Mathematics, Volume 7, Issue 1 (January 2019) – 110 articles

Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic space. We also give the characterizations of the symplectic circular helices as the third- and fourth-order differential equations involving the symplectic curvatures. Full article

The present work gathers an educational experience based on the application of the personalized Kumon Mathematics Method, carried out in the school year 2015–2016, in which 30,849 students and 230 teachers from several educational centers throughout Spain have participated. We start with a theoretical foundation of the Kumon Method and continue with a description of the research methodology used. The empirical analysis carried out has been both in descriptive and correlational terms, using Spearman’s Rho statistic, between the levels at which the students of the sample have started and the Kumon level reached. The results show that the sooner students begin to learn Mathematics with the Kumon Method, the greater the chance of reaching a level of knowledge above their school level, which helps us to demonstrate the potential of this method in the teaching and learning process of Mathematics in the educational levels of Early Childhood and Primary Education. Full article

When facing a multi-period defined contribution (DC) pension plan investment problem during the accumulation phase, the risk aversion attitude of a mean-variance investor may depend on state variables. In this paper, we propose a state-dependent risk aversion model which is a linear function of the current wealth level after contribution. This risk aversion model is reasonable from both the dimensional analysis and the economic point of view. Moreover, we incorporate the wage income factor into our model. In the field of dynamic investment analysis, most studies have irrational situations in their models because of the lack of the positiveness for the wealth process. In view of it, we further improve the work of Wang and Chen by completely eliminating the irrationality of the model. Due to the time-inconsistency of the resulting stochastic control problem, we derive the explicit expressions of the equilibrium control and the corresponding equilibrium value function by adopting the game theoretic framework developed in Björk and Murgoci. Further, two special cases are discussed. Finally, using a more realistic risk aversion coefficient, we provide a series of empirical tests based on the real data from the American market and compare our results with the relevant results in the literature. Full article

Idempotent uninorms are simply defined by fixpointed negations. These uninorms, called here fixpointed idempotent uninorms, have been extensively studied because of their simplicity, whereas logics characterizing such uninorms have not. Recently, fixpointed uninorm mingle logic (fUML) was introduced, and its standard completeness, i.e., completeness on real unit interval $[0,1]$ , was proved by Baldi and Ciabattoni. However, their proof is not algebraic and does not shed any light on the algebraic feature by which an idempotent uninorm is characterized, using operations defined by a fixpointed negation. To shed a light on this feature, this paper algebraically investigates logics based on fixpointed idempotent uninorms. First, several such logics are introduced as axiomatic extensions of uninorm mingle logic (UML). The algebraic structures corresponding to the systems are then defined, and the results of the associated algebraic completeness are provided. Next, standard completeness is established for the systems using an Esteva–Godo-style approach for proving standard completeness. Full article

Calculating nodal voltages and branch current flows in a meshed network is fundamental to electrical engineering. This work demonstrates how such calculations can be performed using the eigenvalues and eigenvectors of the Laplacian matrix which describes the connectivity of the electrical network. These insights should permit the functioning of electrical networks to be understood in the context of spectral analysis. Full article

In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix $A$ and $B$ satisfy certain conditions, the FPPSS iterative method is convergent in the parameter’s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime. Full article

In this survey note, we discuss the notion of completeness for statistical structures. There are at least three connections whose completeness might be taken into account, namely, the Levi-Civita connection of the given metric, the statistical connection, and its conjugate. Especially little is known on the completeness of statistical connections. Full article

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results. Full article

Many factors influence the efficiency and quality of transport works. In particular, consultants and construction contractors of these works play important roles, and critical factors directly affect the quality of traffic works. If the quality of consultancy and construction is good, the project will reduce the total investment; if the contractor is good, the completion time of the new project is guaranteed, thus reducing construction costs. The longer the construction time is, the higher the cost of the project. In this study, the authors used optimal algorithms to evaluate past, present, and future contractors’ technical, technological, and performance effectiveness. Research results show that bidders are divided into three groups: highly effective bidders, stable contractors, and inefficient groups. Research results for this subject will help the government, regulatory agencies, and investors select good contractors as the basis for developing strategies and policies for the development of transport infrastructure. Full article

Medical diagnosis has been recognized as one of the key processes in clinical medicine, which determines diseases from some given symptoms. Nonetheless, previous works about medical diagnosis have some drawbacks because medical data are usually fuzzy, uncertain, incomplete and imprecise. To achieve the optimal medical diagnosis decision by reducing cost and enhancing accuracy, this paper develops a new method named interval three-way concept lattice model. Firstly, we redefine the decision rules and metric function of three-way decision based on interval concept lattice. Secondly, we build a visualized hierarchical structure of relationship between concepts through interval concept construction algorithm which helps us to make decision preferably and clearly. Finally, we establish a dynamic strategy optimization model for medical diagnosis decision making. In addition, a medical case demonstrates the effectiveness and feasibility of this proposed model. Full article

In this paper, we prove some common fixed-point theorems for two self-mappings in the context of a complete b-metric space by proposing a new contractive type condition. Further, we derive a result for three self-mappings in the same setting. We provide two examples to demonstrate the validity of the obtained results. Full article

This paper addresses the problem of global $\mu$ -stability for quaternion-valued neutral-type neural networks (QVNTNNs) with time-varying delays. First, QVNTNNs are transformed into two complex-valued systems by using a transformation to reduce the complexity of the computation generated by the non-commutativity of quaternion multiplication. A new convex inequality in a complex field is introduced. In what follows, the condition for the existence and uniqueness of the equilibrium point is primarily obtained by the homeomorphism theory. Next, the global stability conditions of the complex-valued systems are provided by constructing a novel Lyapunov–Krasovskii functional, using an integral inequality technique, and reciprocal convex combination approach. The gained global $\mu$ -stability conditions can be divided into three different kinds of stability forms by varying the positive continuous function $\mu \left(t\right)$ . Finally, three reliable examples and a simulation are given to display the effectiveness of the proposed methods. Full article

Arumugam and Mathew [Discret. Math. 2012, 312, 1584–1590] introduced the notion of fractional metric dimension of a connected graph. In this paper, a combinatorial technique is devised to compute it. In addition, using this technique the fractional metric dimension of the generalized Jahangir graph $J_{m,k}$ is computed for $k\ge 0$ and $m=5$ . Full article

In real applications, most decisions are fuzzy decisions, and the decision results mainly depend on the choice of aggregation operators. In order to aggregate information more scientifically and reasonably, the Heronian mean operator was studied in this paper. Considering the advantages and limitations of the Heronian mean (HM) operator, four Heronian mean operators for bipolar neutrosophic number (BNN) are proposed: the BNN generalized weighted HM (BNNGWHM) operator, the BNN improved generalized weighted HM (BNNIGWHM) operator, the BNN generalized weighted geometry HM (BNNGWGHM) operator, and the BNN improved generalized weighted geometry HM (BNNIGWGHM) operator. Then, their propositions were examined. Furthermore, two multi-criteria decision methods based on the proposed BNNIGWHM and BNNIGWGHM operator are introduced under a BNN environment. Lastly, the effectiveness of the new methods was verified with an example. Full article

Complex fuzzy sets are characterized by complex-valued membership functions, whose range is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane. In this paper, we define two kinds of entropy measures for complex fuzzy sets, called type-A and type-B entropy measures, and analyze their rotational invariance properties. Among them, two formulas of type-A entropy measures possess the attribute of rotational invariance, whereas the other two formulas of type-B entropy measures lack this characteristic. Full article

In recent years, fuzzy multisets and neutrosophic sets have become a subject of great interest for researchers and have been widely applied to algebraic structures include groups, rings, fields and lattices. Neutrosophic multiset is a generalization of multisets and neutrosophic sets. In this paper, we proposed a algebraic structure on neutrosophic multisets is called neutrosophic multigroups which allow the truth-membership, indeterminacy-membership and falsity-membership sequence have a set of real values between zero and one. This new notation of group as a bridge among neutrosophic multiset theory, set theory and group theory and also shows the effect of neutrosophic multisets on a group structure. We finally derive the basic properties of neutrosophic multigroups and give its applications to group theory. Full article

Chaotic systems with hidden attractors, infinite number of equilibrium points and different closed curve equilibrium have received much attention in the past six years. In this work, we introduce a new family of chaotic systems with different closed curve equilibrium. Using the methods of equilibrium points, phase portraits, maximal Lyapunov exponents, Kaplan–Yorke dimension, and eigenvalues, we analyze the dynamical properties of the proposed systems and extend the general knowledge of such systems. Full article

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is that it is unconditionally stable and convergent on the order $O({\tau }^2+h^4)$ (where $\tau$ is the time step size and h is the mesh size), under the maximum norm for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. The convergence order of this kind of method is also $O({\tau }^2+h^4)$ under the discrete maximum norm when the spatial step size is twice the one of H-OCD, which accelerates the computational process. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in the time direction. The results of numerical experiments are consistent with the theoretical analysis. Full article

Let $G_0$ be a connected graph on n vertices and m edges. The R-graph $R\left(G_0\right)$ of $G_0$ is a graph obtained from $G_0$ by adding a new vertex corresponding to each edge of $G_0$ and by joining each new vertex to the end points of the edge corresponding to it. Let $G_1$ and $G_2$ be graphs on $n_1$ and $n_2$ vertices, respectively. The R-graph double corona $G_0^{\left(R\right)}\circ \{G_1,G_2\}$ of $G_0$ , $G_1$ and $G_2$ , is the graph obtained by taking one copy of $R\left(G_0\right)$ , n copies of $G_1$ and m copies of $G_2$ and then by joining the i-th old-vertex of $R\left(G_0\right)$ to every vertex of the i-th copy of $G_1$ and the j-th new vertex of $R\left(G_0\right)$ to every vertex of the j-th copy of $G_2$ . In this paper, we consider resistance distance in $G_0^{\left(R\right)}\circ \{G_1,G_2\}$ . Moreover, we give an example to illustrate the correction and efficiency of the proposed method. Full article

The q-rung orthopair fuzzy set is a powerful tool for depicting fuzziness and uncertainty, as compared to the Pythagorean fuzzy model. The aim of this paper is to present q-rung orthopair fuzzy competition graphs (q-ROFCGs) and their generalizations, including q-rung orthopair fuzzy k-competition graphs, p-competition q-rung orthopair fuzzy graphs and m-step q-rung orthopair fuzzy competition graphs with several important properties. The study proposes the novel concepts of q-rung orthopair fuzzy cliques and triangulated q-rung orthopair fuzzy graphs with real-life characterizations. In particular, the present work evolves the notion of competition number and m-step competition number of q-rung picture fuzzy graphs with algorithms and explores their bounds in connection with the size of the smallest q-rung orthopair fuzzy edge clique cover. In addition, an application is illustrated in the soil ecosystem with an algorithm to highlight the contributions of this research article in practical applications. Full article

We present a new method to efficiently solve a multi-dimensional linear Partial Differential Equation (PDE) called the quasi-inverse matrix diagonalization method. In the proposed method, the Chebyshev-Galerkin method is used to solve multi-dimensional PDEs spectrally. Efficient calculations are conducted by converting dense equations of systems sparse using the quasi-inverse technique and by separating coupled spectral modes using the matrix diagonalization method. When we applied the proposed method to 2-D and 3-D Poisson equations and coupled Helmholtz equations in 2-D and a Stokes problem in 3-D, the proposed method showed higher efficiency in all cases than other current methods such as the quasi-inverse method and the matrix diagonalization method in solving the multi-dimensional PDEs. Due to this efficiency of the proposed method, we believe it can be applied in various fields where multi-dimensional PDEs must be solved. Full article

We provide a ball comparison between some 4-order methods to solve nonlinear equations involving Banach space valued operators. We only use hypotheses on the first derivative, as compared to the earlier works where they considered conditions reaching up to 5-order derivative, although these derivatives do not appear in the methods. Hence, we expand the applicability of them. Numerical experiments are used to compare the radii of convergence of these methods. Full article

In this work, our focus is to study the Fekete-Szegö functional in a different and innovative manner, and to do this we find its upper bound for certain analytic functions which give hyperbolic regions as image domain. The upper bounds obtained in this paper give refinement of already known results. Moreover, we extend our work by calculating similar problems for the inverse functions of these certain analytic functions for the sake of completeness. Full article

In this study, the value distribution of the differential polynomial $\phi f^2{f}^{\prime 2}-1$ is considered, where f is a transcendental meromorphic function, $\phi (\not\equiv 0)$ is a small function of f by the reduced counting function. This result improves the existed theorems which obtained by Jiang (Bull Korean Math Soc 53: 365-371, 2016) and also give a quantitative inequality of $\phi f{f}^{\prime }-1$ . Full article

Chaos is now recognized as one of three emergent topics of study in the 21c. It is seen as appropriate to examine this in art practice. Accordingly, this paper is written from an art perspective. It does not mimic a traditional mathematical or science format, presenting hypothesis, repeat testing, and a conclusion. The art process operates differently, and chaos is seen in graphic terms, veers more to philosophy, and is obviously subjective. The intent in researching secondary patterns, near the edge of chaos, is to make expressive graphic art images as art works, testing how close they might come to a chaotic state whilst retaining visual coherence. This underpins the author’s current research, but it is recognised as being a very narrow and specialized subset of analogue art activity. The way in which analogue generative art differs from the more common use of digital computers is addressed. Unlike the latter, the work involves designing and making the machines, making the programmers, and writing the algorithms; this is implicit in the text. A brief look at drawing machine history is presented, demonstrating how the author’s machines differ from others. A contextual cross refence is also made, where appropriate, to artists using digital means. The author’s research has documented practitioners who choose an analogue route to make art. However, hardly any of them create programmes to generate coherent images. This shortage creates problems when attempting to cite similar work. Whilst the general principle underlying the work presented is algorithmic, a significant element of quasi-random input is incorporated, consistent with a study of chaos. Emergent facets are implicit, such as the art process, design problem solving, the relationship between quasi-random and determinism, the psychology of evaluation, and the philosophy of how art works. From the author’s Programmable Analogue Drawing Machines, two are selected for this paper which draw Lissajous figures, use X:Y axes, turntables, Direct Current motors, and an asynchronous pen-lift mechanism. Simple instructions generate complex patterns in a similar vein to Alan Turings topics of phyllotaxis and morphogenesis. These aspects will be discussed, presenting two machines that demonstrate these properties. Full article

For a graph G, the resistance distance $r_G(x,y)$ is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index $R^{\ast }(G)={\sum }_{\{x,y\}\subset V(G)}{d}_G(x)d_G(y)r_G(x,y)$ , where $d_G(x)$ is the degree of vertex x, and $V(G)$ denotes the vertex set of G. L. Feng et al. obtained the element in $Cact(n;t)$ with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on $R^{\ast }(G)$ , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively. Full article

In this paper, the notion of moduloid on a nexus is introduced. Moreover, the concept of submoduloid is defined and some different submoduloids are introdused. Also, several interesting facts about submoduloids are proved. Finally, a homomorphism between two moduloids is defined and some related results are investigated. Full article

This article investigates the heat transfer flow of two layers of Phan-Thien-Tanner (PTT) fluids though a cylindrical pipe. The flow is assumed to be steady, incompressible, and stable and the fluid layers do not mix with each other. The fluid flow and heat transfer equations are modeled using the linear PTT fluid model. Exact solutions for the velocity, flow rates, temperature profiles, and stress distributions are obtained. It has also been shown that one can recover the Newtonian fluid results from the obtained results by putting the non-Newtonian parameters to zero. These results match with the corresponding results for Newtonian fluids already present in the literature. Graphical analysis of the behavior of the fluid velocities, temperatures, and stresses is also presented at the end. It is also shown that maximum velocity occurs in the inner fluid layer. Full article