# WSEAS Transactions on Mathematics

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

**Volume 12, 2013**

**Issue 1, Volume 12, January 2013**

Title of the Paper: **Analytical Solution of Two Model Equations for Shallow Water Waves and their Extended Model Equations by Adomian’s Decomposition and He's Variational Iteration Methods **

Authors: **Mehdi Safari **

Abstract: In this paper two model equations for shallow water waves and their extended models were considered. Adomian’s decomposition method (ADM) and variational iteration method (VIM) have been employed to solve them. Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the ADM. The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems. This method efficiently works for initial- value or boundary-value problems and for linear or nonlinear, ordinary or partial differential equations and even for stochastic systems. The variational iteration method (VIM) established in (1999) by He is thoroughly used by many researchers to handle linear and nonlinear models. Finally the results of ADM and VIM methods have been compared and it is shown that the results of the VIM method are in excellent agreement with results of ADM method and the obtained solutions are shown graphically.

Keywords: Adomian's decomposition method, Variational iteration method, shallow water waves

Title of the Paper: **Polar Sets for Slices of Generalized Brownian Sheet **

Authors: **Zhenlong Chen **

Abstract: Let fW be a two-parameter, Rd-valued generalized Brownian sheet. We can view fW as a sequence of interacting generalized Brownian motions or slices. The sample path properties for the slices of fW are studied, and the connections between the polar sets for the slices of fW and capacity are also presented. A common feature of our results is that exhibit phase transition.

Keywords: Generalized Brownian sheet, capacity, polar set, dimension

Title of the Paper: **Key Pre-Distribution using Combinatorial Designs for Wireless Sensor Networks **

Authors: **Wangke Yu, Shuhua Wang **

Abstract: The main requirement in wireless sensor networks is not only the security but also the energy efficient security due to limited resources. In large scale deployment scenarios, there is no priory knowledge of post deployment network configuration since nodes may be randomly scattered over a hostile territory. Thus, shared keys must be distributed before deployment to provide each node some keys. For large sensor networks it is infeasible to store a unique key for all other nodes in the keys of a sensor node. Consequently, for secure communication either two nodes have a key in common in their keys and they have a wireless link between them, or there is a path, among these two nodes where each pair of neighboring nodes on this path have a key in common. We review and examine the appropriateness of combinatorial designs as a tool for building key pre-distribution schemes suitable for such environments. A scalable key pre-distribution scheme based on combinatorial designs is presented. Performance and simulation results show that the combinatorial approach produces better connectivity with smaller key sizes. An important advantage of our schemes is that we can increase the scalability of the network.

Keywords: Key pre-distribution scheme, wireless sensor networks, combinatorial designs, orthogonal arrays, rational normal curves, security

Title of the Paper: **On Explicit Determinants of the RFMLR and RLMFL Circulant Matrices Involving Certain Famous Numbers **

Authors: **Nuo Shen, Zhaolin Jiang, Juan Li **

Abstract: The row first-minus-last right (RFMLR) circulant matrices and the row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the explicit determinants of the two pattern matrices involving Fibonacci, Lucas, Pell and Pell-Lucas sequences in terms of finite many terms of these sequences.

Keywords: Determinant, RFMLR circulant matrix, RLMFL circulant matrix, Fibonacci numbers, Lucas numbers, Pell numbers, Pell-Lucas numbers

Title of the Paper: **A Linearized Finite Difference Scheme with Non-uniform Meshes for Semi-linear Parabolic Equation **

Authors: **Jun Zhou **

Abstract: In the present paper, a linearized difference scheme with non-uniform meshes for semi-linear parabolic equation is proposed. The scheme is constructed according to the change rule of the solution by travelling wave solution theory for partial differential equation. The existence and uniqueness of the numerical solution are derived by linear systems theory, and the convergence and stability of the difference scheme are proved by the discrete energy method. Numerical simulations verify the theoretical analysis, the results show that the numerical solution with non-uniform meshs is more accurate than that with uniform meshes in the sense of not costing much more computing time. It is concluded that our scheme is effective.

Keywords: Semi-linear parabolic equation, non-uniform meshes, difference schemes, convergence, stability

Title of the Paper: **An Isoparametric Finite Element Method for Elliptic Interface Problems with Nonhomogeneous Jump Conditions **

Authors: **Xufa Fang **

Abstract: Numerical solutions of boundary value problems for elliptic equations with discontinuities in the coef- ficients and flux across immersed interface are of special interest. This paper develop a three order isoparametric finite element method for 2D elliptic interface problems. To obtain a high order of accuracy presents some dif- ficulty, especially if the immersed interface does not fit with the elements. For this purpose, based on an initial Cartesian mesh, a body-fitted mesh optimization strategy is proposed by introducing curved boundary elements near the interface, and a quadratic isoparametric finite element basis is constructed on the optimized mesh. Nu- merical examples with immersed interval interface demonstrate that the proposed method is efficient for elliptic interface problems with nonhomogeneous flux jump condition.

Keywords: Isoparametric finite element, Elliptic interface problems, Curved boundary element, Body-fitted mesh, Nonhomogeneous jump conditions

Title of the Paper: **Hybrid Particle Swarm Algorithm for Solving Nonlinear Constraint Optimization Problems **

Authors: **Bingqin Qiao, Xiaoming Chang, Mingwei Cui, Kui Yao **

Abstract: Based on the combination of the particle swarm algorithm and multiplier penalty function method for the constraint conditions, this paper proposes an improved hybrid particle swarm optimization algorithm which is used to solve nonlinear constraint optimization problems. The algorithm converts nonlinear constraint function into no-constraints nonlinear problems by constructing the multiplier penalty function to use the multiplier penalty function values as the fitness of particles. Under the constraint conditions to make the particle swarm track the best particle and fly to the global best, this paper is to redefine p-best which is the particles position last iteration and g-best which refers to the best position of the group last iteration. And, by redefining p-best and g-best, the particle can avoid tracking the p-best and the g-best whose fitness are excellent but violate constraints to great extent, so that particle swarm would finally converge to the optimum position in the feasible domain. Numerical experiments show that the new algorithm is correct and effective.

Keywords: Particle swarm optimization; Multiplier method; Multiplier penalty function; Nonlinear constraint optimization; Nonlinear constraint; Global optimization

Title of the Paper: **Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces **

Authors: **Yuan-Heng Wang, Yan Li **

Abstract: Let C be a closed convex subset of a reflexive and strictly convex Banach space E and F = {T(t); t > 0} be a non-expansive semigroup on the C with the nonempty set of their common fixed points. The purpose of this paper is to study a new viscosity iterative method for a non-expansive semigroup and weakly contraction mappings. And it is proved that the new iterative approximate sequences converge strongly to the solution of a certain variational inequality. These results improve and extend some recent results of the other authors.

Keywords: Non-expansive semigroup, Common fixed point, Uniformly G^ateaux differentiable norm, Weakly contraction, Iterative approximation, Strong convergence

Title of the Paper: **Martingale-Based Computational Liquidity Risk Premium **

Authors: **Zhou Fang, Zhang Wei, Zhou Bing **

Abstract: In this paper, we consider the pricing of liquidity risk in normal market. By employing the no-arbitrage idea of financial calculus and finance engineering, we discuss the pricing of market risk and liquidity risk under martingale measure, and obtain two separate market prices of risk for all tradable assets via the change of equivalent measure to make discounted assets into martingales, and then provide the pricing formula of liquidity risk premium, in which the market price of risk in the same market for all tradable assets and for all the investors is the same, not varying with the levels of risk aversion of investors.

Keywords: Liquidity Risk, Market Risk, Risk Premium, Market Price, Martingale, No-Arbitrage

**Issue 2, Volume 12, February 2013**

Title of the Paper: **An Analysis of the Converging Under-Damped Harmonic Growth of Prime Numbers **

Authors: **Ernest G. Hibbs **

Abstract: This paper presents how the prime number increments converge with respect to their own harmonic double-pole structure and not using our normal harmonic equations or coordinate systems. We examine the key ratio between the forward and reverse oscillating motion of the harmonic elliptical gap growth model. The convergence caused by the ratio of the sums of the two poles over time uses a non-complex version the Riemann zeta function as the primary exponential power that drives the harmonic under-damped exponential decay. The analysis of the gap for the first 2,000 prime numbers results in the conclusion that the prime number increments provide an integer “plug-and-play” framework for creating a harmonic relationship in dipole or double-threaded models in physics, engineering, and bioinformatics. This paper has been expanded to reevaluate the possibility of a tighter and coupled complete solution involving piecewise refinement possibly related to partial derivatives of components of the core function.

Keywords: Prime Number Convergence, Riemann zeta, Harmonic, Under-damped

Title of the Paper: **On Generalized Mixed Equilibrium Problems and Fixed Point Problems with Applications **

Authors: **Jia Wei Chen, Zhong Ping Wan, Yeol Je Cho **

Abstract: In this paper, we introduce and investigate two new generalized mixed equilibrium problems and explore the relationship between them and the properties of their solutions in Banach spaces. Based on the general- ized f-projection, we construct hybrid algorithms to find common fixed points of a countable family of quasi-?- nonexpansive mappings in Banach spaces, a common element of the set of solutions of generalized mixed equi- librium problems and the set of fixed points for quasi-?-nonexpansive mappings and, further, prove some strong convergence theorems for these hybrid algorithms under some suitable assumptions. As some applications of the main results, the strong convergence theorems for the general H-monotone mappings and equilibrium problems are also proven.

Keywords: Generalized mixed equilibrium problem, Quasi-?-nonexpansive mapping, Strong convergence theorem, Fixed point, Generalized f-projection

Title of the Paper: **Interval Oscillation Criteria For A Class Of Nonlinear Fractional Differential Equations **

Authors: **Qinghua Feng, Fanwei Meng **

Abstract: In this work, some new interval oscillation criteria for solutions of a class of nonlinear fractional differential equations are established by using a generalized Riccati function and inequality technique. For illustrating the validity of the established results, we also present some applications for them.

Keywords: Oscillation, Interval criteria, Qualitative properties, Fractional differential equation, Nonlinear equation, Ordinary differential equation

Title of the Paper: **Global Exponential Synchronization of a Class of BAM Neural Networks with Time-Varying Delays **

Authors: **Fengyan Zhou **

Abstract: In this paper, the global exponential synchronization problem is considered for a class of BAM neural networks with time-varying delays. By using Lyapunov functional method and analysis techniques, three sufficient conditions for the global exponential synchronization of the drive-response system are derived. Two numerical examples are given in the end to illustrated the effectiveness of our theoretical results.

Keywords: BAM neural networks, Synchronization, Global exponential synchronization, Lyapunov functional, Halanay inequality, Time-varying delays

Title of the Paper: **Strong Convergence of Modified Gradient-Projection Algorithm for Constrained Convex Minimization Problems **

Authors: **Ming Tian, Lihua Huang **

Abstract: In this article, a modified gradient-projection algorithm (GPA) is introduced, which combines Xu’s idea of an alternative averaged mapping approach to the GPA and the general iterative method for nonexpansive mappings in Hilbert space introduced by Marino and Xu. Under suitable conditions, it is proved that the strong convergence of the sequences generated by implicit and explicit schemes to a solution of a constrained convex minimization problem which also solves a certain variational inequality. Obtained results extend and improve some existed results.

Keywords: Gradient-projection algorithm, Constrained convex minimization, General iterative method, averaged mapping, nonexpansive mapping, fixed point, variational inequality

Title of the Paper: **Exponential Stability of Periodic Solutions for Inertial Cohen-Grossberg-Type BAM Neural Networks with Time Delays **

Authors: **Xuerui Wei **

Abstract: The paper is concerned with the existence and global exponential stability of periodic solutions for inertial Cohen-Grossberg-type BAM neural networks with time delays. With variable transformation the system is transformed to first order differential equations. Some new sufficient conditions ensuring the existence and global exponential stability of periodic solutions for the system are derived by constructing suitable Lyapunov functions, using Weierstrass criteria and boundedness of the solutions. Finally, an example is given to demonstrate the obtained results.

Keywords: Cohen-Grossberg-type BAM neural networks, inertial term, Lyapunov function, Weierstrass criteria, exponential stability

Title of the Paper: **The Effect of Liquidity on Stock Returns: A Style Portfolio Approach **

Authors: **Kuang-Wen Chang, George Yungchih Wang, Chunwei Lu **

Abstract: In stock market, various concepts of stocks, or investment styles, have been raised by fund managers to catch the attention of investors. Style investing was referred to as investing stocks with similar company characteristics to form a style portfolio in order to obtain abnormal returns. Since liquidity in stock trading was important information for investors in investment decision-making, this study examined whether there existed the effect of liquidity, i.e., trading turnover, on stock returns by applying the style portfolio approach to test statistical significance of short-run abnormal returns and long-run cumulative returns of several liquidity-related style portfolios. With the data of Taiwan publicly-listed companies, three findings were concluded: First, the high liquid stocks were found to have higher cumulative returns relative to those of the benchmark portfolio, the market, for the period of 1999-2008. Second, when we integrated stock liquidity into company characteristic and firm size to form two-dimensional style portfolios, stock returns of those style portfolios were significantly higher than those of one-dimensional style portfolios, meaning that the liquidity effect could amplify conventional market anomalies, such as the value effect and the size effect. Third, the returns of the liquidity-related portfolios were also significant in different market conditions. The study therefore concluded that the liquidity effect was a significant investment style in stock market.

Keywords: investment style, style investing, style portfolio, liquidity, value effect, size effect

Title of the Paper: **Products of Volterra-Type Operators and Composition Operators on logarithmic Bloch Space **

Authors: **Shanli Ye **

Abstract: Let D = {z : |z| < 1} be the unit disk in the complex plane C, φ be an analytic self-map of D, and g : D → C is an analytic map. We characterize the boundedness and compactness of the products of Volterra-type operators and composition operators CφUg and UgCφ on the logarithmic Bloch space LB and the little logarithmic space LB0 over the unit disk. Some necessary and sufficient conditions are given for which CφUg or UgCφ is a bounded or a compact operator on LB, or LB0, respectively. The results extend the known results about the composition operator to the logarithmic Bloch space LB.

Keywords: Volterra-type operators, Composition operators, Bloch-type spaces, Analytic functions, Boundedness, Compactness

Title of the Paper: **Learning Rates of Regularized Regression with p-loss **

Authors: **Sheng Baohuai, Yu Wangke, Ye Peixin, Han Yongjie **

Abstract: In the present paper,we investigate the learning rates of regularized regression with sample dependent hypothesis and p-loss. We show the robustness of the solutions with respect to the probability distributions, with which we provide the sample error. Also,we show the approximation error with a kind of K−functional whose convergence rates are described in possibility. Finally, we show the explicit learning rates in cases of the norm regularization and the coefficient regularization. The results show that the parameters p have influences on the learning rates.

Keywords: Regularized regressions,learning rates, reproducing kernel Hilbert spaces, sample dependent, p-loss

Title of the Paper: **Linear Authentication Codes from Free Modules: Bounds and Constructions **

Authors: **Xiuli Wang **

Abstract: In this paper, we principally introduce a new class of unconditionally secure authentication codes, called linear authentication codes(or linear A-codes)from free modules. We then derive an upper bound on the size of source space when other parameters of the system, that is, the sizes of the key space and authenticator space, and the deception probability are fixed. We give constructions that are asymptotically close to the bound and show applications of these codes in constructing distributed authentication systems. We realize the generalization of linear authentication codes from vector space over field to free modules over ring.

Keywords: Authentication codes(A-codes), Distributed A-codes, Multi-receiver A-codes, Multi-sender A-codes, Free modules, Ring

Title of the Paper: **Randomly Mt-decomposable Multigraphs and M2-equipackable Multigraphs **

Authors: **Xuejiao Jiang, Yuqin Zhang **

Abstract: A graph G is called randomly H − decomposable if every maximal H − packing in G uses all edges in G. G is called H − equipackable if every maximal H − packing in G is also a maximum H − packing in G. M2 − decomposable graphs, randomly M2 − decomposable graphs and M2 − equipackable graphs have been characterized. The definitions could be generalized to multigraphs. And M2 − decomposable multigraphs has been characterized. In this paper, all randomly M2 − decomposable multigraphs and M2 − equipackable multigraphs are characterized, and some notes about randomly Mt − decomposable multipraphs are given.

Keywords: Multigraph, packing, decomposable, randomly decomposable, equipackable, matching

Title of the Paper: **Turing Instability for a Two Dimensional Semi-Discrete Gray-Scott System **

Authors: **Li-Li Meng, Guang Zhang, Shu-Xian Xiao, Jian Bao **

Abstract: This paper is concerned with the spatial patterns of the Gray-Scott system which describes a general two- variable kinetic model that represents an activator-substrate scheme, where the space is discrete in two dimensions with the periodic boundary conditions and the time is continuous. Furthermore conditions for producing Turing instability of a general semi-discrete system are obtained through linear stability analysis and this conclusion is applied to the semi-discrete G-S model. Then in the Turing instability region of semi-discrete G-S model, we perform a series of numerical simulations which shown that this system can produce some new Turing patterns such as striped, spotted and lace-liked patterns in the Turing instability region. In particular, the observation of Turing patterns are reported, as a control parameter is varied, from a spatially uniform state to a patterned state. It suggests that the values of parameters make a great impact on Turing patterns.

Keywords: Semi-discrete, Gray-Scott model, Turing instability, Turing pattern, Diffusion, Pattern formation

**Issue 3, Volume 12, March 2013**

Title of the Paper: **Robust Consensus Problem of Data-Sampled Networked Multi-Agent Systems with Delay and Noise **

Authors: **Fang Yan, Dongmei Xie **

Abstract: This paper proposes an observer-based control strategy for networked multi-agent systems with timevarying communication delays and random white noises under both fixed topology and Bernoulli switching topology. First, a queuing mechanism is introduced and thus a team of agents can be modeled as a system with constant delay. Then, using the system transformation method, the robust mean-square consensus problem of multi-agent systems can be converted into the robust mean-square stability problem of an equivalent system, and some equivalent conditions concerning the robust mean-square consensus of networked multi-agent systems are presented, whose related observer-based stabilizability criteria can be established in the form of linear matrix inequalities (LMIs). Furthermore, if the LMIs are feasible, the multi-agent systems achieve robust mean-square consensus if and only if the directed graph has a directed spanning tree (fixed topology) or the union of graphs has a directed spanning tree (Bernoulli switching topology). Finally, numerical simulations are given to illustrate the effectiveness of the obtained theoretical results.

Keywords: Networked multi-agent systems, robust mean-square consensus, delay, sampled control, general linear dynamics, inequalities(LMIs)

Title of the Paper: **Optimal Investment Problem with Taxes, Dividends and Transaction Costs under the Constant Elasticity of Variance (CEV) Model **

Authors: **Danping Li, Ximin Rong, Hui Zhao **

Abstract: This paper studies the optimal investment problem of utility maximization with taxes, dividends and transaction costs under the constant elasticity of variance (CEV) model. The Hamilton-Jacobi-Bellman (HJB) equation associated with the optimization problem is established via stochastic control approach. Applying power transform and variable change technique, we obtain explicit solutions for the logarithmic and exponential utility functions. For the quadratic utility function, we obtain the optimal strategy explicitly via Legendre transform and dual theory. Furthermore, we analyze the properties of the optimal strategies. Finally, a numerical simulation is presented to discuss the effects of market parameters on the strategies.

Keywords: Optimal investment, Transaction cost, Constant elasticity of variance (CEV) model, Utility function, Hamilton-Jacobi-Bellman (HJB) equation, Legendre transform

Title of the Paper: **Global Exponential Stabilization of Uncertain Switched Nonlinear Systems with Time-Varying Delays **

Authors: **Fengwei Yang, Yali Dong **

Abstract: This paper is concerned with the problem of robust exponential stabilization for a class of uncertain hybrid systems with mixed time-varying delays in both the state and control. By using a Lyapunov-Krasovskii functional, a memoryless switching controller design is proposed to guarantee the global exponential stabilization. Based on matrix inequality technique, we establish some new delay-dependent exponential stabilization criteria for the system. Finally, some numerical examples are presented to illustrate the effectiveness of the theoretical results.

Keywords: Uncertain switched system, Time-varying delays, Exponential stabilization, Switching controller design

Title of the Paper: **On Two Variants of Rainbow Connection **

Authors: **Yuefang Sun **

Abstract: A vertex-colored graph G is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. A path P connecting two vertices u and v in a total-colored graph G is called a rainbow total-path between u and v if all elements in V(P) U E(P), except for u and v, are assigned distinct colors. The total-colored graph G is rainbow total-connected if it has a rainbow total-path between every two vertices. The rainbow total-connection number, denoted by rtc(G), of a graph G is the minimum colors such that G is rainbow total-connected. In this paper, we will obtain some results for these two variants of rainbow connection. For rainbow vertex-connection, we will first investigate the rainbow vertex-connection number of a graph according to some structural conditions of its complementary graph G. Next, we will investigate graphs with large rainbow vertex-connection numbers. We then derive a sharp upper bound for rainbow vertex-connection numbers of line graphs. For rainbow total- connection, we will determine the precise values for rainbow total-connection numbers of some special graph classes, including complete graphs, trees, cycles and wheels.

Keywords: vertex-coloring, total-coloring, rainbow vertex-connection number, rainbow total-connection number, rainbow connection number, complementary graph

Title of the Paper: **Positive Periodic Solutions of a Generalized Gilpin-Ayala Competitive System with Time Delays **

Authors: **Chunfang Miao, Yunquan Ke **

Abstract: In this paper, we investigate a generalized Gilpin-Ayala competition system which is more general and more realistic than the classical Lotka-Volterra competition system. By the fixed-point theorem and differential mean value, some sufficient conditions guaranteeing the existence, uniqueness and exponential stability of positive periodic solutions for a generalized Gilpin-Ayala competition system with time delays are given. Two illustrative examples are also given in the end to show the effectiveness of our results.

Keywords: exponential stability, Gilpin-Ayala competition system, periodic solution, fixed-point theorem

Title of the Paper: **A Numerical Investigation of Blow-up in The Moving Heat Source Problems in Two-Dimensions **

Authors: **Hancan Zhu, Kewei Liang **

Abstract: The temperature of a combustible material will rise or even blow up when a heat source moves across it. In this paper, we study the blow-up phenomenon in this kind of moving heat source problems in two-dimensions. First, a two-dimensional heat equation with a nonlinear source term is introduced to model the problem. The nonlinear source is localized around a circle which is allowed to move. By using the coordinate transformation, the equation is simplified to a one-dimensional one. Then it is solved by the moving collocation method. The numerical results show that the blow-up occurs if the speed of the heat source is slow, and the blow-up is avoided when the heat source moves fast enough.

Keywords: Moving heat source, Blow-up, Moving mesh method, Reaction-diffusion equation, Moving collocation method, Local absorbing boundary conditions

Title of the Paper: **Strong Convergence of a Hybrid Projection Algorithm for Approximation of a Common Element of Three Sets in Banach Spaces **

Authors: **Ren-Xing Ni **

Abstract: In this paper, we construct a new iterative scheme by hybrid projection method and prove strong con- vergence theorems for approximation of a common element of set of common fixed points of an infinite family of asymptotically quasi-?-nonexpansive mappings, set of solutions to a variational inequality problem and set of common solutions to a system of generalized mixed equilibrium problems in a uniformly smooth and 2-uniformly convex real Banach space. Our results extend many important recent results in the literature.

Keywords: Asymptotically quasi-φ-nonexpansive mapping, Generalized mixed equilibrium problem, Uniformly smooth, 2-Uniformly convex, Hybrid projection method, Banach space

Title of the Paper: **Empirical Research on the Performance of Circulation Industry Based on Input-Output Analysis – Evidence from Zhejiang Province, China **

Authors: **Diping Zhang **

Abstract: Taking Zhejiang Province as an example and using input-output model, this article adjusts the inputoutput tables over the years and compiles input-output tables of circulation industry. Through analysis of the calculated results and vertical and horizontal comparison (with the economically developed regions in China), it studies industrial properties of Zhejiang circulation industry. The results show that, the position and role of circulation industry in national economy continue to strengthen; Circulation industry has strong ability to absorb labor force, but its basic industrial characteristics are not obvious enough; There is a certain gap between Zhejiang province and the economically developed regions in circulation industrial features; Circulation industry is consumer-driven, investment-driven and export-driven industry; The induced role of final consumption, investment demand and export demand is strong to circulation industry, of which the induced role of consumption is strongest; Currently, to stimulate the final demand, particularly, to positively develop rural consumer market and cultivate new consumption hot in cities and towns can promote circulation industry to rapidly develop.

Keywords: Zhejiang Province, Circulation industry, Performance, Input-output model, Industrial characteristics, Vertical and horizontal comparison

Title of the Paper: **Algorithms for Finding the Minimum Norm Solution of Hierarchical Fixed Point Problems **

Authors: **He Songnian, Guo Jun **

Keywords: Hierarchical fixed point, Iterative algorithm, Variational inequality, Minimum norm, Strong convergence

Title of the Paper: **Calibration Estimation via a Smoothing Newton Method **

Authors: **Weizhe Gu, Jun Zhai, Wei Wang **

Abstract: Calibration estimation is currently the most popular method of estimation using auxiliary information. Its major idea is to use auxiliary information to structure calibration weights, attaching them to survey data, in or- der to improve the accuracy of the gross or mean estimation. Calibration estimation problem with box constraints is equivalently to solve a nonlinear equations system. Mnnich at el proposed a semismooth Newton method for solving this system in [Calibration of estimator-weights via semismooth Newton Method. J. Glob. Optim. 52 (2012):471-485]. In this paper, we give more specific analysis about the semismooth Newton method and some numerical experiments have been reported. On the basis of that, a smoothing Newton method has been proposed and proved to be globally convergent without any assumptions and locally superlinearly convergent under certain assumptions. Numerical results show that both semismooth Newton method and smoothing Newton method are effect for solving the calibration estimation problem.

Keywords: Calibration estimation, Auxiliary information, Sample weights, Semismooth Newton method, Smoothing Newton method, Global convergence

Title of the Paper: **On the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers **

Authors: **Zhaolin Jiang, Juan Li, Nuo Shen **

Abstract: Let A be a r-circulant matrix and B be a left r-circulant matrix whose first rows are (P1, P2, ... , Pn), (Q1, Q2, ... , Qn), (J1, J2, ... , Jn) and (j1, j2, ... , jn) respectively, where Pn is the Pell number, Qn is the Pell- Lucas number, Jn is the Jacobsthal number and jn is the Jacobsthal-Lucas number. In this paper, by using the inverse factorization of polynomial of degree n, the explicit determinants of A and B whose first rows are (P1, P2, ... , Pn) and (Q1, Q2, ... , Qn) are expressed by utilizing only Pell numbers, Pell-Lucas numbers and the parameter r, and the explicit determinants of A and B whose first rows are (J1, J2, ... , Jn) and (j1, j2, ... , jn) are expressed by utilizing only Jacobsthal numbers, Jacobsthal-Lucas numbers and the parameter r. The results not only extend the original results, but also simpler in forms. Also, the singularities of those matrices are discussed. Furthermore, four identities of those famous numbers are given.

Keywords: r-circulant matrix, Left r-circulant matrix, Determinant, Singularity, Pell numbers, Pell-Lucas numbers, Jacobsthal numbers, Jacobsthal-Lucas numbers

Title of the Paper: **A Simple Method to Balanced Procrustes Problem with One Special Constraint **

Authors: **Zhongyang Yuan, Yuyang Qiu **

Abstract: The balanced Procrustes problem with X^T = sX and X X^T = aX +bIn constraints are considered. By one time eigenvalue decomposition or real Schur decomposition of the matrix product generated by the matrices A and B, the constrained solutions are constructed simply. Similar strategy is applied to the problem with the corresponding P-commuting constraints with given symmetric matrix P. And the methods are also suitable for the least squares problem of the extended equations AX = B, XC = D with the same constraints. Numerical examples are presented to show the efficiency of the proposed methods.

Keywords: Balanced Procrustes problem, Eigenvalue decomposition, Real Schur decomposition, Matrix product, Constrained solutions, Numerical examples

**Issue 4, Volume 12, April 2013**

Title of the Paper: **Hedging Strategy for Unit-Linked Life Insurance Contracts in Stochastic Volatility Models **

Authors: **Wei Wang, Linyi Qian, Wensheng Wang **

Abstract: A general class of stochastic volatility model is considered for modeling risky asset. This class of stochastic volatility model contains most of those without jump component which are commonly used in research. We obtain the minimal martingale measure and locally risk minimizing hedging strategy in these models, and employ the results to the unit-linked life insurance contracts. Moreover, we also investigate the locally risk min- imizing hedging strategy for unit-linked life insurance contracts in a Barndorff-Nielsen and Shephard stochastic volatility model.

Keywords: Locally risk minimizing, Stochastic volatility, Unit-linked life insurance contracts

Title of the Paper: **Efficient Algorithms for Finding the Minimal Polynomials and the Inverses of Level-k FLS (r1, ..., rk)-Circulant Matrices **

Authors: **Zhaolin Jiang **

Abstract: The level-k FLS (r1, . . . , rk)-circulant matrix over any field is introduced. The diagonalization and spectral decomposition of level-k FLS (r1, . . . , rk)-circulant matrices over any field are discussed. Algorithms for computing the minimal polynomial of this kind of matrices over any field are presented by means of the algorithm for the Gr¨obner basis of the ideal in the polynomial ring, and two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for the inverse of partitioned matrix with level-k FLS (r1, . . . , rk)-circulant blocks over any field is given by using the Schur complement, which can be realized by CoCoA 4.0, an algebraic system, over the field of rational numbers or the field of residue classes of modulo prime number.

Keywords: Level-k FLS (r1, . . . , rk)-circulant matrix, minimal polynomial, inverse, diagonalization, spectral decomposition, Grobner basis

Title of the Paper: **A Lazy Bureaucrat Scheduling Game **

Authors: **Ling Gai, Yuanchun Song **

Abstract: In this paper we consider the game theoretical issue of the lazy bureaucrat scheduling problem. There are two players working on a pool of tasks, each of them can select a subset of the tasks to execute and spend the corresponding cost. The common choice would introduce the increasing of the task’s cost. Each player has his own budget for these tasks and if the total cost of selected tasks are less than his budget, he can keep the difference part as his “additional” profit. The objective of the players is to make wise selection such that the cost that he spents on the tasks is minimized, while both of them have to obey an assumption called “busy requirement” that as long as there are tasks can be executed by some player (the left budget is more than the cost needed), he must select it to execute. The noncooperative nature and potential interactions between the two players make the problem dynamic and complicated. We prove that Nash equilibrium solutions exist under certain conditions where both players are satisfied with their selection and would not change their mind unilaterally. We also find the method by which we can obtain the Nash equilibrium no matter the player has a single machine or multiple machines to execute on the tasks. Furthermore, we adopt the concept of “price of anarchy” to compare the cost of the worst Nash equilibrium with the social optimum.

Keywords: Game theory, Nash equilibrium, Price of anarchy, Lazy bureaucrat scheduling

Title of the Paper: **Orthogonal Tensor Sparse Neighborhood Preserving Embedding for Two-dimensional Image **

Authors: **Mingming Qi, Yang Xiang **

Abstract: Orthogonal Tensor Neighborhood Preserving Embedding (OTNPE) is an efficient dimensionality reduc- tion algorithm for two-dimensional images. However, insufficiencies of the robustness performance and deficien- cies of supervised discriminant information are remained. Motivated by the sparse learning, an algorithm called Orthogonal Tensor Sparse Neighborhood Embedding (OTSNPE) for two-dimensional images is proposed in the paper. The algorithm firstly regards two-dimensional images as points in the second-order tensor space, then, the neighborhood reconstruction of samples within the same class is achieved with sparse reconstruction. Finally, projections are gotten to preserve local sparse reconstruction relation and neighborhood relation within the same class and spatial relation of pixels in an image. Experiments operated on Yale, YaleB and AR databases show, in contrast to the existing typical tensor dimensionality reduction algorithms, the algorithm can improve the accuracy rate of classification algorithms based on the shortest Euclidean distance.

Keywords: Dimensionality Reduction, Tensor Image, Neighborhood Preserving Embedding, Sparse Reconstruction, Supervised Discriminant Information, Face Recognition

Title of the Paper: **Dissipativity of θ-Methods and One-Leg Methods for Nonlinear Neutral Delay Integro-Differential Equations **

Authors: **Guanghua He, Liping Wen **

Abstract: In this paper we study the dissipativity of a special class of nonlinear neutral delay integro-differential equations. The dissipativity of three kinds of important numerical methods, the linear θ-methods, one-leg θ- methods, and the one-leg methods is obtained when they are applied to these problems. Numerical experiments are presented to check our findings.

Keywords: Linear θ-methods, One-leg θ-methods, One-leg methods, Nonlinear neutral delay integro-differential equations, Dissipativity, Absorbing set

Title of the Paper: **Application of ICA and Contourlet Transform in Image Watermarking **

Authors: **Jia Mo, Zhaofeng Ma, Yixian Yang, Xinxin Niu, Shuhua Xu **

Abstract: Based on independent component analysis (ICA) and contourlet, this paper designs an important robust watermarking scheme that mainly consists of watermark embedding and extracting. The embedding is done by mixing the host image signals and the watermarking signals through , the linear system of ICA model. The extracting is achieved by estimating the demixing matrix and the watermarking signals through ICA model. The scheme takes advantage of human visual system (HVS) to conduct a self-adaptive computation of the scaling factor matrix. The experiment uses PSNR and NCC to evaluate its imperceptibility and robustness, finding that the scheme functions well in resisting frequent attacks including filter, noise, cropping and mirror.

Keywords: Image processing, watermarking, ICA, contourlet transform, HVS, robustness

Title of the Paper: **Numerical Investigation on Shear Driven Cavity Flow by the Constrained Interpolated Profile Lattice Boltzmann Method **

Authors: **C. S. Nor Azwadi, M. H. Al-Mola, S. Agus **

Abstract: In this paper, we proposed a combination of lattice Boltzmann and finite difference schemes to simulate an incompressible fluid flow problem. Our model applies the constrained interpolated profile method to solve the advection term in the governing lattice Boltzmann equation. Compared with the conventional lattice Boltzmann scheme, the current scheme is more accurate. In addition, the proposed model requires less mesh size for the computational at various conditions compared to other lattice Boltzmann models. Simulation of lid-driven cavity flow whose Reynolds number up to 1000 were carried out in order to validate the proposed approach. Numerical results show excellent agreement with those obtained by the conventional computational fluid dynamics approaches. We then extend our computation on the behavior of vortex inside a shallow lid-driven cavity flow at various aspect ratios. We found that the formation, strength and size of primary and secondary vortices are significantly affected by the value of aspect ratio and Reynolds numbers. Good comparisons were obtained when the results are compared with those published in the literature. This indicates that the proposed approach is a reliable alternative computational scheme in predicting various types of fluid flow problem.

Keywords: lattice Boltzmann, distribution function, BGK collision, constrained interpolated profile, lid-driven cavity flow

Title of the Paper: **Introduction to the Elliptical Trigonometry Series Using two Functions Absolute Elliptic Jes (AEjes) and Absolute Elliptic Mar (AEmar) of the First Form **

Authors: **Claude Ziad Bayeh **

Abstract: The Elliptical Trigonometry Series is an original study introduced in mathematical domain, in signal processing and in signal theory; it is a means of representing a periodic signal as a finite or infinite sum of Absolute Elliptic Jes (AEjes) and Absolute Elliptic Mar (AEmar) functions compared to cosine and sine functions in Fourier series. The Elliptical Trigonometry Series is more advanced than the Fourier series. The Fourier series is a particular case of the Elliptical Trigonometry Series when the value of AEjes is equivalent to Cosine and the value of AEmar is equivalent to Sine. The new series has many advantages ahead the Fourier series such as we can reduce the number of parameters for a periodic signal formed by the sum of AEjes and AEmar functions compared to the cosine and sine function in Fourier Series, reduce the circuit size that produce this periodic signal, and reduce the cost of circuits and many other advantages are remarked. In fact, the Elliptical Trigonometry is an original study introduced in Mathematics by the author and it is published by WSEAS journal, and it has enormous applications in mathematics, electronics, signal processing, signal theory and many others domains. This paper emphasizes the importance of this trigonometry in forming what is called the Elliptical Trigonometry Series. In fact, this new Series is introduced for electronics applications in order to reduce as possible the circuit size that form a specific signal and therefore reduce the cost, this is not the case of the Fourier series for the same produced signal. Moreover, we can form from only one circuit an infinite number of combined periodic signals which is not the case of the Fourier series in which one circuit can’t produce more than one signal.

Keywords: Elliptical Trigonometry Series, Fourier series, Signal theory, Signal processing, Mathematics, power electronics, Electrical circuit design

Title of the Paper: **The Sum and Product of Fibonacci Numbers and Lucas Numbers, Pell Numbers and Pell-Lucas Numbers Representation by Matrix Method **

Authors: **Fuliang Lu, Zhaolin Jiang **

Abstract: Denote by {Fn} and {Ln} the Fibonacci numbers and Lucas numbers, respectively. Let Fn = Fn × Ln and Ln = Fn + Ln. Denote by {Pn} and {Qn} the Pell numbers and Pell-Lucas numbers, respectively. Let Pn = Pn × Qn and Qn = Pn + Qn. In this paper, we give some determinants and permanent representations of Pn, Qn, Fn and Ln. Also, complex factorization formulas for those numbers are presented.

Keywords: Fibonacci number, Lucas number, Pell numbers, Pell-Lucas number, matrix

Title of the Paper: **Fuzzy Measures of a System’s Effectiveness - An Application to Problem Solving **

Authors: **Michael Gr. Voskoglou **

Abstract: They appear often situations in a system’s operation characterized by a degree of vagueness and/or uncertainty. In the present paper we use principles of fuzzy logic to develop a general model representing such kind of situations. We also present 3 alternative methods for measuring a fuzzy system’s effectiveness. These methods include the measurement of the system’s total possibilistic uncertainty, the Shannon’s entropy properly modified for use in a fuzzy environment and the “centroid” method in which the coordinates of the center of mass of the graph of the membership function involved provide an alternative measure of the system’s performance An application of the above results is also developed concerning the Problem Solving process and two classroom experiments are presented illustrating the use of our results in practice.

Keywords: Systems Theory, Fuzzy Sets and Logic, Possibility, Uncertainty Theory, Problem solving

Title of the Paper: **On the Determinants and Inverses of Skew Circulant and Skew Left Circulant Matrices with Fibonacci and Lucas Numbers **

Authors: **Yun Gao, Zhaolin Jiang, Yanpeng Gong **

Abstract: In this paper, we consider the skew circulant and skew left circulant matrices with the Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices are also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relation between skew left circulant matrices and skew circulant matrix, respectively.

Keywords: Skew circulant matrix, Skew left circulant matrix, Determinant, Inverse, Fibonacci number, Lucas number

Title of the Paper: **Corporation in a Closed-Loop Supply Chain Based on Remanufacturing **

Authors: **Wenjing Si, Junhai Ma **

Abstract: Remanufacturing is a new manufacture concept which is based on the theory of life cycle of products and focus on the objective to promote reuse of waste products and it has been more and more popular all over the world. This paper presents a theoretical model for remanufacturing system which is a closed-loop supply chain consists of one manufacturer, one dealer and one recycler. Game theory with respect to two pricing strategies is used to analyze the relationshpe between cooperation level and prices, sales quantity and core collection yield. The main results of the research prove that manufacturer will gain more profit by establishing partnership with dealer and recycler. Furthermore, the profits would increase along to the raise of cooperate level. At the same time, the cooperative relationships among manufacturer, dealer and recycler would generate mutual interaction in building sale and product return channel. At the end of this paper, we put forward some coordination suggestions.

Keywords: Closed-loop supply chain, cooperative game, remanufacturing, stackelberg, Supply Chain Coordination

**Issue 5, Volume 12, May 2013**

Title of the Paper: **Learning Optimal Kernel for Pattern Classification **

Authors: **Minmin Ge, Liya Fan **

Abstract: Kernel methods provide an efficient mechanism to derive nonlinear algorithms. Using a kernel function, original data can be implicitly mapped to a very high or even infinite dimensional feature space where the data is approximately linearly separable. For it, a main challenge is to select an appropriate kernel. In this paper, we optimize combinative weight coefficients and combination kernel is constructed by two methods. one method is learning optimal kernel for kernel fisher discriminant analysis (KFDA) for finding optimally combinative weight coefficients. In this method, we treat optimizing combinative weight coefficients as optimization problem over the convex set of finitely many basic kernels. Besides, in order to solve the optimization problem, we use a new iterative method. Another method is a feature space based class separability measure which is introduced in order to further show the efficacy of combination kernel .With this measure, the weight coefficients of combination kernel were optimized. Experiments on five real-words data sets are performed to test and evaluate the efficacy of combination kernel on classification accuracy. The results show that the efficacy of combination kernel is very significant.

Keywords: Fisher discriminant analysis, kernel function, support vector machines, combination kernel, kernel optimization, iterative method

Title of the Paper: **Fast Interpolation and Approximation of Scattered Multidimensional and Dynamic Data Using Radial Basis Functions **

Authors: **Vaclav Skala **

Abstract: Interpolation or approximation of scattered data is very often task in engineering problems. The Radial Basis Functions (RBF) interpolation is convenient for scattered (un-ordered) data sets in k-dimensional space, in general. This approach is convenient especially for a higher dimension k>2 as the conversion to an ordered data set, e.g. using tessellation, is computationally very expensive. The RBF interpolation is not separable and it is based on distance of two points. It leads to a solution of a Linear System of Equations (LSE) Ax=B. There are two main groups of interpolating functions: ‘global” and “local”. Application of “local” functions, called Compactly Supporting RBF (CSFBF), can significantly decrease computational cost as they lead to a system of linear equations with a sparse matrix. In this paper the RBF interpolation theory is briefly introduced at the “application level” including some basic principles and computational issues and an incremental RBF computation is presented and approximation RBF as well. The RBF interpolation or approximation can be used also for image reconstruction, inpainting removal, for solution of Partial Differential Equations (PDE), in GIS systems, digital elevation model DEM etc.

Keywords: Radial basis function, RBF interpolation, image reconstruction, incremental computation, RBF approximation, fast matrix multiplication

Title of the Paper: **Variable-Coefficient Simplest Equation Method For Solving Nonlinear Evolution Equations In Mathematical Physics **

Authors: **Yulu Wang **

Abstract: In this paper, a variable-coefficient simplest equation method is proposed to establish exact solutions for nonlinear evolution equations. For illustrating the validity of this method, we apply it to the asymmetric (2+1)-dimensional NNV system, the (2+1)-dimensional dispersive long wave equations and the (2+1)-dimensional Boussinesq and Kadomtsev-Petviashvili equations. As a result, some new exact solutions and solitary wave solutions involving arbitrary function as coefficients are obtained for them.

Keywords: Simplest equation method, Variable-coefficient, Exact solutions, Nonlinear evolution equations, Traveling wave solutions, Solitary wave solutions

Title of the Paper: **Boundary Problems for Stationary Neutron Transport Equations Solved by Homotopy Perturbation Method **

Authors: **Olga Martin **

Abstract: This paper proposes an algorithm based on the homotopy perturbation method (HPM) for the solving the one-dimensional neutron transport equation with suitable multipoint boundary conditions (MPBC). The homotopy perturbation method is a coupling of traditional perturbation method and the homotopy function from topology, which continuously deforms the given problem to another that can be easily solved. The new version of homotopy method, upon which our algorithm is built, yields rapid convergence of the solution series to exact solution. Usually only two iterations lead to high accuracy solutions. Illustrative numerical examples are provided to prove the efficiency of the proposed algorithm for integral differential equations accompanied by the multipoint boundary conditions.

Keywords: Integral-differential equation, homotopy perturbation method, multipoint boundary value problem

Title of the Paper: **Optimal Control on Lie Groups: Theory and Applications **

Authors: **Karlheinz Spindler **

Abstract: In this paper we review Pontryagin’s Maximum Principle in its classical form, explain its geometric content and formulate it in a way which applies to control problems on arbitrary manifolds. It is then shown how this principle takes a particularly simple form in the case of left- or right-invariant control systems on Lie groups. Finally, we describe various application examples (from areas such as continuum mechanics, spacecraft attitude control and quantum spin systems) which show that the differential geometric version of Pontryagin’s Principle allows one to obtain solutions for concrete problems not easily found in other ways. The presentation is geared towards nonspecialists and strives to convey a feeling for the meaning and the applicability of the Maximum Principle rather than to present technical details.

Keywords: Optimal control, Pontryagin’s Principle, control systems on manifolds, Lie groups

Title of the Paper: **MGD Unstructured Application to a Blunt Body in Two-Dimensions **

Authors: **Edisson Savio De Goes Maciel **

Abstract: In this paper, the Euler and Navier-Stokes equations are solved, according to a finite volume formulation and symmetrical unstructured discretization, applied to the problem of a blunt body in two-dimensions. The work of Gaitonde is the reference one to present the fluid dynamics and Maxwell equations of electromagnetism based on a conservative and finite volume formalisms. The Jameson and Mavriplis symmetrical scheme is applied to solve the conserved equations. Two types of numerical dissipation models are applied, namely: Mavriplis and Azevedo. A spatially variable time step procedure is employed aiming to accelerate the convergence of the numerical schemes to the steady state solution. Effective gains in terms of convergence acceleration are observed with this technique (see Maciel). The results have proved that, when the Jameson and Mavriplis scheme is employed with an unstructured alternated discretization, better contours of proprieties are obtained (see Maciel). Moreover, an increase in the shock standoff distance is observed, which guarantees a minor increase in the temperature at the blunt body nose (minor armour problems), and a minor increase in the drag aerodynamic coefficient.

Keywords: Euler and Navier-Stokes equations, Magnetogasdynamics formulation, Jameson and Mavriplis algorithm, Unstructured spatial discretization, Finite volumes, Two-dimensional space

Title of the Paper: **A New Fractional Sub-Equation Method For Fractional Partial Differential Equations **

Authors: **Chuanbao Wen, Bin Zheng **

Abstract: In this paper, a new fractional sub-equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann-Liouville derivative, which is the fractional version of the known (G’/G) method. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KDV equations and the space-time fractional (2+1)-dimensional breaking soliton equations. As a result, some new exact solutions for them are successfully established.

Keywords: Fractional sub-equation method, Fractional partial differential equation, Exact solution, Fractional complex transformation, Fractional generalized Hirota-Satsuma coupled KDV equations, Fractional (2+1)-dimensional breaking soliton equations

Title of the Paper: **Axiological Impact Analysis of Legal Regulations and Policies. A Seminal Case-Study from Land Law for a Mathematical Theory **

Authors: **Massimiliano Ferrara, Angelo Roberto Gaglioti **

Abstract: This paper offers a seminal case-study of an elementary legal micro-system composed of estate property right and right to privacy. We intend to decompose the structure of such legal positions, identifying their most basic factors and logical relations and adopting a mathematical model for the quantitative analysis of law that we have recently introduced, still to be validated by the scientific community. We will describe the conflicts among identical legal values (property right vs. property right) and different legal values (property right vs. privacy right), originating by a claim to open a window next to the land boundary. Comparing two possible regulatory options for this pattern of conflict, we will be ultimately able to unveil the truly legal nature of each option (status personarum policy vs. axiological balancing policy), as well as the axiological (in)-efficiency of every devised solution. This method might lead the way to more extensive implementations of mathematical analysis in the realm of axiological impact deriving from legal regulations and policies.

Keywords: Law, Mathematics, Value, Number, Conflict, Property, Privacy, Impact, Regulation, Policy

Title of the Paper: **Resolution of Stokes Equations with the Ca,b Boundary Condition Using Mixed Finite Element Method **

Authors: **Jaouad El-Mekkaoui, Ahmed Elkhalfi, Abdeslam Elakkad **

Abstract: In this paper, we introduce the Stokes equations with a new boundary condition. In this context, we show the existence and uniqueness of the solution of the weak formulation associated with the proposed problem. To solve this problem, we use the discretization by mixed finite element method. In addition, two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true error. In order to evaluate the performance of the method, the numerical results are compared with some previously published works and with others coming from commercial code like ADINA system.

Keywords: Stokes equations, Ca,b Boundary condition, Mixed finite element method, Residual error estimator, Adina system

Title of the Paper: **BS-Patch: Constrained Bezier Parametric Patch **

Authors: **Vaclav Skala, Vit Ondracka **

Abstract: Bezier parametric patches are used in engineering practice quite often, especially in CAD/CAM systems oriented to mechanical design. In many cases quadrilateral meshes are used for tessellation of parameters domain. We propose a new modification of the Bezier cubic rectangular patch, the BS-patch, which is based on the requirement that diagonal curves must be of degree 3 instead of degree 6 as it is in the case of the Bezier patch. Theoretical derivation of conditions is presented and some experimental results as well. The BS-Patch is convenient for applications where for different tessellation of the u-v domain different degrees of diagonal curves are not acceptable.

Keywords: Parametric surface, geometric modeling, computer graphics, spline, cubic surface, Bezier

Title of the Paper: **TVD and ENO Applications to Supersonic Flows in 3D – Part I **

Authors: **Edisson Sávio De Góes Maciel **

Abstract: In this work, first part of this study, the high resolution numerical schemes of Lax and Wendroff, of Yee, Warming and Harten, of Yee, and of Harten and Osher are applied to the solution of the Euler and Navier-Stokes equations in three-dimensions. With the exception of the Lax and Wendroff and of the Yee schemes, which are symmetrical ones, all others are flux difference splitting algorithms. All schemes are second order accurate in space and first order accurate in time. The Euler and Navier-Stokes equations, written in a conservative and integral form, are solved, according to a finite volume and structured formulations. A spatially variable time step procedure is employed aiming to accelerate the convergence of the numerical schemes to the steady state condition. It has proved excellent gains in terms of convergence acceleration as reported by Maciel. The physical problems of the supersonic flows along a compression corner and along a ramp are solved, in the inviscid case. For the viscous case, the transonic flow along a convergent-divergent nozzle is solved. In the inviscid case, an implicit formulation is employed to marching in time, whereas in the viscous case, a time splitting approach is used. The results have demonstrated that the Harten and Osher algorithm, in its ENO version, presents the best solutions in the inviscid compression corner and ramp problems; whereas the Lax and Wendroff algorithm has presented the best solution to the nozzle problem.

Keywords: Lax and Wendroff algorithm, Yee, Warming and Harten algorithm, Yee algorithm, Harten and Osher algorithm, TVD and ENO flux splitting, Euler and Navier-Stokes equations, Finite volume, Three-dimensions

Title of the Paper: **Global Dynamics of an SEIRS Epidemic Model with Constant Immigration and Immunity **

Authors: **Li Juan Zhang, Yingqiu Li, Qingqing Ren, Zhenxiang Huo **

Abstract: An SEIRS model for disease transmission that includes immigration of the infective, susceptible, ex- posed, and recovered has been constructed and analyzed. For the reason that the immunity of the recovered is temporary, a proportion δ1 of recovered will come back to susceptible. We also consider vaccine injection to the susceptible with a proportion c. The model also incorporates a population size dependent contact rate and a disease- related death. As the infected fraction cannot be eliminated from the population, this kind of model has only one unique endemic equilibrium that is globally asymptotically stable. In a special case where the new members of immigration are all susceptible, the model shows a threshold phenomenon. In order to prove the global asymptot- ical stability of the endemic equilibrium, we change our system to a three-dimensional asymptotical autonomous system with limit equation. Finally, we discussed syphilis as a case to predict the development in China. Computer simulation shows that the model can reflect the dynamic and immigration behaviour for disease transmission.

Keywords: SEIRS model, population size dependent contact rate, syphilis, compound matrix

**Issue 6, Volume 12, June 2013**

Title of the Paper: **The Existence of Solution for the Nonstationary Two Dimensional Microflow Boundary Layer System **

Authors: **Xia Ye, Huashui Zhan **

Abstract: The paper concerns with the nonstationary two dimensional microflow boundary layer system. By posing some restrictions on the viscous function, the existence and the uniqueness of local solutions to the system are got. The main technique we used in the paper is Oleinik line method based on a successive approximation, which is used in the study of Prandtl system. However, the corresponding calculations in our paper are much more complicated.

Keywords: Two dimensional microflow boundary layer system, Prandtl system, Existence, Local solution

Title of the Paper: **Maximum Likelihood Estimation of Burr Type V Distribution under Left Censored Samples **

Authors: **Navid Feroze, Muhammad Aslam **

Abstract: The paper deals with the maximum likelihood estimation of the parameters of the Burr type V distribution based on left censored samples. The maximum likelihood estimators (MLE) of the parameters have been derived. The Fisher information matrix for the parameters of the said distribution has been obtained explicitly. The confidence intervals for the parameters have also been discussed. A simulation study has been conducted to investigate the performance of the point and interval estimates.

Keywords: Fisher information matrix, confidence intervals, censoring

Title of the Paper: **Complexity and Control of a Cournot Duopoly Game in Exploitation of a Renewable Resource with Bounded Rationality Players **

Authors: **Hongliang Tu, Junhai Ma **

Abstract: Based on the related literatures, a new dynamic Cournot duopoly game model in exploitation of a re- newable resource with bounded rationality players is built up. The local stable region of Nash equilibrium point is obtained through using the theory of bifurcations of dynamical systems. It is found that increasing the output ad- justment speed parameters of the system can affect the stability of Nash equilibrium point and lead chaos to occur. Its complex dynamics is demonstrated by the way of plotting the bifurcation diagrams, computing and plotting the Lyapunov exponents, plotting phase portraits and calculating the fractal dimension. Furthermore, the chaos can be respectively controlled by making use of the straight-line stabilization method, parameters adjustment method and time-delayed feedback method. The derived results have important theoretical and practical significance to the exploitation of renewable resource.

Keywords: Complexity, Bifurcation, Chaos, Chaos control, Cournot duopoly game

Title of the Paper: **Heuristic Solution for the p-hub Problem **

Authors: **Ivan Derpich, Remigio Contreras, Juan Sepulveda **

Abstract: The hub location problem is important in the selection of technological networks, such as computer, cellular, or wireless sensor networks. These modern communication networks must be dynamically set as triggered by changes in external conditions; the nodes deplete their batteries and go out of service. For this reason, it is necessary to update the available data in order to determine which nodes can be used as hubs. The dynamic location problem requires a short solution time in despite of optimality. Heuristic methods are used for their simplicity and they are easy to package in the firmware. The central aim of this work is to design a heuristic method that will obtain a good feasible solution in a reasonable amount of time. The methodology proposed for the heuristic method consists of obtaining the optimum solution of the relaxed problem, followed by rounding this solution to a 0 or 1 value. The strategy developed for rounding the calculations is to first use a measure, called attractive force, for each node and then to define those nodes more attractive as hubs. Finally, an integer programming model is solved for assigning the nodes to the selected hubs. An interesting result is that the hubs selected by the optimal solution of the relaxed problem are always between the nodes that have the major attractive force. The heuristic algorithm is well established for problems with 10, 20, 25, 50 and 100 nodes. So, mixing two levels of difficulty we obtain four problems.

Keywords: Hubs, location, networks, integer programming, heuristics

Title of the Paper: **A Review of Studies on Fluid Flow Passing through an Ideal Aggregate Part I: Models with an Uniformly Permeable Porous Layer **

Authors: **Rong Yuan **

Abstract: This is a review of studies on fluid flow passing through an isolated ideal aggregate with a porous layer with the uniformly permeability. Fluid flows are governed by the Brinkman’s extension of Darcy’s law and continuity equation. These equations with appropriate boundary conditions are analytically solved by introducing stream functions, respectively. The comparisons between Kim & Yuan’s cell model with a porous layer with the uniformly permeability and other previous works are investigated.

Keywords: Ideal aggregate; porous media; uniformly permeability; drag force

Title of the Paper: **Slowly Changing Function Oriented Growth Analysis of Differential Monomials and Differential Polynomials **

Authors: **Sanjib Kumar Datta, Tanmay Biswas, Golok Kumar Mondal **

Abstract: In this paper, we establish some new results depending on the comparative growth properties of composite entire or meromorphic functions using L∗-order and L∗-type and differential monomials, differential polynomials generated by one of the factors.

Keywords: Meromorphic function, transcendental entire function, composition, growth, L*-order, L*-type, differential monomials, differential polynomial, slowly changing function

Title of the Paper: **On the Harmonic Index of the Unicyclic and Bicyclic Graphs **

Authors: **Yumei Hu, Xingyi Zhou **

Abstract: The harmonic index is one of the most important indices in chemical and mathematical fields. It’s a variant of the Randi´c index which is the most successful molecular descriptor in structure-property and structure- activity relationships studies. The harmonic index gives somewhat better correlations with physical and chemical properties comparing with the well known Randi´c index. The harmonic index H(G) of a graph G is defined as the sum of the weights 2 d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we present the unicyclic and bicyclic graphs with minimum and maximum harmonic index, and also characterize the corresponding extremal graphs. The unicyclic and bicyclic graphs with minimum harmonic index are S+ n , S1 n respectively, and the unicyclic and bicyclic graphs with maximum are Cn, Bn or B′ n respectively. As a simple result, we present a short proof of one theorem in Applied Mathematics Letters 25 (2012) 561-566, that the trees with maximum and minimum harmonic index are the path Pn and the star Sn, respectively. Moreover, we give a further discussion about the property of the graphs with the maximum harmonic index, and show that the regular or almost regular graphs have the maximum harmonic index in connected graphs with n vertices and m edges.

Keywords: The harmonic index; minimum; maximum; unicyclic graphs; bicyclic graphs

Title of the Paper: **A New, Robust and Applied Model for Approximation of Huge Data **

Authors: **Mehdi Zamani **

Abstract: A powerful and accurate model for interpolation of data is piecewise cubic spline method. In this model the obtained final curve consists of a series of local cubic spline curves which combine together with suitable continuity along their boundaries. The continuity at the boundaries of each local curve is c2. The attraction of this research is to apply cubic spline method for approximation and estimation of data. Therefore; each local spline curve satisfies the minimization of sum of square errors along its length in addition of obtaining c2 continuity at its edges or boundaries. Because in the local scale the approximation of any section of data by cubic spline is accurate; therefore, the presented model is applicable to any kind of data with highly nonlinear distributions.

Keywords: Approximation; estimation; cubic spline; optimization; B-spline; sum of square error

**Issue 7, Volume 12, July 2013**

Title of the Paper: **Blowup Analysis for a Nonlocal Reaction Diffusion Equation with Potential **

Authors: **Yulan Wang, Jiqin Chen, Bin Zhou **

Abstract: In this paper we investigate a nonlocal reaction diffusion equation with potential, under Neumann boundary. We obtain the complete classification of the parameters for which the solution blows up in finite time or exists globally. Moreover, we study the blowup rate and the blowup set for the blowup solution.

Keywords: Nonlocal diffusion, Blow up, Blowup rate, Blowup set

Title of the Paper: **A General Approach for Computing Residues of Partial-Fraction Expansion of Transfer Matrices **

Authors: **M. I. Garcia-Planas, J. L. Dominguez-Garcia **

Abstract: This paper deals with the description of a general method for calculating the residues of a linear system. Considering, physical models, it is well-assumed that the system described only presents simple eigenvalues, or at least simple-complex eigenvalues. However, as demonstrated in this paper, it is not completely true for all the real systems, and a method to evaluate the residues for these cases is required. In this paper, a methodology for computing the residues, even with the existence of multiple eigenvalues (described by their Jordan normal form) is developed and presented. Moreover, the calculation of the residues is applied to analyze the output-controllability of dynamic systems. Finally, some real examples are presented to validate the methodologies proposed.

Keywords: Eigenvalues, Jordan normal form, output-controllability, residues

Title of the Paper: **Generalized Delay Integral Inequalities In Two Independent Variables On Time Scales **

Authors: **Qinghua Feng, Bosheng Fu **

Abstract: In this paper, we discuss some new Gronwall-Bellman type delay integral inequalities in two independent variables on time scales, which contain arbitrary nonlinear items on unknown functions outside the integrals as well as inside the integrals. The inequalities established have different forms compared with the existing Gronwall- Bellman type inequalities so far in the literature. As applications, we research the upper bounds of one certain delay dynamic equation on time scales. New explicit bounds for the solutions are derived with the results established.

Keywords: Delay integral inequality, Time scales, Delay dynamic equations, Qualitative analysis, Bounded

Title of the Paper: **An Exponentially Fitted Non Symmetric Finite Difference Method for Singular Perturbation Problems **

Authors: **Gbsl Soujanya, K. Phaneendra, Y. N. Reddy **

Abstract: In this paper, we have presented an exponentially fitted non symmetric numerical method for singularly perturbed differential equations with layer behaviour. We have introduced a fitting factor in a non symmetric finite difference scheme which takes care of the rapid changes occur that in the boundary layer. This fitting factor is obtained from the theory of singular perturbations. The discrete invariant imbedding algorithm is used to solve the tridiagonal system of the fitted method. This method controls the rapid changes that occur in the boundary layer region and it gives good results in both cases i.e., h ≤ " and " << h. The existence and uniqueness of the discrete problem along with stability estimates are discussed. Also we have discussed the convergence of the method. Maximum absolute errors in numerical results are presented to illustrate the proposed method for " << h.

Keywords: Singularly perturbed two point boundary value problem, Boundary layer, Taylor series, Fitting factor, Maximum absolute error

Title of the Paper: **Positive Periodic Solutions of Delayed Dynamic Equations with Feedback Control on Time Scales **

Authors: **Meng Hu, Pingli Xie **

Abstract: In this paper, based on the theory of calculus on time scales, by using Avery-Peterson fixed point theorem for cones, some criteria are established for the existence of three positive periodic solutions of delayed dynamic equations with feedback control on time scales.

Keywords: Positive periodic solution; Dynamic equations; Feedback control; Time scale

Title of the Paper: **Self-Synchronization of Networks with a Strong Kernel of Integrate and Fire Excitatory Neurons **

Authors: **Eleonora Catsigeras **

Abstract: We study the global dynamics of networks of pulsed-coupled neurons that are modeled as integrate and fire oscillators. We focus on excitatory networks with a strong kernel. We prove the synchronization of the whole network from any initial state, and find a bound from above of the transients until the full synchronization is achieved. The methodology of research is by exact mathematical definitions and statements and by deductive proofs, from standard arguments of the mathematical theory of abstract dynamical systems. We include examples of applications to diverse fields, and also a brief review of other mathematical methods of research on general networks of dynamically interacting units.

Keywords: Pulse-coupled networks, synchronization, neural networks, integrate and fire oscillators

Title of the Paper: **Robust Mixed H2=H∞ Output Tracking Control of Uncertain Discrete-Time Switched Systems with State Time-Delay **

Authors: **Yitao Yang **

Abstract: This paper focuses on the problem of mixed H2/H∞ output tracking control for uncertain discrete-time switched systems with state time-delay. By using single Lyapunov function theory, a state feedback controller is presented to guarantee the closed-loop error system robust asymptotically stable with mixed H2/H∞ performance is developed. The controller gains are obtained by a set of linear matrix inequalities(LMIs). The corresponding stabilizing switching rule is provided. A numerical example is given to demonstrate the effectiveness of the pro- posed approach.

Keywords: Discrete-time switched systems, time-delay systems, tacking control, mixed H2/H∞control, single Lyapunov function

**Issue 8, Volume 12, August 2013**

Title of the Paper: **Multi-Objective Integer Programming Model and Algorithm of the Crew Pairing Problem in a Stochastic Environment **

Authors: **Deyi Mou, Yingnan Zhang **

Abstract: Crew scheduling is an important production planning of airlines. Being optimized crew scheduling could make full use of human resources, and reduce operating costs. The traditional airline crew scheduling model is deterministic and does not include potential disruptions due to weather, air traffic control, etc. To take into account of effects of random factors such as weather, air traffic control, passenger demand, etc., we develop a stochastic chance-constrained programming model (SCCPM) for minimizing the crew cost and maximizing the passenger satisfaction. Based on Monte Carlo method, Back Propagation (BP) neural network and genetic algorithm, we develop a hybrid intelligent algorithm to solve the model. To evaluate the robustness of the model, the signal to noise ratio (SNR) method is included in this paper. We present computational results which show the effectiveness of our SCCPM and the hybrid intelligent algorithm.

Keywords: Airline operations, crew scheduling, passenger satisfaction, stochastic chance-constrained program-ming, hybrid intelligent algorithm

Title of the Paper: **On Four-Point Nonlocal Boundary Value Problems of Nonlinear Impulsive Equations of Fractional Order **

Authors: **Dehong Ji, Weigao Ge **

Abstract: This paper is motivated from some recent papers treating the boundary value problems for impulsive fractional differential equations. We first give some notations, recall some concepts and preparation results. Sec- ond, we establish a general framework to find the solutions for impulsive fractional boundary value problems, which will provide an effective way to deal with such problems. Third, some sufficient conditions for the existence of the solutions are established by applying fixed point methods. Our results complements previous work in the area of four-point boundary value problems of fractional order.

Keywords: Fractional differential equation; Impulse; Four-point boundary value problem; Fixed point theorem

Title of the Paper: **The Spectrum of A Parallel Repairable System with Warm Standby **

Authors: **Wenlong Wang, Zhiying Li **

Abstract: In this paper, the spectrum of a parallel repairable system with warm standby is investigated. Firstly, we formulate the problem into a suitable Banach space. Then we carry out a detailed spectral analysis of the system operator. Based on the spectral analysis and C0 semigroup theory, we prove the existence of positive solution and finite expansion of the solution corresponding to its eigenvector. As a consequence we get that its dynamic solutions converges exponentially to the stead-state solution. Finally, we obtain the finite expansion of solution and derive some reliability indices of the system.

Keywords: C0 semigroup theory; Spectrum; Parallel Repairable system; Stead-state; Availability

Title of the Paper: **Non-Collocated Feedback Stabilization of a Kind of System Described by Wave Equations **

Authors: **Yan Ni Guo, Ling Ling Zhang, Ya Xuan Zhang **

Abstract: In this paper, we study the stabilization problem of a three-edge network system described by variable coefficients wave equations. With the root node fixed and a tip mass attached on the common vertex, we design two non-collocated controllers. Then we show that the closed-loop system is well-posed and satisfies spectrum- determined growth condition while the feedback gain constants fulfill some requirements. Moreover, we prove that the system is exponentially stable by applying Riesz basis method and utilizing some tricks of inequalities.

Keywords: Wave equation, variable coefficient, non-collocated control, exponential stabilization

Title of the Paper: **A New Type of Sequence Space of Non-Absolute Type and Matrix Transformation **

Authors: **Neyaz Ahmad Sheik, Ab. Hamid Ganie **

Keywords: Sequence space of non-absolute type; paranormed sequence space; α-, β- and γ-duals ; matrix transformations

Title of the Paper: **Stability and Bifurcation Analysis for an Improved HIV Model with Time Delay and Cure Rate **

Authors: **Kejun Zhuang, Hailong Zhu **

Abstract: In this paper, a modified delayed mathematical model for the dynamics of HIV with cure rate is consid- ered. By regarding the time delay as bifurcation parameter, stability and existence of local Hopf bifurcation are studied by analyzing the transcendental characteristic equation. Then the global existence of bifurcating periodic solutions is established with the assistance of global Hopf bifurcation theory. Finally, some numerical examples are given.

Keywords: HIV, Hopf bifurcation, stability, delay

**Issue 9, Volume 12, September 2013
Special Issue: Network Reliability and Vulnerability Models and their Applications
Invited Editors: Louis Petingi, Charles Suffel**

Title: **An Introduction to the Special Issue on Network Reliability and Vulnerability Models and their Applications **

Authors: **Louis Petingi, Charles Suffel **

Title of the Paper: **On Component Order Edge Reliability and the Existence of Uniformly Most Reliable Unicycles **

Authors: **Daniel Gross, Lakshmi Iswara, L. William Kazmierczak, Kristi Luttrell, John T. Saccoman, Charles Suffel **

Keywords: Component Order edge reliability, uniformly most reliable, unreliability, unicycle

Title of the Paper: **Diameter-Related Properties of Graphs and Applications to Network Reliability Theory **

Authors: **Louis Petingi **

Abstract: Given an undirected graph G = (V;E), two distinguished vertices s and t of G, and a diameter bound D, a D-s; t-path is a path between s and t composed of at most D edges. An edge e is called D-irrelevant if does not belong to any D-s; t-path of G. In this paper we study the problem of efficiently detecting D-irrelevant edges and also study the computational complexity of diameter-related problems in graphs. Detection and subsequent deletion of D-irrelevant edges have been shown to be fundamental in reducing the computational effort to evaluate the Source-to-terminal Diameter-Constrained reliability of a graph G, R{s;t}(G;D), which is defined as the probability that at least a path between s and t, with at most D edges, survives after deletion of the failed edges (under the assumption that edges fail independently and nodes are perfectly reliable). Among other results, we present sufficient conditions to efficiently recognize irrelevant edges and we present computational results illustrating the importance of embedding a procedure to detect irrelevant edges based on these conditions, within the frame of an algorithm to calculate R{s;t}(G;D), built on a theorem of Moskowitz. These results yield a research path for the theoretical study of the problem of determining families of topologies in which R{s;t}(G;D) can be computed in polynomial time, as the general problem of evaluating this reliability measure is NP-Hard.

Keywords: Network reliability, diameter constraint, paths, factoring, topological reductions

Title of the Paper: **A Survey of Component Order Connectivity Models of Graph Theoretic Networks**

Authors: **Daniel Gross, Monika Heinig, Lakshmi Iswara, L. William Kazmierczak, Kristi Luttrell, John T. Saccoman, Charles Suffel**** **

Abstract: The traditional vulnerability parameter connectivity is the minimum number of nodes needed to be removed to disconnect a network. Likewise, edge connectivity is the minimum number of edges needed to be removed to disconnect. A disconnected network may still be viable if it contains a sufficiently large component. Component order connectivity and component order edge connectivity are the minimum number of nodes, respectively edges needed to be removed so that all components of the resulting network have order less than some preassigned threshold value. In this paper we survey some results of the component order connectivity models.

Keywords: Connectivity, edge connectivity, component order connectivity, component order edge connectivity, component order neighbor connectivity

Title of the Paper: **A Framework for Faults in Detectors within Network Monitoring Systems **

Authors: **Peter J. Slater **

Abstract: Various different types of detectors can be used to identify malfunctioning elements in a network. The detectors themselves might not function properly. Four types of possible detector faults are identified, where detectors are located at vertices and are used to identify malfunctioning vertex locations. This leads to a sequence of four detector-failure parameters for various (domination-related) parameters.

Keywords: Intruder detection, locating-dominating sets, identifying codes, open-locating-dominating sets, detector faults

Title of the Paper: **A Modified Measure of Covert Network Performance **

Authors: **Lynne L. Doty **

Abstract: In a covert network the need for secrecy is at odds with the desire for easy transmission of information. Lindelauf et al. [7] have defined several measures that can be used to evaluate the total performance of a covert social network by using a product of individual measures of secrecy and of information transmission. As one of their simplest measures of secrecy, Lindelauf et al. use a modified version of the idea of neighbor connectivity. Using this measure, the optimal network structure is a star graph. In this paper we modify the Lindelauf measure of secrecy to include information about the connectedness (not just the order) of the survival subgraph using a measure based on ideas related to Chv´atal’s toughness parameter. We determine an upper bound on performance of trees and conjecture that a specific class of spider graphs achieves maximum performance. We also describe several opportunities for further research.

Keywords: Covert networks, neighbor connectivity, spider graphs, toughness

Title of the Paper: **An Update on Supereulerian Graphs **

Authors: **Hong-Jian Lai, Yehong Shao, Huiya Yan **

Abstract: A graph is supereulerian if it has a spanning Eulerian subgraph. Motivated by the Chinese Postman Problem, Boesch, Suffel, and Tindell ([2]) in 1997 proposed the supereulerian problem, which seeks a characterization of graphs that have spanning Eulerian subgraphs, and they indicated that this problem would be very difficult. Pulleyblank ([71]) later in 1979 proved that determining whether a graph is supereulerian, even within planar graphs, is NP-complete. Since then, there have been lots of researches on this topic. Catlin ([7]) in 1992 presented the first survey on supereulerian graphs. This paper is intended as an update of Catlin’s survey article and will focus on the developments in the study of supereulerian graphs and the related problems over the past 20 years.

Keywords: Eulerian graphs, Supereulerian graphs, Collapsible graphs, Catlin’s reduction method, Line graphs, Claw-free graphs

Title of the Paper: **Network Theory Tools for RNA Modeling **

Authors: **Namhee Kim, Louis Petingi, Tamar Schlick **

Abstract: An introduction into the usage of graph or network theory tools for the study of RNA molecules is presented. By using vertices and edges to define RNA secondary structures as tree and dual graphs, we can enumerate, predict, and design RNA topologies. Graph connectivity and associated Laplacian eigenvalues relate to biological properties of RNA and help understand RNA motifs as well as build, by computational design, various RNA target structures. Importantly, graph theoretical representations of RNAs reduce drastically the conformational space size and therefore simplify modeling and prediction tasks. Ongoing challenges remain regarding general RNA design, representation of RNA pseudoknots, and tertiary structure prediction. Thus, developments in network theory may help advance RNA biology.

Keywords: Network Theory, RNA-As-Graphs, RNA Prediction, In Vitro Selection

**Issue 10, Volume 12, October 2013**

Title of the Paper: **Multiple Periodic Solutions for a General Class of Delayed Cooperative Systems on Time Scales **

Authors: **Kaihong Zhao, Liang Ding **

Abstract: In this paper, we consider a general class of delayed nonautonomous logistic Lotka-Volterra type multi- species cooperative system with harvesting terms on time scales. The model invovles the intraspecific cooperative terms defined by functions which depend on population densities. An existence theorem of at least 2n periodic solutions is established by using the coincidence degree theory. An example is given to illustrate the effectiveness of our result.

Keywords: Time scales; Periodic solutions; Delayed cooperative system; Coincidence degree; Harvesting term

Title of the Paper: **Some New Dynamic Inequalities and Their Applications in the Qualitative Analysis of Dynamic Equations **

Authors: **Hongxia Wang, Bin Zheng **

Abstract: In this paper, we establish some new Gronwall-Bellman-type dynamic inequalities in two independent variables containing integration on infinite intervals on time scales, which can be used as a handy tool in the boundedness analysis for solutions to some certain dynamic equations containing integration on infinite intervals on time scales. The presented inequalities are of new forms so far in the literature to our best knowledge.

Keywords: Gronwall-Bellman-type inequality; Time scales; Dynamic equation; Qualitative analysis; Quantitative analysis; Bound

Title of the Paper: **On Further Ordering Bicyclic Graphs with Respect to the Laplacian Spectra Radius **

Authors: **Guangqing Jin, Liancui Zuo **

Abstract: Among all the n-vertex bicyclic graphs, the first eight largest Laplacian spectra radii had been obtained in the past, all of which are no smaller than n − 1. In this paper, it is first obtained that all the bicyclic graphs on n vertices with Laplacian spectra radii at least n − 2 must contain two adjacent vertices which cover all the vertices except possibly two and one of the two adjacent vertices must have the degree at least n−3. Then the total forty-two such graphs are further ordered, and the ninth to the forty-first largest Laplacian spectra radii among all the n-vertex bicyclic graphs are finally determined in this way.

Keywords: Bicyclic graph; the Laplacian spectra radius; the Laplacian characteristic polynomial

Title of the Paper: **Simulation of Sierpinski-Type Fractals and their Geometric Constructions in Matlab Environment **

Authors: **Zhiyong Zhu, Enemi Dong **

Abstract: Study on properties of Sierpinski-type fractals, including dimension, measure, connectedness, Lipschitz equivalence, etc are very interesting. Although there have been some very nice results were obtained, there is still a long way to go to solve all the problems. In order to facilitate understanding of these results and further study, in this paper, we simulate this kind of fractals and their geometric constructions in Matlab environment that is more easily understood and mastered for researcher base on the recursive and iterative algorithms that are used to simulate fractals. Furthermore, our results are also interesting results to enrich the theoretical and applied research of fractal simulation.

Keywords: Sierpinski gasket-type fractal, Sierpinski carpet-type fractal, Fractal simulation, Recursive algorithm, Iterative algorithm, Matlab

Title of the Paper: **Exponential Stabilization of 1-d Wave Equation with Input Delay **

Authors: **Han Wang, Gen Qi Xu **

Abstract: In this paper, we consider the stabilization problem of 1-d wave equation with input delay. Suppose that the wave system is fixed at one end whereas a control force is applied at other end. Here we consider the control force of the form αu(t) + βu(t − τ ) where τ is the time delay. In this paper we find a feedback control law that stabilizes exponentially the system for any |α| ΜΈ= |β| and τ > 0.

Keywords: Wave system; input delay; feedback control; exponential stabilization

Title of the Paper: **A Modified Power Spectrum Estimator for Minimizing the Prediction Error Energies **

Authors: **Xiao Liu, Huihui Liu, Ying Li, Zhanjie Song **

Abstract: The power spectrum estimation plays a significant role in the marine monitoring and the quality of spectrum will seriously affect the results of marine monitoring. There have been various algorithms on power spectrum estimation, one of which is Burg algorithm. Though this method can effectively improve the resolution of spectral estimation, it will cause the shifting of peak frequency and the splitting of spectral lines accompanying spurious frequency components. In this paper, a novel method for minimizing the prediction error energies is presented, which is based on the optimum tapered Burg algorithm. The proposed method is implemented in two aspects. Firstly, the first order reflection coefficient is calculated by minimizing the averaged optimum tapered forward and backward prediction error energies of the third order filter. Then, the residual factor is introduced to estimate the model order. In the end of the paper, the experimental results demonstrate that in contrast to Burg algorithm and optimum tapered Burg algorithm, the proposed algorithm can not only effectively improve the spectrum resolution, lower the degree of peak frequency shifting and spectrum lines splitting, but also be able to alleviate the generation of spurious peak frequencies and the phenomenon of spectral leakage.

Keywords: Burg algorithm, Spectrum estimation, Prediction error energy, Levinson-Durbin recursion

Title of the Paper: **The Construction of A3-Code from Projective Spaces over Finite Fields **

Authors: **Gao You, Liu Yanqin **

Abstract: A3-code has a dishonest arbiter who may disturb the communication compared with A2-code, and the arbiter also has some secret key information used to arbitrate in the case of dispute between the senders and receivers. This paper firstly introduces the model of A3-code and the seven types of possible cheating attacks as well as their computational formula. And then a construction of A3-code is presented using the incidence relation of flats from projective spaces over finite fields. The parameters of the code and the probabilities of success in different attacks are also computed, assuming that the probability distributions of source states and participants keys are uniform.

Keywords: A3-codes, projective spaces , finite fields

**Issue 11, Volume 12, November 2013**

Title of the Paper: **Fast Algorithms for Solving RFPrLR Circulant Linear Systems **

Authors: **Zhaolin Jiang, Jing Wang **

Abstract: In this paper, fast algorithms for solving RFPrLR circulant linear systems are presented by the fast algorithm for computing polynomials. The unique solution is obtained when the RFPrLR circulant matrix over the complex field C is nonsingular, and the special solution and general solution are obtained when the RFPrLR circulant matrix over the complex field C is singular. The extended algorithms is used to solve the RLPrFL circulant linear systems. Examples show the effectiveness of the algorithms.

Keywords: RFPrLR circulant matrix, Linear system, Fast algorithm

Title of the Paper: **New Generalized Ostrowski-Grüss Type Inequalities in Two Independent Variables on Time Scales **

Authors: **Qinghua Feng, Xilian Fu **

Abstract: Some new generalized Ostrowski-Gr¨uss type integral inequalities in two independent variables on time scales are established in this paper. The established results generalize some known results in the literature, and unify corresponding continuous and discrete analysis. New bounds for the related Ostrowski-Gr¨uss type inequali- ties are derived, and some of these bounds are sharp.

Keywords: Ostrowski type inequality, Gr¨uss type inequality, Time scales, Bounds, Sharp inequalities

Title of the Paper: **Minimal Spanning Tree From a Minimum Dominating Set **

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

Abstract: In this article, we provide a constructive procedure to generate a spanning tree for any graph from its dominating set, γ - set. We introduce a new kind of minimum dominating set and hence generate a minimum weighted spanning tree from a γ - set for G. We also provide a method for generating a minimum weighted spanning tree using adjacency matrix of G.

Keywords: Spanning tree, Minimum dominating set, Minimum weighted spanning tree

Title of the Paper: **Optimal Investment and Consumption Decisions Under the Ho-Lee Interest Rate Model **

Authors: **Hao Chang, Xi-Min Rong, Hui Zhao **

Abstract: In this paper, we consider an investment and consumption problem with stochastic interest rate, in which risk-free interest rate dynamics is driven by the Ho-Lee model,while risky asset price is supposed to follow a geo- metric Brownian motion and be correlated with interest rate dynamics. Our goal is to seek an optimal investment and consumption strategy to maximize the expected discounted utility of consumption and terminal wealth in the finite horizon. Firstly, we apply dynamic programming principle to derive Hamilton-Jacobi-Bellman(HJB) equa- tion for the value function and take power utility and logarithm utility for our analysis. Secondly, by conjecturing the form of a solution and solving partial differential equations, we obtain the closed-form solutions to the optimal investment and consumption strategies. Finally, we provide a numerical example to demonstrate the impact of market parameters on the optimal investment and consumption strategy.

Keywords: Investment and consumption problem, the Ho-Lee model, dynamic programming principle, HJB equation, closed-form solution

Title of the Paper: **The Dynamic Game between Imported Luxury Car and Domestic Car and Competition Strategy Comparison in Chinese Car Market **

Authors: **Fang Wu, Junhai Ma **

Abstract: Chinese car market has became a differentiated market with the increasing income gap in China. Ac- cording to the differentiation characteristic of Chinese car market, two models are built in this paper: One is output model and the other is price model. Then, the complex dynamic behaviors are analyzed by numerical simulations such as the local stability, three-dimensional dynamic evolutionary map, bifurcation, chaos, Lyapunov exponents, and initial conditions sensitivity analysis. Finally, two models are compared, and the complex dynamic behaviors are explained from the economic viewpoint. The comparison results have theoretical and practical significance for the development of Chinese car industry.

Keywords: Repeated Game, Complex Dynamics, Output Game, Price Game, Product Differentiation, Chaos, Chinese Car Market

Title of the Paper: **Several Projection Algorithms for Solving the Split Equality Problem **

Authors: **Qiao-Li Dong, Songnian He, Han-Bao Yuan **

Abstract: The split equality problem (SEP) has extraordinary utility and broad applicability in many areas of applied mathematics. Many researchers studied the SEP and proposed algorithms to solve it. However, there are only convergence results of the algorithms in their results and is no estimate on the rate of the convergence. In this paper, we introduce three projection algorithms for solving the split equality problem (SEP), two of which are self-adaptive. The global rate of convergence is firstly investigated. One algorithm is proved to have a global convergence rate O(1/k) and two other algorithms have a global convergence rate O(1/k2).

Keywords: Split equality problem, Split feasibility problem, Self-adaptive algorithm, Global rate of convergence, Fast algorithm

Title of the Paper: **Numerical Solving Two-dimensional Variable-Order Fractional Advection-dispersion Equation **

Authors: **Liping Wen, Xiaolin Tang **

Abstract: In this paper, a two-dimensional variable-order fractional advection-dispersion equation with variable coefficient is considered. The numerical method with first order temporal accuracy and first order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by using energy method. Finally, the results of a numerical example supports the theoretical analysis.

Keywords: two-dimensional variable-order fractional advection-dispersion equation, finite difference methods, stability, convergence

Title of the Paper: **On s-Quasinormally Embedded or Weakly s-Permutable Subgroups of Finite Groups **

Authors: **Ping Kang, Hong Pan **

Abstract: Suppose that G is a finite group and H is a subgroup of G. H is said to be s-quasinormally embedded in G if for each prime p dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G; H is said to be weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that every subgroup H of P with |H| = |D| is either s-quasinormally embedded or weakly s-permutable in G. Some recent results are generalized and unified.

Keywords: s-quasinormally embedded subgroup, Weakly s-permutable subgroup, Solvable groups, Saturated formation, Finite groups

Title of the Paper: **A Novel Inexact Smoothing Method for Second-Order Cone Complementarity Problems **

Authors: **Xiaoni Chi, Zhongping Wan, Jiawei Chen **

Abstract: A novel inexact smoothing method is presented for solving the second-order cone complementarity problems (SOCCP). Our method reformulates the SOCCP as an equivalent nonlinear system of equations by in- troducing a regularized Chen-Harker-Kanzow-Smale smoothing function. At each iteration, Newton’s method is adopted to solve the system of equations approximately, which saves computation work compared to the cal- culations of exact search directions. Under rather weak assumptions, the algorithm is proved to possess global convergence and local quadratic convergence.

Keywords: Second-order cone complementarity problem, Smoothing Newton method, Inexact search direction, Global convergence, Local quadratic convergence

Title of the Paper: **Almost Periodic Solution of Predator-Prey System with Beddington-DeAngelis Functional Response on Time Scales **

Authors: **Lili Wang, Meng Hu **

Abstract: This paper is concerned with a predator-prey system with Beddington-DeAngelis functional response on time scales. Based on the theory of calculus on time scales, by using the properties of almost periodic functions and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally attractive positive almost periodic solution of the system are obtained. Finally, an example and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

Keywords: Permanence, Almost periodic solution, Global attractivity, Time scale

**Issue 12, Volume 12, December 2013**

Title of the Paper: **The Spectral Decomposition of Some Tridiagonal Matrices **

Authors: **Zhaolin Jiang, Nuo Shen, Juan Li **

Abstract: Some properties of near-Toeplitz tridiagonal matrices with specific perturbations in the first and last main diagonal entries are considered. Applying the relation between the determinant and Chebyshev polynomial of the second kind, we first give the explicit expressions of determinant and characteristic polynomial, then eigenvalues are shown by finding the roots of the characteristic polynomial, which is due to the zeros of Chebyshev polynomial of the second kind, and the eigenvectors are obtained by solving symmetric tridiagonal linear systems in terms of Chebyshev polynomial of the third kind or the fourth kind. By constructing the inverse of the transformation matrices, we give the spectral decomposition of this kind of tridiagonal matrices. Furthermore, the inverse (if the matrix is invertible), powers and a square root are also determined.

Keywords: Tridiagonal matrices, Spectral decomposition, Powers, Inverses, Chebyshev polynomials

Title of the Paper: **Eulers Method for Fractional Differential Equations **

Authors: **Ping Tong, Yuqiang Feng, Hengjin Lv **

Abstract: This paper presents a numerical method for solving fractional differential equations in the Riemann- Liouville sense. The approach is based on the Eulers method. The main characteristic behind the approach is that Euler method has intuitive geometric meaning. The algorithm is presented and the convergence of the algorithm is proved. As applications of main results, three specific numerical examples are given.

Keywords: Fractional differential equations, Initial value problem, Solution, Existence, Eulers method

Title of the Paper: **Positive Solutions for Nonlocal Boundary Value Problems of Fractional Differential Equation **

Authors: **Yitao Yang, Fanwei Meng **

Abstract: In this paper, we investigate the existence, uniqueness and multiplicity of positive solutions for nonlocal boundary value problems of fractional differential equation. Firstly, we reduce the problem considered to the equivalent integral equation. Secondly, the existence and uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions. Thirdly, by means of some fixed point theorems, some results on the multiplicity of positive solutions are obtained.

Keywords: Riemann-Liouville fractional derivative, Nonlocal boundary value problem, Fixed point theorem, successive iteration, Positive solution

Title of the Paper: **Total Dominating Set Based Algorithm for Connected Dominating Set in Ad Hoc Wireless Networks **

Authors: **S. Balaji, K. Kannan, Y. B. Venkatakrishnan **

Abstract: In an efficient design of routing protocols in ad hoc wireless networks, the connected dominating set (CDS) is widely used as a virtual backbone. To construct the CDS with its size as minimum, many heuristic, meta-heuristic, greedy, approximation and distributed algorithmic approaches have been proposed in the recent years. These approaches mostly concentrated on deriving independent set and then constructing the CDS using Steiner tree and also these algorithms perform well only for the graphs having smaller number of nodes and also for the networks that are generated in an one fixed 2D simulation area. This paper provides a novel approach for constructing the CDS, based on the concept of total dominating set and bipartite theory of graphs. Since the total dominating set is the best lower bound for the CDS, the proposed approach reduces the computational complexity to construct the CDS through the number of iterations. Moreover the conducted simulation reveals that the proposed approach finds better solution than the recently developed approaches when all the three important factors of ad hoc network such as number of nodes, transmission radio range and area of network density varies.

Keywords: connected dominating set, total dominating set, adhoc, algorithms

Title of the Paper: **Numerical Solution of Integro-Differential Equations of Fractional Order by Laplace Decomposition Method **

Authors: **Changqing Yang, Jianhua Hou **

Abstract: In this paper, Laplace decomposition method is developed to solve linear and nonlinear fractional integro- differential equations. The proposed method is based on the application of Laplace transform to nonlinear frac- tional integro-differential equation. The nonlinear term can easily be handled with the help of Adomian polyno- mials. The fractional derivative is described in the Caputo sense. The Laplace decomposition method is found to be fast and accurate. Illustrative examples are included to demonstrate the validity and applicability of presented technique and comparison is made with exacting results.

Keywords: integro-differential equations, Laplace transform, fractional derivative, Adomian polynomials, Pade approximants

Title of the Paper: **Spectral Norms of Circulant-Type Matrices Involving Some Well-Known Numbers **

Authors: **Jianwei Zhou, Zhaolin Jiang **

Abstract: In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skew- circulant and g-circulant matrices. The entries are product of binomial coefficients with Fibonacci numbers and Lucas numbers, respectively. We obtain identity estimations for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.

Keywords: Binomial coefficients, Circulant type, Fibonacci numbers, Lucas numbers, Spectral norms

Title of the Paper: **A Superior Attraction Bacterial Foraging Optimizer for Global Optimization **

Authors: **Xianghua Chu, Ben Niu, Qiang Lu, Jun Ding **

Abstract: In order to improve the performance of basic bacterial foraging optimization (BFO) for various global optimization problems, a superior attraction bacterial foraging optimizer (SABFO) is proposed in this paper. In SABFO, a novel movement guiding technique termed as superior attraction strategy is introduced to make use of all bacteria historical experience as potential exemplars to lead individuals direction. This strategy enables the bacteria in population to exchange information and collaborate with the superior individuals to search better solutions for different dimensions. Two variants of SABFO are studied and tested on a set of sixteen benchmark functions including various properties, such as unimodal, multimodal, shifted and inseparable characteristics. Four state-of-the-art evolutionary algorithms are adopted for comparison. Experimental study demonstrates remarkable improvement of the proposed algorithm for global optimization problems in terms of solution accuracy and convergence speed.

Keywords: Global optimization, Bacterial foraging optimization, Swarm intelligence, Engineering optimization, Movement updating, Meta-heuristic, Evolutionary algorithms

Title of the Paper: **Semilinear Nonlocal Differential Inclusions in Banach Spaces **

Authors: **Shaochun Ji, Gang Li **

Abstract: This paper is concerned with the existence of mild solutions to a class of semilinear differential inclu- sions with nonlocal conditions. By using the fixed point theory for multivalued maps, we get some general results on nonlocal differential inclusions, which include some recent results on nonlocal problems as special cases. An example of partial differential equations is provided to illustrate our results.

Keywords: Differential inclusions, Nonlocal conditions, Fixed point theorems, Multivalued analysis, Mild solutions

Title of the Paper: **Research on Supply Chain Coordination for Substitutable Products in Competitive Models **

Authors: **Junhai Ma, Aiwen Ma **

Abstract: This paper studies four substitutable products in a two level supply chain which consists of two suppliers and two manufacturers. We develop two competitive models that decisions are made on the decentralized and centralized decisions respectively. Then we get the optimal pricing for four products in each condition. Through detail numerical analyses, we discuss the price, product demand and the benefit of supply chain under the influences of the price sensitive coefficient, auxiliary material cost and combination of price sensitive coefficient and production cost. Besides that, we also discuss the impact of price adjustment speed parameter on the strategy of pricing for products. We find that the optimal material providing strategies for manufacturers are changing with the value of price sensitive coefficient in the decentralized and centralized decisions. And the competition among substitutable products can slow down the effect of production cost on the benefit of supply chain. We also conclude that the supply chain with coordination mechanism can be more stable under the repeated game between manufacturers.

Keywords: substitutable product, supply chain coordination, price sensitive coefficient, repeated game, chaos

Title of the Paper: **Existence and Nonexistence of Global Solution for a Reaction-Diffusion Equation with Exponential Nonlinearity **

Authors: **Hongwei Zhang, Donghao Li, Qingying Hu **

Abstract: In this work we consider the existence and decay estimate and nonexistence of global solution of reaction-diffusion equation with nonlinear exponential growth reaction terms.

Keywords: reaction-diffusion equation, stable set, exponential reaction term, existence, global nonexistence