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Mathematics, Volume 7, Issue 4 (April 2019)

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Cover Story (view full-size image) The figure says that all polynomials are related to each other and can represent each by finite [...] Read more.
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Open AccessArticle
Optimal Repeated Measurements for Two Treatment Designs with Dependent Observations: The Case of Compound Symmetry
Mathematics 2019, 7(4), 378; https://doi.org/10.3390/math7040378
Received: 18 February 2019 / Revised: 9 April 2019 / Accepted: 13 April 2019 / Published: 25 April 2019
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Abstract
In this paper, we construct optimal repeated measurement designs of two treatments for estimating direct effects, and we examine the case of compound symmetry dependency. We present the model and the design that minimizes the variance of the estimated difference of the two [...] Read more.
In this paper, we construct optimal repeated measurement designs of two treatments for estimating direct effects, and we examine the case of compound symmetry dependency. We present the model and the design that minimizes the variance of the estimated difference of the two treatments. The optimal designs with dependent observations in a compound symmetry model are the same as in the case of independent observations. Full article
(This article belongs to the Special Issue Applied and Computational Statistics)
Open AccessArticle
Solving the Systems of Equations of Lane-Emden Type by Differential Transform Method Coupled with Adomian Polynomials
Mathematics 2019, 7(4), 377; https://doi.org/10.3390/math7040377
Received: 3 February 2019 / Revised: 14 April 2019 / Accepted: 23 April 2019 / Published: 25 April 2019
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Abstract
In this work, we applied the improved differential transform method to find the solutions of the systems of equations of Lane-Emden type arising in various physical models. With our proposed scheme, the desired solutions take the form of a convergent series with easily [...] Read more.
In this work, we applied the improved differential transform method to find the solutions of the systems of equations of Lane-Emden type arising in various physical models. With our proposed scheme, the desired solutions take the form of a convergent series with easily computable components. The results disclosing the relation between the differential transforms of multi-variables and the corresponding Adomian polynomials are proven. One can see that both the differential transforms and the Adomian polynomials of those nonlinearities have the same mathematical structure merely with constants instead of variable components. By using this relation, we computed the differential transforms of nonlinear functions given in the systems. The validity and applicability of the proposed method are illustrated through several homogeneous and nonhomogeneous nonlinear systems. Full article
(This article belongs to the Section Mathematics and Computers Science)
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Open AccessArticle
Generalized Solutions of the Third-Order Cauchy-Euler Equation in the Space of Right-Sided Distributions via Laplace Transform
Mathematics 2019, 7(4), 376; https://doi.org/10.3390/math7040376
Received: 3 April 2019 / Revised: 12 April 2019 / Accepted: 17 April 2019 / Published: 25 April 2019
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Abstract
Using the Laplace transform technique, we investigate the generalized solutions of the third-order Cauchy-Euler equation of the form t3y(t)+at2y(t)+by(t [...] Read more.
Using the Laplace transform technique, we investigate the generalized solutions of the third-order Cauchy-Euler equation of the form t 3 y ( t ) + a t 2 y ( t ) + b y ( t ) + c y ( t ) = 0 , where a , b , and c Z and t R . We find that the types of solutions in the space of right-sided distributions, either distributional solutions or weak solutions, depend on the values of a, b, and c. At the end of the paper, we give some examples showing the types of solutions. Our work improves the result of Kananthai (Distribution solutions of the third order Euler equation. Southeast Asian Bull. Math. 1999, 23, 627–631). Full article
Open AccessArticle
On Opial’s Type Integral Inequalities
Mathematics 2019, 7(4), 375; https://doi.org/10.3390/math7040375
Received: 17 March 2019 / Revised: 17 April 2019 / Accepted: 18 April 2019 / Published: 25 April 2019
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Abstract
In the article we establish some new Opial’s type inequalities involving higher order partial derivatives. These new inequalities, in special cases, yield Agarwal-Pang’s, Pachpatte’s and Das’s type inequalities. Full article
Open AccessArticle
New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator
Mathematics 2019, 7(4), 374; https://doi.org/10.3390/math7040374
Received: 3 March 2019 / Revised: 16 April 2019 / Accepted: 17 April 2019 / Published: 24 April 2019
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Abstract
In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t [...] Read more.
In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration. Full article
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Open AccessArticle
Nadler and Kannan Type Set Valued Mappings in M-Metric Spaces and an Application
Mathematics 2019, 7(4), 373; https://doi.org/10.3390/math7040373
Received: 27 February 2019 / Revised: 16 April 2019 / Accepted: 18 April 2019 / Published: 24 April 2019
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Abstract
This article intends to initiate the study of Pompeiu–Hausdorff distance induced by an M-metric. The Nadler and Kannan type fixed point theorems for set-valued mappings are also established in the said spaces. Moreover, the discussion is supported with the aid of competent [...] Read more.
This article intends to initiate the study of Pompeiu–Hausdorff distance induced by an M-metric. The Nadler and Kannan type fixed point theorems for set-valued mappings are also established in the said spaces. Moreover, the discussion is supported with the aid of competent examples and a result on homotopy. This approach improves the current state of art in fixed point theory. Full article
(This article belongs to the Special Issue Recent Advances on Quasi-Metric Spaces)
Open AccessArticle
A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems
Mathematics 2019, 7(4), 372; https://doi.org/10.3390/math7040372
Received: 9 March 2019 / Revised: 17 April 2019 / Accepted: 17 April 2019 / Published: 24 April 2019
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Abstract
In this paper, we introduce the notion of higher-order weak adjacent epiderivative for a set-valued map without lower-order approximating directions and obtain existence theorem and some properties of the epiderivative. Then by virtue of the epiderivative and Benson proper efficiency, we establish the [...] Read more.
In this paper, we introduce the notion of higher-order weak adjacent epiderivative for a set-valued map without lower-order approximating directions and obtain existence theorem and some properties of the epiderivative. Then by virtue of the epiderivative and Benson proper efficiency, we establish the higher-order Mond-Weir type dual problem for a set-valued optimization problem and obtain the corresponding weak duality, strong duality and converse duality theorems, respectively. Full article
(This article belongs to the Special Issue Applied Functional Analysis and Its Applications)
Open AccessArticle
On an Exact Relation between ζ″(2) and the Meijer G -Functions
Mathematics 2019, 7(4), 371; https://doi.org/10.3390/math7040371
Received: 25 March 2019 / Revised: 12 April 2019 / Accepted: 15 April 2019 / Published: 24 April 2019
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Abstract
In this paper we consider some integral representations for the evaluation of the coefficients of the Taylor series for the Riemann zeta function about a point in the complex half-plane (s)>1. Using the standard approach based upon [...] Read more.
In this paper we consider some integral representations for the evaluation of the coefficients of the Taylor series for the Riemann zeta function about a point in the complex half-plane ( s ) > 1 . Using the standard approach based upon the Euler-MacLaurin summation, we can write these coefficients as Γ ( n + 1 ) plus a relatively smaller contribution, ξ n . The dominant part yields the well-known Riemann’s zeta pole at s = 1 . We discuss some recurrence relations that can be proved from this standard approach in order to evaluate ζ ( 2 ) in terms of the Euler and Glaisher-Kinkelin constants and the Meijer G -functions. Full article
Open AccessArticle
p-Regularity and p-Regular Modification in ⊤-Convergence Spaces
Mathematics 2019, 7(4), 370; https://doi.org/10.3390/math7040370
Received: 9 March 2019 / Revised: 19 April 2019 / Accepted: 22 April 2019 / Published: 24 April 2019
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Abstract
Fuzzy convergence spaces are extensions of convergence spaces. ⊤-convergence spaces are important fuzzy convergence spaces. In this paper, p-regularity (a relative regularity) in ⊤-convergence spaces is discussed by two equivalent approaches. In addition, lower and upper p-regular modifications in ⊤-convergence spaces [...] Read more.
Fuzzy convergence spaces are extensions of convergence spaces. ⊤-convergence spaces are important fuzzy convergence spaces. In this paper, p-regularity (a relative regularity) in ⊤-convergence spaces is discussed by two equivalent approaches. In addition, lower and upper p-regular modifications in ⊤-convergence spaces are further investigated and studied. Particularly, it is shown that lower (resp., upper) p-regular modification and final (resp., initial) structures have good compatibility. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications)
Open AccessArticle
A Probabilistic Proof for Representations of the Riemann Zeta Function
Mathematics 2019, 7(4), 369; https://doi.org/10.3390/math7040369
Received: 4 April 2019 / Revised: 18 April 2019 / Accepted: 19 April 2019 / Published: 24 April 2019
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Abstract
In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic [...] Read more.
In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic approach. Full article
Open AccessArticle
Computing Degree Based Topological Properties of Third Type of Hex-Derived Networks
Mathematics 2019, 7(4), 368; https://doi.org/10.3390/math7040368
Received: 20 February 2019 / Revised: 17 April 2019 / Accepted: 17 April 2019 / Published: 23 April 2019
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Abstract
In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical [...] Read more.
In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical network. Simonraj et al. derived a new type of graphs, which is named a third type of hex-derived networks. In our work, we discuss the third type of hex-derived networks H D N 3 ( r ) , T H D N 3 ( r ) , R H D N 3 ( r ) , C H D N 3 ( r ) , and compute exact results for topological indices which are based on degrees of end vertices. Full article
(This article belongs to the Special Issue Computational Methods in Analysis and Applications)
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Open AccessArticle
Solving ODEs by Obtaining Purely Second Degree Multinomials via Branch and Bound with Admissible Heuristic
Mathematics 2019, 7(4), 367; https://doi.org/10.3390/math7040367
Received: 6 February 2019 / Revised: 12 April 2019 / Accepted: 16 April 2019 / Published: 22 April 2019
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Abstract
Probabilistic evolution theory (PREVTH) forms a framework for the solution of explicit ODEs. The purpose of the paper is two-fold: (1) conversion of multinomial right-hand sides of the ODEs to purely second degree multinomial right-hand sides by space extension; (2) decrease the computational [...] Read more.
Probabilistic evolution theory (PREVTH) forms a framework for the solution of explicit ODEs. The purpose of the paper is two-fold: (1) conversion of multinomial right-hand sides of the ODEs to purely second degree multinomial right-hand sides by space extension; (2) decrease the computational burden of probabilistic evolution theory by using the condensed Kronecker product. A first order ODE set with multinomial right-hand side functions may be converted to a first order ODE set with purely second degree multinomial right-hand side functions at the expense of an increase in the number of equations and unknowns. Obtaining purely second degree multinomial right-hand side functions is important because the solution of such equation set may be approximated by probabilistic evolution theory. A recent article by the authors states that the ODE set with the smallest number of unknowns can be found by searching. This paper gives the details of a way to search for the optimal space extension. As for the second purpose of the paper, the computational burden can be reduced by considering the properties of the Kronecker product of vectors and how the Kronecker product appears within the recursion of PREVTH: as a Cauchy product structure. Full article
(This article belongs to the Section Mathematics and Computers Science)
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Open AccessArticle
Reformulated Zagreb Indices of Some Derived Graphs
Mathematics 2019, 7(4), 366; https://doi.org/10.3390/math7040366
Received: 15 March 2019 / Revised: 13 April 2019 / Accepted: 15 April 2019 / Published: 22 April 2019
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Abstract
A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. [...] Read more.
A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Open AccessArticle
B-Spline Solutions of General Euler-Lagrange Equations
Mathematics 2019, 7(4), 365; https://doi.org/10.3390/math7040365
Received: 28 March 2019 / Revised: 13 April 2019 / Accepted: 15 April 2019 / Published: 22 April 2019
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Abstract
The Euler-Lagrange equations are useful for solving optimization problems in mechanics. In this paper, we study the B-spline solutions of the Euler-Lagrange equations associated with the general functionals. The existing conditions of B-spline solutions to general Euler-Lagrange equations are given. As part of [...] Read more.
The Euler-Lagrange equations are useful for solving optimization problems in mechanics. In this paper, we study the B-spline solutions of the Euler-Lagrange equations associated with the general functionals. The existing conditions of B-spline solutions to general Euler-Lagrange equations are given. As part of this work, we present a general method for generating B-spline solutions of the second- and fourth-order Euler-Lagrange equations. Furthermore, we show that some existing techniques for surface design, such as Coons patches, are exactly the special cases of the generalized Partial differential equations (PDE) surfaces with appropriate choices of the constants. Full article
(This article belongs to the Special Issue Discrete and Computational Geometry)
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Open AccessArticle
Certain Chebyshev-Type Inequalities Involving Fractional Conformable Integral Operators
Mathematics 2019, 7(4), 364; https://doi.org/10.3390/math7040364
Received: 2 April 2019 / Revised: 15 April 2019 / Accepted: 16 April 2019 / Published: 21 April 2019
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Abstract
Since an interesting functional by P.L. Chebyshev was presented in the year 1882, many results, which are called Chebyshev-type inequalities, have been established. Some of these inequalities were obtained by using fractional integral operators. Very recently, a new variant of the fractional conformable [...] Read more.
Since an interesting functional by P.L. Chebyshev was presented in the year 1882, many results, which are called Chebyshev-type inequalities, have been established. Some of these inequalities were obtained by using fractional integral operators. Very recently, a new variant of the fractional conformable integral operator was introduced by Jarad et al. Motivated by this operator, we aim at establishing novel inequalities for a class of differentiable functions, which are associated with Chebyshev’s functional, by employing a fractional conformable integral operator. We also aim at showing important connections of the results here with those including Riemann–Liouville fractional and classical integrals. Full article
(This article belongs to the Special Issue Special Functions and Applications)
Open AccessArticle
Numerical Approximation for Nonlinear Noisy Leaky Integrate-and-Fire Neuronal Model
Mathematics 2019, 7(4), 363; https://doi.org/10.3390/math7040363
Received: 1 March 2019 / Revised: 14 April 2019 / Accepted: 15 April 2019 / Published: 21 April 2019
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Abstract
We consider a noisy leaky integrate-and-fire (NLIF) neuron model. The resulting nonlinear time-dependent partial differential equation (PDE) is a Fokker-Planck Equation (FPE) which describes the evolution of the probability density. The finite element method (FEM) has been proposed to solve the governing PDE. [...] Read more.
We consider a noisy leaky integrate-and-fire (NLIF) neuron model. The resulting nonlinear time-dependent partial differential equation (PDE) is a Fokker-Planck Equation (FPE) which describes the evolution of the probability density. The finite element method (FEM) has been proposed to solve the governing PDE. In the realistic neural network, the irregular space is always determined. Thus, FEM can be used to tackle those situations whereas other numerical schemes are restricted to the problems with only a finite regular space. The stability of the proposed scheme is also discussed. A comparison with the existing Weighted Essentially Non-Oscillatory (WENO) finite difference approximation is also provided. The numerical results reveal that FEM may be a better scheme for the solution of such types of model problems. The numerical scheme also reduces computational time in comparison with time required by other schemes. Full article
(This article belongs to the Section Mathematics and Computers Science)
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Open AccessArticle
Graphs Based on Hoop Algebras
Mathematics 2019, 7(4), 362; https://doi.org/10.3390/math7040362
Received: 14 March 2019 / Revised: 13 April 2019 / Accepted: 17 April 2019 / Published: 21 April 2019
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Abstract
In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. By using [...] Read more.
In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. By using zero divisor elements, we construct a productive graph and prove that it is connected and both complete and a tree under some conditions. Full article
(This article belongs to the Special Issue General Algebraic Structures)
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Open AccessArticle
Goodness of Fit Tests for the Log-Logistic Distribution Based on Cumulative Entropy under Progressive Type II Censoring
Mathematics 2019, 7(4), 361; https://doi.org/10.3390/math7040361
Received: 24 March 2019 / Revised: 11 April 2019 / Accepted: 16 April 2019 / Published: 20 April 2019
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Abstract
In this paper, we propose two new methods to perform goodness-of-fit tests on the log-logistic distribution under progressive Type II censoring based on the cumulative residual Kullback-Leibler information and cumulative Kullback-Leibler information. Maximum likelihood estimation and the EM algorithm are used for statistical [...] Read more.
In this paper, we propose two new methods to perform goodness-of-fit tests on the log-logistic distribution under progressive Type II censoring based on the cumulative residual Kullback-Leibler information and cumulative Kullback-Leibler information. Maximum likelihood estimation and the EM algorithm are used for statistical inference of the unknown parameter. The Monte Carlo simulation is conducted to study the power analysis on the alternative distributions of the hazard function monotonically increasing and decreasing. Finally, we present illustrative examples to show the applicability of the proposed methods. Full article
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Open AccessArticle
A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation
Mathematics 2019, 7(4), 360; https://doi.org/10.3390/math7040360
Received: 31 January 2019 / Revised: 6 April 2019 / Accepted: 16 April 2019 / Published: 20 April 2019
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Abstract
In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the [...] Read more.
In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems - Theory and Applications)
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Open AccessArticle
A Hybrid Framework Combining Genetic Algorithm with Iterated Local Search for the Dominating Tree Problem
Mathematics 2019, 7(4), 359; https://doi.org/10.3390/math7040359
Received: 29 March 2019 / Revised: 11 April 2019 / Accepted: 11 April 2019 / Published: 19 April 2019
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Abstract
Given an undirected, connected and edge-weighted graph, the dominating tree problem consists of finding a tree with minimum total edge weight such that for each vertex is either in the tree or adjacent to a vertex in the tree. In this paper, we [...] Read more.
Given an undirected, connected and edge-weighted graph, the dominating tree problem consists of finding a tree with minimum total edge weight such that for each vertex is either in the tree or adjacent to a vertex in the tree. In this paper, we propose a hybrid framework combining genetic algorithm with iterated local search (GAITLS) for solving the dominating tree problem. The main components of our framework are as follows: (1) the score functions D s c o r e and W s c o r e applied in the initialization and local search phase; (2) the initialization procedure with restricted candidate list (RCL) by controlling the parameter to balance the greediness and randomness; (3) the iterated local search with three phases, which is used to intensify the individuals; (4) the mutation with high diversity proposed to perturb the population. The experimental results on the classical instances show that our method performs much better than the-state-of-art algorithms. Full article
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Open AccessArticle
A Fast and Robust Spectrogram Reassignment Method
Mathematics 2019, 7(4), 358; https://doi.org/10.3390/math7040358
Received: 17 March 2019 / Revised: 5 April 2019 / Accepted: 14 April 2019 / Published: 19 April 2019
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Abstract
The improvement of the readability of time-frequency transforms is an important topic in the field of fast-oscillating signal processing. The reassignment method is often used due to its adaptivity to different transforms and nice formal properties. However, it strongly depends on the selection [...] Read more.
The improvement of the readability of time-frequency transforms is an important topic in the field of fast-oscillating signal processing. The reassignment method is often used due to its adaptivity to different transforms and nice formal properties. However, it strongly depends on the selection of the analysis window and it requires the computation of the same transform using three different but well-defined windows. The aim of this work is to provide a simple method for spectrogram reassignment, named FIRST (Fast Iterative and Robust Reassignment Thinning), with comparable or better precision than classical reassignment method, a reduced computational effort, and a near independence of the adopted analysis window. To this aim, the time-frequency evolution of a multicomponent signal is formally provided and, based on this law, only a subset of time-frequency points is used to improve spectrogram readability. Those points are the ones less influenced by interfering components. Preliminary results show that the proposed method can efficiently reassign spectrograms more accurately than the classical method in the case of interfering signal components, with a significant gain in terms of required computational effort. Full article
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Open AccessArticle
An Enhanced Partial Search to Particle Swarm Optimization for Unconstrained Optimization
Mathematics 2019, 7(4), 357; https://doi.org/10.3390/math7040357
Received: 18 February 2019 / Revised: 7 April 2019 / Accepted: 9 April 2019 / Published: 17 April 2019
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Abstract
Particle swarm optimization (PSO) is a population-based optimization technique that has been applied extensively to a wide range of engineering problems. This paper proposes a variation of the original PSO algorithm for unconstrained optimization, dubbed the enhanced partial search particle swarm optimizer (EPS-PSO), [...] Read more.
Particle swarm optimization (PSO) is a population-based optimization technique that has been applied extensively to a wide range of engineering problems. This paper proposes a variation of the original PSO algorithm for unconstrained optimization, dubbed the enhanced partial search particle swarm optimizer (EPS-PSO), using the idea of cooperative multiple swarms in an attempt to improve the convergence and efficiency of the original PSO algorithm. The cooperative searching strategy is particularly devised to prevent the particles from being trapped into the local optimal solutions and tries to locate the global optimal solution efficiently. The effectiveness of the proposed algorithm is verified through the simulation study where the EPS-PSO algorithm is compared to a variety of exiting “cooperative” PSO algorithms in terms of noted benchmark functions. Full article
(This article belongs to the Section Mathematics and Computers Science)
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Open AccessArticle
A Hierarchical Approach for Joint Parameter and State Estimation of a Bilinear System with Autoregressive Noise
Mathematics 2019, 7(4), 356; https://doi.org/10.3390/math7040356
Received: 7 March 2019 / Revised: 31 March 2019 / Accepted: 8 April 2019 / Published: 17 April 2019
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Abstract
This paper is concerned with the joint state and parameter estimation methods for a bilinear system in the state space form, which is disturbed by additive noise. In order to overcome the difficulty that the model contains the product term of the system [...] Read more.
This paper is concerned with the joint state and parameter estimation methods for a bilinear system in the state space form, which is disturbed by additive noise. In order to overcome the difficulty that the model contains the product term of the system input and states, we make use of the hierarchical identification principle to present new methods for estimating the system parameters and states interactively. The unknown states are first estimated via a bilinear state estimator on the basis of the Kalman filtering algorithm. Then, a state estimator-based recursive generalized least squares (RGLS) algorithm is formulated according to the least squares principle. To improve the parameter estimation accuracy, we introduce the data filtering technique to derive a data filtering-based two-stage RGLS algorithm. The simulation example indicates the efficiency of the proposed algorithms. Full article
(This article belongs to the Section Engineering Mathematics)
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Open AccessArticle
An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization
Mathematics 2019, 7(4), 355; https://doi.org/10.3390/math7040355
Received: 3 February 2019 / Revised: 1 April 2019 / Accepted: 10 April 2019 / Published: 16 April 2019
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Abstract
The B-spline function representation is commonly used for data approximation and trajectory definition, but filter-based methods for nonlinear weighted least squares (NWLS) approximation are restricted to a bounded definition range. We present an algorithm termed nonlinear recursive B-spline approximation (NRBA) for an iterative [...] Read more.
The B-spline function representation is commonly used for data approximation and trajectory definition, but filter-based methods for nonlinear weighted least squares (NWLS) approximation are restricted to a bounded definition range. We present an algorithm termed nonlinear recursive B-spline approximation (NRBA) for an iterative NWLS approximation of an unbounded set of data points by a B-spline function. NRBA is based on a marginalized particle filter (MPF), in which a Kalman filter (KF) solves the linear subproblem optimally while a particle filter (PF) deals with nonlinear approximation goals. NRBA can adjust the bounded definition range of the approximating B-spline function during run-time such that, regardless of the initially chosen definition range, all data points can be processed. In numerical experiments, NRBA achieves approximation results close to those of the Levenberg–Marquardt algorithm. An NWLS approximation problem is a nonlinear optimization problem. The direct trajectory optimization approach also leads to a nonlinear problem. The computational effort of most solution methods grows exponentially with the trajectory length. We demonstrate how NRBA can be applied for a multiobjective trajectory optimization for a battery electric vehicle in order to determine an energy-efficient velocity trajectory. With NRBA, the effort increases only linearly with the processed data points and the trajectory length. Full article
(This article belongs to the Special Issue Recent Trends in Multiobjective Optimization and Optimal Control)
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Stability, Existence and Uniqueness of Boundary Value Problems for a Coupled System of Fractional Differential Equations
Mathematics 2019, 7(4), 354; https://doi.org/10.3390/math7040354
Received: 20 February 2019 / Revised: 10 April 2019 / Accepted: 10 April 2019 / Published: 16 April 2019
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Abstract
The current article studies a coupled system of fractional differential equations with boundary conditions and proves the existence and uniqueness of solutions by applying Leray-Schauder’s alternative and contraction mapping principle. Furthermore, the Hyers-Ulam stability of solutions is discussed and sufficient conditions for the [...] Read more.
The current article studies a coupled system of fractional differential equations with boundary conditions and proves the existence and uniqueness of solutions by applying Leray-Schauder’s alternative and contraction mapping principle. Furthermore, the Hyers-Ulam stability of solutions is discussed and sufficient conditions for the stability are developed. Obtained results are supported by examples and illustrated in the last section. Full article
Open AccessArticle
Refined Neutrosophy and Lattices vs. Pair Structures and YinYang Bipolar Fuzzy Set
Mathematics 2019, 7(4), 353; https://doi.org/10.3390/math7040353
Received: 4 March 2019 / Revised: 30 March 2019 / Accepted: 1 April 2019 / Published: 16 April 2019
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Abstract
In this paper, we present the lattice structures of neutrosophic theories. We prove that Zhang-Zhang’s YinYang bipolar fuzzy set is a subclass of the Single-Valued bipolar neutrosophic set. Then we show that the pair structure is a particular case of refined neutrosophy, and [...] Read more.
In this paper, we present the lattice structures of neutrosophic theories. We prove that Zhang-Zhang’s YinYang bipolar fuzzy set is a subclass of the Single-Valued bipolar neutrosophic set. Then we show that the pair structure is a particular case of refined neutrosophy, and the number of types of neutralities (sub-indeterminacies) may be any finite or infinite number. Full article
(This article belongs to the Special Issue New Challenges in Neutrosophic Theory and Applications)
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Open AccessArticle
Value Distribution and Arbitrary-Order Derivatives of Meromorphic Solutions of Complex Linear Differential Equations in the Unit Disc
Mathematics 2019, 7(4), 352; https://doi.org/10.3390/math7040352
Received: 14 February 2019 / Revised: 3 April 2019 / Accepted: 5 April 2019 / Published: 15 April 2019
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Abstract
In this paper, we investigate the value distribution of meromorphic solutions and their arbitrary-order derivatives of the complex linear differential equation f+A(z)f+B(z)f=F(z) in [...] Read more.
In this paper, we investigate the value distribution of meromorphic solutions and their arbitrary-order derivatives of the complex linear differential equation f + A ( z ) f + B ( z ) f = F ( z ) in Δ with analytic or meromorphic coefficients of finite iterated p-order, and obtain some results on the estimates of the iterated exponent of convergence of meromorphic solutions and their arbitrary-order derivatives taking small function values. Full article
Open AccessArticle
Some Liouville Theorems on Finsler Manifolds
Mathematics 2019, 7(4), 351; https://doi.org/10.3390/math7040351
Received: 22 January 2019 / Revised: 6 April 2019 / Accepted: 11 April 2019 / Published: 15 April 2019
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Abstract
We give some Liouville type theorems of Lp harmonic (resp. subharmonic, superharmonic) functions on a complete noncompact Finsler manifold. Using the geometric relationship between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility. These improve the recent [...] Read more.
We give some Liouville type theorems of L p harmonic (resp. subharmonic, superharmonic) functions on a complete noncompact Finsler manifold. Using the geometric relationship between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility. These improve the recent literature (Zhang and Xia, 2014). Full article
Open AccessArticle
Global Stability Analysis of Two-Stage Quarantine-Isolation Model with Holling Type II Incidence Function
Mathematics 2019, 7(4), 350; https://doi.org/10.3390/math7040350
Received: 28 February 2019 / Revised: 27 March 2019 / Accepted: 10 April 2019 / Published: 15 April 2019
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Abstract
A new two-stage model for assessing the effect of basic control measures, quarantine and isolation, on a general disease transmission dynamic in a population is designed and rigorously analyzed. The model uses the Holling II incidence function for the infection rate. First, the [...] Read more.
A new two-stage model for assessing the effect of basic control measures, quarantine and isolation, on a general disease transmission dynamic in a population is designed and rigorously analyzed. The model uses the Holling II incidence function for the infection rate. First, the basic reproduction number ( R 0 ) is determined. The model has both locally and globally asymptotically stable disease-free equilibrium whenever R 0 < 1 . If R 0 > 1 , then the disease is shown to be uniformly persistent. The model has a unique endemic equilibrium when R 0 > 1 . A nonlinear Lyapunov function is used in conjunction with LaSalle Invariance Principle to show that the endemic equilibrium is globally asymptotically stable for a special case. Full article
(This article belongs to the Section Mathematics and Computers Science)
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Open AccessArticle
Best Proximity Point Theorems for Two Weak Cyclic Contractions on Metric-Like Spaces
Mathematics 2019, 7(4), 349; https://doi.org/10.3390/math7040349
Received: 3 April 2019 / Revised: 10 April 2019 / Accepted: 11 April 2019 / Published: 14 April 2019
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Abstract
In this paper, we establish two best proximity point theorems in the setting of metric-like spaces that are based on cyclic contraction: Meir–Keeler–Kannan type cyclic contractions and a generalized Ćirić type cyclic φ-contraction via the MT-function. We express some examples to [...] Read more.
In this paper, we establish two best proximity point theorems in the setting of metric-like spaces that are based on cyclic contraction: Meir–Keeler–Kannan type cyclic contractions and a generalized Ćirić type cyclic φ -contraction via the MT -function. We express some examples to indicate the validity of the presented results. Our results unify and generalize a number of best proximity point results on the topic in the corresponding recent literature. Full article
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