Axioms

27 pages, 344 KiB

Open AccessEditor’s ChoiceArticle

Euclidean Space Controllability Conditions for Singularly Perturbed Linear Systems with Multiple State and Control Delays

by Valery Y. Glizer

Axioms 2019, 8(1), 36; https://doi.org/10.3390/axioms8010036 - 21 Mar 2019

Cited by 4 | Viewed by 2694

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution)

16 pages, 346 KiB

Open AccessArticle

Operator Ordering and Solution of Pseudo-Evolutionary Equations

by Nicolas Behr, Giuseppe Dattoli and Ambra Lattanzi

Axioms 2019, 8(1), 35; https://doi.org/10.3390/axioms8010035 - 16 Mar 2019

Viewed by 2853

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Fractional Differential Equations)

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12 pages, 234 KiB

Open AccessArticle

Fixed Point Theorems for Geraghty Contraction Type Mappings in b-Metric Spaces and Applications

by Hamid Faraji, Dragana Savić and Stojan Radenović

Axioms 2019, 8(1), 34; https://doi.org/10.3390/axioms8010034 - 14 Mar 2019

Cited by 34 | Viewed by 3737

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

17 pages, 306 KiB

Open AccessArticle

Functional Boundedness of Balleans: Coarse Versions of Compactness

by Taras Banakh and Igor Protasov

Axioms 2019, 8(1), 33; https://doi.org/10.3390/axioms8010033 - 14 Mar 2019

Cited by 4 | Viewed by 2339

Abstract

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 [...] Read more.

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. Full article

14 pages, 436 KiB

Open AccessArticle

A Quantum Adiabatic Algorithm for Multiobjective Combinatorial Optimization

by Benjamín Barán and Marcos Villagra

Axioms 2019, 8(1), 32; https://doi.org/10.3390/axioms8010032 - 09 Mar 2019

Cited by 1 | Viewed by 3228

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Foundations of Quantum Computing)

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20 pages, 15560 KiB

Open AccessArticle

A New Set Theory for Analysis

by Juan Pablo Ramírez

Axioms 2019, 8(1), 31; https://doi.org/10.3390/axioms8010031 - 06 Mar 2019

Cited by 2 | Viewed by 6115

Abstract

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, [...] Read more.

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’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. Full article

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22 pages, 385 KiB

Open AccessArticle

On the Linear Quadratic Optimal Control for Systems Described by Singularly Perturbed Itô Differential Equations with Two Fast Time Scales

by Vasile Drăgan

Axioms 2019, 8(1), 30; https://doi.org/10.3390/axioms8010030 - 05 Mar 2019

Cited by 7 | Viewed by 2361

Abstract

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 Itô differential equations with two fast time scales is considered. The asymptotic structure of the stabilizing solution [...] Read more.

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 Itô 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. Full article

(This article belongs to the Special Issue Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution)

17 pages, 1121 KiB

Open AccessArticle

Using Ramsey Theory to Measure Unavoidable Spurious Correlations in Big Data

by Micheal Pawliuk and Michael Alexander Waddell

Axioms 2019, 8(1), 29; https://doi.org/10.3390/axioms8010029 - 05 Mar 2019

Cited by 1 | Viewed by 4043

Abstract

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’s theorem. By combining statistical tools with Ramsey theory [...] Read more.

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’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. Full article

(This article belongs to the Special Issue Perspectives on Big Data and Data Sciences)

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5 pages, 374 KiB

Open AccessLetter

Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

by Metod Saniga and Edyta Bartnicka

Axioms 2019, 8(1), 28; https://doi.org/10.3390/axioms8010028 - 04 Mar 2019

Viewed by 3205

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Foundations of Quantum Computing)

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20 pages, 321 KiB

Open AccessArticle

Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov’s Regularization Method

by Abduhafiz Bobodzhanov, Valeriy Safonov and Vasiliy Kachalov

Axioms 2019, 8(1), 27; https://doi.org/10.3390/axioms8010027 - 01 Mar 2019

Cited by 11 | Viewed by 2281

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution)

21 pages, 7922 KiB

Open AccessArticle

Optimization of Fuzzy Controller Using Galactic Swarm Optimization with Type-2 Fuzzy Dynamic Parameter Adjustment

by Emer Bernal, Oscar Castillo, José Soria and Fevrier Valdez

Axioms 2019, 8(1), 26; https://doi.org/10.3390/axioms8010026 - 25 Feb 2019

Cited by 31 | Viewed by 3592

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Type-2 Fuzzy Logic: Theory, Algorithms and Applications)

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10 pages, 379 KiB

Open AccessArticle

A Logic for Quantum Register Measurements

by Andrea Masini and Margherita Zorzi

Axioms 2019, 8(1), 25; https://doi.org/10.3390/axioms8010025 - 24 Feb 2019

Cited by 4 | Viewed by 3051

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Foundations of Quantum Computing)

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9 pages, 250 KiB

Open AccessArticle

General Relativity with a Positive Cosmological Constant Λ as a Gauge Theory

by Marta Dudek and Janusz Garecki

Axioms 2019, 8(1), 24; https://doi.org/10.3390/axioms8010024 - 21 Feb 2019

Viewed by 2579

Abstract

In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein–Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein–Palatini (EP) action, and then we present how [...] Read more.

In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein–Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein–Palatini (EP) action, and then we present how Einstein fields equations can be derived from it. In the next section, we study Einstein–Palatini action integral for general relativity with a positive cosmological constant

Λ

in terms of the corrected curvature

Ω_{c o r}

. We see that in terms of

Ω_{c o r}

this action takes the form typical for a gauge field. Finally, we give a geometrical interpretation of the corrected curvature

Ω_{c o r}

. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

11 pages, 275 KiB

Open AccessArticle

First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model

by João Fialho and Feliz Minhós

Axioms 2019, 8(1), 23; https://doi.org/10.3390/axioms8010023 - 16 Feb 2019

Cited by 3 | Viewed by 3018

Abstract

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. [...] Read more.

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 Arzèla Ascoli theorem and Schauder’s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

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14 pages, 272 KiB

Open AccessArticle

Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

by Feliz Minhós

Axioms 2019, 8(1), 22; https://doi.org/10.3390/axioms8010022 - 15 Feb 2019

Cited by 2 | Viewed by 2130

Abstract

In this paper, we consider the second order discontinuous differential equation in the real line, [...] Read more.

In this paper, we consider the second order discontinuous differential equation in the real line,

{(a (t, u) ϕ (u^{'}))}^{'} = f (t, u, u^{'}), a . e . t \in R, u (- \infty) = ν^{-}, u (+ \infty) = ν^{+},

with

ϕ

an increasing homeomorphism such that

ϕ (0) = 0

and

ϕ (R) = R

,

a \in C (R^{2}, R)

with

a (t, x) > 0

for

(t, x) \in R^{2}

,

f : R^{3} \to R

a

L^{1}

-Carathéodory function and

ν^{-}, ν^{+} \in R

such that

ν^{-} < ν^{+}

. 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^{'} (t))}^{'} = f (t, u (t), u^{'} (t)), a . e . t \in R

. Moreover, these results can be applied to classical and singular

ϕ

-Laplacian equations and to the mean curvature operator. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

8 pages, 238 KiB

Open AccessArticle

3-Lie Superalgebras Induced by Lie Superalgebras

by Viktor Abramov

Axioms 2019, 8(1), 21; https://doi.org/10.3390/axioms8010021 - 06 Feb 2019

Cited by 5 | Viewed by 2424

Abstract

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 [...] Read more.

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. Full article

22 pages, 317 KiB

Open AccessArticle

Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

by Michael Gil’

Axioms 2019, 8(1), 20; https://doi.org/10.3390/axioms8010020 - 06 Feb 2019

Cited by 6 | Viewed by 2654

Abstract

The paper is devoted to the discrete Lyapunov equation

X - 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 [...] Read more.

The paper is devoted to the discrete Lyapunov equation

X - 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. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

10 pages, 327 KiB

Open AccessEditor’s ChoiceArticle

Note on Limit-Periodic Solutions of the Difference Equation x_{t + 1} − [h(x_t) + λ]x_t = r_t, λ > 1

by Jan Andres and Denis Pennequin

Axioms 2019, 8(1), 19; https://doi.org/10.3390/axioms8010019 - 05 Feb 2019

Cited by 2 | Viewed by 2776

Abstract

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar [...] Read more.

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, 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. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

14 pages, 40736 KiB

Open AccessEditor’s ChoiceArticle

Complex Soliton Solutions to the Gilson–Pickering Model

by Haci Mehmet Baskonus

Axioms 2019, 8(1), 18; https://doi.org/10.3390/axioms8010018 - 01 Feb 2019

Cited by 57 | Viewed by 4673

Abstract

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D [...] Read more.

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

(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)

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13 pages, 652 KiB

Open AccessArticle

Best Proximity Point Theorems on Rectangular Metric Spaces Endowed with a Graph

by Nizar Souayah, Hassen Aydi, Thabet Abdeljawad and Nabil Mlaiki

Axioms 2019, 8(1), 17; https://doi.org/10.3390/axioms8010017 - 01 Feb 2019

Cited by 16 | Viewed by 3509

Abstract

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

(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

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3 pages, 200 KiB

Open AccessEditorial

Advanced Numerical Methods in Applied Sciences

by Luigi Brugnano and Felice Iavernaro

Axioms 2019, 8(1), 16; https://doi.org/10.3390/axioms8010016 - 31 Jan 2019

Cited by 1 | Viewed by 2678

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

11 pages, 239 KiB

Open AccessArticle

Interval Methods with Fifth Order of Convergence for Solving Nonlinear Scalar Equations

by Ivan G. Ivanov and Tonya Mateva

Axioms 2019, 8(1), 15; https://doi.org/10.3390/axioms8010015 - 31 Jan 2019

Cited by 3 | Viewed by 2847

Abstract

In this paper, based on Kou’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 [...] Read more.

In this paper, based on Kou’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’s interval method, Ostrowski’s interval method and Ostrowski’s modified interval method. We conclude that the proposed interval schemes are effective and they are comparable to the classical interval methods. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)

21 pages, 4553 KiB

Open AccessArticle

PSO with Dynamic Adaptation of Parameters for Optimization in Neural Networks with Interval Type-2 Fuzzy Numbers Weights

by Fernando Gaxiola, Patricia Melin, Fevrier Valdez, Juan R. Castro and Alain Manzo-Martínez

Axioms 2019, 8(1), 14; https://doi.org/10.3390/axioms8010014 - 24 Jan 2019

Cited by 22 | Viewed by 4339

Abstract

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. [...] Read more.

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. Full article

(This article belongs to the Special Issue Type-2 Fuzzy Logic: Theory, Algorithms and Applications)

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15 pages, 258 KiB

Open AccessArticle

Fixed Point Results in Partial Symmetric Spaces with an Application

by Mohammad Asim, A. Rauf Khan and Mohammad Imdad

Axioms 2019, 8(1), 13; https://doi.org/10.3390/axioms8010013 - 22 Jan 2019

Cited by 13 | Viewed by 2932

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

9 pages, 217 KiB

Open AccessArticle

A New Identity for Generalized Hypergeometric Functions and Applications

by Mohammad Masjed-Jamei and Wolfram Koepf

Axioms 2019, 8(1), 12; https://doi.org/10.3390/axioms8010012 - 18 Jan 2019

Cited by 5 | Viewed by 2979

Abstract

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 [...] Read more.

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). Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

10 pages, 228 KiB

Open AccessArticle

Relations between Shadowing and Inverse Shadowing in Dynamical Systems

by Alexey A. Petrov

Axioms 2019, 8(1), 11; https://doi.org/10.3390/axioms8010011 - 17 Jan 2019

Cited by 1 | Viewed by 2506

Abstract

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. [...] Read more.

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. Full article

(This article belongs to the Special Issue Shadowing in Dynamical Systems)

21 pages, 349 KiB

Open AccessArticle

Some Features of Rank One Real Solvable Cohomologically Rigid Lie Algebras with a Nilradical Contracting onto the Model Filiform Lie Algebra Q_n

by Rutwig Campoamor-Stursberg and Francisco Oviaño García

Axioms 2019, 8(1), 10; https://doi.org/10.3390/axioms8010010 - 16 Jan 2019

Cited by 3 | Viewed by 2632

Abstract

The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum [...] Read more.

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, \dots, n + k - 3, n + 2 k - 3)

for

k \geq 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 \leq 5

, some series of cohomologically rigid algebras for arbitrary values of k are determined. Full article

15 pages, 336 KiB

Open AccessEditor’s ChoiceArticle

Harrod–Domar Growth Model with Memory and Distributed Lag

by Vasily E. Tarasov and Valentina V. Tarasova

Axioms 2019, 8(1), 9; https://doi.org/10.3390/axioms8010009 - 15 Jan 2019

Cited by 10 | Viewed by 7778

Abstract

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–Domar growth model. Fractional differential equations of this generalized model with memory [...] Read more.

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–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. Full article

(This article belongs to the Special Issue Fractional Differential Equations)

35 pages, 9964 KiB

Open AccessArticle

Optimal Genetic Design of Type-1 and Interval Type-2 Fuzzy Systems for Blood Pressure Level Classification

by Juan Carlos Guzmán, Ivette Miramontes, Patricia Melin and German Prado-Arechiga

Axioms 2019, 8(1), 8; https://doi.org/10.3390/axioms8010008 - 15 Jan 2019

Cited by 42 | Viewed by 5036

Abstract

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 [...] Read more.

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 “Design of an optimized fuzzy classifier for the diagnosis of blood pressure with a new computational method for expert rule optimization” with 98%. In addition, we also obtained a better classification rate than the other architectures proposed in this work. Full article

(This article belongs to the Special Issue Type-2 Fuzzy Logic: Theory, Algorithms and Applications)

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15 pages, 280 KiB

Open AccessArticle

The Laplacian Flow of Locally Conformal Calibrated G₂-Structures

by Marisa Fernández, Victor Manero and Jonatan Sánchez

Axioms 2019, 8(1), 7; https://doi.org/10.3390/axioms8010007 - 11 Jan 2019

Cited by 1 | Viewed by 2362

Abstract

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 [...] Read more.

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

(- \infty, T)

, where

T > 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

- \infty

, and they blow-up at a finite-time singularity. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

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