surrounding the robot. To reach the target and for avoiding obstacles efficiently under different shapes of obstacle in an environment, an IT2FIS is designed to generate a path. A simulation of the path planning technique compared with other methods is performed. We tested the algorithm within various scenarios. Experiment results showed the efficiency of the generated path using an overhead camera for a mobile robot.

]]>Axioms doi: 10.3390/axioms8020057

Authors: Tariq Qawasmeh Abdalla Tallafha Wasfi Shatanawi

In this manuscript, we utilize the concept of modified &omega; -distance mapping, which was introduced by Alegre and Marin [Alegre, C.; Marin, J. Modified &omega; -distance on quasi metric spaces and fixed point theorems on complete quasi metric spaces. Topol. Appl. 2016, 203, 120&ndash;129] in 2016 to introduce the notions of ( &omega; , &phi; ) -Suzuki contraction and generalized ( &omega; , &phi; ) -Suzuki contraction. We employ these notions to prove some fixed point results. Moreover, we introduce an example to show the novelty of our results. Furthermore, we introduce some applications for our results.

]]>Axioms doi: 10.3390/axioms8020056

Authors: Galina Kurina

Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil&rsquo;eva and Butuzov (Differ. Uravn. 1970, 6(4), 650&ndash;664 (in Russian); English transl.: Differential Equations 1971, 6, 499&ndash;510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix A ( t ) standing in front of the unknown function. In the present paper, the orthogonal projectors onto k e r A ( t ) and k e r A ( t ) &prime; (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction.

]]>Axioms doi: 10.3390/axioms8020055

Authors: Francisco I. Chicharro Alicia Cordero Neus Garrido Juan R. Torregrosa

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.

]]>Axioms doi: 10.3390/axioms8020054

Authors: Mauricio Achigar

For an &alpha; -expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the &alpha; -shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system.

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Authors: Fevrier Valdez Oscar Castillo Camilo Caraveo Cinthia Peraza

Currently, we are in the digital era, where robotics, with the help of the Internet of Things (IoT), is exponentially advancing, and in the technology market we can find multiple devices for achieving these systems, such as the Raspberry Pi, Arduino, and so on. The use of these devices makes our work easier regarding processing information or controlling physical mechanisms, as some of these devices have microcontrollers or microprocessors. One of the main challenges in speed control applications is to make the decision to use a fuzzy logic control (FLC) system instead of a conventional controller system, such as a proportional integral (PI) or a proportional integral-derivative (PID). The main contribution of this paper is the design, integration, and comparative study of the use of these three types of controllers—FLC, PI, and PID—for the speed control of a robot built using the Lego Mindstorms EV3 kit. The root mean square error (RMSE) and the settling time were used as metrics to validate the performance of the speed control obtained with the controllers proposed in this paper.

]]>Axioms doi: 10.3390/axioms8020052

Authors: Jean-Pierre Antoine Camillo Trapani

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z : = Z ( X , &mu; ), where ( X , &mu; ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.

]]>Axioms doi: 10.3390/axioms8020051

Authors: Alicia Cordero Javier G. Maimó Juan R. Torregrosa María P. Vassileva

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.

]]>Axioms doi: 10.3390/axioms8020050

Authors: Paolo Emilio Ricci

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.

]]>Axioms doi: 10.3390/axioms8020049

Authors: Atiya Perveen Idrees A. Khan Mohammad Imdad

In this paper, by introducing the concept of generalized Ćirić-type weak ( ϕ g , R ) -contraction, we prove some common fixed point results in partial metric spaces endowed with binary relation R . We also deduce some useful consequences showing the usability of our results. Finally, we present an application to establish the solution of a system of integral equations.

]]>Axioms doi: 10.3390/axioms8020048

Authors: Luciano Stefanini Barnabas Bede

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.

]]>Axioms doi: 10.3390/axioms8020047

Authors: Mama Foupouagnigni Salifou Mboutngam

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the &ldquo;left&rdquo; inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution&mdash;non polynomial solution&mdash;of a second-order divided-difference equation of hypergeometric type.

]]>Axioms doi: 10.3390/axioms8020046

Authors: Alfonsina Tartaglione

The displacement problem of linear elastostatics in bounded and exterior domains with a non-regular boundary datum a is considered. Precisely, if the elastic body is represented by a domain of class C k ( k &ge; 2 ) of R 3 and a &isin; W 2 &minus; k &minus; 1 / q , q ( &part; &Omega; ) , q &isin; ( 1 , + &infin; ) , then it is proved that there exists a solution which is of class C &infin; in the interior and takes the boundary value in a well-defined sense. Moreover, it is unique in a natural function class.

]]>Axioms doi: 10.3390/axioms8020045

Authors: Tatyana V. Redkina Robert G. Zakinyan Arthur R. Zakinyan Olesya B. Surneva Olga S. Yanovskaya

In this work, new B&auml;cklund transformations (BTs) for generalized Liouville equations were obtained. Special cases of Liouville equations with exponential nonlinearity that have a multiplier that depends on the independent variables and first-order derivatives from the function were considered. Two- and three-dimensional cases were considered. The BTs construction is based on the method proposed by Clairin. The solutions of the considered equations have been found using the BTs, with a unified algorithm. In addition, the work develops the Clairin&rsquo;s method for the system of two third-order equations related to the integrable perturbation and complexification of the Korteweg-de Vries (KdV) equation. Among the constructed BTs an analog of the Miura transformations was found. The Miura transformations transfer the initial system to that of perturbed modified KdV (mKdV) equations. It could be shown on this way that, considering the system as a link between the real and imaginary parts of a complex function, it is possible to go to the complexified KdV (cKdV) and here the analog of the Miura transformations transforms it into the complexification of the mKdV.

]]>Axioms doi: 10.3390/axioms8020044

Authors: Igor Kondrashuk Eduardo Notte-Cuello Mariano Poblete-Cantellano Marko Rojas-Medar

We show, using the spectral Galerkin method together with compactness arguments, the existence and uniqueness of the periodic strong solutions for the magnetohydrodynamic-type equations with inhomogeneous boundary conditions. Furthermore, we study the asymptotic stability for the time periodic solution for this system. In particular, when the magnetic field h ( x , t ) is zero, we obtain the existence, uniqueness, and asymptotic behavior of the strong solutions to the Navier–Stokes equations with inhomogeneous boundary conditions.

]]>Axioms doi: 10.3390/axioms8020043

Authors: Yoshihiro Sugimoto

In this paper, we prove that on any contact manifold ( M , &xi; ) there exists an arbitrary C &infin; -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C &infin; -small contactomorphism which is not &ldquo;autonomous&rdquo;. This paper is the first step to study the topology of C o n t 0 ( M , &xi; ) ∖ Aut ( M , &xi; ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M.

]]>Axioms doi: 10.3390/axioms8020042

Authors: Yuri N. Lovyagin Nikita Y. Lovyagin

This paper lies in the framework of axiomatic non-standard analysis based on the non-standard arithmetic axiomatic theory. This arithmetic includes actual infinite numbers. Unlike the non-standard model of arithmetic, this approach does not take models into account but uses an axiomatic research method. In the axiomatic theory of non-standard arithmetic, hyperrational numbers are defined as triplets of hypernatural numbers. Since the theory of hyperrational numbers and axiomatic non-standard analysis is mainly published in Russian, in this article we give a brief review of its basic concepts and required results. Elementary hyperrational analysis includes defining and evaluating such notions as continuity, differentiability and integral calculus. We prove that a bounded monotonic sequence is a Cauchy sequence. Also, we solve the task of line segment measurement using hyperrational numbers. In fact, this allows us to approximate real numbers using hyperrational numbers, and shows a way to model real numbers and real functions using hyperrational numbers and functions.

]]>Axioms doi: 10.3390/axioms8020041

Authors: Vladimir I. Semenov

In this article, I consider local solutions of the 3D Navier&ndash;Stokes equations and its properties such as an existence of global and smooth solution, uniform boundedness. The basic role is assigned to a special invariant class of solenoidal vector fields and three parameters that are invariant with respect to the scaling procedure. Since in spaces of even dimensions the scaling procedure is a conformal mapping on the Heisenberg group, then an application of invariant parameters can be considered as the application of conformal invariants. It gives the possibility to prove the sufficient and necessary conditions for existence of a global regular solution. This is the main result and one among some new statements. With some compliments, the rest improves well-known classical results.

]]>Axioms doi: 10.3390/axioms8020040

Authors: Modjtaba Ghorbani Matthias Dehmer Vahid Taghvayi-Yazdelli Frank Emmert-Streib

In this paper, we present four product operations to construct cryptographic boolean functions from smaller ones with predictable Walsh spectrum. A lot of cryptographic properties of boolean functions can be presented by their Walsh spectrum. In our method, we use the product of Cayley graphs to present new boolean functions with desired Walsh spectrum and investigate their non-linearity, algebraic and correlation immunity.

]]>Axioms doi: 10.3390/axioms8020039

Authors: Foad Shokrollahi

This study deals with the problem of pricing compound options when the underlying asset follows a mixed fractional Brownian motion with jumps. An analytic formula for compound options is derived under the risk neutral measure. Then, these results are applied to value extendible options. Moreover, some special cases of the formula are discussed, and numerical results are provided.

]]>Axioms doi: 10.3390/axioms8020038

Authors: Mohsen Maleki Javier E. Contreras-Reyes Mohammad R. Mahmoudi

In this paper, we examine the finite mixture (FM) model with a flexible class of two-piece distributions based on the scale mixtures of normal (TP-SMN) family components. This family allows the development of a robust estimation of FM models. The TP-SMN is a rich class of distributions that covers symmetric/asymmetric and light/heavy tailed distributions. It represents an alternative family to the well-known scale mixtures of the skew normal (SMSN) family studied by Branco and Dey (2001). Also, the TP-SMN covers the SMN (normal, t, slash, and contaminated normal distributions) as the symmetric members and two-piece versions of them as asymmetric members. A key feature of this study is using a suitable hierarchical representation of the family to obtain maximum likelihood estimates of model parameters via an EM-type algorithm. The performances of the proposed robust model are demonstrated using simulated and real data, and then compared to other finite mixture of SMSN models.

]]>Axioms doi: 10.3390/axioms8020037

Authors: Parimala Sivakumar Jayakumar Jayaraman

This manuscript presents a new two-step weighted Newton&rsquo;s algorithm with convergence order five for approximating solutions of system of nonlinear equations. This algorithm needs evaluation of two vector functions and two Frechet derivatives per iteration. Furthermore, it is improved into a general multi-step algorithm with one more vector function evaluation per step, with convergence order 3 k + 5 , k &ge; 1 . Error analysis providing order of convergence of the algorithms and their computational efficiency are discussed based on the computational cost. Numerical implementation through some test problems are included, and comparison with well-known equivalent algorithms are presented. To verify the applicability of the proposed algorithms, we have implemented them on 1-D and 2-D Bratu problems. The presented algorithms perform better than many existing algorithms and are equivalent to a few available algorithms.

]]>Axioms doi: 10.3390/axioms8010036

Authors: Valery Y. Glizer

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.

]]>Axioms doi: 10.3390/axioms8010035

Authors: Nicolas Behr Giuseppe Dattoli Ambra Lattanzi

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.

]]>Axioms doi: 10.3390/axioms8010034

Authors: Hamid Faraji Dragana Savić Stojan Radenović

In this paper, some new results are given on fixed and common fixed points of Geraghty type contractive mappings defined in b-complete b-metric spaces. Moreover, two examples are represented to show the compatibility of our results. Some applications for nonlinear integral equations are also given.

]]>Axioms doi: 10.3390/axioms8010033

Authors: Taras Banakh Igor Protasov

We survey some results and pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Special attention is paid to balleans on groups. The boundedness of functions that respect the coarse structure of a ballean could be considered as a coarse counterpart of pseudo-compactness.

]]>Axioms doi: 10.3390/axioms8010032

Authors: Benjamín Barán Marcos Villagra

In this work we show how to use a quantum adiabatic algorithm to solve multiobjective optimization problems. For the first time, we demonstrate a theorem proving that the quantum adiabatic algorithm can find Pareto-optimal solutions in finite-time, provided some restrictions to the problem are met. A numerical example illustrates an application of the theorem to a well-known problem in multiobjective optimization. This result opens the door to solve multiobjective optimization problems using current technology based on quantum annealing.

]]>Axioms doi: 10.3390/axioms8010031

Authors: Juan Pablo Ramírez

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf&rsquo;s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

]]>Axioms doi: 10.3390/axioms8010030

Authors: Vasile Drăgan

In this paper a stochastic optimal control problem described by a quadratic performance criterion and a linear controlled system modeled by a system of singularly perturbed It&ocirc; differential equations with two fast time scales is considered. The asymptotic structure of the stabilizing solution (satisfying a prescribed sign condition) to the corresponding stochastic algebraic Riccati equation is derived. Furthermore, a near optimal control whose gain matrices do not depend upon small parameters is discussed.

]]>Axioms doi: 10.3390/axioms8010029

Authors: Micheal Pawliuk Michael Alexander Waddell

Given a dataset, we quantify the size of patterns that must always exist in the dataset. This is done formally through the lens of Ramsey theory of graphs, and a quantitative bound known as Goodman&rsquo;s theorem. By combining statistical tools with Ramsey theory of graphs, we give a nuanced understanding of how far away a dataset is from correlated, and what qualifies as a meaningful pattern. This method is applicable to a wide range of datasets. As examples, we analyze two very different datasets. The first is a dataset of repeated voters ( n = 435 ) in the 1984 US congress, and we quantify how homogeneous a subset of congressional voters is. We also measure how transitive a subset of voters is. Statistical Ramsey theory is also used with global economic trading data ( n = 214 ) to provide evidence that global markets are quite transitive. While these datasets are small relative to Big Data, they illustrate the new applications we are proposing. We end with specific calls to strengthen the connections between Ramsey theory and statistical methods.

]]>Axioms doi: 10.3390/axioms8010028

Authors: Metod Saniga Edyta Bartnicka

In this paper, it is shown that there exists a particular associative ring with unity of order 16 such that the relations between non-unimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.

]]>Axioms doi: 10.3390/axioms8010027

Authors: Abduhafiz Bobodzhanov Valeriy Safonov Vasiliy Kachalov

We consider a singularly perturbed integral equation with weakly and rapidly varying kernels. The work is a continuation of the studies carried out previously, but these were focused solely on rapidly changing kernels. A generalization for the case of two kernels, one of which is weakly, and the other rapidly varying, has not previously been carried out. The aim of this study is to investigate the effects introduced into the asymptotics of the solution of the problem by a weakly varying integral kernel. In the second part of the work, the problem of constructing exact (more precise, pseudo-analytic) solutions of singularly perturbed problems is considered on the basis of the method of holomorphic regularization developed by one of the authors of this paper. The power series obtained with the help of this method for the solutions of singularly perturbed problems (in contrast to the asymptotic series constructed in the first part of this paper) converge in the usual sense.

]]>Axioms doi: 10.3390/axioms8010026

Authors: Emer Bernal Oscar Castillo José Soria Fevrier Valdez

Galactic swarm optimization (GSO) is a recently created metaheuristic which is inspired by the motion of galaxies and stars in the universe. This algorithm gives us the possibility of finding the global optimum with greater precision since it uses multiple exploration and exploitation cycles. In this paper we present a modification to galactic swarm optimization using type-1 (T1) and interval type-2 (IT2) fuzzy systems for the dynamic adjustment of the c3 and c4 parameters in the algorithm. In addition, the modification is used for the optimization of the fuzzy controller of an autonomous mobile robot. First, the galactic swarm optimization is tested for fuzzy controller optimization. Second, the GSO algorithm with the dynamic adjustment of parameters using T1 fuzzy systems is used for the optimization of the fuzzy controller of an autonomous mobile robot. Finally, the GSO algorithm with the dynamic adjustment of parameters using the IT2 fuzzy systems is applied to the optimization of the fuzzy controller. In the proposed approaches, perturbation (noise) was added to the plant in order to find out if our approach behaves well under perturbation to the autonomous mobile robot plant; additionally, we consider our ability to compare the results obtained with the approaches when no perturbation is considered.

]]>Axioms doi: 10.3390/axioms8010025

Authors: Andrea Masini Margherita Zorzi

We know that quantum logics are the most prominent logical systems associated to the lattices of closed Hilbert subspaces. However, what happen if, following a quantum computing perspective, we want to associate a logic to the process of quantum registers measurements? This paper gives an answer to this question, and, quite surprisingly, shows that such a logic is nothing else that the standard propositional intuitionistic logic.

]]>Axioms doi: 10.3390/axioms8010024

Authors: Marta Dudek Janusz Garecki

In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein&ndash;Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein&ndash;Palatini (EP) action, and then we present how Einstein fields equations can be derived from it. In the next section, we study Einstein&ndash;Palatini action integral for general relativity with a positive cosmological constant &Lambda; in terms of the corrected curvature &Omega; c o r. We see that in terms of &Omega; c o r this action takes the form typical for a gauge field. Finally, we give a geometrical interpretation of the corrected curvature &Omega; c o r.

]]>Axioms doi: 10.3390/axioms8010023

Authors: João Fialho Feliz Minhós

The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arz&egrave;la Ascoli theorem and Schauder&rsquo;s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions.

]]>Axioms doi: 10.3390/axioms8010022

Authors: Feliz Minhós

In this paper, we consider the second order discontinuous differential equation in the real line, a t , u ϕ u &prime; &prime; = f t , u , u &prime; , a . e . t &isin; R , u ( &minus; &infin; ) = &nu; &minus; , u ( + &infin; ) = &nu; + , with ϕ an increasing homeomorphism such that ϕ ( 0 ) = 0 and ϕ ( R ) = R , a &isin; C ( R 2 , R ) with a ( t , x ) &gt; 0 for ( t , x ) &isin; R 2 , f : R 3 &rarr; R a L 1 -Carath&eacute;odory function and &nu; &minus; , &nu; + &isin; R such that &nu; &minus; &lt; &nu; + . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities ϕ and f . To the best of our knowledge, this result is even new when ϕ ( y ) = y , that is for equation a t , u ( t ) u &prime; ( t ) &prime; = f t , u ( t ) , u &prime; ( t ) , a . e . t &isin; R . Moreover, these results can be applied to classical and singular ϕ -Laplacian equations and to the mean curvature operator.

]]>Axioms doi: 10.3390/axioms8010021

Authors: Viktor Abramov

We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of ( m , n ) -block matrices taking a supertrace of a matrix as the element of dual space. Then we also apply this approach to commutative superalgebra and the Lie superalgebra of its derivations to construct 3-Lie superalgebra. The graded Lie brackets are constructed by means of a derivation and involution of commutative superalgebra, and we use them to construct 3-Lie superalgebras.

]]>Axioms doi: 10.3390/axioms8010020

Authors: Michael Gil’

The paper is devoted to the discrete Lyapunov equation X &minus; A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.

]]>Axioms doi: 10.3390/axioms8010019

Authors: Jan Andres Denis Pennequin

As a nontrivial application of the abstract theorem developed in our recent paper titled &ldquo;Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions&rdquo;, 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.

]]>Axioms doi: 10.3390/axioms8010018

Authors: Haci Mehmet Baskonus

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson&ndash;Pickering model is applied. A set of new complex soliton solutions to the Gilson&ndash;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.

]]>Axioms doi: 10.3390/axioms8010017

Authors: Nizar Souayah Hassen Aydi Thabet Abdeljawad Nabil Mlaiki

In this paper, we ensure the existence and uniqueness of a best proximity point in rectangular metric spaces endowed with a graph structure.

]]>Axioms doi: 10.3390/axioms8010016

Authors: Luigi Brugnano Felice Iavernaro

The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.

]]>Axioms doi: 10.3390/axioms8010015

Authors: Ivan G. Ivanov Tonya Mateva

In this paper, based on Kou&rsquo;s classical iterative methods with fifth-order of convergence, we propose new interval iterative methods for computing a real root of the nonlinear scalar equations. Some numerical experiments have executed with the program INTLAB in order to confirm the theoretical results. The computational results have described and compared with Newton&rsquo;s interval method, Ostrowski&rsquo;s interval method and Ostrowski&rsquo;s modified interval method. We conclude that the proposed interval schemes are effective and they are comparable to the classical interval methods.

]]>Axioms doi: 10.3390/axioms8010014

Authors: Fernando Gaxiola Patricia Melin Fevrier Valdez Juan R. Castro Alain Manzo-Martínez

A dynamic adjustment of parameters for the particle swarm optimization (PSO) utilizing an interval type-2 fuzzy inference system is proposed in this work. A fuzzy neural network with interval type-2 fuzzy number weights using S-norm and T-norm is optimized with the proposed method. A dynamic adjustment of the PSO allows the algorithm to behave better in the search for optimal results because the dynamic adjustment provides good synchrony between the exploration and exploitation of the algorithm. Results of experiments and a comparison between traditional neural networks and the fuzzy neural networks with interval type-2 fuzzy numbers weights using T-norms and S-norms are given to prove the performance of the proposed approach. For testing the performance of the proposed approach, some cases of time series prediction are applied, including the stock exchanges of Germany, Mexican, Dow-Jones, London, Nasdaq, Shanghai, and Taiwan.

]]>Axioms doi: 10.3390/axioms8010013

Authors: Mohammad Asim A. Rauf Khan Mohammad Imdad

In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of Fredholm integral equations. Furthermore, we introduce an analogue of the Hausdorff metric in the context of partial symmetric spaces and utilize the same to prove an analogue of the Nadler contraction principle in such spaces. Our results extend and improve many results in the existing literature. We also give some examples exhibiting the utility of our newly established results.

]]>Axioms doi: 10.3390/axioms8010012

Authors: Mohammad Masjed-Jamei Wolfram Koepf

We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38).

]]>Axioms doi: 10.3390/axioms8010011

Authors: Alexey A. Petrov

In this paper, we study relations between shadowing and inverse shadowing for homeomorphisms of a compact space. We present an example of a smooth diffeomorphism of a compact three-dimensional manifold that has the shadowing property and does not have the inverse shadowing property. For some classes of inverse shadowing, we construct examples of homeomorphisms that have the inverse shadowing property but do not have the shadowing property.

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Authors: Rutwig Campoamor-Stursberg Francisco Oviaño García

The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum spec ( t ) = 1 , k , k + 1 , ⋯ , n + k &minus; 3 , n + 2 k &minus; 3 for k &ge; 2 are analyzed, with special emphasis on the resulting Lie algebras for which the second Chevalley cohomology space vanishes. From the detailed inspection of the values k &le; 5 , some series of cohomologically rigid algebras for arbitrary values of k are determined.

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Authors: Vasily E. Tarasov Valentina V. Tarasova

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&ndash;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.

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Authors: Juan Carlos Guzmán Ivette Miramontes Patricia Melin German Prado-Arechiga

The use of artificial intelligence techniques such as fuzzy logic, neural networks and evolutionary computation is currently very important in medicine to be able to provide an effective and timely diagnosis. The use of fuzzy logic allows to design fuzzy classifiers, which have fuzzy rules and membership functions, which are designed based on the experience of an expert. In this particular case a fuzzy classifier of Mamdani type was built, with 21 rules, with two inputs and one output and the objective of this classifier is to perform blood pressure level classification based on knowledge of an expert which is represented in the fuzzy rules. Subsequently different architectures were made in type-1 and type-2 fuzzy systems for classification, where the parameters of the membership functions used in the design of each architecture were adjusted, which can be triangular, trapezoidal and Gaussian, as well as how the fuzzy rules are optimized based on the ranges established by an expert. The main contribution of this work is the design of the optimized interval type-2 fuzzy system with triangular membership functions. The final type-2 system has a better classification rate of 99.408% than the type-1 classifier developed previously in &ldquo;Design of an optimized fuzzy classifier for the diagnosis of blood pressure with a new computational method for expert rule optimization&rdquo; with 98%. In addition, we also obtained a better classification rate than the other architectures proposed in this work.

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Authors: Marisa Fernández Victor Manero Jonatan Sánchez

We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( &minus; &infin; , T ) , where T &gt; 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to &minus; &infin; , and they blow-up at a finite-time singularity.

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Authors: Domenico Perrone

There is one-to-one correspondence between contact semi-Riemannian structures ( &eta; , &xi; , &phi; , g ) and non-degenerate almost CR structures ( H , &thetasym; , 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 &minus; i J X , X &isin; 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.

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Authors: Axioms Editorial Office

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