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Axioms, Volume 8, Issue 2 (June 2019)

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Open AccessBook Review
Review of “The Significance of the New Logic” Willard Van Orman Quine. Edited and Translated by Walter Carnielli, Frederique Janssen-Lauret, and William Pickering. Cambridge University Press, Cambridge, UK, 2018, pp. 1–200. ISBN-10: 1107179025 ISBN-13: 978-1107179028
Axioms 2019, 8(2), 64; https://doi.org/10.3390/axioms8020064 (registering DOI)
Received: 30 April 2019 / Revised: 15 May 2019 / Accepted: 17 May 2019 / Published: 22 May 2019
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Abstract
In this review, I will discuss the historical importance of “The Significance of the New Logic” by Quine. This is a translation of the original “O Sentido da Nova Lógica” in Portuguese by Carnielli, Janssen-Lauret, and Pickering. The American philosopher wrote this book [...] Read more.
In this review, I will discuss the historical importance of “The Significance of the New Logic” by Quine. This is a translation of the original “O Sentido da Nova Lógica” 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. Full article
(This article belongs to the Special Issue Deductive Systems)
Open AccessArticle
Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions
Axioms 2019, 8(2), 63; https://doi.org/10.3390/axioms8020063 (registering DOI)
Received: 28 March 2019 / Revised: 16 May 2019 / Accepted: 17 May 2019 / Published: 21 May 2019
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Abstract
In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–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<ℜ( [...] Read more.
In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–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<ℜ(s)<1) to 0<ℜ(s)<μ. 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–Lerch zeta functions as special cases. Fractional derivatives and the relationship of the generalized Bose–Einstein and Fermi–Dirac functions with Apostol–Euler–Nörlund polynomials are established to prove new identities. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
Open AccessArticle
A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions
Received: 23 April 2019 / Revised: 16 May 2019 / Accepted: 17 May 2019 / Published: 19 May 2019
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Abstract
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’s ordinary and [...] Read more.
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’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. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
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Open AccessArticle
Oscillation of Fourth-Order Functional Differential Equations with Distributed Delay
Received: 20 April 2019 / Revised: 8 May 2019 / Accepted: 10 May 2019 / Published: 18 May 2019
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Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Special Functions and Their Applications)
Open AccessArticle
Unification Theories: New Results and Examples
Received: 3 May 2019 / Revised: 16 May 2019 / Accepted: 17 May 2019 / Published: 18 May 2019
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Abstract
This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler’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 [...] Read more.
This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler’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 “modified” Yang–Baxter type equation, which unifies several problems in mathematics. Full article
(This article belongs to the Special Issue Non-associative Structures and Other Related Structures)
Open AccessAddendum
Corrigendum to “On a Class of Conjugate Symplectic Hermite–Obreshkov One-Step Methods with Continuous Spline Extension” [Axioms 7(3), 58, 2018]
Received: 7 May 2019 / Revised: 11 May 2019 / Accepted: 13 May 2019 / Published: 16 May 2019
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Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle
Visual-Servoing Based Global Path Planning Using Interval Type-2 Fuzzy Logic Control
Received: 1 March 2019 / Revised: 19 April 2019 / Accepted: 1 May 2019 / Published: 10 May 2019
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Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Type-2 Fuzzy Logic: Theory, Algorithms and Applications)
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Open AccessArticle
Fixed Point Theorems through Modified ω-Distance and Application to Nontrivial Equations
Received: 27 April 2019 / Revised: 27 April 2019 / Accepted: 5 May 2019 / Published: 8 May 2019
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Abstract
In this manuscript, we utilize the concept of modified ω-distance mapping, which was introduced by Alegre and Marin [Alegre, C.; Marin, J. Modified ω-distance on quasi metric spaces and fixed point theorems on complete quasi metric spaces. Topol. Appl. 2016, [...] Read more.
In this manuscript, we utilize the concept of modified ω -distance mapping, which was introduced by Alegre and Marin [Alegre, C.; Marin, J. Modified ω -distance on quasi metric spaces and fixed point theorems on complete quasi metric spaces. Topol. Appl. 2016, 203, 120–129] in 2016 to introduce the notions of ( ω , φ ) -Suzuki contraction and generalized ( ω , φ ) -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. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
Open AccessArticle
Projector Approach to Constructing Asymptotic Solution of Initial Value Problems for Singularly Perturbed Systems in Critical Case
Received: 13 February 2019 / Revised: 20 April 2019 / Accepted: 21 April 2019 / Published: 8 May 2019
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Abstract
Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with [...] Read more.
Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) 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 ) (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction. Full article
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Open AccessArticle
Generating Root-Finder Iterative Methods of Second Order: Convergence and Stability
Received: 1 April 2019 / Revised: 30 April 2019 / Accepted: 30 April 2019 / Published: 6 May 2019
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Abstract
In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are [...] Read more.
In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are members of this family for particular choices of the weight function. The dynamical behavior of one of these choices is presented, analyzing the stability of the fixed points and the critical points of the rational function obtained when the iterative expression is applied on low degree polynomials. Several numerical tests are given to compare different elements of the proposed family on non-polynomial problems. Full article
Open AccessArticle
A Note on Anosov Homeomorphisms
Received: 9 April 2019 / Revised: 25 April 2019 / Accepted: 29 April 2019 / Published: 1 May 2019
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Abstract
For an α-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 α-shadowing property for one-jump [...] Read more.
For an α -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 α -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. Full article
(This article belongs to the Special Issue Shadowing in Dynamical Systems)
Open AccessArticle
Comparative Study of the Conventional Mathematical and Fuzzy Logic Controllers for Velocity Regulation
Received: 26 March 2019 / Revised: 17 April 2019 / Accepted: 26 April 2019 / Published: 1 May 2019
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Abstract
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 [...] Read more.
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. Full article
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Open AccessArticle
PIP-Space Valued Reproducing Pairs of Measurable Functions
Received: 16 January 2019 / Revised: 16 April 2019 / Accepted: 18 April 2019 / Published: 30 April 2019
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Abstract
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 [...] Read more.
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 , μ ), where ( X , μ ) 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. Full article
(This article belongs to the Special Issue Harmonic Analysis and Applications)
Open AccessArticle
Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence
Received: 22 February 2019 / Revised: 11 April 2019 / Accepted: 19 April 2019 / Published: 25 April 2019
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Abstract
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 [...] Read more.
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. Full article
Open AccessReview
Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey
Received: 8 March 2019 / Revised: 19 April 2019 / Accepted: 20 April 2019 / Published: 25 April 2019
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Abstract
By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
Open AccessArticle
Relation Theoretic Common Fixed Point Results for Generalized Weak Nonlinear Contractions with an Application
Received: 11 March 2019 / Revised: 17 April 2019 / Accepted: 18 April 2019 / Published: 24 April 2019
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Abstract
In this paper, by introducing the concept of generalized Ćirić-type weak (ϕg,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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
Open AccessArticle
A New gH-Difference for Multi-Dimensional Convex Sets and Convex Fuzzy Sets
Received: 19 March 2019 / Revised: 15 April 2019 / Accepted: 16 April 2019 / Published: 24 April 2019
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Abstract
In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space [...] Read more.
In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space of compact convex sets in terms of a pseudo-norm, i.e., the magnitude of the difference set. A computational procedure for two dimensional sets is outlined and some examples of the new difference are given. Full article
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Open AccessArticle
On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices
Received: 31 January 2019 / Revised: 26 March 2019 / Accepted: 10 April 2019 / Published: 21 April 2019
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Abstract
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, [...] Read more.
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 “left” 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—non polynomial solution—of a second-order divided-difference equation of hypergeometric type. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
A Note on the Displacement Problem of Elastostatics with Singular Boundary Values
Received: 18 March 2019 / Revised: 12 April 2019 / Accepted: 16 April 2019 / Published: 19 April 2019
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Abstract
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 Ck(k2) of R3 and a [...] Read more.
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 2 ) of R 3 and a W 2 k 1 / q , q ( Ω ) , q ( 1 , + ) , then it is proved that there exists a solution which is of class C in the interior and takes the boundary value in a well-defined sense. Moreover, it is unique in a natural function class. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
Open AccessArticle
Bäcklund Transformations for Nonlinear Differential Equations and Systems
Received: 5 February 2019 / Revised: 4 April 2019 / Accepted: 7 April 2019 / Published: 11 April 2019
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Abstract
In this work, new Bäcklund 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 [...] Read more.
In this work, new Bäcklund 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’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. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
Periodic Solution and Asymptotic Stability for the Magnetohydrodynamic Equations with Inhomogeneous Boundary Condition
Received: 6 December 2018 / Revised: 29 March 2019 / Accepted: 6 April 2019 / Published: 11 April 2019
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Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Applications of Differential Equations and Dynamical Systems)
Open AccessArticle
Applications of Square Roots of Diffeomorphisms
Received: 9 March 2019 / Revised: 8 April 2019 / Accepted: 9 April 2019 / Published: 11 April 2019
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Abstract
In this paper, we prove that on any contact manifold (M,ξ) there exists an arbitrary C-small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C-small contactomorphism which is not [...] Read more.
In this paper, we prove that on any contact manifold ( M , ξ ) there exists an arbitrary C -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C -small contactomorphism which is not “autonomous”. This paper is the first step to study the topology of C o n t 0 ( M , ξ ) Aut ( M , ξ ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
Open AccessArticle
The Monotonic Sequence Theorem and Measurement of Lengths and Areas in Axiomatic Non-Standard Hyperrational Analysis
Received: 24 February 2019 / Revised: 31 March 2019 / Accepted: 4 April 2019 / Published: 10 April 2019
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Abstract
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. [...] Read more.
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. Full article
Open AccessArticle
The 3D Navier–Stokes Equations: Invariants, Local and Global Solutions
Received: 31 January 2019 / Revised: 7 March 2019 / Accepted: 1 April 2019 / Published: 7 April 2019
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Abstract
In this article, I consider local solutions of the 3D Navier–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 [...] Read more.
In this article, I consider local solutions of the 3D Navier–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. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
A Graph Theoretic Approach to Construct Desired Cryptographic Boolean Functions
Received: 7 January 2019 / Revised: 15 March 2019 / Accepted: 26 March 2019 / Published: 3 April 2019
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Abstract
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 [...] Read more.
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. Full article
Open AccessArticle
Pricing Compound and Extendible Options under Mixed Fractional Brownian Motion with Jumps
Received: 6 February 2019 / Revised: 22 March 2019 / Accepted: 30 March 2019 / Published: 3 April 2019
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Abstract
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 [...] Read more.
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. Full article
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Open AccessArticle
Robust Mixture Modeling Based on Two-Piece Scale Mixtures of Normal Family
Received: 18 February 2019 / Revised: 15 March 2019 / Accepted: 27 March 2019 / Published: 1 April 2019
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Abstract
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 [...] Read more.
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. Full article
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Open AccessArticle
Efficient Two-Step Fifth-Order and Its Higher-Order Algorithms for Solving Nonlinear Systems with Applications
Received: 20 February 2019 / Revised: 20 March 2019 / Accepted: 26 March 2019 / Published: 1 April 2019
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Abstract
This manuscript presents a new two-step weighted Newton’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 [...] Read more.
This manuscript presents a new two-step weighted Newton’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 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. Full article
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Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
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