Research

Open AccessArticle

Generation of Julia and Mandelbrot Sets via Fixed Points

by

Mujahid Abbas

,

Hira Iqbal

and

Manuel De la Sen

Symmetry 2020, 12(1), 86; https://doi.org/10.3390/sym12010086 - 02 Jan 2020

Cited by 13 | Viewed by 3029

Abstract

The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form [...] Read more.

The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form

T (x) = x^{n} + m x + r

where

m, r \in C

and

n \geq 2

. Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Mandelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessArticle

The Asymmetric Alpha-Power Skew-t Distribution

by

Roger Tovar-Falón

,

Heleno Bolfarine

and

Guillermo Martínez-Flórez

Symmetry 2020, 12(1), 82; https://doi.org/10.3390/sym12010082 - 02 Jan 2020

Cited by 3 | Viewed by 2105

Abstract

In this paper, we propose a new asymmetric and heavy-tail model that generalizes both the skew-t and power-t models. Properties of the model are studied in detail. The score functions and the elements of the observed information matrix are given. The [...] Read more.

In this paper, we propose a new asymmetric and heavy-tail model that generalizes both the skew-t and power-t models. Properties of the model are studied in detail. The score functions and the elements of the observed information matrix are given. The process to estimate the parameters in model is discussed by using the maximum likelihood approach. Also, the observed information matrix is shown to be non-singular at the whole parametric space. Two applications to real data sets are reported to demonstrate the usefulness of this new model. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessArticle

Parametric Jensen-Shannon Statistical Complexity and Its Applications on Full-Scale Compartment Fire Data

by

Flavia-Corina Mitroi-Symeonidis

,

Ion Anghel

and

Nicușor Minculete

Symmetry 2020, 12(1), 22; https://doi.org/10.3390/sym12010022 - 20 Dec 2019

Cited by 7 | Viewed by 2161

Abstract

The order/disorder characteristics of a compartment fire are researched based on experimental data. From our analysis performed by new, pioneering methods, we claim that the parametric Jensen-Shannon complexity can be successfully used to detect unusual data, and that one can use it also [...] Read more.

The order/disorder characteristics of a compartment fire are researched based on experimental data. From our analysis performed by new, pioneering methods, we claim that the parametric Jensen-Shannon complexity can be successfully used to detect unusual data, and that one can use it also as a means to perform relevant analysis of fire experiments. Thoroughly comparing the performance of different algorithms (known as permutation entropy and two-length permutation entropy) to extract the probability distribution is an essential step. We discuss some of the theoretical assumptions behind each step and stress that the role of the parameter is to fine-tune the results of the Jensen-Shannon statistical complexity. Note that the Jensen-Shannon statistical complexity is symmetric, while its parametric version displays a symmetric duality due to the a priori probabilities used. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessArticle

On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives

by

Janak Raj Sharma

,

Sunil Kumar

and

Lorentz Jäntschi

Symmetry 2019, 11(12), 1452; https://doi.org/10.3390/sym11121452 - 26 Nov 2019

Cited by 29 | Viewed by 1853

Abstract

Many optimal order multiple root techniques involving derivatives have been proposed in literature. On the contrary, optimal order multiple root techniques without derivatives are almost nonexistent. With this as a motivational factor, here we develop a family of optimal fourth-order derivative-free iterative schemes [...] Read more.

Many optimal order multiple root techniques involving derivatives have been proposed in literature. On the contrary, optimal order multiple root techniques without derivatives are almost nonexistent. With this as a motivational factor, here we develop a family of optimal fourth-order derivative-free iterative schemes for computing multiple roots. The procedure is based on two steps of which the first is Traub–Steffensen iteration and second is Traub–Steffensen-like iteration. Theoretical results proved for particular cases of the family are symmetric to each other. This feature leads us to prove the general result that shows the fourth-order convergence. Efficacy is demonstrated on different test problems that verifies the efficient convergent nature of the new methods. Moreover, the comparison of performance has proven the presented derivative-free techniques as good competitors to the existing optimal fourth-order methods that use derivatives. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

Open AccessArticle

Volume Preserving Maps Between p-Balls

by

Adrian Holhoş

and

Daniela Roşca

Symmetry 2019, 11(11), 1404; https://doi.org/10.3390/sym11111404 - 14 Nov 2019

Viewed by 1444

Abstract

We construct a volume preserving map

U_{p}

from the p-ball

B_{p} (r) = \{x \in R^{3}, {∥ x ∥}_{p} \leq r\}

to the regular octahedron

B_{1} (r^{'})

, for arbitrary [...] Read more.

We construct a volume preserving map

U_{p}

from the p-ball

B_{p} (r) = \{x \in R^{3}, {∥ x ∥}_{p} \leq r\}

to the regular octahedron

B_{1} (r^{'})

, for arbitrary

p > 0

. Then we calculate the inverse

U_{p}^{- 1}

and we also deduce explicit expressions for

U_{\infty}

and

U_{\infty}^{- 1} .

This allows us to construct volume preserving maps between arbitrary balls

B_{p} (r)

and

B_{p^{'}} (\tilde{r})

, and also to map uniform and refinable grids between them. Finally we list some possible applications of our maps. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessArticle

One-Dimensional Optimal System for 2D Rotating Ideal Gas

by

Andronikos Paliathanasis

Symmetry 2019, 11(9), 1115; https://doi.org/10.3390/sym11091115 - 03 Sep 2019

Cited by 20 | Viewed by 1586

Abstract

We derive the one-dimensional optimal system for a system of three partial differential equations, which describe the two-dimensional rotating ideal gas with polytropic parameter

γ > 2 .

The Lie symmetries and the one-dimensional optimal system are determined for the nonrotating and rotating [...] Read more.

We derive the one-dimensional optimal system for a system of three partial differential equations, which describe the two-dimensional rotating ideal gas with polytropic parameter

γ > 2 .

The Lie symmetries and the one-dimensional optimal system are determined for the nonrotating and rotating systems. We compare the results, and we find that when there is no Coriolis force, the system admits eight Lie point symmetries, while the rotating system admits seven Lie point symmetries. Consequently, the two systems are not algebraic equivalent as in the case of

γ = 2

, which was found by previous studies. For the one-dimensional optimal system, we determine all the Lie invariants, while we demonstrate our results by reducing the system of partial differential equations into a system of first-order ordinary differential equations, which can be solved by quadratures. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

Open AccessArticle

Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory

by

,

,

and

Symmetry 2019, 11(9), 1085; https://doi.org/10.3390/sym11091085 - 29 Aug 2019

Cited by 5 | Viewed by 2027

Abstract

To address issues involving inconsistencies, this paper proposes a stochastic multi-criteria group decision making algorithm based on neutrosophic soft sets, which includes a pair of asymmetric functions: Truth-membership and false-membership, and an indeterminacy-membership function. For integrating an inherent stochastic, the algorithm expresses the [...] Read more.

To address issues involving inconsistencies, this paper proposes a stochastic multi-criteria group decision making algorithm based on neutrosophic soft sets, which includes a pair of asymmetric functions: Truth-membership and false-membership, and an indeterminacy-membership function. For integrating an inherent stochastic, the algorithm expresses the weights of decision makers and parameter subjective weights by neutrosophic numbers instead of determinate values. Additionally, the algorithm is guided by the prospect theory, which incorporates psychological expectations of decision makers into decision making. To construct the prospect decision matrix, this research establishes a conflict degree measure of neutrosophic numbers and improves it to accommodate the stochastic multi-criteria group decision making. Moreover, we introduce the weighted average aggregation rule and weighted geometric aggregation rule of neutrosophic soft sets. Later, this study presents an algorithm for neutrosophic soft sets in the stochastic multi-criteria group decision making based on the prospect theory. Finally, we perform an illustrative example and a comparative analysis to prove the effectiveness and feasibility of the proposed algorithm. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessArticle

On a Reduced Cost Higher Order Traub-Steffensen-Like Method for Nonlinear Systems

by

Janak Raj Sharma

,

Deepak Kumar

and

Lorentz Jäntschi

Symmetry 2019, 11(7), 891; https://doi.org/10.3390/sym11070891 - 08 Jul 2019

Cited by 4 | Viewed by 1637

Abstract

We propose a derivative-free iterative method with fifth order of convergence for solving systems of nonlinear equations. The scheme is composed of three steps, of which the first two steps are that of third order Traub-Steffensen-type method and the last is derivative-free modification [...] Read more.

We propose a derivative-free iterative method with fifth order of convergence for solving systems of nonlinear equations. The scheme is composed of three steps, of which the first two steps are that of third order Traub-Steffensen-type method and the last is derivative-free modification of Chebyshev’s method. Computational efficiency is examined and comparison between the efficiencies of presented technique with existing techniques is performed. It is proved that, in general, the new method is more efficient. Numerical problems, including those resulting from practical problems viz. integral equations and boundary value problems, are considered to compare the performance of the proposed method with existing methods. Calculation of computational order of convergence shows that the order of convergence of the new method is preserved in all the numerical examples, which is not so in the case of some of the existing higher order methods. Moreover, the numerical results, including the CPU-time consumed in the execution of program, confirm the accurate and efficient behavior of the new technique. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessEditor’s ChoiceArticle

A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested

by

Lorentz Jäntschi

Symmetry 2019, 11(6), 835; https://doi.org/10.3390/sym11060835 - 25 Jun 2019

Cited by 36 | Viewed by 6088

Abstract

One of the pillars of experimental science is sampling. Based on the analysis of samples, estimations for populations are made. There is an entire science based on sampling. Distribution of the population, of the sample, and the connection among those two (including sampling [...] Read more.

One of the pillars of experimental science is sampling. Based on the analysis of samples, estimations for populations are made. There is an entire science based on sampling. Distribution of the population, of the sample, and the connection among those two (including sampling distribution) provides rich information for any estimation to be made. Distributions are split into two main groups: continuous and discrete. The present study applies to continuous distributions. One of the challenges of sampling is its accuracy, or, in other words, how representative the sample is of the population from which it was drawn. To answer this question, a series of statistics have been developed to measure the agreement between the theoretical (the population) and observed (the sample) distributions. Another challenge, connected to this, is the presence of outliers - regarded here as observations wrongly collected, that is, not belonging to the population subjected to study. To detect outliers, a series of tests have been proposed, but mainly for normal (Gauss) distributions—the most frequently encountered distribution. The present study proposes a statistic (and a test) intended to be used for any continuous distribution to detect outliers by constructing the confidence interval for the extreme value in the sample, at a certain (preselected) risk of being in error, and depending on the sample size. The proposed statistic is operational for known distributions (with a known probability density function) and is also dependent on the statistical parameters of the population—here it is discussed in connection with estimating those parameters by the maximum likelihood estimation method operating on a uniform U(0,1) continuous symmetrical distribution. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessArticle

A Continuous Coordinate System for the Plane by Triangular Symmetry

by

Benedek Nagy

and

Khaled Abuhmaidan

Symmetry 2019, 11(2), 191; https://doi.org/10.3390/sym11020191 - 09 Feb 2019

Cited by 11 | Viewed by 5132

Abstract

The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only [...] Read more.

The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this paper, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems. The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [−1, 1], which gives many other vital properties of this coordinate system. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Graphical abstract

Open AccessArticle

Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations

by

M. Umar Farooq

Symmetry 2019, 11(2), 162; https://doi.org/10.3390/sym11020162 - 01 Feb 2019

Cited by 4 | Viewed by 1639

Abstract

Coupled systems of Lane–Emden equations are of considerable interest as they model several physical phenomena, for instance population evolution, pattern formation, and chemical reactions. Assuming a complex variational structure, we classify the generalized system of Lane–Emden type equations in relation to Noether-like operators [...] Read more.

Coupled systems of Lane–Emden equations are of considerable interest as they model several physical phenomena, for instance population evolution, pattern formation, and chemical reactions. Assuming a complex variational structure, we classify the generalized system of Lane–Emden type equations in relation to Noether-like operators and associated first integrals. Various forms of functions appearing in the considered system are taken, and it is observed that the Noether-like operators form an Abelian algebra for the corresponding Euler–Lagrange-type systems. Interestingly, we find that in many cases, the Noether-like operators satisfy the classical Noether symmetry condition and become the Noether symmetries. Moreover, we observe that the classical Noetherian integrals and the first integrals we determine using the complex Lagrangian approach turn out to be the same for the underlying system of Lane–Emden equations. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

Open AccessArticle

Minimal Energy Configurations of Finite Molecular Arrays

by

Pablo V. Negrón-Marrero

and

Melissa López-Serrano

Symmetry 2019, 11(2), 158; https://doi.org/10.3390/sym11020158 - 31 Jan 2019

Viewed by 1994

Abstract

In this paper, we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attractive or repulsive forces given by a certain intermolecular potential. We limit ourselves to the cases of three particles [...] Read more.

In this paper, we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attractive or repulsive forces given by a certain intermolecular potential. We limit ourselves to the cases of three particles arranged in a triangular array and that of four particles in a tetrahedral array. The minimization is constrained to a fixed area in the case of the triangular array, and to a fixed volume in the tetrahedral case. For a general class of intermolecular potentials we give conditions for the homogeneous configuration (either an equilateral triangle or a regular tetrahedron) of the array to be stable that is, a minimizer of the potential energy of the system. To determine whether or not there exist other stable states, the system of first-order necessary conditions for a minimum is treated as a bifurcation problem with the area or volume variable as the bifurcation parameter. Because of the symmetries present in our problem, we can apply the techniques of equivariant bifurcation theory to show that there exist branches of non-homogeneous solutions bifurcating from the trivial branch of homogeneous solutions at precisely the values of the parameter of area or volume for which the homogeneous configuration changes stability. For the triangular array, we construct numerically the bifurcation diagrams for both a Lennard–Jones and Buckingham potentials. The numerics show that there exist non-homogeneous stable states, multiple stable states for intervals of values of the area parameter, and secondary bifurcations as well. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessArticle

Facility Location Problem Approach for Distributed Drones

by

,

,

and

Symmetry 2019, 11(1), 118; https://doi.org/10.3390/sym11010118 - 20 Jan 2019

Cited by 20 | Viewed by 4565

Abstract

Currently, industry and academia are undergoing an evolution in developing the next generation of drone applications. Including the development of autonomous drones that can carry out tasks without the assistance of a human operator. In spite of this, there are still problems left [...] Read more.

Currently, industry and academia are undergoing an evolution in developing the next generation of drone applications. Including the development of autonomous drones that can carry out tasks without the assistance of a human operator. In spite of this, there are still problems left unanswered related to the placement of drone take-off, landing and charging areas. Future policies by governments and aviation agencies are inevitably going to restrict the operational area where drones can take-off and land. Hence, there is a need to develop a system to manage landing and take-off areas for drones. Additionally, we proposed this approach due to the lack of justification for the initial location of drones in current research. Therefore, to provide a foundation for future research, we give a justified reason that allows predetermined location of drones with the use of drone ports. Furthermore, we propose an algorithm to optimally place these drone ports to minimize the average distance drones must travel based on a set of potential drone port locations and tasks generated in a given area. Our approach is derived from the Facility Location problem which produces an efficient near optimal solution to place drone ports that reduces the overall drone energy consumption. Secondly, we apply various traveling salesman algorithms to determine the shortest route the drone must travel to visit all the tasks. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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Open AccessArticle

Khovanov Homology of Three-Strand Braid Links

by

,

,

,

,

and

Symmetry 2018, 10(12), 720; https://doi.org/10.3390/sym10120720 - 05 Dec 2018

Viewed by 2140

Abstract

Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some [...] Read more.

Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links

Δ^{2 k + 1} = x_{1}^{2 k + 2} x_{2} x_{1}^{2} x_{2}^{2} x_{1}^{2} \dots x_{2}^{2} x_{1}^{2} x_{1}^{2}

,

Δ^{2 k + 1} x_{2}

, and

Δ^{2 k + 1} x_{1}

, where

Δ

is the Garside element

x_{1} x_{2} x_{1}

, and which are three out of all six classes of the general braid

x_{1} x_{2} x_{1} x_{2} \dots

with n factors. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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