Mathematics doi: 10.3390/math6120317

Authors: David W. Roberts

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.

]]>Mathematics doi: 10.3390/math6120316

Authors: Peter J. Zeitsch

Riemann&rsquo;s method is one of the definitive ways of solving Cauchy&rsquo;s problem for a second order linear hyperbolic partial differential equation in two variables. The first review of Riemann&rsquo;s method was published by E.T. Copson in 1958. This study extends that work. Firstly, three solution methods were overlooked in Copson&rsquo;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&rsquo;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-B&auml;cklund symmetry algebra. Combining the results from several methods, a new class of Riemann functions is then derived which admits no symmetries whatsoever.

]]>Mathematics doi: 10.3390/math6120315

Authors: Beáta Oborny

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.

]]>Mathematics doi: 10.3390/math6120314

Authors: Alessandra Bernardi Enrico Carlini Maria Virginia Catalisano Alessandro Gimigliano Alessandro Oneto

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.

]]>Mathematics doi: 10.3390/math6120313

Authors: Tingting Wang Guohui Chen

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 &equiv; 1 mod 12 ) to give an exact evaluation formula for it.

]]>Mathematics doi: 10.3390/math6120312

Authors: Aqeel Ketab AL-khafaji Waggas Galib Atshan Salwa Salman Abed

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.

]]>Mathematics doi: 10.3390/math6120311

Authors: Songting Yin Pan Zhang

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.

]]>Mathematics doi: 10.3390/math6120310

Authors: Fiza Zafar Alicia Cordero Juan R. Torregrosa

Finding a repeated zero for a nonlinear equation f ( x ) = 0 , f : I &sube; R &rarr; R has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton&rsquo;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.

]]>Mathematics doi: 10.3390/math6120309

Authors: James Tilley

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.

]]>Mathematics doi: 10.3390/math6120308

Authors: Resat Yilmazer Mustafa Inc Mustafa Bayram

In this article, we obtain new fractional solutions of the general class of non-Fuchsian differential equations by using discrete fractional nabla operator &nabla; &eta; ( 0 &lt; &eta; &lt; 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.

]]>Mathematics doi: 10.3390/math6120307

Authors: Yılmaz Çeven

In this paper, firstly, as a generalization of derivations on a lattice, the notion of n-derivation is introduced and some fundamental properties are investigated. Secondly, the concept of (n,m)-derivation-homomorphism on lattices is described and important and characteristic properties are given.

]]>Mathematics doi: 10.3390/math6120306

Authors: Juan Liang Linke Hou Xiaowu Li Feng Pan Taixia Cheng Lin Wang

Orthogonal projection a point onto a parametric curve, three classic first order algorithms have been presented by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter, H-H-H method). In this research, we give a proof of the approach’s first order convergence and its non-dependence on the initial value. For some special cases of divergence for the H-H-H method, we combine it with Newton’s second order method (hereafter, Newton’s method) to create the hybrid second order method for orthogonal projection onto parametric curve in an n-dimensional Euclidean space (hereafter, our method). Our method essentially utilizes hybrid iteration, so it converges faster than current methods with a second order convergence and remains independent from the initial value. We provide some numerical examples to confirm robustness and high efficiency of the method.

]]>Mathematics doi: 10.3390/math6120305

Authors: Aydin Secer Neslihan Ozdemir Mustafa Bayram

In this paper, the problem of the spread of a non-fatal disease in a population is solved by using the Hermite collocation method. Mathematical modeling of the problem corresponds to a three-dimensional system of nonlinear ODEs. The presented scheme reduces the problem to a nonlinear algebraic equation system by expanding the approximate solutions by using Hermite polynomials with unknown coefficients. These coefficients of the Hermite polynomials are computed by using the matrix operations of derivatives together with the collocation method. Maple software is used to carry out the computations. In addition, comparison of our method with the Homotopy perturbation method (HPM) and Laplece-Adomian decomposition method (LADM) proves accuracy of solution.

]]>Mathematics doi: 10.3390/math6120304

Authors: Juan L. G. Guirao Sarfraz Ahmad Muhammad Kamran Siddiqui Muhammad Ibrahim

A graph labeling is the task of integers, generally spoken to by whole numbers, to the edges or vertices, or both of a graph. Formally, given a graph G = ( V , E ) a vertex labeling is a capacity from V to an arrangement of integers. A graph with such a capacity characterized is known as a vertex-labeled graph. Similarly, an edge labeling is an element of E to an arrangement of labels. For this situation, the graph is called an edge-labeled graph. We examine an edge irregular reflexive k-labeling for the disjoint association of the cycle related graphs and decide the correct estimation of the reflexive edge strength for the disjoint association of s isomorphic duplicates of the cycle related graphs to be specific Generalized Peterson graphs.

]]>Mathematics doi: 10.3390/math6120303

Authors: Jun He Yanmin Liu Junkang Tian Zhuanzhou Zhang

In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufficient condition for the positive definiteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efficiency of our results.

]]>Mathematics doi: 10.3390/math6120302

Authors: Chia-Nan Wang Van Thanh Nguyen Hoang Tuyet Nhi Thai Ngoc Nguyen Tran Thi Lan Anh Tran

Today, business organizations are facing increasing pressure from a variety of sources to operate using sustainable processes. Thus, most companies need to focus on their supply chains to enhance sustainability to meet customer demands and comply with environmental legislation. To achieve these goals, companies must focus on criteria that include CO2 (carbon footprint) and toxic emissions, energy use and efficiency, wastage generations, and worker health and safety. As in other industries, the food processing industry requires large inputs of resources, which results in several negative environmental effects; thus, decision-makers have to evaluate qualitative and quantitative factors. This work identifies the best supplier for edible oil production in the small and medium enterprise (SME) food processing industry in Vietnam. This study also processes a hybrid multicriteria decision-making (MCDM) model using a fuzzy analytical hierarchy process (FAHP) and green data envelopment analysis (GDEA) model to identify the weight of all criteria of a supplier&rsquo;s selection process based on opinions from company procurement experts. Subsequently, GDEA is applied to rank all potential supplier lists. The primary objective of this work is to present a novel approach which integrates FAHP and DEA for supplier selection and also consider the green issue in edible oil production in uncertain environments. The aim of this research is also to provide a useful guideline for supplier selection based on qualitative and quantitative factors to improve the efficiency of supplier selection in the food industry and other industries. The results reveal that Decision-Making Unit 1 (DMU 1), DMU 3, DMU 7, and DMU 9 are identified as extremely efficient for five DEA models, which are the optimal suppliers for edible oil production. The contributions of this research include a proposed MCDM model using a hybrid FAHP and GDEA model for supplier selection in the SME food processing industry under a fuzzy environment conditions in Vietnam. This research also is part of an evolution of a new hybrid model that is flexible and practical for decision-makers. In addition, the research also provides a useful guideline in supplier selection in the food processing industry and a guideline for supplier selection in other industries.

]]>Mathematics doi: 10.3390/math6120301

Authors: Kashif Elahi Ali Ahmad Roslan Hasni

Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this paper we discuss eccentric topological indices of zero divisor graphs of commutative rings Z p 1 p 2 &times; Z q , where p 1 , p 2 , and q are primes. To enhance the importance of these indices a construction algorithm is also devised for zero divisor graphs of commutative rings Z p 1 p 2 &times; Z q .

]]>Mathematics doi: 10.3390/math6120300

Authors: Guohui Chen Li Chen

In this paper, we first introduce a new second-order non-linear recursive polynomials U h , i ( x ) , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini polynomials, Euler polynomials and Euler numbers.

]]>Mathematics doi: 10.3390/math6120299

Authors: Aditi Khanna Aakanksha Kishore Biswajit Sarkar Chandra K. Jaggi

The present model develops a three-echelon supply chain, in which the manufacturer offers full permissible delay to the whole seller, while the latter, in turn, adopts distinct trade credit policies for his subsequent downstream retailers. The type of credit policy being offered to the retailers is decided on the basis of their past profiles. Hence, the whole seller puts forth full and partial permissible delays to his old and new retailers respectively. This study considers bad debts from the portion of new retailers who fail to make up for the delayed part of the partial payment. The analysis shows that it is beneficial for the whole seller to make shorter contracts, particularly with new retailers, along with the fetching of a higher fraction of initial purchase cost from them. In addition to the above-described scenario, the lot received by the whole seller from the manufacturer is not perfect, and it contains some defects for which he employs an inspection process before selling the items to the retailers. In order to make the study more realistic, Type-I, as well as Type-II misclassification errors, and the case of out-of-stock are considered. The impact of Type-I error has been found to be crucial in the study. The present paper determines the optimal policy for the whole seller by maximizing the expected total profit per unit time. For the optimality of the solution, theoretical results are provided. Finally, a numerical example and a sensitivity analysis are done to validate the model.

]]>Mathematics doi: 10.3390/math6120298

Authors: Sarfraz Nawaz Malik Shahid Mahmood Mohsan Raza Sumbal Farman Saira Zainab

In the theory of analytic and univalent functions, coefficients of functions&rsquo; Taylor series representation and their related functional inequalities are of major interest and how they estimate functions&rsquo; growth in their specified domains. One of the important and useful functional inequalities is the Fekete-Szeg&ouml; inequality. In this work, we aim to analyze the Fekete-Szeg&ouml; functional and to find its upper bound for certain analytic functions which give parabolic and petal type regions as image domains. Coefficient inequalities and the Fekete-Szeg&ouml; inequality of inverse functions to these certain analytic functions are also established in this work.

]]>Mathematics doi: 10.3390/math6120297

Authors: Hongyan Xu Hua Wang

By using the Tsuji characteristic of meromorphic function in an angular domain, we investigate two meromorphic functions partially sharing some values in an angle region, and obtain one main result and a series of corollaries that are improvements and generalization of the previous results given by Zheng, Cao-Yi, Li-Yi and Xuan.

]]>Mathematics doi: 10.3390/math6120296

Authors: Ramandeep Behl Alicia Cordero Juan R. Torregrosa Ali Saleh Alshomrani

In this manuscript, a new type of study regarding the iterative methods for solving nonlinear models is presented. The goal of this work is to design a new fourth-order optimal family of two-step iterative schemes, with the flexibility through weight function/s or free parameter/s at both substeps, as well as small residual errors and asymptotic error constants. In addition, we generalize these schemes to nonlinear systems preserving the order of convergence. Regarding the applicability of the proposed techniques, we choose some real-world problems, namely chemical fractional conversion and the trajectory of an electron in the air gap between two parallel plates, in order to study the multi-factor effect, fractional conversion of species in a chemical reactor, Hammerstein integral equation, and a boundary value problem. Moreover, we find that our proposed schemes run better than or equal to the existing ones in the literature.

]]>Mathematics doi: 10.3390/math6120295

Authors: Wildi

When striving for the ordination methods best predicting independently measured site factors, the following questions arise: does the optimal choice depend on the kind of biological community analysed? Are there different ordination methods needed to address different site factors? Simultaneously, I explore alternative similarity approaches of entire ordinations, as well as the role of the transformations applied to the scale used in measuring species performance. The combination of methods and data transformations results in 96 alternative solutions for any one data set. These are compared by a graphical display, that is, an ordination of ordinations. The goodness-of-fit of independently measured site factors is assessed by two alternative methods. The resulting 96 performance values serve as independent variables in trend surfaces overlaid to the ordination of ordinations. The results from two real-world data sets indicate that some ordination methods greatly vary with data transformation. Scores close to a binary scale perform best in almost all ordination methods. Methods that intrinsically constrain the range of species scores, such as principal components analysis based on correlation, correspondence analysis (including its detrended version), nonmetric multidimensional scaling, as well as principal coordinates analysis based on the Bray-Curtis distance, always figure among the most successful methods, irrespective of data used.

]]>Mathematics doi: 10.3390/math6120294

Authors: Liangping Wu Guiwu Wei Hui Gao Yu Wei

In this paper, we expand the Hamy mean (HM) operator and Dombi operations with interval-valued intuitionistic fuzzy numbers (IVIFNs) to propose the interval-valued intuitionistic fuzzy Dombi Hamy mean (IVIFDHM) operator, interval-valued intuitionistic fuzzy weighted Dombi Hamy mean (IVIFWDHM) operator, interval-valued intuitionistic fuzzy dual Dombi Hamy mean (IVIFDDHM) operator, and interval-valued intuitionistic fuzzy weighted dual Dombi Hamy mean (IVIFWDDHM) operator. Then the MADM models are designed with IVIFWDHM and IVIFWDDHM operators. Finally, we gave an example for evaluating the elderly tourism service quality in tourism destination to show the proposed models.

]]>Mathematics doi: 10.3390/math6120293

Authors: Muhammad Gulistan Feng Feng Madad Khan Aslıhan Sezgin

Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of [ 0 , 1 ] and a number from [ 0 , 1 ] . Generalized cubic sets generalized the cubic sets with the help of cubic point. On the other hand Soft sets were proved to be very effective tool for handling imprecision. Semigroups are the associative structures have many applications in the theory of Automata. In this paper we blend the idea of cubic sets, generalized cubic sets and semigroups with the soft sets in order to develop a generalized approach namely generalized cubic soft sets in semigroups. As the ideal theory play a fundamental role in algebraic structures through this we can make a quotient structures. So we apply the idea of neutrosophic cubic soft sets in a very particular class of semigroups namely weakly regular semigroups and characterize it through different types of ideals. By using generalized cubic soft sets we define different types of generalized cubic soft ideals in semigroups through three different ways. We discuss a relationship between the generalized cubic soft ideals and characteristic functions and cubic level sets after providing some basic operations. We discuss two different lattice structures in semigroups and show that in the case when a semigroup is regular both structures coincides with each other. We characterize right weakly regular semigroups using different types of generalized cubic soft ideals. In this characterization we use some classical results as without them we cannot prove the inter relationship between a weakly regular semigroups and generalized cubic soft ideals. This generalization leads us to a new research direction in algebraic structures and in decision making theory.

]]>Mathematics doi: 10.3390/math6120292

Authors: Sandeep Kaur Chouhan Jatinderdeep Kaur Satvinder Singh Bhatia

In this paper, the extensions of classes S, C and BV are made by defining the classes Sr, Cr and BVr,r = 0,1,2,... . It is also shown that class Sr is a subclass of Cr ∩ BVr. Moreover, the results on L1-convergence of r times differentiated trigonometric sine series have been obtained by considering the rth (r = 0,1,2,...) derivative of modified sine sum under the new extended class Cr ∩ BVr.

]]>Mathematics doi: 10.3390/math6120291

Authors: Yang-Hui He

D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of standard techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, conformality and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is a trichotomy in set theory of finite, tame and wild representation types. At the intersection of the above lies the theory of quivers. We briefly review some of the terminology standard to the physics and to the mathematics. Then, we utilise certain results from graph theory and axiomatic representation theory of path algebras to address physical issues such as the implication of graph additivity to finiteness of gauge theories, the impossibility of constructing completely IR free string orbifold theories and the unclassifiability of N &lt; 2 Yang-Mills theories in four dimensions.

]]>Mathematics doi: 10.3390/math6120290

Authors: Arash Ghaani Farashahi Gregory S. Chirikjian

This paper presents a systematic study of the analytic aspects of Fourier&ndash;Zernike series of convolutions of functions supported on disks. We then investigate different aspects of the presented theory in the cases of zero-padded functions.

]]>Mathematics doi: 10.3390/math6120289

Authors: Matt Visser

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica&rsquo;s conjecture: &forall; n &ge; 1 , is p n + 1 &minus; p n &le; 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 / ln p n + 1 &minus; p n / ln p n &lt; 11 / 25 ; ( n &ge; 1 ) . Then, by considering more general m t h roots, again assuming the Riemann hypothesis, I show that p n + 1 m &minus; p n m &lt; 44 / ( 25 e [ m &minus; 2 ] ) ; ( n &ge; 3 ; m &gt; 2 ) . In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln 2 p n + 1 &minus; ln 2 p n &lt; 9 ; ln 3 p n + 1 &minus; ln 3 p n &lt; 52 ; ln 4 p n + 1 &minus; ln 4 p n &lt; 991 ; ( n &ge; 1 ) . I shall also update the region on which Andrica&rsquo;s conjecture is unconditionally verified.

]]>Mathematics doi: 10.3390/math6120288

Authors: Songting Yin

We obtain a Rellich type inequality on the sphere and give the corresponding best constant. The result complements some related inequalities in recent literatures.

]]>Mathematics doi: 10.3390/math6120287

Authors: Xin Zhang Dexuan Zou Xin Shen

In order to overcome the several shortcomings of Particle Swarm Optimization (PSO) e.g., premature convergence, low accuracy and poor global searching ability, a novel Simple Particle Swarm Optimization based on Random weight and Confidence term (SPSORC) is proposed in this paper. The original two improvements of the algorithm are called Simple Particle Swarm Optimization (SPSO) and Simple Particle Swarm Optimization with Confidence term (SPSOC), respectively. The former has the characteristics of more simple structure and faster convergence speed, and the latter increases particle diversity. SPSORC takes into account the advantages of both and enhances exploitation capability of algorithm. Twenty-two benchmark functions and four state-of-the-art improvement strategies are introduced so as to facilitate more fair comparison. In addition, a t-test is used to analyze the differences in large amounts of data. The stability and the search efficiency of algorithms are evaluated by comparing the success rates and the average iteration times obtained from 50-dimensional benchmark functions. The results show that the SPSO and its improved algorithms perform well comparing with several kinds of improved PSO algorithms according to both search time and computing accuracy. SPSORC, in particular, is more competent for the optimization of complex problems. In all, it has more desirable convergence, stronger stability and higher accuracy.

]]>Mathematics doi: 10.3390/math6120286

Authors: Michael D. Marcozzi

We demonstrate the existence, uniqueness and Galerkin approximatation of linear ultraparabolic terminal value/infinite-horizon problems on unbounded spatial domains. Furthermore, we provide a probabilistic interpretation of the solution in terms of the expectation of an associated ultradiffusion process.

]]>Mathematics doi: 10.3390/math6120285

Authors: Michel Riguidel

This article proposes a morphogenesis interpretation of the zeta function by computational approach by relying on numerical approximation formulae between the terms and the partial sums of the series, divergent in the critical strip. The goal is to exhibit structuring properties of the partial sums of the raw series by highlighting their morphogenesis, thanks to the elementary functions constituting the terms of the real and imaginary parts of the series, namely the logarithmic, cosine, sine, and power functions. Two essential indices of these sums appear: the index of no return of the vagrancy and the index of smothering of the function before the resumption of amplification of its divergence when the index tends towards infinity. The method consists of calculating, displaying graphically in 2D and 3D, and correlating, according to the index, the angles, the terms and the partial sums, in three nested domains: the critical strip, the critical line, and the set of non-trivial zeros on this line. Characteristics and approximation formulae are thus identified for the three domains. These formulae make it possible to grasp the morphogenetic foundations of the Riemann hypothesis (RH) and sketch the architecture of a more formal proof.

]]>Mathematics doi: 10.3390/math6120284

Authors: Lina Zhang Xuesi Ma

The polynomial bounds of Jordan’s inequality, especially the cubic and quartic polynomial bounds, have been studied and improved in a lot of the literature; however, the linear and quadratic polynomial bounds can not be improved very much. In this paper, new refinements and improvements of Jordan’s inequality are given. We present new lower bounds and upper bounds for strengthened Jordan’s inequality using polynomials of degrees 1 and 2. Our bounds are tighter than the previous results of polynomials of degrees 1 and 2. More importantly, we give new improvements of Jordan’s inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.

]]>Mathematics doi: 10.3390/math6120283

Authors: Li Zhang Jing Zhao Jia-Bao Liu Micheal Arockiaraj

In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G 1 , G 2 , &hellip; , G k . Recall that H is an arbitrary graph with V ( H ) = { 1 , 2 , &hellip; , k } , and G 1 , G 2 , &hellip; , G k are disjoint graphs. Then, the H-join of graphs G 1 , G 2 , &hellip; , G k , denoted by ⋁ H { G 1 , G 2 , &hellip; , G k } , is a graph formed by taking G 1 , G 2 , &hellip; , G k and joining every vertex of G i to every vertex of G j whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of ⋁ H { G 1 , G 2 , &hellip; , G k } , and then give a { 1 } -inverse L ( ⋁ H { G 1 , G 2 , &hellip; , G k } ) { 1 } or group inverse L ( ⋁ H { G 1 , G 2 , &hellip; , G k } ) # of L ( ⋁ H { G 1 , G 2 , &hellip; , G k } ) . It is well know that, there exists a relationship between resistance distance and entries of { 1 } -inverse or group inverse. Therefore, we can easily obtain resistance distance in ⋁ H { G 1 , G 2 , &hellip; , G k } . In addition, some applications are presented in this paper.

]]>Mathematics doi: 10.3390/math6120282

Authors: Yang-Hi Lee Soon-Mo Jung

We prove general stability theorems for n-dimensional quartic-cubic-quadratic-additive type functional equations of the form &sum; i = 1 ℓ c i f a i 1 x 1 + a i 2 x 2 + ⋯ + a i n x n = 0 . by applying the direct method. These stability theorems can save us the trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations.

]]>Mathematics doi: 10.3390/math6120281

Authors: Erhan Güler

We consider a family of higher degree Enneper minimal surface E m for positive integers m in the three-dimensional Euclidean space E 3 . We compute algebraic equation, degree and integral free representation of Enneper minimal surface for m = 1 , 2 , 3 . Finally, we give some results and relations for the family E m .

]]>Mathematics doi: 10.3390/math6120280

Authors: Harish Garg Gagandeep Kaur

Probabilistic dual hesitant fuzzy set (PDHFS) is an enhanced version of a dual hesitant fuzzy set (DHFS) in which each membership and non-membership hesitant value is considered along with its occurrence probability. These assigned probabilities give more details about the level of agreeness or disagreeness. By emphasizing the advantages of the PDHFS and the aggregation operators, in this manuscript, we have proposed several weighted and ordered weighted averaging and geometric aggregation operators by using Einstein norm operations, where the preferences related to each object is taken in terms of probabilistic dual hesitant fuzzy elements. Several desirable properties and relations are also investigated in details. Also, we have proposed two distance measures and its based maximum deviation method to compute the weight vector of the different criteria. Finally, a multi-criteria group decision-making approach is constructed based on proposed operators and the presented algorithm is explained with the help of the numerical example. The reliability of the presented decision-making method is explored with the help of testing criteria and by comparing the results of the example with several prevailing studies.

]]>Mathematics doi: 10.3390/math6120279

Authors: Erhan Güler Ömer Kişi Christos Konaxis

Considering the Weierstrass data as ( &psi; , f , g ) = ( 2 , 1 - z - m , z n ) , we introduce a two-parameter family of Henneberg-type minimal surface that we call H m , n for positive integers ( m , n ) by using the Weierstrass representation in the four-dimensional Euclidean space E 4 . We define H m , n in ( r , &theta; ) coordinates for positive integers ( m , n ) with m &ne; 1 , n &ne; - 1 , - m + n &ne; - 1 , and also in ( u , v ) coordinates, and then we obtain implicit algebraic equations of the Henneberg-type minimal surface of values ( 4 , 2 ) .

]]>Mathematics doi: 10.3390/math6120278

Authors: Muhammad Akram Jawaria Mohsan Dar Adeel Farooq

Graph theory plays a substantial role in structuring and designing many problems. A number of structural designs with crossings can be found in real world scenarios. To model the vagueness and uncertainty in graphical network problems, many extensions of graph theoretical ideas are introduced. To deal with such uncertain situations, the present paper proposes the concept of Pythagorean fuzzy multigraphs and Pythagorean fuzzy planar graphs with some of their eminent characteristics by investigating Pythagorean fuzzy planarity value with strong, weak and considerable edges. A close association is developed between Pythagorean fuzzy planar and dual graphs. This paper also includes a brief discussion on non-planar Pythagorean fuzzy graphs and explores the concepts of isomorphism, weak isomorphism and co-weak isomorphism for Pythagorean fuzzy planar graphs. Moreover, it presents a problem that shows applicability of the proposed concept.

]]>Mathematics doi: 10.3390/math6120277

Authors: Feng Qi Pietro Cerone

In the paper, the authors express the Fuss&ndash;Catalan numbers as several forms in terms of the Catalan&ndash;Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss&ndash;Catalan numbers, and derive a double inequality for bounding the Fuss&ndash;Catalan numbers.

]]>Mathematics doi: 10.3390/math6120276

Authors: Taekyun Kim Dae San Kim Lee-Chae Jang Gwan-Woo Jang

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sums of finite products of polynomials as linear combinations of Bernoulli polynomials.

]]>Mathematics doi: 10.3390/math6120275

Authors: Jong Ryul Kim

The warped product structure of a gradient Yamabe soliton and a Ricci soliton with a concircular potential field is proved in another way.

]]>Mathematics doi: 10.3390/math6120274

Authors: Ioannis K. Argyros Daniel González

We use Newton&rsquo;s method to solve previously unsolved problems, expanding the applicability of the method. To achieve this, we used the idea of restricted domains which allows for tighter Lipschitz constants than previously seen, this in turn led to a tighter convergence analysis. The new developments were obtained using special cases of functions which had been used in earlier works. Numerical examples are used to illustrate the superiority of the new results.

]]>Mathematics doi: 10.3390/math6120273

Authors: Maurice R. Kibler

The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) quantum bits or qubits (and more generally qudits), the analogues of the usual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantum mechanics, namely, linearity, which manifests itself through the superposition of qubits and the action of unitary operators on qubits, and entanglement of certain multi-qubit states, a resource that is specific to quantum mechanics. A, second, focus is on some mathematical problems related to the so-called mutually unbiased bases used in quantum computing and quantum information processing. In this direction, the construction of mutually unbiased bases is presented via two distinct approaches: one based on the group SU(2) and the other on Galois fields and Galois rings.

]]>Mathematics doi: 10.3390/math6110272

Authors: José Luis Usó-Doménech Josué Antonio Nescolarde-Selva Hugh Gash

There is a fairly widespread belief that the problem of existence is not an essential issue for logic. Logic, though formal, must deal with the problem of existence. However, logic should be limited to describing &ldquo;formal existence&rdquo; or &ldquo;existence of a formal system&rdquo;. However, the logical problem of existence and how to treat and resolve this problem differ completely from the corresponding metaphysical problem. It is possible to deduce that formal existence is nothing other than belonging to the universe of discourse, so proposing a solution to the logical problem of existence in an epistemological, rather than a metaphysical, context. In this paper, we conclude, from a formal point of view, no universe of discourse is given in advance; any universe of discourse that satisfies the necessary conditions can be used. The extended epistemological belief that there is a universe of discourse defined rigorously, which would be the true and should be &ldquo;the universe of discourse of logic&rdquo;, cannot be justified.

]]>Mathematics doi: 10.3390/math6110271

Authors: Fang Gao Xiaoxin Li Kai Zhou Jia-Bao Liu

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than m .

]]>Mathematics doi: 10.3390/math6110270

Authors: Ali Sadeghi Mansour Saraj Nezam Mahdavi Amiri

In this article, a methodology is developed to solve an interval and a fractional interval programming problem by converting into a non-interval form for second order cone constraints, with the objective function and constraints being interval valued functions. We investigate the parametric and non-parametric forms of the interval valued functions along with their convexity properties. Two approaches are developed to obtain efficient and properly efficient solutions. Furthermore, the efficient solutions or Pareto optimal solutions of fractional and non-fractional programming problems over R + n ⋃ { 0 } are also discussed. The main idea of the present article is to introduce a new concept for efficiency, called efficient space, caused by the lower and upper bounds of the respective intervals of the objective function which are shown in different figures. Finally, some numerical examples are worked through to illustrate the methodology and affirm the validity of the obtained results.

]]>Mathematics doi: 10.3390/math6110269

Authors: Sergio Camiz Valério Pillar

The identification of a reduced dimensional representation of the data is among the main issues of exploratory multidimensional data analysis and several solutions had been proposed in the literature according to the method. Principal Component Analysis (PCA) is the method that has received the largest attention thus far and several identification methods—the so-called stopping rules—have been proposed, giving very different results in practice, and some comparative study has been carried out. Some inconsistencies in the previous studies led us to try to fix the distinction between signal from noise in PCA—and its limits—and propose a new testing method. This consists in the production of simulated data according to a predefined eigenvalues structure, including zero-eigenvalues. From random populations built according to several such structures, reduced-size samples were extracted and to them different levels of random normal noise were added. This controlled introduction of noise allows a clear distinction between expected signal and noise, the latter relegated to the non-zero eigenvalues in the samples corresponding to zero ones in the population. With this new method, we tested the performance of ten different stopping rules. Of every method, for every structure and every noise, both power (the ability to correctly identify the expected dimension) and type-I error (the detection of a dimension composed only by noise) have been measured, by counting the relative frequencies in which the smallest non-zero eigenvalue in the population was recognized as signal in the samples and that in which the largest zero-eigenvalue was recognized as noise, respectively. This way, the behaviour of the examined methods is clear and their comparison/evaluation is possible. The reported results show that both the generalization of the Bartlett’s test by Rencher and the Bootstrap method by Pillar result much better than all others: both are accounted for reasonable power, decreasing with noise, and very good type-I error. Thus, more than the others, these methods deserve being adopted.

]]>Mathematics doi: 10.3390/math6110268

Authors: Kuddusi Kayaduman Fevzi Yaşar

In 1978, the domain of the N&ouml;rlund matrix on the classical sequence spaces lp and l&infin; was introduced by Wang, where 1 &le; p &lt; &infin;. Tuğ and Başar studied the matrix domain of N&ouml;rlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the N&ouml;rlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space b s ( N t ) and c s ( N t ) and examined the domain of the N&ouml;rlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their &alpha;-, &beta;-, &gamma;-duals, and characterized their matrix transformations on this space and into this space.

]]>Mathematics doi: 10.3390/math6110267

Authors: Yeol Je Cho Shin Min Kang Peyman Salimi

In this paper, we introduce the concepts of an &alpha; -admissible nonself-mapping, an &alpha; -F-contractive nonself-mapping, a weak &alpha; -F-contractive nonself-mapping, and a generalized &alpha; -F-contractive nonself-mapping and prove some P P F (past-present-future)-dependent fixed point theorems for the proposed contractive nonself-mappings in certain Razumikhin classes. By using our results, we derive some P P F -dependent fixed point theorems for an &alpha; -F-contractive nonself-mapping endowed with a graph or a partial order. Finally, we give some applications to illustrate the main results.

]]>Mathematics doi: 10.3390/math6110266

Authors: Mamoru Nunokawa Janusz Sokół Nak Eun Cho

Let g be an analytic function with the normalization in the open unit disk. Let L ( r ) be the length of g ( { z : | z | = r } ) . In this paper we present a correspondence between g and L ( r ) for the case when g is not necessary univalent. Furthermore, some other results related to the length of analytic functions are also discussed.

]]>Mathematics doi: 10.3390/math6110265

Authors: Şeyda Gür Tamer Eren

Background: People want to be able to evaluate different kinds of information in a good way. There are various methods that they develop in such situations. Among the optimization methods, the goal programming method is often used when there are multiple objectives that decision makers want to accomplish. Because scheduling and planning problems have multiple objectives that are desired to be achieved, the goal programming method helps the researcher in contradictory situations between these goals. Methods: This study includes, examines, and analyzes recent research on service scheduling and planning. In the literature, service scheduling and planning studies have been examined using goal programming method from past to today. Results: The studies are detailed according to the type of goal programming, according to scheduling types, the purpose used in the studies, and the methods integrated with the goal programming. There are 142 studies in Emerald, Science Direct, Jstor, Springer, Taylor and Francis, Google Scholar, etc. databases that are examined in detail. For readers, diversification has been made to facilitate the identification of these studies and a detailed overview has been presented. Conclusion: As a result of the study, studies with the goal programming in the literature have been seen. The readers&rsquo; perspectives for planning and scheduling are discussed.

]]>Mathematics doi: 10.3390/math6110264

Authors: Dagmar Markechová

This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order q , where q is a positive real number not equal to 1, of fuzzy partitions are introduced and their mathematical behavior is described. As an important result, we showed that the Tsallis entropy of fuzzy partitions of order q &gt; 1 satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order q &gt; 1 of a fuzzy dynamical system. It was shown that Tsallis entropy is an invariant under isomorphisms of fuzzy dynamical systems; thus, we acquired a tool for distinguishing some non-isomorphic fuzzy dynamical systems. Finally, we formulated a version of the Kolmogorov&ndash;Sinai theorem on generators for the case of the Tsallis entropy of a fuzzy dynamical system. The obtained results extend the results provided by Markechov&aacute; and Riečan in Entropy, 2016, 18, 157, which are particularized to the case of logical entropy.

]]>Mathematics doi: 10.3390/math6110263

Authors: Nak Eun Cho Virendra Kumar Ji Hyang Park

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for large values of the parameters involved therein. Further, sharp estimate on the generalized Fekete-Szeg&ouml; functional is also derived. Relevant connections of our results with the existing ones are also made.

]]>Mathematics doi: 10.3390/math6110262

Authors: Chayut Kongban Poom Kumam Juan Martínez-Moreno

In this paper, we introduce a new concept of random &alpha; -proximal admissible and random &alpha; - Z -contraction. Then we establish random best proximity point theorems for such mapping in complete separable metric spaces.

]]>Mathematics doi: 10.3390/math6110261

Authors: Wasfi Shatanawi

In this paper, we introduce the notion of ( &alpha; , &beta; , &psi; ) -contraction for a pair of mappings ( S , T ) defined on a set X. We use our new notion to create and prove a common fixed point theorem for two mappings defined on a metric space ( X , d ) under a set of conditions. Furthermore, we employ our main result to get another new result. Our results are modifications of many existing results in the literature. An example is included in order to show the authenticity of our main result.

]]>Mathematics doi: 10.3390/math6110260

Authors: Janak Raj Sharma Ioannis K. Argyros Sunil Kumar

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

]]>Mathematics doi: 10.3390/math6110259

Authors: Chul Woo Lee Jae Won Lee

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections &nabla; and &nabla; * in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant &phi; -sectional curvature. In addition, we give examples to show that the total space is a sphere.

]]>Mathematics doi: 10.3390/math6110258

Authors: Subuhi Khan Tabinda Nahid

The present paper intends to introduce the hybrid form of q-special polynomials, namely q-Hermite-Appell polynomials by means of generating function and series definition. Some significant properties of q-Hermite-Appell polynomials such as determinant definitions, q-recurrence relations and q-difference equations are established. Examples providing the corresponding results for certain members belonging to this q-Hermite-Appell family are considered. In addition, graphs of certain q-special polynomials are demonstrated using computer experiment. Thereafter, distribution of zeros of these q-special polynomials is displayed.

]]>Mathematics doi: 10.3390/math6110256

Authors: Erdal Karapinar Ravi Agarwal Hassen Aydi

By giving a counter-example, we point out a gap in the paper by Karapinar (Adv. Theory Nonlinear Anal. Its Appl. 2018, 2, 85&ndash;87) where the given fixed point may be not unique and we present the corrected version. We also state the celebrated fixed point theorem of Reich&ndash;Rus&ndash;Ćirić in the framework of complete partial metric spaces, by taking the interpolation theory into account. Some examples are provided where the main result in papers by Reich (Can. Math. Bull. 1971, 14, 121&ndash;124; Boll. Unione Mat. Ital. 1972, 4, 26&ndash;42 and Boll. Unione Mat. Ital. 1971, 4, 1&ndash;11.) is not applicable.

]]>Mathematics doi: 10.3390/math6110257

Authors: Huancheng Zhang Yunhua Qu Yongfu Su

This paper uses the viscosity implicit midpoint rule to find common points of the fixed point set of a nonexpansive mapping and the zero point set of an accretive operator in Banach space. Under certain conditions, this paper obtains the strong convergence results of the proposed algorithm and improves the relevant results of researchers in this field. In the end, this paper gives numerical examples to support the main results.

]]>Mathematics doi: 10.3390/math6110255

Authors: Hsien-Chung Wu

Cooperative games endowed with interval-valued payoffs are studied in this paper. Based on the interval-valued payoff and the different types of orderings, we can propose many types of so-called interval-valued cores and interval-valued dominance cores. The main issue of this paper is to establish the equalities of different types of interval-valued cores and interval-valued dominance cores under a mild assumption. Without considering the individual rationality, we also establish the equalities of different types of interval-valued pre-cores and interval-valued dominance pre-cores without any extra assumptions.

]]>Mathematics doi: 10.3390/math6110254

Authors: Mir Mohammad Reza Alavi Milani Sahereh Hosseinpour Huseyin Pehlivan

There are situations in which one needs to write various kinds of mathematical expressions, such as practicing tests and school exams. There is a variety of methods to produce such expressions, but they are usually based on a database. This paper addresses the production of new expressions using the template ones that can be derived from the evaluation process or entered by users. With special limitations on the values of parameters, some templates can be dynamically constructed for the automatic generation of mathematical expressions and represented in the form of classes. For this purpose, a new type of grammar is proposed. This grammar is similar to Context-Free Grammar, but it empowers the producer to gain control over the generation of rules for different expressions. Our work mainly focuses on generating mathematical expressions in a user-oriented way, using a predefined set of templates of production rules. The production of expressions is not completely random, and is based on the defined subject.

]]>Mathematics doi: 10.3390/math6110253

Authors: Aditya Kamath Sergei Manzhos

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schr&ouml;dinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large, and for a range of IMQ exponents. The IMQ functions are; however, advantageous when a small number of functions is used or with a small number of collocation points (e.g., when using square collocation).

]]>Mathematics doi: 10.3390/math6110252

Authors: Toshikazu Kuniya

I have found an error in Equation (17) in my paper [1]&nbsp;[...]

]]>Mathematics doi: 10.3390/math6110251

Authors: Ralph Høibakk Dag Lukkassen Annette Meidell Lars-Erik Persson

The main aim of this paper is to contribute to the recently initiated research concerning geometric constructions of means, where the variables are appearing as line segments. The present study shows that all Lehmer means of two variables for integer power k and for k = m 2 , where m is an integer, can be geometrically constructed, that Lehmer means for power k = 0 , 1 and 2 can be geometrically constructed for any number of variables and that Lehmer means for power k = 1 / 2 and &minus; 1 can be geometrically constructed, where the number of variables is n = 2 m and m is a positive integer.

]]>Mathematics doi: 10.3390/math6110250

Authors: János Podani Sandrine Pavoine Carlo Ricotta

Community structure as summarized by presence&ndash;absence data is often evaluated via diversity measures by incorporating taxonomic, phylogenetic and functional information on the constituting species. Most commonly, various dissimilarity coefficients are used to express these aspects simultaneously such that the results are not comparable due to the lack of common conceptual basis behind index definitions. A new framework is needed which allows such comparisons, thus facilitating evaluation of the importance of the three sources of extra information in relation to conventional species-based representations. We define taxonomic, phylogenetic and functional beta diversity of species assemblages based on the generalized Jaccard dissimilarity index. This coefficient does not give equal weight to species, because traditional site dissimilarities are lowered by taking into account the taxonomic, phylogenetic or functional similarity of differential species in one site to the species in the other. These, together with the traditional, taxon- (species-) based beta diversity are decomposed into two additive fractions, one due to taxonomic, phylogenetic or functional excess and the other to replacement. In addition to numerical results, taxonomic, phylogenetic and functional community structure is visualized by 2D simplex or ternary plots. Redundancy with respect to taxon-based structure is expressed in terms of centroid distances between point clouds in these diagrams. The approach is illustrated by examples coming from vegetation surveys representing different ecological conditions. We found that beta diversity decreases in the following order: taxon-based, taxonomic (Linnaean), phylogenetic and functional. Therefore, we put forward the beta-redundancy hypothesis suggesting that this ordering may be most often the case in ecological communities, and discuss potential reasons and possible exceptions to this supposed rule. Whereas the pattern of change in diversity may be indicative of fundamental features of the particular community being studied, the effect of the choice of functional traits&mdash;a more or less subjective element of the framework&mdash;remains to be investigated.

]]>Mathematics doi: 10.3390/math6110249

Authors: Songnian He Qiao-Li Dong

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some x * &isin; C : = &cap; i = 1 m { x &isin; H | c i ( x ) &le; 0 } , where m is a positive integer, H is a real Hilbert space, and { c i } i = 1 m are convex functions defined as H . The key of the CPM is that, for the current iterate x k , the CPM firstly constructs a new level set H k through a convex combination of some of { c i } i = 1 m in an appropriate way, and then updates the new iterate x k + 1 only by using the projection P H k . We also introduce the combination relaxation projection methods (CRPM) to project onto half-spaces to make CPM easily implementable. The simplicity and easy implementation are two advantages of our methods since only one projection is used in each iteration and the projections are also easy to calculate. The weak convergence theorems are proved and the numerical results show the advantages of our methods.

]]>Mathematics doi: 10.3390/math6110248

Authors: Ghulam Farid Waqas Nazeer Muhammad Shoaib Saleem Sajid Mehmood Shin Min Kang

In this article, we establish bounds of sum of the left and right sided Riemann Liouville (RL) fractional integrals and related inequalities in general form. A new and novel approach is followed to obtain these results for general Riemann Liouville (RL) fractional integrals. Monotonicity and convexity of functions are used with some usual and straight forward inequalities. The presented results are also have connection with some known and already published results. Applications and motivations of presented results are briefly discussed.

]]>Mathematics doi: 10.3390/math6110247

Authors: Alessandro De Paris

We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds.

]]>Mathematics doi: 10.3390/math6110246

Authors: Yan Zhao Wenjie Wang Ximin Liu

Let M be a three-dimensional trans-Sasakian manifold of type ( &alpha; , &beta; ) . In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an &alpha; -Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature. Applying this, we give a new characterization of proper trans-Sasakian 3-manifolds.

]]>Mathematics doi: 10.3390/math6110245

Authors: Ali Akgül Esra Karatas Akgül Dumitru Baleanu Mustafa Inc

In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables.

]]>Mathematics doi: 10.3390/math6110244

Authors: Yixue Zhang Zhuoyu Chen

In this paper, firstly, we introduced a second order non-linear recursive sequence, then we use this sequence and the combinatorial methods to perform a deep study on the computational problem concerning one kind sums, which includes the Chebyshev polynomials. This makes it possible to simplify a class of complex computations involving the second type Chebyshev polynomials into a very simple problem. Finally, we give a new and interesting identity for it.

]]>Mathematics doi: 10.3390/math6110243

Authors: Jia-Bao Liu Zohaib Zahid Ruby Nasir Waqas Nazeer

Consider an undirected and connected graph G = ( V G , E G ) , where V G and E G represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly resolving sets is based on the distances of edges in a graph. In this paper, we find the edge version of metric dimension and doubly resolving sets for the necklace graph.

]]>Mathematics doi: 10.3390/math6110242

Authors: Wee Chin Wong Ewan Chee Jiali Li Xiaonan Wang

The pharmaceutical industry has witnessed exponential growth in transforming operations towards continuous manufacturing to increase profitability, reduce waste and extend product ranges. Model predictive control (MPC) can be applied to enable this vision by providing superior regulation of critical quality attributes (CQAs). For MPC, obtaining a workable system model is of fundamental importance, especially if complex process dynamics and reaction kinetics are present. Whilst physics-based models are desirable, obtaining models that are effective and fit-for-purpose may not always be practical, and industries have often relied on data-driven approaches for system identification instead. In this work, we demonstrate the applicability of recurrent neural networks (RNNs) in MPC applications in continuous pharmaceutical manufacturing. RNNs were shown to be especially well-suited for modelling dynamical systems due to their mathematical structure, and their use in system identification has enabled satisfactory closed-loop performance for MPC of a complex reaction in a single continuous-stirred tank reactor (CSTR) for pharmaceutical manufacturing.

]]>Mathematics doi: 10.3390/math6110241

Authors: Chayut Kongban Poom Kumam Somayya Komal Kanokwan Sitthithakerngkiet

In this work, we introduced new notions of a new contraction named S -weakly contraction; after that, we obtained the p-common best proximity point results for different types of contractions in the setting of complete metric spaces by using weak P p -property and proved the uniqueness of these points. Also, we presented some examples to prove the validity of our results.

]]>Mathematics doi: 10.3390/math6110240

Authors: Jonathan Aaron Azlan Mosiun Suzeini Abdul Halim

Let f ( z ) = z + &sum; n = 2 &infin; a n z n and g p , b , c ( z ) = z + &sum; n = 2 &infin; ( &minus; c 4 ) n &minus; 1 ( 3 2 ) n &minus; 1 ( k ) n &minus; 1 z n with p , b , c &isin; ℂ , k = p + b + 2 2 &ne; 0 , &minus; 1 , &minus; 2 , &hellip; be two analytic functions in the unit disk U = { z : | z | &lt; 1 } . This paper gives conditions so that the function T p , b , c ( z ) = ( f &lowast; g ) ( z ) , a function associated with the Struve function, is univalent, starlike, or convex in the unit disk.

]]>Mathematics doi: 10.3390/math6110239

Authors: Sung-Soo Pyo Taekyun Kim Seog-Hoon Rim

In this paper, we define new Daehee numbers, the degenerate Daehee numbers of the third kind, using the degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee, and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.

]]>Mathematics doi: 10.3390/math6110238

Authors: Aydin Secer Selvi Altun

This paper introduces a new numerical approach to solving a system of fractional differential equations (FDEs) using the Legendre wavelet operational matrix method (LWOMM). We first formulated the operational matrix of fractional derivatives in some special conditions using some notable characteristics of Legendre wavelets and shifted Legendre polynomials. Then, the system of fractional differential equations was transformed into a system of algebraic equations by using these operational matrices. At the end of this paper, several examples are presented to illustrate the effectivity and correctness of the proposed approach. Comparing the methodology with several recognized methods demonstrates that the advantages of the Legendre wavelet operational matrix method are its accuracy and the understandability of the calculations.

]]>Mathematics doi: 10.3390/math6110237

Authors: Ali N. A. Koam

Koam and Pirashivili developed the equivariant version of Hochschild cohomology by mixing the standard chain complexes computing group with associative algebra cohomologies to obtain the bicomplex C &tilde; G * ( A , X ). In this paper, we form a new bicomplex F ˘ G * ( A , X ) by deleting the first column and the first row and reindexing. We show that H ˘ G 1 ( A , X ) classifies the singular extensions of oriented algebras.

]]>Mathematics doi: 10.3390/math6110236

Authors: Xiumei Deng Jie Wang Guiwu Wei Mao Lu

The Hamy mean (HM) operator, as a useful aggregation tool, can capture the correlation between multiple integration parameters, and the 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) are a special kind of Pythagorean fuzzy numbers (PFNs), which can easily describe the fuzziness in actual decision making by 2-tuple linguistic terms (2TLTs). In this paper, to consider both Hamy mean (HM) operator and 2TLPFNs, we combine the HM operator, weighted HM (WHM) operator, dual HM (DHM) operator, and dual WHM (DWHM) operator with 2TLPFNs to propose the 2-tuple linguistic Pythagorean fuzzy HM (2TLPFHM) operator, 2-tuple linguistic Pythagorean fuzzy WHM (2TLPFWHM) operator, 2-tuple linguistic Pythagorean fuzzy DHM (2TLPFDHM) operator and 2-tuple linguistic Pythagorean fuzzy DWHM (2TLPFDWHM) operator. Then some multiple attribute decision making (MADM) procedures are developed based on these operators. At last, an applicable example for green supplier selection is given.

]]>Mathematics doi: 10.3390/math6110235

Authors: Feng Feng Hee Sik Kim Joseph Neggers

There are several equivalent axioms, which can be used to characterize the positive implicativity in B C K -algebras. In this paper, we investigate interrelationships among such axioms in a more general setting of groupoids, and several aspects regarding their differences in the theory of groupoids.

]]>Mathematics doi: 10.3390/math6110234

Authors: Muhammad Akram Hina Gulzar Florentin Smarandache Said Broumi

The concept of neutrosophic set from philosophical point of view was first considered by Smarandache. A single-valued neutrosophic set is a subclass of the neutrosophic set from a scientific and engineering point of view and an extension of intuitionistic fuzzy sets. In this research article, we apply the notion of single-valued neutrosophic sets to K-algebras. We introduce the notion of single-valued neutrosophic topological K-algebras and investigate some of their properties. Further, we study certain properties, including C 5 -connected, super connected, compact and Hausdorff, of single-valued neutrosophic topological K-algebras. We also investigate the image and pre-image of single-valued neutrosophic topological K-algebras under homomorphism.

]]>Mathematics doi: 10.3390/math6110233

Authors: Ioannis K. Argyros Santhosh George

The aim of this article is to present a unified semi-local convergence analysis for a k-step iterative method containing the inverse of a flexible and frozen linear operator for Banach space valued operators. Special choices of the linear operator reduce the method to the Newton-type, Newton&rsquo;s, or Stirling&rsquo;s, or Steffensen&rsquo;s, or other methods. The analysis is based on center, as well as Lipschitz conditions and our idea of the restricted convergence region. This idea defines an at least as small region containing the iterates as before and consequently also a tighter convergence analysis.

]]>Mathematics doi: 10.3390/math6110232

Authors: Adnan Daraghmeh Naji Qatanani

In this article, we present up to date results on the balanced model reduction techniques for linear control systems, in particular the singular perturbation approximation. One of the most important features of this method is it allows for an a priori L 2 and H &infin; bounds for the approximation error. This method has been successfully applied for systems with homogeneous initial conditions, however, the main focus in this work is to derive an L 2 error bound for singular perturbation approximation for system with inhomogeneous initial conditions, extending the work by Antoulas et al. The theoretical results are validated numerically.

]]>Mathematics doi: 10.3390/math6110231

Authors: Jae Won Lee Chul Woo Lee

The main purpose of this article is to construct inequalities between a main intrinsic invariant (the normalized scalar curvature) and an extrinsic invariant (the Casorati curvature) for some submanifolds in a Sasakian manifold with a zero C-Bochner tensor.

]]>Mathematics doi: 10.3390/math6110230

Authors: Yang Qi

This note is a short survey of nonnegative tensors, primarily from the geometric point of view. In addition to basic definitions, we discuss properties of and questions about nonnegative tensors, which may be of interest to geometers.

]]>Mathematics doi: 10.3390/math6110229

Authors: Ravi Agarwal Snezhana Hristova Donal O’Regan

One approach to study various stability properties of solutions of nonlinear Caputo fractional differential equations is based on using Lyapunov like functions. A basic question which arises is the definition of the derivative of the Lyapunov like function along the given fractional equation. In this paper, several definitions known in the literature for the derivative of Lyapunov functions among Caputo fractional differential equations are given. Applications and properties are discussed. Several sufficient conditions for stability, uniform stability and asymptotic stability with respect to part of the variables are established. Several examples are given to illustrate the theory.

]]>Mathematics doi: 10.3390/math6110228

Authors: Zengxian Li Guiwu Wei Hui Gao

Interval-valued Pythagorean fuzzy numbers (IVPFNs) can easily describe the incomplete and indeterminate information by degrees of membership and non-membership, and the Hamy mean (HM) operator and dual HM (DHM) operators are a good tool for dealing with multiple attribute decision making (MADM) problems because it can capture the interrelationship among the multi-input arguments. Motivated by the studies regarding the HM operator and dual HM operator, we expand the HM operator and dual HM (DMM) operator to process the interval-valued Pythagorean fuzzy numbers (IVPFNs) and then to solve the MADM problems. Firstly, we propose some HM and DHM operators with IVPFNs. Moreover, we present some new methods to solve MADM problems with the IVPFNs. Finally, an applicable example is given.

]]>Mathematics doi: 10.3390/math6110227

Authors: Shaohui Wang Chunxiang Wang Lin Chen Jia-Bao Liu Zehui Shao

Given a (molecular) graph, the first multiplicative Zagreb index &Pi; 1 is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index &Pi; 2 is expressed as the product of endvertex degree of each edge over all edges. We consider a set of graphs G n , k having n vertices and k cut edges, and explore the graphs subject to a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs in G n , k are provided. We also provide these graphs with the largest and smallest &Pi; 1 ( G ) and &Pi; 2 ( G ) in G n , k .

]]>Mathematics doi: 10.3390/math6110226

Authors: Erhan Güler

We consider a new kind of helicoidal surface for natural numbers ( m , n ) in the three-dimensional Euclidean space. We study a helicoidal surface of value ( m , n ) , which is locally isometric to a rotational surface of value ( m , n ) . In addition, we calculate the Laplace&ndash;Beltrami operator of the rotational surface of value ( 0 , 1 ) .

]]>Mathematics doi: 10.3390/math6110225

Authors: Takashi Yamauchi Jinsil Hwaryoung Seo Annie Sungkajun

Using a multisensory interface system, we examined how people&rsquo;s emotional experiences change as their tactile sense (touching a plant) was augmented with visual sense (&ldquo;seeing&rdquo; their touch). Our system (the Interactive Plant system) senses the electrical capacitance of the human body and visualizes users&rsquo; tactile information on a flat screen (when the touch is gentle, the program draws small and thin roots around the pot; when the touch is more harsh or abrupt, big and thick roots are displayed). We contrasted this multimodal combination (touch + vision) with a unimodal interface (touch only or watch only) and measured the impact of the multimodal interaction on participants&rsquo; emotion. We found significant emotional gains in the multimodal interaction. Participants&rsquo; self-reported positive affect, joviality, attentiveness and self-assurance increased dramatically in multimodal interaction relative to unimodal interaction; participants&rsquo; electrodermal activity (EDA) increased in the multimodal condition, suggesting that our plant-based multisensory visual-tactile interaction raised arousal. We suggest that plant-based tactile interfaces are advantageous for emotion generation because haptic perception is by nature embodied and emotional.

]]>Mathematics doi: 10.3390/math6110224

Authors: Zühal Küçükarslan Yüzbaşı Dae Won Yoon

In this paper, we study inextensible flows of a curve on a lightlike surface in Minkowski three-space and give a necessary and sufficient condition for inextensible flows of the curve as a partial differential equation involving the curvatures of the curve on a lightlike surface. Finally, we classify lightlike ruled surfaces in Minkowski three-space and characterize an inextensible evolution of a lightlike curve on a lightlike tangent developable surface.

]]>Mathematics doi: 10.3390/math6110223

Authors: Ye Shuang Feng Qi

In the paper, the authors set up an identity for a function whose third derivative is integrable, establish by the H&ouml;lder inequality some new integral inequalities of the Hermite&ndash;Hadamard type for extended s-convex functions in the second sense, and apply these integral inequalities to construct inequalities for several special means.

]]>Mathematics doi: 10.3390/math6110222

Authors: Shrideh K. Q. Al-Omari Ghalib Jumah Jafar Al-Omari Deepali Saxena

This article deals with some variants of Kr&auml;tzel integral operators involving Fox&rsquo;s H-function and their extension to classes of distributions and spaces of Boehmians. For real numbers a and b &gt; 0 , the Fr&eacute;chet space H a , b of testing functions has been identified as a subspace of certain Boehmian spaces. To establish the Boehmian spaces, two convolution products and some related axioms are established. The generalized variant of the cited Kr&auml;tzel-Fox integral operator is well defined and is the operator between the Boehmian spaces. A generalized convolution theorem has also been given.

]]>Mathematics doi: 10.3390/math6110220

Authors: Tianhua Jiang Chao Zhang Huiqi Zhu Jiuchun Gu Guanlong Deng

Under the current environmental pressure, many manufacturing enterprises are urged or forced to adopt effective energy-saving measures. However, environmental metrics, such as energy consumption and CO2 emission, are seldom considered in the traditional production scheduling problems. Recently, the energy-related scheduling problem has been paid increasingly more attention by researchers. In this paper, an energy-efficient job shop scheduling problem (EJSP) is investigated with the objective of minimizing the sum of the energy consumption cost and the completion-time cost. As the classical JSP is well known as a non-deterministic polynomial-time hard (NP-hard) problem, an improved whale optimization algorithm (IWOA) is presented to solve the energy-efficient scheduling problem. The improvement is performed using dispatching rules (DR), a nonlinear convergence factor (NCF), and a mutation operation (MO). The DR is used to enhance the initial solution quality and overcome the drawbacks of the random population. The NCF is adopted to balance the abilities of exploration and exploitation of the algorithm. The MO is employed to reduce the possibility of falling into local optimum to avoid the premature convergence. To validate the effectiveness of the proposed algorithm, extensive simulations have been performed in the experiment section. The computational data demonstrate the promising advantages of the proposed IWOA for the energy-efficient job shop scheduling problem.

]]>Mathematics doi: 10.3390/math6110221

Authors: Azhar Hussain Tanzeela Kanwal Muhammad Adeel Stojan Radenovic

Based on the concepts of contractive conditions due to Suzuki (Suzuki, T., A generalized Banach contraction principle that characterizes metric completeness, Proceedings of the American Mathematical Society, 2008, 136, 1861&ndash;1869) and Jleli (Jleli, M., Samet, B., A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014, 2014, 38), our aim is to combine the aforementioned concepts in more general way for set valued and single valued mappings and to prove the existence of best proximity point results in the context of b-metric spaces. Endowing the concept of graph with b-metric space, we present some best proximity point results. Some concrete examples are presented to illustrate the obtained results. Moreover, we prove the existence of the solution of nonlinear fractional differential equation involving Caputo derivative. Presented results not only unify but also generalize several existing results on the topic in the corresponding literature.

]]>Mathematics doi: 10.3390/math6110219

Authors: Hsien-Chung Wu

The main aim of this paper is to propose the concept of so-called near fixed point and establish many types of near fixed point theorems in the set of all bounded and closed intervals in R . The concept of null set will be proposed in order to interpret the additive inverse element in the set of all bounded closed intervals. Based on the null set, the concepts of metric interval space and normed interval space are proposed, which are not the conventional metric and normed spaces. The concept of near fixed point is also defined based on the null set. In this case, we shall establish many types of near fixed point theorems in the metric and normed interval spaces.

]]>Mathematics doi: 10.3390/math6110218

Authors: Rujira Ouncharoen Nichaphat Patanarapeelert Thanin Sitthiwirattham

In this paper, we prove the sufficient conditions for the existence results of a solution of a nonlocal q-symmetric integral boundary value problem for a sequential q-symmetric integrodifference equation by using the Banach&rsquo;s contraction mapping principle and Krasnoselskii&rsquo;s fixed point theorem. Some examples are also presented to illustrate our results.

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