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# Mathematics, Volume 6, Issue 12 (December 2018) – 65 articles

Cover Story (view full-size image): We present a new family of optimal eighth order convergent iterative methods for finding multiple roots of nonlinear equations with known multiplicity. Since, root finding iterative methods have wide applications in many fields of science and engineering. So, the numerical performance and efficiency of the proposed methods is demonstrated by applying the new methods for some real life models from Life Science, Engineering and Physics. The dynamical performance of these methods is also analyzed through basins of attraction which proves their good behavior in terms of dependence on the initial estimations. View this paper.
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11 pages, 256 KiB
Article
Mean Values of Products of L-Functions and Bernoulli Polynomials
by Abdelmejid Bayad and Daeyeoul Kim
Mathematics 2018, 6(12), 337; https://doi.org/10.3390/math6120337 - 19 Dec 2018
Cited by 2 | Viewed by 2338
Abstract
Let $m 1 , ⋯ , m r$ be nonnegative integers, and set: $M r = m 1 + ⋯ + m r .$ In this paper, first we establish an explicit linear decomposition of: [...] Read more.
Let $m 1 , ⋯ , m r$ be nonnegative integers, and set: $M r = m 1 + ⋯ + m r .$ In this paper, first we establish an explicit linear decomposition of: $∏ i = 1 r B m i ( x ) m i !$ in terms of Bernoulli polynomials $B k ( x )$ with $0 ≤ k ≤ M r$ . Second, for any integer $q ≥ 2$ , we study the mean values of the Dirichlet L-functions at negative integers: $∑ χ 1 , ⋯ , χ r ( mod q ) ; χ 1 ⋯ χ r = 1 ∏ i = 1 r L ( − m i , χ i )$ where the summation is over Dirichlet characters $χ i$ modulo q. Incidentally, a part of our work recovers Nielsen’s theorem, Nörlund’s formula, and its generalization by Hu, Kim, and Kim. Full article
14 pages, 2577 KiB
Article
Aggregating a Plankton Food Web: Mathematical versus Biological Approaches
by Ferenc Jordán, Anett Endrédi, Wei-chung Liu and Domenico D’Alelio
Mathematics 2018, 6(12), 336; https://doi.org/10.3390/math6120336 - 19 Dec 2018
Cited by 10 | Viewed by 3744
Abstract
Species are embedded in a web of intricate trophic interactions. Understanding the functional role of species in food webs is of fundamental interests. This is related to food web position, so positional similarity may provide information about functional overlap. Defining and quantifying similar [...] Read more.
Species are embedded in a web of intricate trophic interactions. Understanding the functional role of species in food webs is of fundamental interests. This is related to food web position, so positional similarity may provide information about functional overlap. Defining and quantifying similar trophic functioning can be addressed in different ways. We consider two approaches. One is of mathematical nature involving network analysis where unique species can be defined as those whose topological position is very different to others in the same food web. A species is unique if it has very different connection pattern compared to others. The second approach is of biological nature, based on trait-based aggregations. Unique species are not easy to aggregate with others because their traits are not in common with the ones of most others. Our goal here is to illustrate how mathematics can provide an alternative perspective on species aggregation, and how this is related to its biological counterpart. We illustrate these approaches using a toy food web and a real food web and demonstrate the sensitive relationships between those approaches. The trait-based aggregation focusing on the trait values of size (sv) can be best predicted by the mathematical aggregation algorithms. Full article
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19 pages, 430 KiB
Article
Four Particular Cases of the Fourier Transform
by Jens V. Fischer
Mathematics 2018, 6(12), 335; https://doi.org/10.3390/math6120335 - 18 Dec 2018
Cited by 8 | Viewed by 13666
Abstract
In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula [...] Read more.
In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense. Full article
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8 pages, 223 KiB
Article
Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers
by Yuankui Ma and Wenpeng Zhang
Mathematics 2018, 6(12), 334; https://doi.org/10.3390/math6120334 - 18 Dec 2018
Cited by 17 | Viewed by 4349
Abstract
The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using [...] Read more.
The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using this new sequence, the properties of the power series, and the combinatorial methods. Full article
14 pages, 338 KiB
Article
Dynamics of the Almost Periodic Discrete Mackey–Glass Model
by Zhijian Yao, Jehad Alzabut and Debaldev Jana
Mathematics 2018, 6(12), 333; https://doi.org/10.3390/math6120333 - 17 Dec 2018
Cited by 4 | Viewed by 2382
Abstract
This paper is concerned with a class of the discrete Mackey–Glass model that describes the process of the production of blood cells. Prior to proceeding to the main results, we prove the boundedness and extinction of its solutions. By means of the contraction [...] Read more.
This paper is concerned with a class of the discrete Mackey–Glass model that describes the process of the production of blood cells. Prior to proceeding to the main results, we prove the boundedness and extinction of its solutions. By means of the contraction mapping principle and under appropriate assumptions, we prove the existence of almost periodic positive solutions. Furthermore and by the implementation of the discrete Lyapunov functional, sufficient conditions are established for the exponential convergence of the almost periodic positive solution. Examples, as well as numerical simulations are illustrated to demonstrate the effectiveness of the theoretical findings of the paper. Our results are new and generalize some previously-reported results in the literature. Full article
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10 pages, 267 KiB
Article
Some Symmetric Identities Involving the Stirling Polynomials Under the Finite Symmetric Group
by Dongkyu Lim and Feng Qi
Mathematics 2018, 6(12), 332; https://doi.org/10.3390/math6120332 - 17 Dec 2018
Cited by 1 | Viewed by 2965
Abstract
In the paper, the authors present some symmetric identities involving the Stirling polynomials and higher order Bernoulli polynomials under all permutations in the finite symmetric group of degree n. These identities extend and generalize some known results. Full article
10 pages, 2147 KiB
Article
Exact Solution of Ambartsumian Delay Differential Equation and Comparison with Daftardar-Gejji and Jafari Approximate Method
by Huda O. Bakodah and Abdelhalim Ebaid
Mathematics 2018, 6(12), 331; https://doi.org/10.3390/math6120331 - 17 Dec 2018
Cited by 31 | Viewed by 2775
Abstract
The Ambartsumian equation, a linear differential equation involving a proportional delay term, is used in the theory of surface brightness in the Milky Way. In this paper, the Laplace-transform was first applied to this equation, and then the decomposition method was implemented to [...] Read more.
The Ambartsumian equation, a linear differential equation involving a proportional delay term, is used in the theory of surface brightness in the Milky Way. In this paper, the Laplace-transform was first applied to this equation, and then the decomposition method was implemented to establish a closed-form solution. The present closed-form solution is reported for the first time for the Ambartsumian equation. Numerically, the calculations have demonstrated a rapid rate of convergence of the obtained approximate solutions, which are displayed in several graphs. It has also been shown that only a few terms of the new approximate solution were sufficient to achieve extremely accurate numerical results. Furthermore, comparisons of the present results with the existing methods in the literature were introduced. Full article
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11 pages, 2211 KiB
Article
Fractional Modeling for Quantitative Inversion of Soil-Available Phosphorus Content
by Chengbiao Fu, Heigang Xiong and Anhong Tian
Mathematics 2018, 6(12), 330; https://doi.org/10.3390/math6120330 - 14 Dec 2018
Cited by 4 | Viewed by 2377
Abstract
The study of field spectra based on fractional-order differentials has rarely been reported, and traditional integer-order differentials only perform the derivative calculation for 1st-order or 2nd-order spectrum signals, ignoring the spectral transformation details between 0th-order to 1st-order and 1st-order to 2nd-order, resulting in [...] Read more.
The study of field spectra based on fractional-order differentials has rarely been reported, and traditional integer-order differentials only perform the derivative calculation for 1st-order or 2nd-order spectrum signals, ignoring the spectral transformation details between 0th-order to 1st-order and 1st-order to 2nd-order, resulting in the problem of low-prediction accuracy. In this paper, a spectral quantitative analysis model of soil-available phosphorus content based on a fractional-order differential is proposed. Firstly, a fractional-order differential was used to perform a derivative calculation of original spectral data from 0th-order to 2nd-order using 0.2-order intervals, to obtain 11 fractional-order spectrum data. Afterwards, seven bands with absolute correlation coefficient greater than 0.5 were selected as sensitive bands. Finally, a stepwise multiple linear regression algorithm was used to establish a spectral estimation model of soil-available phosphorus content under different orders, then the prediction effect of the model under different orders was compared and analyzed. Simulation results show that the best order for a soil-available phosphorus content regression model is a 0.6 fractional-order, the coefficient of determination ( $R 2$ ), root mean square error (RMSE), and ratio of performance to deviation (RPD) of the best model are 0.7888, 3.348878, and 2.001142, respectively. Since the RPD value is greater than 2, the optimal fractional model established in this study has good quantitative predictive ability for soil-available phosphorus content. Full article
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14 pages, 306 KiB
Article
Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials
by Yuan He, Serkan Araci, Hari M. Srivastava and Mahmoud Abdel-Aty
Mathematics 2018, 6(12), 329; https://doi.org/10.3390/math6120329 - 14 Dec 2018
Cited by 15 | Viewed by 2557
Abstract
In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the [...] Read more.
In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas. Full article
12 pages, 6526 KiB
Article
Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies
by Yanli Ma, Jia-Bao Liu and Haixia Li
Mathematics 2018, 6(12), 328; https://doi.org/10.3390/math6120328 - 14 Dec 2018
Cited by 26 | Viewed by 4371
Abstract
In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number $R 0$ , which [...] Read more.
In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number $R 0$ , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when $R 0 < 1$ , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when $R 0 > 1$ . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when $R 0 > 1$ . Finally, some numerical simulations are presented to illustrate the analysis results. Full article
(This article belongs to the Special Issue Mathematical Models in Epidemiology )
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12 pages, 1357 KiB
Article
Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations
by Phatiphat Thounthong, Muhammad Nawaz Khan, Iltaf Hussain, Imtiaz Ahmad and Poom Kumam
Mathematics 2018, 6(12), 327; https://doi.org/10.3390/math6120327 - 14 Dec 2018
Cited by 29 | Viewed by 3772
Abstract
In this paper, the symmetric radial basis function method is utilized for the numerical solution of two- and three-dimensional elliptic PDEs. Numerical results are obtained by using a set of uniform or random points. Numerical tests are accomplished to demonstrate the efficacy and [...] Read more.
In this paper, the symmetric radial basis function method is utilized for the numerical solution of two- and three-dimensional elliptic PDEs. Numerical results are obtained by using a set of uniform or random points. Numerical tests are accomplished to demonstrate the efficacy and accuracy of the method on both regular and irregular domains. Furthermore, the proposed method is tested for the solution of elliptic PDE in the case of various frequencies. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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15 pages, 270 KiB
Article
A Parameter-Based Ostrowski–Grüss Type Inequalities with Multiple Points for Derivatives Bounded by Functions on Time Scales
by Seth Kermausuor and Eze R. Nwaeze
Mathematics 2018, 6(12), 326; https://doi.org/10.3390/math6120326 - 14 Dec 2018
Cited by 4 | Viewed by 2189
Abstract
In this paper, we present some Ostrowski–Grüss-type inequalities on time scales for functions whose derivatives are bounded by functions for k points via a parameter. The 2D versions of these inequalities are also presented. Our results generalize some of the results in the [...] Read more.
In this paper, we present some Ostrowski–Grüss-type inequalities on time scales for functions whose derivatives are bounded by functions for k points via a parameter. The 2D versions of these inequalities are also presented. Our results generalize some of the results in the literature. As a by-product, we apply our results to the continuous and discrete calculus to obtain some interesting inequalities in this direction. Full article
19 pages, 895 KiB
Article
A Model and an Algorithm for a Large-Scale Sustainable Supplier Selection and Order Allocation Problem
by Jong Soo Kim, Eunhee Jeon, Jiseong Noh and Jun Hyeong Park
Mathematics 2018, 6(12), 325; https://doi.org/10.3390/math6120325 - 13 Dec 2018
Cited by 14 | Viewed by 3568
Abstract
We consider a buyer’s decision problem of sustainable supplier selection and order allocation (SSS & OA) among multiple heterogeneous suppliers who sell multiple types of items. The buyer periodically orders items from chosen suppliers to refill inventory to preset levels. Each supplier is [...] Read more.
We consider a buyer’s decision problem of sustainable supplier selection and order allocation (SSS & OA) among multiple heterogeneous suppliers who sell multiple types of items. The buyer periodically orders items from chosen suppliers to refill inventory to preset levels. Each supplier is differentiated from others by the types of items supplied, selling price, and order-related costs, such as transportation cost. Each supplier also has a preset requirement for minimum order quantity or minimum purchase amount. In the beginning of each period, the buyer constructs an SSS & OA plan considering various information from both parties. The buyer’s planning problem is formulated as a mathematical model, and an efficient algorithm to solve larger instances of the problem is developed. The algorithm is designed to take advantage of the branch-and-bound method, and the special structure of the model. We perform computer experiments to test the accuracy of the proposed algorithm. The test result confirmed that the algorithm can find a near-optimal solution with only 0.82 percent deviation on average. We also observed that the use of the algorithm can increase solvable problem size by about 2.4 times. Full article
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14 pages, 2904 KiB
Article
New Existence of Fixed Point Results in Generalized Pseudodistance Functions with Its Application to Differential Equations
by Sujitra Sanhan, Winate Sanhan and Chirasak Mongkolkeha
Mathematics 2018, 6(12), 324; https://doi.org/10.3390/math6120324 - 12 Dec 2018
Cited by 4 | Viewed by 2601
Abstract
The purpose of this article is to prove some existences of fixed point theorems for generalized $F$ -contraction mapping in metric spaces by using the concept of generalized pseudodistance. In addition, we give some examples to illustrate our main results. As the application, [...] Read more.
The purpose of this article is to prove some existences of fixed point theorems for generalized $F$ -contraction mapping in metric spaces by using the concept of generalized pseudodistance. In addition, we give some examples to illustrate our main results. As the application, the existence of the solution of the second order differential equation is given. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)
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8 pages, 226 KiB
Article
Soft Neutrosophic Modules
by Mikail Bal and Necati Olgun
Mathematics 2018, 6(12), 323; https://doi.org/10.3390/math6120323 - 12 Dec 2018
Cited by 4 | Viewed by 1937
Abstract
In this study, for the first time, we study some basic definitions of soft neutrosophic modules in algebra being generalized and its several related properties, structural characteristics are investigated with suitable examples. In this paper, we utilized neutrosophic soft sets and neutrosophic modules. [...] Read more.
In this study, for the first time, we study some basic definitions of soft neutrosophic modules in algebra being generalized and its several related properties, structural characteristics are investigated with suitable examples. In this paper, we utilized neutrosophic soft sets and neutrosophic modules. As a result, we defined soft neutrosophic modules. After weak soft neutrosophic modules and strong soft neutrosophic modules are described and illustrated by examples. Finally soft neutrosophic module homomorphism is defined and soft neutrosophic module isomorphism is explained. Full article
17 pages, 2055 KiB
Article
Coopetitive Games for Management of Marine Transportation Activity: A Study Case
by Alessia Donato, David Carfì and Beatrice Blandina
Mathematics 2018, 6(12), 322; https://doi.org/10.3390/math6120322 - 12 Dec 2018
Cited by 3 | Viewed by 2682
Abstract
In this paper, we will use coopetitive game theory to analyze a case of real coopetition among port companies, for what concerns loading and unloading of goods, within a competitive management scenario of marine transportation activities. Our research consists of the analysis of [...] Read more.
In this paper, we will use coopetitive game theory to analyze a case of real coopetition among port companies, for what concerns loading and unloading of goods, within a competitive management scenario of marine transportation activities. Our research consists of the analysis of a study case involving coopetition between two real companies from which we obtained the financial and contractual data allowing us to define two modeling payoff functions, both of them based on real agreements and tariffs. We recognize actual coopetition and an asymmetric R&D alliance in this type of agreement, where a bigger enterprise deals with a smaller competitor, in order to capture more value from their activities. In particular, our model will show a precise coopetitive bi-dimensional trajectory within which we suggest, after a quantitative analysis, different kinds of solutions: the purely coopetitive solution, a Kalai-Smorodinsky solution and, finally, a transferable utility Kalai-Smorodinsky solution. Our methods provide specific strategy procedures determining win-win solutions for both. Full article
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20 pages, 370 KiB
Article
Serret-Frenet Frame and Curvatures of Bézier Curves
by Esra Erkan and Salim Yüce
Mathematics 2018, 6(12), 321; https://doi.org/10.3390/math6120321 - 12 Dec 2018
Cited by 19 | Viewed by 5869
Abstract
The aim of this study is to view the role of Bézier curves in both the Euclidean plane $E 2$ and Euclidean space $E 3$ with the help of the fundamental algorithm which is commonly used in Computer Science and Applied Mathematics and [...] Read more.
The aim of this study is to view the role of Bézier curves in both the Euclidean plane $E 2$ and Euclidean space $E 3$ with the help of the fundamental algorithm which is commonly used in Computer Science and Applied Mathematics and without this algorithm. The Serret-Frenet elements of non-unit speed curves in the Euclidean plane $E 2$ and Euclidean space $E 3$ are given by Gray et al. in 2016. We used these formulas to find Serret-Frenet elements of planar Bézier curve at the end points and for every parameter t. Moreover, we reconstruct these elements for a planar Bézier curve, which is defined by the help of the algorithm based on intermediate points. Finally, in the literature, the spatial Bézier curve only mentioned at the end points, so we improve these elements for all parameters t. Additionally, we calculate these elements for all parameters t using algorithm above mentioned for spatial Bézier curve. Full article
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10 pages, 269 KiB
Article
Double Controlled Metric Type Spaces and Some Fixed Point Results
by Thabet Abdeljawad, Nabil Mlaiki, Hassen Aydi and Nizar Souayah
Mathematics 2018, 6(12), 320; https://doi.org/10.3390/math6120320 - 12 Dec 2018
Cited by 111 | Viewed by 5838
Abstract
In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions $α ( x , y )$ and $μ ( x , y )$ on the right-hand side of the $b -$ triangle [...] Read more.
In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions $α ( x , y )$ and $μ ( x , y )$ on the right-hand side of the $b -$ triangle inequality, that is, $d ( x , y ) ≤ α ( x , z ) d ( x , z ) + μ ( z , y ) d ( z , y ) , for all x , y , z ∈ X .$ Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and $ϕ$ -nonlinear type contractions in the setting of double controlled metric type spaces. Full article
26 pages, 350 KiB
Article
Bayesian and Non-Bayesian Inference for the Generalized Pareto Distribution Based on Progressive Type II Censored Sample
by Xuehua Hu and Wenhao Gui
Mathematics 2018, 6(12), 319; https://doi.org/10.3390/math6120319 - 11 Dec 2018
Cited by 1 | Viewed by 2469
Abstract
In this paper, first we consider the maximum likelihood estimators for two unknown parameters, reliability and hazard functions of the generalized Pareto distribution under progressively Type II censored sample. Next, we discuss the asymptotic confidence intervals for two unknown parameters, reliability and hazard [...] Read more.
In this paper, first we consider the maximum likelihood estimators for two unknown parameters, reliability and hazard functions of the generalized Pareto distribution under progressively Type II censored sample. Next, we discuss the asymptotic confidence intervals for two unknown parameters, reliability and hazard functions by using the delta method. Then, based on the bootstrap algorithm, we obtain another two pairs of approximate confidence intervals. Furthermore, by applying the Markov Chain Monte Carlo techniques, we derive the Bayesian estimates of the two unknown parameters, reliability and hazard functions under various balanced loss functions and the corresponding confidence intervals. A simulation study was conducted to compare the performances of the proposed estimators. A real dataset analysis was carried out to illustrate the proposed methods. Full article
18 pages, 285 KiB
Article
Classification Theorems of Ruled Surfaces in Minkowski Three-Space
by Miekyung Choi and Young Ho Kim
Mathematics 2018, 6(12), 318; https://doi.org/10.3390/math6120318 - 11 Dec 2018
Cited by 2 | Viewed by 2066
Abstract
By generalizing the notion of the pointwise 1-type Gauss map, the generalized 1-type Gauss map has been recently introduced. Without any assumption, we classified all possible ruled surfaces with the generalized 1-type Gauss map in a 3-dimensional Minkowski space. In particular, null scrolls [...] Read more.
By generalizing the notion of the pointwise 1-type Gauss map, the generalized 1-type Gauss map has been recently introduced. Without any assumption, we classified all possible ruled surfaces with the generalized 1-type Gauss map in a 3-dimensional Minkowski space. In particular, null scrolls do not have the proper generalized 1-type Gauss map. In fact, it is harmonic. Full article
12 pages, 419 KiB
Article
Statistical Analysis of Maximally Similar Sets in Ecological Research
by David W. Roberts
Mathematics 2018, 6(12), 317; https://doi.org/10.3390/math6120317 - 11 Dec 2018
Viewed by 2559
Abstract
Maximally similar sets (MSSs) are sets of elements that share a neighborhood in a high-dimensional space defined by a symmetric, reflexive similarity relation. Each element of the universe is employed as the kernel of a neighborhood of a given size (number of members), [...] Read more.
Maximally similar sets (MSSs) are sets of elements that share a neighborhood in a high-dimensional space defined by a symmetric, reflexive similarity relation. Each element of the universe is employed as the kernel of a neighborhood of a given size (number of members), and elements are added to the neighborhood in order of similarity to the current members of the set until the desired neighborhood size is achieved. The set of neighborhoods is then reduced to the set of unique, maximally similar sets by eliminating all sets that are permutations of an existing set. Subsequently, the within-MSS variability of candidate explanatory variables associated with the elements is compared to random sets of the same size to estimate the probability of obtaining variability as low as was observed. Explanatory variables can be compared for effect size by the rank order of within-MSS variability and random set variability, correcting for statistical power as necessary. The analyses performed identify constraints, as opposed to determinants, in the triangular distribution of pair-wise element similarity. In the example given here, the variability in spring temperature, summer temperature, and the growing degree days of forest vegetation sample units shows the greatest constraint on forest composition of a large set of candidate environmental variables. Full article
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31 pages, 442 KiB
Review
On the Riemann Function
by Peter J. Zeitsch
Mathematics 2018, 6(12), 316; https://doi.org/10.3390/math6120316 - 10 Dec 2018
Cited by 3 | Viewed by 4455
Abstract
Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in two variables. The first review of Riemann’s method was published by E.T. Copson in 1958. This study extends that work. Firstly, [...] Read more.
Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in two variables. The first review of Riemann’s method was published by E.T. Copson in 1958. This study extends that work. Firstly, three solution methods were overlooked in Copson’s original paper. Secondly, several new approaches for finding Riemann functions have been developed since 1958. Those techniques are included here and placed in the context of Copson’s original study. There are also numerous equivalences between Riemann functions that have not previously been identified in the literature. Those links are clarified here by showing that many known Riemann functions are often equivalent due to the governing equation admitting a symmetry algebra isomorphic to $S L ( 2 , R )$ . Alternatively, the equation admits a Lie-Bäcklund symmetry algebra. Combining the results from several methods, a new class of Riemann functions is then derived which admits no symmetries whatsoever. Full article
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13 pages, 782 KiB
Review
Scaling Laws in the Fine-Scale Structure of Range Margins
by Beáta Oborny
Mathematics 2018, 6(12), 315; https://doi.org/10.3390/math6120315 - 9 Dec 2018
Cited by 5 | Viewed by 2377
Abstract
Margins of the geographic distributions of species are important regions in terms of ecological and evolutionary processes, including the species’ response to climate change. This paper reviews some spatially explicit metapopulation models of range margins across environmental gradients (e.g., across latitudes or altitudes). [...] Read more.
Margins of the geographic distributions of species are important regions in terms of ecological and evolutionary processes, including the species’ response to climate change. This paper reviews some spatially explicit metapopulation models of range margins across environmental gradients (e.g., across latitudes or altitudes). These models share some robust results, which allow for generalizations within a broad variety of species and environments: (1) sharp edges can emerge even across relatively smooth environmental gradients; (2) intraspecific competition combined with dispersal limitation is a sufficient condition for the sharpening; (3) at the margin, the “mainland” of continuous occurrence splits into “islands”. Computer simulations pointed out some characteristic scaling laws in the size distribution of the islands, and in the structure of the hull of the mainland. The hull is a fractal with a dimension 7/4. Its width and length scale with the gradient according to characteristic scaling laws (with exponents 3/7 and 4/7, respectively). These general features follow from a second-order phase transition from a connected to a fragmented state. The results contribute to understanding the origin of vegetation zones and the spatial pattern of ecotones. Full article
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86 pages, 894 KiB
Article
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition
by Alessandra Bernardi, Enrico Carlini, Maria Virginia Catalisano, Alessandro Gimigliano and Alessandro Oneto
Mathematics 2018, 6(12), 314; https://doi.org/10.3390/math6120314 - 8 Dec 2018
Cited by 24 | Viewed by 6114
Abstract
We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only [...] Read more.
We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject. Full article
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6 pages, 209 KiB
Article
A Note on the Classical Gauss Sums
by Tingting Wang and Guohui Chen
Mathematics 2018, 6(12), 313; https://doi.org/10.3390/math6120313 - 8 Dec 2018
Cited by 5 | Viewed by 2446
Abstract
The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums $mod p$ (a prime with [...] Read more.
The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums $mod p$ (a prime with $p ≡ 1 mod 12$ ) to give an exact evaluation formula for it. Full article
9 pages, 253 KiB
Article
On the Generalization of a Class of Harmonic Univalent Functions Defined by Differential Operator
by Aqeel Ketab AL-khafaji, Waggas Galib Atshan and Salwa Salman Abed
Mathematics 2018, 6(12), 312; https://doi.org/10.3390/math6120312 - 7 Dec 2018
Cited by 4 | Viewed by 2543
Abstract
In this article, a new class of harmonic univalent functions, defined by the differential operator, is introduced. Some geometric properties, like, coefficient estimates, extreme points, convex combination and convolution (Hadamard product) are obtained. Full article
7 pages, 252 KiB
Article
A Lichnerowicz–Obata–Cheng Type Theorem on Finsler Manifolds
by Songting Yin and Pan Zhang
Mathematics 2018, 6(12), 311; https://doi.org/10.3390/math6120311 - 7 Dec 2018
Cited by 1 | Viewed by 2127
Abstract
Let $( M , F , d μ )$ be a Finsler manifold with the Ricci curvature bounded below by a positive number and constant S-curvature. We prove that, if the first eigenvalue of the Finsler–Laplacian attains its lower bound, then M [...] Read more.
Let $( M , F , d μ )$ be a Finsler manifold with the Ricci curvature bounded below by a positive number and constant S-curvature. We prove that, if the first eigenvalue of the Finsler–Laplacian attains its lower bound, then M is isometric to a Finsler sphere. Moreover, we establish a comparison result on the Hessian trace of the distance function. Full article
16 pages, 1028 KiB
Article
An Efficient Family of Optimal Eighth-Order Multiple Root Finders
by Fiza Zafar, Alicia Cordero and Juan R. Torregrosa
Mathematics 2018, 6(12), 310; https://doi.org/10.3390/math6120310 - 7 Dec 2018
Cited by 13 | Viewed by 2880
Abstract
Finding a repeated zero for a nonlinear equation $f ( x ) = 0$ , $f : I ⊆ R → R$ has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified [...] Read more.
Finding a repeated zero for a nonlinear equation $f ( x ) = 0$ , $f : I ⊆ R → R$ has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton’s method is usually applied to solve this kind of problems. Keeping in view that very few optimal higher-order convergent methods exist for multiple roots, we present a new family of optimal eighth-order convergent iterative methods for multiple roots with known multiplicity involving a multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from life science, engineering, and physics are considered for the sake of comparison. The numerical experiments and dynamical analysis show that our proposed methods are efficient for determining multiple roots of nonlinear equations. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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15 pages, 721 KiB
Article
Kempe-Locking Configurations
by James Tilley
Mathematics 2018, 6(12), 309; https://doi.org/10.3390/math6120309 - 7 Dec 2018
Cited by 1 | Viewed by 4118
Abstract
The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored [...] Read more.
The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored in this article is that the connectivity and coloring properties are incompatible. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. All Kempe-locked triangulations that we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $x y$ , and (2) they have a Birkhoff diamond with endpoints x and y as a subgraph. On the strength of our investigations, we formulate a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample are indeed incompatible. It would also imply the appealing conclusion that the Birkhoff diamond configuration alone is responsible for the 4-colorability of planar triangulations. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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9 pages, 271 KiB
Article
On Discrete Fractional Solutions of Non-Fuchsian Differential Equations
by Resat Yilmazer, Mustafa Inc and Mustafa Bayram
Mathematics 2018, 6(12), 308; https://doi.org/10.3390/math6120308 - 7 Dec 2018
Cited by 2 | Viewed by 2058
Abstract
In this article, we obtain new fractional solutions of the general class of non-Fuchsian differential equations by using discrete fractional nabla operator $∇ η ( 0 < η < 1 )$ . This operator is applied to homogeneous and nonhomogeneous linear ordinary differential [...] Read more.
In this article, we obtain new fractional solutions of the general class of non-Fuchsian differential equations by using discrete fractional nabla operator $∇ η ( 0 < η < 1 )$ . This operator is applied to homogeneous and nonhomogeneous linear ordinary differential equations. Thus, we obtain new solutions in fractional forms by a newly developed method. Full article
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