# WSEAS Transactions on Mathematics

**Print ISSN: 1109-2769
E-ISSN: 2224-2880**

**Volume 15, 2016**

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

**Volume 15, 2016**

Title of the Paper: **Stability Analysis of Gompertz’s Logistic Growth Equation Under Strong, Weak and No Allee Effects **

Authors: **J. Leonel Rocha **

Abstract: The interest and the relevance of the study of population dynamics and extinction phenomenon are the main motivation to investigate the induction of Allee effects in Gompertz’s logistic growth equation. The stability analysis of the equilibrium points of Gompertz’s logistic growth equation under strong, weak and no Allee effects is presented. Properties and sufficient conditions for the existence of strong, weak and no Allee effects for these new continuous population growth models are provided and discussed. It is established a sufficient condition to prove that the time evolution of the population density to the stable equilibria gets larger, as the Allee effects get stronger. These continuous population growth models subjected to Allee effects take longer time to reach its equilibrium states. The developed models are validated using the Icelandic herring population, with GPDD Id.1765.

Keywords: Gompertz logistic growth equation, stability analysis, strong and weak Allee effects

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #54, pp. 578-587

Title of the Paper: **Matrix Inequalities in the Theory of Mixed Quermassintegrals and the Lp-Brunn-Minkowski Theory **

Authors: **John A. Gordon **

Abstract: The Brunn-Minkowski theory is central to convex geometric analysis, and mixed quermassintegrals and mixed p-quermassintegrals play a very important role in this theory. During the past quarter of a century both duals and Lp extensions of this theory have been developed. It is the aim of this present work to continue the development of analogues, for positive definite symmetric matrices, of some of the fundamental notations, invariants, and inequalities of mixed quermassintegrals, mixed p-quermassintegrals and Lp Brunn-Minkowski theory.

Keywords: Mixed determinant, Ordinary Quermassintegral, Mixed Quermassintegral, Mixed p-Quermassintegral, Matrix Lp-sum, Minkowski inequality, Lp-Brunn Minkowski inequality

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #53, pp. 566-577

Title of the Paper: **On Probability Algebra: Classic Theory of Probability Revisited **

Authors: **Yingxu Wang **

Abstract: It is recognized that the classic probability theory is cyclically defined among a small set of coupled operations. The sample space of probability is not merely 1-D invariant structures rather than n-D variant hyperstructure where the types of probability events encompassing those of joint or disjoint as well as dependent, independent, or mutually-exclusive ones. These fundamental properties of probability lead to a three-dimensional dynamic model of probability structures constrained by types of sample spaces, relations and dependencies of events. A reductive framework of general probability theory is rigorously derived from the independently defined model of conditional probability. This basic study reveals that the Bayes’ law needs to be extended in order to fit more general contexts of variant sample spaces and complex event properties. The revisited probability theory enables an extended mathematical structure known as probability algebra for rigorous manipulating uncertainty events and causations in formal inference, qualification, quantification, and semantic analysis in contemporary fields such as cognitive informatics, computational intelligence, cognitive robots, complex systems, soft computing, semantic computing, and brain informatics.

Keywords: Denotational mathematics, probability theory, probability algebra, statistics, formal inference, cognitive informatics, cognitive computing, computational intelligence, semantic computing, soft computing, brain informatics, cognitive systems

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #52, pp. 550-565

Title of the Paper: **Modeling of a Transport Network and its Spectral Analysis **

Authors: **Xuan Li, Genqi Xu, Yingfeng Shang **

Abstract: In the present paper, we study a regular triangle bi-directed transport network by the approach of the operator semigroup theory and linear operator spectral theory. First we established a model of the partial differential equations for the transport problem. And then we prove the well-posedness of the network system. By the detailed spectral analysis of the system operator, we prove that the spectrum of the system operator is composed of isolated eigenvalue of finite multiplicity, all root vectors are incomplete in the state space. Finally, we discuss some operation strategies for the transport networks based on the spectral distribution.

Keywords: triangle transport network, semigroup theory, spectral analysis, operation strategies

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #51, pp. 535-549

Title of the Paper: **Characterization of Lemniscate-Like Curves Generated by Linkages **

Authors: **Hee Seok Nam, Gaspar Porta, Hyung Ju Nam **

Abstract: Cassini Ovals is a well-known family of curves characterized by the product of distances from two fixed points called foci. One special case is the Lemniscate of Bernoulli, a figure-8 curve which can be drawn by many ways. This paper focuses on the family of lemniscate-like curves especially generated by three-bar linkage systems where the marker is the midpoint of the middle rod. An algebraic characterization of such family of curves is investigated using the distances from the foci. It turns out that the corresponding Cartesian equation is of the form of the Hippopede defined as the intersection of a torus and a plane. A geometric construction showing the connection between three-bar linkages and lemniscate-like curves leads to another representation using polar equations. A parametric representation is also given to illustrate the family of curves using an EXCEL worksheet. Finally, another interesting family of curves so called skewed lemniscate-like curves is constructed and characterized algebraically as a natural generalization where the marker can be located on any fixed point on the middle rod.

Keywords: Lemniscate of Bernoulli, lemniscate-like curve, Hippopede, characterization, three-bar linkage system, skewed lemniscate-like curve, Cassini Ovals

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #50, pp. 528-534

Title of the Paper: **A New Complete Irresoluteness Function **

Authors: **Aynur Keskin Kaymakci **

Abstract: In this presentation, first of all we definite a new type of function by using delta-b-open sets. Then, we obtain some characterizations and some properties of this function. Besides, we give their relationships with other types of functions between topological spaces.

Keywords: δ-b-open sets, b-open sets, δ-semi-open sets, semi open sets, completely δ-b-irresolute functions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #49, pp. 523-527

Title of the Paper: **Media Access Over PLC Technology Using Shapley-Shubik Power Index **

Authors: **Juan C. Vesga, Javier E. Sierra, Gerardo Granados **

Abstract: Due to the growing demand of new services, IEEE 1901 has embedded a mechanism of Access to the hybrid means, supported in CSMA/CA and TDMA in order to offer accurate levels of quality of service (QoS). Nevertheless, such strategy has shown low efficiency, as the number of nodes increases. In the view of the above, this paper proposes a mechanism of media access over OFDMA (Orthogonal Frequency-Division Multiple Access), based on the weighted voting games, supported in the Shapley-Shubik’s power index in order to optimize the allocation of resources in the time and frequency domain. This optimizing will, thus, improve the network performance and allow multiple nodes to be able to transmit simultaneously. According to the results, it was possible to evidence that the usage of weighted voting games supported in the Shapley-Shubik’s power index can be considered as an alternative to establish a mechanism of access to the adequate media in PLC networks and its feasibility of being implemented in low-cost embedded systems due to the fact that such alternative does not require a high computational complexity for its development.

Keywords: Resource management, Computer networks, Cooperative Game Theory, Optimization methods, Shapley-Shubik’s power index, Power Line Communications

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #48, pp. 511-522

Title of the Paper: **On Rank One Perturbations of Hamiltonian System with Periodic Coefficients **

Authors: **Mouhamadou Dosso, Arouna G. Y. Traore, Jean-Claude Koua Brou **

Abstract: From a theory developed by C. Mehl, et al., a theory of the rank one perturbation of Hamiltonian systems with periodic coefficients is proposed. It is shown that the rank one perturbation of the fundamental solution of Hamiltonian system with periodic coefficients is solution of its rank one perturbation. Some results on the consequences of the strong stability of these types of systems on their rank one perturbation is proposed. Two numerical examples are given to illustrate this theory.

Keywords: Eigenvalue, symplectic matrix, Hamiltonian system, Fundamental solutions, Perturbation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #47, pp. 502-510

Title of the Paper: **Application of the Fractional Derivative Kelvin-Voigt Model for the Analysis of Impact Response of a Kirchhoff-Love Plate **

Authors: **Y. Rossikhin, M. Shitikova, Phan Thanh Trung **

Abstract: In the present paper, we consider the problem on a transverse impact of an elastic sphere upon an elastic Kirchhoff-Love plate in a viscoelastic medium, the viscoelastic features of which are described by the fractional derivative Kelvin-Voigt model. Within the contact domain the contact force is defined by the Hertzian contact. The functional equation for determining the contact force is derived and its approximate analytical solution is obtained.

Keywords: Fractional derivative Kelvin-Voigt model, Impact, Kirchhoff-Love Plate

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #46, pp. 498-501

Title of the Paper: **A Modified Cubic Equation of State Model and Parameter Optimization in Modeling Supercritical Fluid Solubility **

Authors: **Jinghuan Li, Li Xu, Zhen Huang **

Abstract: Cubic equation of state (EOS) is one of most attracting models in modeling solute solubility in supercritical fluid. The traditional implementation in EOS models requires critical parameters and acentric factor of the solute, however, estimating these physical parameters is not a trivia. As a modification of the traditional EOS method, in this paper, we do not estimate critical properties and acentric factor, but consider the energy parameter of solute and the binary interaction parameter in the co-volume term as adjustable parameters. These adjustable parameters are optimized using pattern search (PS) method by minimizing average absolute relative deviation (AARD) between calculated and experimental solubility data. To illustrate the efficiency of our modified model, a comparison with traditional EOS model is presented. The result shows that our modified model gives better performance reflected by lower AARD. Moreover, the optimization method, PS, is compared with the genetic algorithm(GA), and it is found that the former gives better optimizations and more significant reduction of computing times than the latter. Finally, the rationality of the modified model is further discussed.

Keywords: Supercritical fluid, Solubility, Equation of state, Critical property, Parameter optimization, Pattern search method

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #45, pp. 486-497

Title of the Paper: **The Nakagami and its Related Distributions **

Authors: **Li-Fei Huang **

Abstract: This paper introduces the Nakagami distribution which is usually used to simulate the ultrasound image. The Nakagami distribution is related to the gamma distribution, the Rayleigh distribution, the weibull distribution, the chi-square distribution and the exponential distribution. Through the gamma distribution, it’s much easier to derive the moments of a Nakagami random variable. The confidence interval for the ratio of medians from two independent Nakagami distributions is constructed.

Keywords: The Nakagami distribution, The gamma distribution, The Rayleigh distribution, TheWeibull distribution, The chi-square distribution, The exponential distribution, The ratio of median

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #44, pp. 477-485

Title of the Paper: **Simple Equation Method for Nonlinear Evolution Equations in Mathematical Physics **

Authors: **Khaled A. Gepreel, Taher A. Nofal, Khulood O. Alweil **

Abstract: In this article, we use a direct method namely simple equation method to construct the traveling wave solutions for some nonlinear evolution equations in mathematical physics via, the nonlinear Schrodinger equation with dual power law nonlinearity, the nonlinear variant Boussinesq differential equations, the nonlinear generalized- Zakharov equations and the nonlinear coupled Maccaris equations. In the simple equation method, we suppose the trial equation satisfies the first order Bernoulli differential equation or the first order Riccati differential equation. It has been shown that this method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.

Keywords: Simple equation method, Nonlinear evaluation equations, Traveling wave solutions, Exact solutions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #43, pp. 462-476

Title of the Paper: **Practical Notes on Obtaining Reliability Polynomials **

Authors: **Alexey S. Rodionov, Olga Rodionova **

Abstract: General technical problems of obtaining coefficients of reliability polynomials for different reliability indices, such as all-terminal reliability, average pairwise reliability and average size of a connected sub-graph containing a special node are discussed in the paper. It is shown that one of possible forms of a reliability polynomial presentation helps in considerable speeding up of its obtaining by using an intentional meaning of coefficients. Procedure of calculations in this case can be reduced to summarizing parts of vectors of binomial coefficients.

Keywords: Network reliability, Reliability polynomials

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #42, pp. 450-461

Title of the Paper: **Restricted Edge Connectivity and Restricted Connectivity of Graphs **

Authors: **Litao Guo, Xiaofeng Guo **

Keywords: 3-Restricted edge connectivity, Super-λ3, Super-κ3

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #41, pp. 441-449

Title of the Paper: **The Assets-Claims on Assets Equivalence in the Axiomatic Method **

Authors: **Fernando Juárez **

Abstract: The purpose of the study is to analyze the assets-claims on assets equivalence based on the dual concept of monetary units and the axiomatic method. The methodology is analytical, rationalistic and deductive; it uses axiomatic theory with set theory and predicate logic to test set equivalence. The axiomatic theory involves a set of axioms, which are used in combination with accounting axioms to develop a proof of the assets-claims on the assets considered as finite sets. The analysis uses a bijective function based on the dual concept of monetary units, and proof by contraposition to test the fulfillment of the requirement of a bijective function. Results show that assets cardinality is not equal to claims on assets cardinality when taking into account the dual concept of monetary units, and as a consequence assets and claims on assets are not equivalent.

Keywords: Dual concept, axiomatic method, cardinality, equivalence, balance sheet

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #40, pp. 431-440

Title of the Paper: **The Balance Sheet and the Assets-Claims on Assets Relationship in the Axiomatic Method **

Authors: **Fernando Juárez **

Abstract: The purpose of this study is to analyze the set structure of the balance sheet and assets-claims on assets relationship, considering the dual concept of monetary units, the axiomatic theory and accounting-specific axioms. The structure of the balance sheet and assets-claims on assets relationship are examined using a rationalistic, analytical and deductive method; this method uses the axiomatic set theory and predicate logic to define a set of axioms and the logical rationale to apply them to any deductive proof. The method includes accounting primitives and axioms to use in combination with those of the axiomatic theory. A direct proof is applied to test the balance sheet fit to a hereditary set structure according to the axiomatic theory, and proof by contraposition is employed to examine the assets-claims on assets equality by comparing their elements. Results show that balance sheet has a set structure that can be defined and analyzed with the axiomatic method and fits a hereditary set structure. Also, by comparing the elements of assets and claims on assets and considering their financial classification, it is shown that these sets do not contain the same elements and, consequently, they are not equal under the postulates of the axiomatic method.

Keywords: Dual concept, accounting transactions, axiomatic method, assets, claims on assets, balance sheet

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #39, pp. 420-430

Title of the Paper: **Predictive Associative Search Models in Variable Structure Control Systems **

Authors: **Natalia N. Bakhtadze, Ekaterina A. Sakrutina, Elena F. Jharko **

Abstract: Predictive associative search models found a wide application in advanced control systems, decision making systems, play a fundamental role in any activity concerned with the signal processing process. In the paper, associative search identification models in variable structure control systems are considered. The application of this approach is especially effective whilst compensating for insufficient lab data for model development. In such a case, fuzzy specification of certain process variables using process knowledge base is practiced. A methodology of the synthesis of a variable structure control system is presented, as well issues of the stability of a model built by use of the associative search are considered, in the aspect of the spectrum analysis of the multi-scale wavelet expansion.

Keywords: system identification, control system, information model, virtual model, associative search, fuzzy model, knowledge base, wavelets

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #38, pp. 407-419

Title of the Paper: **Regression Analysis of Octal Rings as Mechanical Force Transducers **

Authors: **Khaled A. Abuhasel, Essam Soliman **

Abstract: This paper presents analytical approach for octal rings as mechanical force transducers. The work first analyzes the characteristics of octal rings as mechanical force transducers, using a finite element model of the ring to determine its state of strain upon the application of load. It also correlates correlate ring design parameters and performance measures using an L9 orthogonal array of finite element simulations. Design parameters include height, thickness, width, and edge curvature. Performance measures include sensitivity and stiffness. Model simulation results showed considerable variation in strain along ring face with considerable difference in maximum values between tensile and compressive strains. Also, revealed a region of a relatively large tensile strain within the ring not addressed in the literature. The relation between strain gauge length and average strain revealed an optimal gauge length that improves ring performance. Moreover, simulation results showed that increasing ring height and decreasing its thickness increases its sensitivity and decreases its stiffness. Finally, results of the regression analysis indicated that there is enough statistical evidence to say that the parameters, height and thickness, influences the stiffness response and the parameters, height, thickness, and width, influences the sensitivity response.

Keywords: Octal rings, strain gauge, average strain, sensitivity, stiffness, force transducer, regression analysis

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #37, pp. 398-406

Title of the Paper: **The Complexity Analysis of Dynamic Output Game with Risk-Averse Consideration **

Authors: **Qiuxiang Li, Junhai Ma **

Abstract: Risk aversion is prevalent behaviors of decision makers when the external environment is uncertain. In this paper, the dynamic output game strategies in two-tier supply chains under uncertain demand and competitive environment are discussed, where the supply chains include a risk-averse manufacturer and customers respectively. Two game models with a mean-variance framework are developed under decentralized and team game respectively. The complex dynamics characteristics and influences of parameters on the dynamics behaviors of two game strategies are investigated using parameter basin plots, bifurcation diagrams and the largest Lyapunov exponent etal. The results show that game strategy will affect the stability of the system, and risk reference has little effect on stability of the system while the weight parameter of the products and uncertain demand have. The instability of the system causes an unfavorable outcome and the parameter adjustment mechanism can be applied by manufacturers to eliminate the chaotic behaviors.

Keywords: Risk-averse behavior, Team decision, Parameter basin, Mean-variance method

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #36, pp. 385-397

Title of the Paper: **The k-Path Vertex Cover of Some Product Graphs **

Authors: **Bitao Zhang, Liancui Zuo **

Abstract: For a graph G and a positive integer k, a subset S of vertices of G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. The cardinality of a minimum k-path vertex cover is denoted by k(G). In this paper, we give some bounds and the exact values in special cases for k of the Cartesian, and lexicographic products of some graphs.

Keywords: Vertex cover, k-path vertex cover, Cartesian product, lexicographic product

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #35, pp. 374-384

Title of the Paper: **Analysis of Dynamical Behaviors of a Double Belt Friction-Oscillator Model **

Authors: **Ge Chen, Jinjun Fan **

Abstract: In this paper, analytical results of complex motions of a double belt friction-oscillator are investigated using the flow switchability theory of the discontinuous dynamical systems. The friction-oscillator is composed of a mass connected by viscoelastic element and linear spring-loading and interacting with two moving belts by means of dry friction. Different domains and boundaries for such system are defined according to the friction discontinuity, which exhibits multiple discontinuous boundaries in the phase space. The necessary and sufficient conditions of the stick motions, non-stick motions and grazing motions of such system are given in the form of theorem mathematically. The switching planes and basic mappings will be defined to study grazing motions and periodic motions. The results of computer simulation of the stick motions and grazing motions for different parameters are submitted in the present paper. With appropriate mapping structures, the simulation of the stick and non-stick periodic motions for such an oscillator are also given.

Keywords: double belt friction-oscillator, discontinuous dynamical system, switchability, stick motion, grazing motion

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #34, pp. 357-373

Title of the Paper: **On The Fekete-Szegö Problem for Certain Subclasses of Bi-Univalent Functions Involving Fractional q− Calculus Operators **

Authors: **G. Murugusundaramoorthy, T. Janani **

Abstract: In this paper, we introduce and investigate new subclasses of the function class Σ of bi-univalent functions defined in the open unit disk, which are associated with fractional q? calculus operators satisfying subordinate conditions. Estimates on the coefficients |a2| and |a3| are obtained and Fekete-Szeg¨o inequalities for the function class are determined.

Keywords: Analytic functions, Univalent functions, Bi-univalent functions, Bi-starlike functions, Bi-convex functions, Bi-Mocanu-convex functions, Subordination, q-calculus operator

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #33, pp. 345-356

Title of the Paper: **Almost Periodic Solution for a Predator-Prey System with Impulses on Time Scales **

Authors: **Lili Wang, Limin Wang **

Abstract: This paper is concerned with a predator-prey system with impulses on time scales. Based on the theory of calculus on time scales and the properties of almost periodic functions as well as Razumikhin type theorem, sufficient conditions which guarantee the existence of a unique uniformly asymptotic stable almost periodic solution of the system are obtained, by the relation between the solutions of impulsive system and the corresponding non-impulsive system. Finally, an example and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

Keywords: Permanence, Almost periodic solution, Uniformly asymptotic stable, Impulse, Time scale

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #32, pp. 335-344

Title of the Paper: **Dynamical Analysis of a Delayed Hepatitis B Virus Model with Immune Responses **

Authors: **Kejun Zhuang **

Abstract: In this paper, a novel hepatitis B virus infection model with time delay and CTL immune responses is proposed. The threshold parameters are derived and existence of equilibria is discussed. Then, the uniform boundedness and nonnegativeness of solutions are obtained. By analyzing the transcendental characteristic equations, asymptotic stabilities of nonnegative equilibria are investigated. In particular, the occurrence of Hopf bifurcation phenomena is considered by regarding time delay as a bifurcation parameter. Finally, some numerical simulations are given to confirm the theoretical findings.

Keywords: HBV infection, CTL immune responses, stability, Hopf bifurcation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #31, pp. 325-334

Title of the Paper: **G* Properties -Without G* Construction **

Authors: **M. Yamuna, K. Karthika **

Abstract: In any planar graph G there is a one to one correspondence between edges in G, its dual G∗. As the number of vertices and edges increase construction of G* is complicated, when graph G* is not required and only properties of G* is a need, determining properties of G* without actual construction of G* is comfortable. In this paper we determine the domination number of G*, G*, chromatic polynomial of G*, spanning tree of G*, number of spanning trees of G* from G. Determining the domination number of G*, G* using matrix representation is presented. MATLAB code for determining the same is also provided.

Keywords: Planar graph, γ-stable, Chromatic polynomial, Spanning tree

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #30, pp. 312-324

Title of the Paper: **Dynamics of an Epidemic Model with Saturation Recovery and Delays **

Authors: **Juan Liu, Changjin Xu **

Abstract: This paper discusses the dynamical behaviors of an SIRS epidemic model with saturation recovery and two delays. The main results are given in terms of local stability and Hopf bifurcation. By choosing the diverse delay as a bifurcation parameter, we show that the complex Hopf bifurcation phenomenon at the positive equilibrium of the model can occur as the diverse delay crosses the corresponding critical value. Particularly, the direction and stability of the local Hopf bifurcation are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations supporting our theoretical results are presented.

Keywords: Epidemic model, Delays, Hopf bifurcation, Stability, Periodic solution

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #29, pp. 300-311

Title of the Paper: **Existence and Iteration of Positive Solutions for a Mixed-Order Four-Point Boundary Value Problem with p-Laplacian **

Authors: **Yitao Yang **

Keywords: Iteration, Mixed-order four-point boundary value problem, Positive solutions, p-Laplacian

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #28, pp. 290-299

Title of the Paper: **The Modes of Some Distributions in Independent Trials **

Authors: **Jiguang Shao, Qiuyuan Wang, Sheng Fu **

Abstract: By the methods of multivariate transition probability flow graphs, we define the multivariate geometric distribution and the multi-parameter geometric distribution in independent trials, derive their generating functions, probability distributions and the exact modes. For the multinomial distribution, we only get its mode in some cases. It is still an open question for all other cases. Following the multinomial distribution, we state the definition of the multivariate Poisson distribution, and then employ the joint probability generating function of multivariate Poisson distribution to discuss its modes. We also prove that k is the unique mode of the geometric distribution of order k. Finally, we propose some open questions to the interested readers as an objective and challenge for further study.

Keywords: mode, multivariate geometric distribution, multivariate Poisson distribution, success run, MTPFG, probability generating function

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #27, pp. 280-289

Title of the Paper: **Thermosolutal MHD Marangoni Convective Flow of a Nanofluid Past a Flat Plate with Viscous Dissipation and Radiation Effects **

Authors: **D. R. V. S. R. K. Sastry **

Abstract: The radiation and viscous effects on a steady two dimensional boundary layer flow of a nanofluid over a flat plate is analyzed. Two types of nanofluids namely Cu and Ag are considered with water as base fluid (Prandtl number 6.2). The governing non linear partial differential equations are transformed into a system of non linear ordinary differential equations by similarity transformation. A numerical solution is obtained by the help of Matlab ’bvp4c’ function. The effects of volume fraction, radiation parameter, Eckert number, Schmidt number and Soret number on velocity, temperature and concentration profiles as well as on heat and mass transfer are studied. The results are presented through plots and tables.

Keywords: Viscous dissipation, radiation, nano fluid, heat and mass transfer, MHD

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #26, pp. 271-279

Title of the Paper: **Potential Eventual Positivity of One Specific Tree Sign Pattern **

Authors: **Ber-Lin Yu, Jie Cui **

Abstract: A sign pattern is a matrix whose entries belong to the set {+;−; 0}. An n-by-n sign pattern A is said to allow an eventually positive matrix if there exist some real matrices A with the same sign pattern as A and a positive integer k0 such that Ak > 0 for all k ≥ k0. Identifying and classifying the n-by-n sign patterns that allow an eventually positive matrix were posed as two open problems by Berman, Catral, Dealba, et al. [Sign patterns that allow eventual positivity, ELA, 19(2010): 108-120]. In this article, a tree sign pattern A obtained from the double star sign pattern S3;2 by adding an pendent edge is investigated. Some necessary or sufficient conditions for the sign pattern A to allow an eventually positive matrix are established first. Then all the minimal tree sign patterns that allow an eventually positive matrix are identified as five specific tree sign patterns. Finally, all the tree sign patterns that allow an eventually positive matrix is classified.

Keywords: Sign pattern, Tree, Eventually positive matrix; Checkerboard block matrix

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #25, pp. 261-270

Title of the Paper: **Uniform Persistence and Global Attractivity of A Predator-Prey Model with Mutual Interference and Delays **

Authors: **Kai Wang, Yanling Zhu **

Abstract: In this paper, a predator-prey model with mutual interference and delays is studied by utilizing the comparison theorem and Lyapunov second method. Some sufficient conditions for uniform persistence and global attractivity of positive periodic solution of the model are obtained. It is very interesting that the results obtained are related to delays, especially based on the mutual interference constant. Furthermore, it is the first time that such a model is considered. An example is illustrated to verify the feasibility of the results in the last part.

Keywords: Mutual interference, Delays, Uniform persistence, Lyapunov second method, Global attractivity

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #24, pp. 251-260

Title of the Paper: **The Influences of Smooth Approximation Functions for SPTSVM **

Authors: **Xinxin Zhang, Liya Fan **

Abstract: The recently proposed smooth projection twin support vector machine(SPTSVM) gains a good generalization ability and is suitable for many binary classification problems. But we know that different smooth approximation functions may bring different classification accuracies. In order to study the influence of smooth approximation functions for SPTSVM, in this paper, we first overview eight known smooth approximation functions and describe their differentiability and error ranges by five lemmas and one theorem. Then, we perform a series of comparative experiments on classification accuracy and running time by using SPTSVM with Newton-Armijo method on 10 UCI datasets and 6 NDC datasets. From experiment results, we can get a choice order of the eight approximation functions in generally.

Keywords: Smooth projection TSVM, plus function, smooth approximation function, choice order

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #23, pp. 241-250

Title of the Paper: **Optimal Reinsurance and Investment Problem with Stochastic Interest Rate and Stochastic Volatility in the Mean-Variance Framework **

Authors: **Wei Wei, Shunhou Fan, Hao Chang **

Abstract: This paper studied an optimal reinsurance and investment problem for insurers under the mean-variance criterion in the stochastic interest rate and stochastic volatility environment, where the financial market consists of two assets: one is the risk-free asset (i.e bond) and the other is the risky-asset (i.e stock) whose volatility satisfying the Heston model. Assume that the interest rate is driven by Vasicek interest rate model and the surplus process is approximated by diffusion approximation model. In order to hedge the risk of the insurance, proportional reinsurance is considered. And the insurer wishes to look for the optimal reinsurance and investment strategies to minimize the variance of the terminal wealth for a given expected terminal wealth. By employing dynamic programming principle and Lagrange duality theorem, the optimal reinsurance and investment strategies and the efficient frontier are explicitly obtained. Finally, some special cases and sensitivity analysis are provided to illustrate our results.

Keywords: Optimal reinsurance and investment strategy, Vasicek interest rate, Stochastic volatility, mean-variance framework, the efficient frontier

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #22, pp. 226-240

Title of the Paper: **Exact Determinants of the RFPrLrR Circulant Involving Jacobsthal, Jacobsthal-Lucas, Perrin and Padovan Numbers **

Authors: **Tingting Xu, Zhaolin Jiang **

Abstract: The row first-plus-rlast r-right (RFPrLrR) circulant matrix and row last-plus-rfirst r-left (RLPrFrL) circulant matrices are two special pattern matrices. On the basis of the beautiful properties of famous numbers and the inverse factorization of polynomial, we give the exact determinants of the two pattern matrices involving Jacobsthal, Jacobsthal-Lucas, Perrin and Padovan numbers.

Keywords: Determinant, RFPrLrR circulant matrix, RLPrFrL circulant matrix, Jacobsthal number, Jacobsthal-Lucas number, Perrin number, Padovan number

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #21, pp. 215-225

Title of the Paper: **Time-Fractional Klein-Gordon Equation: Formulation and Solution Using Variational Methods **

Authors: **Youwei Zhang **

Abstract: This paper presents the formulation of time-fractional Klein-Gordon equation using the Euler-Lagrange variational technique in the Riesz derivative sense and derives an approximate solitary wave solution. Our results witness that He’s variational iteration method was very efficient and powerful technique in finding the solution of the proposed equation. The basic idea described in this paper is efficient and powerful in solving wide classes of nonlinear fractional high order evolution equations.

Keywords: Riesz fractional derivative, Euler-Lagrange equation, Klein-Gordon equation, He’s variational iteration method, Solitary wave

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #20, pp. 206-214

Title of the Paper: **Dual-Channel Supply Chain Coordination Strategy with Dynamic Cooperative Advertising **

Authors: **Qinghua Li, Bo Li **

Abstract: With the rapid development of electronic commerce, many manufacturers choose to establish dual-channel supply chains. To promote sales, the manufacturer in a dual-channel supply chain usually invests in national advertising and provides support to the retailer who promotes the goodwill through local advertisement. In our research, we use a Stackelberg differential game theoretic model to show that the manufacturer who establishes an e-channel can adopt the cooperative advertising strategy with the retailer, so that the profits of both the manufacturer and the retailer will increase. We analyze the advantages brought by the e-channel which is established by the manufacturer. Further, a revenue sharing contract is given to coordinate the dual-channel supply chain.

Keywords: Electronic Commerce, Dual-channel Supply Chain, Cooperative Advertising, Stackelberg Differential Game, Nerlove-Arrow Model, Revenue Sharing Contract

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #19, pp. 196-205

Title of the Paper: **Pricing Forward Starting Options Under Regime Switching Jump Diffusion Models **

Authors: **Wei Wang, Liegang Dong, Xiaonan Su **

Abstract: This paper studies the pricing of forward starting options under regime switching jump diffusion models. We suppose that a market economy has only two states, one is a stable state, the other is a high volatility state. The dynamics of a risky asset is modeled by a geometry Brownian motion when the market state is stable, otherwise, it follows a jump diffusion model. We propose two types of regime switching jump diffusion models: one is a two-state regime switching Merton jump diffusion model, and the other is a two-state regime switching double exponential jump diffusion model. Finally, some analytic formulas for pricing forward starting options are derived under these regime switching jump diffusion models.

Keywords: Jump diffusion, Option pricing, Regime switching

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #18, pp. 185-195

Title of the Paper: **New Construction of Deterministic Compressed Sensing Matrices via Singular Linear Spaces over Finite Fields **

Authors: **Xuemei Liu, Yingmo Jie **

Abstract: As an emerging approach of signal processing, not only has compressed sensing (CS) successfully compressed and sampled signals with few measurements, but also has owned the capabilities of ensuring the exact recovery of signals. However, the above-mentioned properties are based on the (compressed) sensing matrices. Hence the construction of sensing matrices is the key problem. Compared with the intensive study of random sensing matrices, only a few deterministic constructions are known. In this paper, we provide a family of new construction of deterministic sensing matrices via singular linear spaces over finite fields, and show its better performance than Devore’s construction using polynomials over finite fields.

Keywords: Compressed sensing matrices, Singular linear spaces, Coherence, Restricted isometry property (RIP)

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #17, pp. 176-184

Title of the Paper: **Global State-Feedback Stabilization of High-Order Nonholonomic Systems with Time-Varying Delays **

Authors: **Deheng Hou, Fangzheng Gao **

Abstract: This paper investigates the problem of global stabilization by state-feedback for a class of high-order nonholonomic systems in power chained form with time-varying delays. By using input-state-scaling, and adding a power integrator techniques, and choosing an appropriate Lyapunov-Krasoviskii functional, a state-feedback controller is constructed. Based on switching strategy to eliminate the phenomenon of uncontrollability, the proposed controller could ensure that the closed-loop system is globally asymptotically regulated at origin. A simulation example is provided to illustrate the effectiveness of the proposed method.

Keywords: High-order nonholonomic systems, Time-varying delays, Input-state-scaling, Adding a power integrator

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #16, pp. 166-175

Title of the Paper: **Positive Periodic Solutions in Shifts δ± for a Neutral Delay Logistic Equation on Time Scales **

Authors: **Lili Wang, Meng Hu **

Keywords: Positive periodic solution, Neutral delay logistic equation, Strict-set-contraction, Shift operator, Time scale

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #15, pp. 156-165

Title of the Paper: **Some Generalizations of Geometric Distribution in Bernoulli Trials by TPFG Methods **

Authors: **Jiguang Shao, Sheng Fu **

Abstract: In this paper, by the transition probability flow graphs theory, we obtain the transition probability flow graphs of the random variables distributed as Gk(p) and NBk(r; p), the generalizations of usual geometric distribution. We derive their probability generating functions, means, variances and their exact probability distributions, in addition, we reveal the correlation between them. For success runs in n Bernoulli trials, we find the new concise formulae for the probability distributions of Ln and N(k)n . Finally, proceed from the parameter n = 2; 3; 4, we derive the generalized geometric distribution denoted by G(p1, • • • , pn) and discuss its properties base on multivariate transition probability flow graphs methods.

Keywords: geometric distribution, negative binomial distribution, success run, TPFG methods, probability generating function, probability distribution

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #14, pp. 146-155

Title of the Paper: **The Spectrum Distribution of Transport Operator with Abstract Boundary Conditions in Slab Geometry **

Authors: **Hongxing Wu, Shenghua Wang, Dengbin Yuan **

Abstract: In this paper, transport equation with continuous energy, nonhomogeneous medium and abstract boundary conditions is studied in slab geometry. It is to prove that |Imλ| âˆ¥ K(λI − BH)−1K âˆ¥ (|Imλ| → +∞) is bounded in the trip Γε, and the spectrum of transport operator AH consists of only finite isolated eigenvalues with a finite algebraic multiplicities in trip Γε. The main methods rely on operators theory, resolvent operators and comparison operators approach.

Keywords: Transport operator, abstract boundary condition, boundedness, isolated eigenvalues

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #13, pp. 136-145

Title of the Paper: **Optimal Investment and Risk Management for Open-Ended Funds Under Heston’s SV Model **

Authors: **Zanyong Huang, Ximin Rong **

Abstract: This paper considers the optimal investment and risk management strategies for a manager of open-ended funds under Heston’s stochastic volatility model. The manager is allowed to invest the fund in a financial market, which consists of one risk-free asset and one risky asset whose price process satisfies Heston’s SV model. The objective of the fund manager is to maximize the expected exponential utility of the terminal wealth of the fund assets. We obtain the optimal strategies and value function via stochastic optimal control approach explicitly. Moreover, a verification theorem is provided and the properties of the optimal strategies are discussed. Finally, sensitivity analysis is presented to illustrate the influences of parameters on the optimal investment strategy and redemption limit.

Keywords: Open-ended funds, Subscribe and redeem, Compound Poisson process, Stochastic volatility, Hamilton-Jacobi-Bellman equations, Stochastic optimal control

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #12, pp. 126-135

Title of the Paper: **The Bullwhip Effect Caused by Information Distortion in a Complex Supply Chain Under Exponential Smoothing Forecast **

Authors: **Junhai Ma, Weihua Lai, Xinaogang Ma **

Abstract: The bullwhip effect is the phenomenon in which information on demand is transferred in the form of orders between the nodes of a supply chain tends to be distorted when it moves from downstream to upstream. In this paper, we measure the impact of bullwhip effect under Exponential Smoothing Forecast for a simple, two-stage supply chain which consists of one supplier and two retailers. And it is a simple replenishment system where a first-order autoregressive process describes the customer demand and an order-up-to inventory policy characterizes the replenishment decision. We get the influence of information distortion on the bullwhip effect through investigating the impacts of autoregressive coefficient, the lead-time, the smoothing parameter, market competition degree, and the consistency of demand volatility on the bullwhip effect by using algebraic analysis and numerical simulation. And, we also find the ways in which these parameters affect the bullwhip effect are different. Finally, we discuss some measures to mitigate the influence of information distortion on the bullwhip effect.

Keywords: Information distortion, Bullwhip effect, Complex Supply chain, Two retailers model, Exponential Smoothing

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #11, pp. 110-125

Title of the Paper: **Optimal Consumption and Portfolio Decisions with Stochastic Affine Interest Rate Model **

Authors: **Hao Chang **

Abstract: This article is concerned with an investment and consumption problem with stochastic affine interest rate model, which includes the CIR model and the Vasicek model as special cases. The financial market is composed of three assets: one cash account, one stock and one zero-coupon bond. Moreover, the price dynamics of the stock and zero-coupon bond is affected by the dynamics of interest rate. Our objective is to seek an optimal consumption and portfolio decisions to maximize the expected discounted utility of intermediate consumption and terminal wealth in the finite horizon. By applying stochastic dynamic programming principle and variable change techniques, we obtain the explicit expressions of the optimal consumption and portfolio decisions in the power utility and logarithm utility cases. In order to analyze the impact of the parameters of interest rate on the optimal consumption and portfolio decisions, we provide a numerical example to illustrate our results.

Keywords: Investment and consumption, stochastic affine interest rate model, stochastic dynamic programming principle, the closed-form solution

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #10, pp. 96-109

Title of the Paper: **The Projected Generalized Sylvester Equations: Numerical Solution and Applications **

Authors: **Yiqin Lin, Liang Bao **

Abstract: In this paper we consider the numerical solution of large-scale projected generalized continuous-time and discrete-time Sylvester equations with low-rank right-hand sides. First, we present the results on the sufficient conditions for the existence, uniqueness, and analytic formula of the solutions of these equations. Second, we review the low-rank alternating direction implicit method and the low-rank cyclic Smith method for solving the projected generalized continuous-time Sylvester equation, and propose a numerical method for the projected generalized discrete-time Sylvester equation. Third, we show that the solutions of these two equations are useful for computing the HL2 inner product of two descriptor systems. Finally, we present some numerical experiments.

Keywords: Projected generalized Sylvester equation, Matrix pencil, Alternating direction implicit method, Smith method, Descriptor system, HL2 inner product

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #9, pp. 83-95

Title of the Paper: **Information Value in a Dual-channel Supply Chain Under Revenue Sharing Contract **

Authors: **Bo Li, Zhizhong Tang **

Abstract: This paper assesses the impact of sharing customers’ channel preference information on pricing and performance in a dual channel supply chain, where the supplier opens a direct channel and a traditional retailer owns a retail channel. Both the supplier and the retailer perform their forecasting demands based on the ratio of consumers’ preferences in the direct channel when we consider the characteristics of consumers’ purchase in electronic commerce, while the supplier as a Stackelberg leader designs a revenue sharing contract with the retailer to share the profits in his direct channel. The two scenarios are analyzed under complete information and incomplete information on the ratio of consumers’ preferences in the direct channel. The paper finds that information sharing always benefits the supplier, but may hurt the retailer. Therefore, the supplier should design the revenue sharing contract to incent the retailer to share her information. Moreover, the paper obtains the conditions under which the retailer is willing to share her information with the supplier.

Keywords: Supply chain management, Online channel, Revenue sharing contract, Information sharing, Consumer channel preference, Stackelberge game

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #8, pp. 73-82

Title of the Paper: **The Boundary Value Condition of a Degenerate Parabolic Equation **

Authors: **Huashui Zhan **

Abstract: By Fichera-Oleinik Rule, how to give a homogeneous boundary condition to assure the posedness of the equation ∂xxu + u∂yu − ∂tu = f(x, y, t, u), (x, y, t) ∈ QT = Ω × (0, T), is researched. By introducing a new kind of entropy solution, in which the trace γ(∂u/∂xi), xi = x or y, on the boundary of Ω is avoided. By the parabolic regularization method, the uniformly estimate of the gradient is obtained, and using Kolmogoroff’s theorem, the solvability of the equation in BV (QT ) is obtained.

Keywords: Degenerate parabolic equation, Fichera-Oleinik Rule, boundary value condition, entropy solution, trace.

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #7, pp. 63-72

Title of the Paper: **The Vertex Linear Arboricity of Integer Distance Graph G(Dm,1,4) **

Authors: **Liancui Zuo, Chunhong Shang, Yanyun Niu **

Abstract: An integer distance graph is a graph G(D) with the set Z of all integers as vertex set and two vertices u, v ∈ Z are adjacent if and only if |u − v| ∈ D, where the distance set D is a subset of positive integers. A k-vertex coloring of a graph G is a mapping f from V (G) to [0, k − 1]. A path k-vertex coloring of a graph G is a k-vertex coloring such that every connected component is a path in the induced subgraph of Vi(1 ≤ i ≤ k), where the vertex set Vi is the subset of vertices assigned color i. The vertex linear arboricity of a graph G is the minimum positive integer k such that G has a path k-vertex coloring. In this paper, we studied the vertex linear arboricity of the integer distance graph G(Dm,1,4), where Dm,1,4 = [1,m] | [1, 4], and proved that vla (G(Dm,1,4)) =⌈m/7⌉+1 for every integer m ≥ 6.

Keywords: Integer distance graph, Vertex linear arboricity, Path coloring

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #6, pp. 52-62

Title of the Paper: **Exact Solutions for Fractional Partial Differential Equations by a New Generalized Fractional Sub-Equation Method **

Authors: **Qinghua Feng **

Abstract: In this paper, combining with a fractional complex transformation, we propose a new generalized fractional sub-equation method named fractional (DaG/G ) method to seek exact solutions for fractional partial differential equations. This method is the fractional version of the improved (G’/G) method. Based on this method, some new exact solutions for the space-time fractional (2+1)-dimensional breaking soliton equations and the space-time fractional Fokas equation are successfully found. Under some special cases, we get solitary wave solutions for them.

Keywords: Fractional sub-equation method, Fractional partial differential equations, Exact solutions, Fractional complex transformation, Fractional breaking soliton equations, Fractional Fokas equation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #5, pp. 44-51

Title of the Paper: **On the Norms and Spreads of Fermat, Mersenne and Gaussian Fibonacci RFMLR Circulant Matrices **

Authors: **Yao Yu, Zhaolin Jiang **

Abstract: In this paper, we consider norms and spreads of RFMLR circulant matrices involving the Fermat, Mersenne sequences and Gaussian Fibonacci number, respectively. Firstly, we reviewed some properties of the Fermat, Mersenne sequences, Gaussian Fibonacci number and RFMLR circulant matrices. Furthermore, we give lower and upper bounds for the spectral norms and spread of these special matrices. Finally, we give several corollaries related to norms of Hadamard and Kronecker products of these matrices.

Keywords: RFMLR circulant matrix, Norm, Spread, Fermat sequence, Mersenne sequence, Gaussian Fibonacci number

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #4, pp. 34-43

Title of the Paper: **Differential Pricing and Recycling Decisions Based on Market Segmentation **

Authors: **Lisha Wang, Yongzhao Wang, Huaming Song, Fu Huang **

Abstract: This paper studies differential pricing and recycling decisions in a closed supply chain with one manufacturer and one retailer. The manufacturer makes the new product directly from raw material and remanufactured products with recycling used one. Considering the market segmentation and the consumer preference, we establish four decision models, which contain a centralized decision model and three Stankelberg game models, to explore the chain members’ optimal strategies on price and profit. By using game theory, the firms optimal strategies and some managerial insights are obtained.

Keywords: Differential pricing, Game theory, Recycling decisions, Market segmentation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #3, pp. 23-33

Title of the Paper: **Exponential Stabilization of 1-d Wave Network with One Circuit **

Authors: **Xiu Fang Liu, Gen Qi Xu **

Abstract: In this paper, we consider a complex network of strings. Suppose that the network is comprised of eight strings with a fixed vertex, and other exterior vertices that are imposed velocity feedback controller. The displacement is not continuous at one interior node, while at the other interior nodes continuity holds and the force is not balanced at all interior nodes. We design controllers for the nodes with discontinuous displacement and with unbalanced force. We show that the operator determined by the closed loop system has a compact resolvent and generates C0 semigroup in an appropriate Hilbert space. Under certain condition, we prove that the closed loop system is asymptotically stable. We also show that there is a sequence of generalized eigenvectors of the system operator, which forms a Riesz basis. Hence the spectrum determined growth condition holds. If the imaginary axis is not an asymptote of the spectrum, then the closed loop system is exponentially stable.

Keywords: String, Network, Stabilization, Feedback control, C0 Semigroup

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #2, pp. 13-22

Title of the Paper: **A Novel Vague Set Based Score Function for Multi-Criteria Fuzzy Decision Making **

Authors: **Kuo-Sui Lin **

Abstract: For handling multi-criteria fuzzy decision-making problems, the degree of satisfiability and non-satisfiability of each alternative with respect to a set of criteria is often represented by a set of vague values. In multi-criteria fuzzy decision-making problems, vague set based score functions have become more and more popular for building models that concern the evaluation and comparison of alternatives in the decision-making process. However, several deficiencies remain evident when using these vague based score functions to handle multi-criteria decision-making problems. Therefore, the main objective of this study is propose a novel score function for a more effective and reasonable method for measuring the degree of suitability to which an alternative satisfied the decision maker’s requirement. An illustrative example is provided, which shows that the proposed vague set based score function is more effective and reasonable than other existing score functions in handling multi-criteria decision-making problems.

Keywords: Multi-Criteria Fuzzy Decision Making, Score Function, Type-2 Fuzzy Sets, Vague Sets

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #1, pp. 1-12