# WSEAS Transactions on Mathematics

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

**Volume 14, 2015**

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

**Volume 14, 2015**

Title of the Paper: **The Learning Rate of Vector-Valued Ranking with Least Square Loss **

Authors: **Liu Huanxiang, Sheng Baohuai, Ye Peixin **

Abstract: In the present paper, we give an investigation on the quantitative convergence analysis of the kernel regularized vector ranking with least square loss. We present with Gȃteaux derivative the qualitative relation between the solution and the hiding distribution and quantitatively show the robustness for the solution. Finally, we provide a learning rate in terms of the approximation ability and capacity of the involved vector-valued RKHS.

Keywords: Vector ranking, least square loss, Gȃteaux derivative, vector-valued RKHS

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #35, pp. 364-375

Title of the Paper: **Stochastic Analysis of Stock Market Price Models: A Case Study of the Nigerian Stock Exchange (NSE) **

Authors: **M. E. Adeosun, S. O. Edeki, O. O. Ugbebor **

Abstract: In this paper, stochastic analysis of the behaviour of stock prices is considered using a proposed log-normal distribution model. To test this model, stock prices for a period of 19 years were taken from the Nigerian Stock Exchange (NSE) for simulation, and the results reveal that the proposed model is efficient for the prediction of stock prices. Better accuracy of results via this model can be improved upon when the drift and the volatility parameters are structured as stochastic functions of time instead of constants parameters.

Keywords: Stochastic model, lognormal distribution, random walk, option pricing, stock exchange market

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #34, pp. 353-363

Title of the Paper: **Integro-Differential Splines of Two Variables **

Authors: **I. G. Burova, S. V. Poluyanov, Iu. V. Shirokova **

Abstract: Here we construct the basic splines of two variables which can be used for approximation functions. The approximation can be constructed in every elementary rectangular separately if the values of the function in nodes and the values of the integrals over elementary rectangles are known. The purpose of the article is to describe representation of surfaces using the local basic splines of two variables. We discuss the construction of surfaces with given accuracy. As a result we present examples and suggest directions for further investigations.

Keywords: Polynomial splines, Exponential splines, Integro-Differential Splines, Interpolation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #33, pp. 345-352

Title of the Paper: **Independence Tests Based on the Conditional Expectation **

Authors: **Sergio Ortobelli, Tommaso Lando **

Abstract: In this paper we propose a new procedure for testing independence of random variables, which is based on the conditional expectation. As it is well known, the behaviour of the conditional expectation may determine a necessary condition for stochastic independence, that is, the so called mean independence. We provide a necessary and sufficient condition for independence in terms of conditional expectation and propose an alternative method to test independence based on this result. Consequently, we provide general class of tests. Observe that generally some non-parametric methods are needed to approximate the conditional expectation, since its exact expression (given the joint distribution) is usually unknown, except for few trivial cases (e.g. Gaussian): we generalize this well known result to the family of elliptical distributions. In order to obtain a sufficiently accurate approximation of the conditional expectation, we propose to use the kernel method or, alternatively, the recently introduced OLP method.

Keywords: Independence test, conditional expectation, Kernel, Non Parametric test

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #32, pp. 335-344

Title of the Paper: **Efficient Class Matrix Congruential Generator Easily Implemented from Multiple Recursive Generators **

Authors: **Gwei-Hung Tsai, Der-Jin Chen, Chiou-Hua Lin **

Abstract: Classical random number generators such as Linear Congruential Generators (LCGs) and Multiple Recursive Generators (MRGs) are commonly employed for modern simulation studies. LCGs have a too short period and a too coarse structure to be used as reliable random number generators. Most Matrix Congruential Generators (MCGs) are less applicable due to the inefficiency and the difficulty of deciding the too many coefficients. We propose a class of Efficient Matrix Congruent Generators (EMCGs) including EMCG-1, EMCG-2, EMCG-D and EMCG-G that simultaneously combine k LCGs to form the multiplier matrix of MCGs. We can easily derive the multiple-matrices for all four EMCGs from the corresponding full fixed minimal primitive characteristic polynomials. This makes them easy to implement the results from MRGs without any restrictions on the dimension of a multiple-matrix. We also provide the portable and efficient MAPLE algorithms without using any matrix multiplications for generating the four EMCGs with some selected coefficients that can achieve the high-dimensional uniformity with extreme long cycles.

Keywords: Linear Congruential Generators, Multiple Recursive Generators, Matrix Congruential Generators, Fast Matrix Congruential Generators

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #31, pp. 324-334

Title of the Paper: **Application of Mixtures of von Mises-Fisher Model to Investigate the Statistical Characteristics of the Wind Direction Data **

Authors: **Nurulkamal Masseran, Ahmad Mahir Razali, Kamarulzaman Ibrahim **

Abstract: Wind direction has a substantial effect on the environment and human lives. Wind direction influences the dispersion of particulate matter in the air and affects the construction of engineering structures, such as towers, bridges, and tall buildings. In fact, knowledge of the wind direction and wind speed can be used to obtain information about the energy potential. This study investigates the characteristics of the wind regime involving the wind direction in Kudat, Malaysia using a mixture of von Mises-Fisher model (mvMF). The suitability of each mvMF was judged based on a graphical representation and goodness-of-fit statistics. In addition, the best-fit mvMF model was compared with the circular distribution based on nonnegative trigonometric sums to determine the best model. The results found that the mvMF model with components is the best model. Additionally, the circular density plots of the suitable model clearly show the dominant wind directions in the Kudat region.

Keywords: Circular distribution based on nonnegative trigonometric sums, directional statistics; directional distribution, mixture of von Mises-Fisher distributions, wind direction modelling

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #30, pp. 313-323

Title of the Paper: **k-Segments Classifier - A Non-Linear Approach for the Classification of Sampling Data **

Authors: **Zaudir Dal Cortivo, Jair Mendes Marques **

Abstract: This paper proposes a method to classify sampling data based on the k-segments algorithm. The data classification efficiency is relevant for this multivariate statistical analysis technique. The method consists of adjusting an a priori defined polygonal line for each class. A new observation is then classified into the class for which polygonal line it has the smallest orthogonal distance. Experimentally, the algorithm is applied to several sets of sampling data and the results are compared with the apparent error rate, which demonstrates the good performance of this methodology.

Keywords: Discriminant analysis, k-segments algorithm, principal curves

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #29, pp. 301-312

Title of the Paper: **The Least Norm Solution of the Linear System via a Quadratic Penalty Function **

Authors: **Suleyman Safak **

Abstract: In this paper, the linear system of equations in unknowns is formulated as a quadratic programming problem, and the least norm solution for the consistent or inconsistent system of the linear equations is investigated using the optimality conditions of the quadratic penalty function (QPF). In addition, several algebraic characterizations of the equivalent cases of the QPFs are given using the orthogonal decompositions and the generalized inverses of the coefficient matrices obtained from optimality conditions. It is seen that the least norm solution of the consistent or inconsistent system of the linear equations can be found with the penalty method. In addition, it is shown that the method can be applied to all systems of the linear equations to find the least norm solution. Numerical examples are presented by using the analytic results that were obtained.

Keywords: Least norm solution, Linear systems, Quadratic programming, Penalty function

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #28, pp. 292-300

Title of the Paper: **On Some Properties of Certain Subclasses of Analytic Functions Defined by Using the Subordination Principle **

Authors: **Rabha El-Ashwah, Alaa Hassan **

Abstract: In this paper, we introduce some new subclasses of analytic functions related to starlike, convex, close-to-convex and quasi-convex functions defined by using a generalized operator and the differential subordination principle. Inclusion relationships for these subclasses are established. Moreover, we introduce some integral-preserving properties.

Keywords: Starlike function, Convex function, Close-to-convex function, Quasi-convex function, Subordination principle

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #27, pp. 281-291

Title of the Paper: **Time-Dependent Axial Velocity and Temperature-Dependent Viscosity Effects on MHD Free Convection Flow and Heat Transfer of a Dissipative Fluid over a Vertical Moving Cylinder **

Authors: **Rasool Alizadeh, Komeil Rahmdel **

Abstract: Numerical solutions are obtained for the Temperature-Dependent Viscosity Effects on MHD free convection flow of a dissipative fluid over a vertical moving cylinder with time-dependent axial velocity,the surface of which is exposed to a constant wall temperatures. Numerical solutions of the Navier–Stokes equations, concentration equation and energy equation are derived in this problem. A reduction of these equations is obtained by use of appropriate transformations. The semi-similar solution of the Navier-Stokes equations, concentration equation and energy equation has been obtained numerically using by the implicit finite difference scheme of Crank–Nicolson’s type. The velocity, concentration and temperature profiles for different values of cylinder axial velocity are plotted.The influence of the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscosity-variation parameter and magnetic parameter for different values of cylinder axial velocity on free convection flow and heat transfer are discussed. It is observed that, when Prandtl number increases the velocity and temperature decrease in the boundary layer. Also, it is found that as increase in the magnetic parameter leads to decrease in the velocity feld and rise in the thermal boundary thickness.

Keywords: Time-Dependent Axial Velocity, Moving Cylinder, Temperature-Dependent Viscosity, Dissipative Fluid, MHD Free Convection

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #26, pp. 271-280

Title of the Paper: **On a Two-Variable Functional Equation Arising from Databases **

Authors: **El-Sayed El-Hady, Wolfgang Förg-Rob, Mohamed Mahmoud **

Abstract: Functional equations offer a tool for narrowing the models used to describe many phenomena. Recently, a certain class of functional equations stems when obtaining the generating functions of queueing systems distributions. This paper has been motivated by an issue considered by L. Flatto and S. Hahn in [SIAM Journal on Applied Mathematics 44(5), (1984), 1041–1053]. The functional equation obtained there has been converted into a set of conditions on the two-unknowns, which in turn lead to the determination of the main unknown. Another motivation of solving such functional equation comes from the fact that the underlying queueing system has applications in the inventory control of database systems. Unfortunately that solution seems to be a bit too general with many technical assumptions. In this paper we introduce a solution using boundary value problem approach. Our solution is validated, obtained by assuming full symmetry on the system under study, and by the reduction to a Riemann-Hilbert boundary value problem.

Keywords: Queueing systems, Functional equations, Boundary value problems, Inventory control of database systems

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #25, pp. 265-270

Title of the Paper: **Post-Newtonian Equations of Motion for Inertial Guided Space APT Systems**

Authors: **Jose M. Gambi, Maria L. Garcia Del Pino**

Abstract: The equations of motion derived in this paper are aimed to determine according to the post-Newtonian framework the relative motion of Earth satellites with respect to inertial guided space Acquisition, Pointing and Tracking systems. The equations are suitable for satellites that are even far from the systems. The tool used to derive them is Synge’s world-function for the Earth surrounding space. Hence the equations are written in terms of the quasi-Cartesian coordinates tied to the systems.

Keywords: Earth satellites, Intersatellite laser links, Acquisition, Pointing and Tracking systems, Earth post-Newtonian framework

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #24, pp. 256-264

Title of the Paper: **Modal Paraconsistent Tableau Systems of Logic**

Authors: **Tomasz Jarmuzek, Marcin Tkaczyk**

Abstract: The aim of this paper is to show how to simply define paraconsistent tableaus by liberalization of construction of complete tableaus. The presented notions allow us to list all tableau inconsistencies that appear in a complete tableau. Then we can easily choose these inconsistencies that are effects of interactions between premises and a conclusion, simultaneously excluding other inconsistencies. A general technique we describe is presented here for the case of modal logic, as one of the most interesting, and with many applications in communication theory and data mining. In case we have an inconsistent set of information and apply a logic that includes classically accepted patterns of reasoning, we may correctly conclude any arbitrary proposition. To avoid this we propose paraconsistent tableau proofs that list all relevant premises which guarantee truth of conclusion under consideration, but at the same time are consistent.

Keywords: blind rule, modal logic, paraconsistent tableaus, t-inconsistency, tableau rules, tableaus Typing manuscripts, LATEX

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #23, pp. 248-255

Title of the Paper: **On the Approximation of a Conditional Expectation**

Authors: **Tommaso Lando, Sergio Ortobelli**

Abstract: In this paper, we discuss how to approximate the conditional expectation of a random variable Y given a random variable X, i.e. E(Y|X). We propose and compare two different non parametric methodologies to approximate E(Y|X). The first approach (namely the OLP method) is based on a suitable approximation of the σ-algebra generated by X. A second procedure is based on the well known kernel non-parametric regression method. We analyze the convergence properties of the OLP estimator and we compare the two approaches with a simulation study.

Keywords: Conditional Expectation, Kernel, Non Parametric methods, Regression, Approximation, Simulation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #22, pp. 237-247

Title of the Paper: **On Adaptive Processing of Discrete Flow**

Authors: **I. G. Burova, Yu. K. Dem’yanovich**

Abstract: An algorithm of approximation for discrete initial numerical flow f by the first order splines, associated with the adaptive grid, is proposed. Quantitative evaluations for volumes of using data are obtained under different characteristics of regularity of initial flow. Limit characteristics of mentioned volumes are represented. Difference representations for deviation of data flow from approximate flow are done. There are discussions for pseudo-equidistant grid, grid of adaptive type and comparison of their effectiveness under condition of the same approximation. The conditions, in which the algorithm is more affective than the analogous algorithm with equidistant grid, are done.

Keywords: plines, Interpolation, Approximation, Numerical flow, Adaptive grid, Limit characteristics, Data Volume

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #21, pp. 226-236

Title of the Paper: **Optimal Time-Consistent Investment in the Dual Risk Model with Diffusion **

Authors: **Lidong Zhang, Ximin Rong, Ziping Du **

Abstract: The objective of this paper is to investigate optimal investment strategy for a pharmaceutical or petroleum company under mean-variance criterion. The surplus of the company is modeled as a dual risk model. We assume that the company can invest into a risk-free asset and n risky assets. Short-selling and borrowing money are allowed. Since this problem is time-inconsistent, we study it within a game theoretical framework. The subgame perfect Nash equilibrium strategies (namely time-consistent strategies) are derived by solving the Extended Hamilton-Jacobi-Bellman(HJB) equations in the classic diffusion dual model and in the diffusion dual approximation model, separately. Surprisingly when a coefficient parameter ρ = 0, optimal time-consistent investment strategies and the value functions have the same expressions in both cases. Finally, We present economics implications and provide sensitivity analysis for our results by numerical examples.

Keywords: Optimal time-consistent investment, Extended Hamilton-Jacobi-Bellman equation, Mean-variance criterion

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #20, pp. 213-225

Title of the Paper: **Index Pricing Model Embedding Investor Sentiment: Based on Path Converged Design **

Authors: **Yanyun Yao, Bing Xu **

Abstract: This paper proposes an index pricing method to model the risk premium of market, industry and area for individual stocks. The individual stocks of Shanghai Index 50 in China’s stock market are selected. Path converged design is employed to construct three time-varying coefficient semi-parameter regression models for each stock. The three models are base model excluding investor sentiment, path model I including investor sentiment of each individual stock and path model II including investor sentiments of indexes. As the empirical result, the path models explain the fluctuation of individual stocks better than the base model, which means that investor sentiment is an important factor for asset pricing. Furthermore, pricing errors analysis is carried out. The result shows that index pricing model plays an important role in the determination of individual stock’s rational prices and it may contribute to trading strategy construction for programming trading. The paper is an effective modeling trial of asset pricing in the nonparametric frame.

Keywords: Investor sentiment, Index pricing model, Path converged design, Pricing error analysis, Nonparametric estimation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #19, pp. 202-212

Title of the Paper: **Exponential Stabilization of 1-dWave Equation with Distributed Disturbance **

Authors: **Qi Hong Fu, Gen Qi Xu **

Abstract: In this paper, we consider the stabilization problem of a one-dimensional wave equation with unknown disturbance. In order to stabilize the system with disturbance, we design a distributed feedback controller by employing the idea of sliding mode control technology. For the resulted nonlinear closed-loop system, we prove its solvability by using the maximal monotone operator. Further we prove the exponentially stable of the closed-loop system.

Keywords: 1-d wave equation, distributed control, distributed disturbance, exponential stabilization

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #18, pp. 192-201

Title of the Paper: **Mathematical Modelling and Stability Analysis for Diabetes Predicting System **

Authors: **Xiuling Cao, Danhong Liu, Suzhi Li, Yu Wang **

Abstract: In this paper, a model consisting of four states of diabetics is proposed for predicting the development of diabetes. Suppose that all diabetics in the model are with type 2 diabetes and that there is no input or output. While the states of diabetics can alter between each other. Under these assumptions, a model of partial differential equations is derived. The general well-posedness result of the system is obtained and the exponential stability of dynamic solution converging to the steady-state solution is proved. Herein, the steady-state solution gives the stable distribution probability of diabetics (the predicted result). Furthermore, some reliability indices are discussed. The results of some numerical simulations for the cost problem are illustrated to conclude that, if diabetics want to live a higher standard and cost less, they need to follow doctor’s advice, learn more knowledge about diabetes, pay attention to their diet and so on which can make the recovery rate larger.

Keywords: Diabetes, Modelling, Dynamic solution, Stability, Predicting

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #17, pp. 178-191

Title of the Paper: **Multiplier Algorithm Based on a New Augmented Lagrangian Function **

Authors: **Xunzhi Zhu, Jingyong Tang, Jian Wang, Shujiang Ding **

Abstract: In this paper, for nonconvex optimization problem with both equality and inequality constrains, we introduce a new augmented Lagrangian function and propose the corresponding multiplier algorithm. The globalconvergence is established without requiring the boundedness of multiplier sequences. In particular, if the algorithm terminates in finite steps, then we obtain a KKT point of the primal problem; otherwise the iterative sequence{x^k} generalized by algorithm converges to optimal solutions. Even if {x^k} is divergent we also present a necessary and sufficient condition for the convergence of {f(x^k)} to the optimal value. Finally, preliminary numerical experience is reported.

Keywords: Constrained global optimization, multiplier algorithm, global convergence

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #16, pp. 169-177

Title of the Paper: **On the Validity of Thompson’s Conjecture for Alternating Groups Ap+4 of degree p + 4 **

Authors: **Yong Yang, Shitian Liu **

Abstract: Let G be a group. Let π(G) be the set of prime divisor of |G|. Let GK(G) denote the graph with vertex set π(G) such that two primes p and q in π(G) are joined by an edge if G has an element of order p · q. We use s(G) to denote the number of connected components of the prime graph GK(G). Let N(G) be the set of nonidentity orders of conjugacy classes of elements in G. Some authors have proved that the groups An where n = p, p + 1, p + 2 with s(G) ≥ 2, are characterized by N(G). Then if s(G) = 1, we know that Liu and Yang proved that alternating groups Ap+3 are characterized by N(G). As the development of this topics, we will prove that if G is a finite group with trivial center and N(G) = N(Ap+4) with p + i composite and 1 ≤ i ≤ 4, then G is isomorphic to Ap+4.

Keywords: Element order, Alternating group, Thompson’s conjecture, Conjugacy classes, Simple group

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #15, pp. 159-168

Title of the Paper: **The Solvability of Finite Groups with Four Conjugacy Class Sizes **

Authors: **Ping Kang **

Abstract: Let G be a finite group and let G* be the set of elements of primary and biprimary orders of G. We show that when the conjugacy class sizes of G* are {1, m, n, mn}, then G is solvable.

Keywords: Conjugacy class sizes, Finite groups, Nilpotent groups, Solvable groups, Sylow p-subgroup

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #14, pp. 150-158

Title of the Paper: **Measure of Bullwhip Effect Considering Stochastic Disturbance Based on Price Fluctuations in a Supply Chain with Two Retailers based on price fluctuations in a supply chain with two retailers **

Authors: **Junhai Ma, Jing Zhang **

Abstract: This paper establishes a new price-sensitive demand model which considers stochastic disturbance similar to ARMA(1,1) model. We examine the impact of two forecasting methods on the bullwhip effect in a two-stage supply chain with two retailers. It is assumed that two retailers face the same demand model and an order-up-to inventory policy is employed. The lead-time demand is forecasted respectively by the moving average (MA) and exponential smoothing(ES) methods. The effect of various parameters is investigated by numerical simulation and the bullwhip effect under two forecasting methods is compared. The results show that the MA forecasting method is better than the ES method based on our demand process. Besides, conclusions indicate that both the extent of consumers concerning about the historical price volatility and the lead time play significant roles on reducing the bullwhip effect, and stochastic disturbance impacts the bullwhip effect differently based on the lead time. The larger the variance of stochastic disturbance of the retailer which has a longer lead time, the greater the bullwhip effect in the supply chain. The moving average coefficient of stochastic disturbance generally has a little different impact on the bullwhip effect under the different relationship of the lead time between two retailers. Moreover, some proposals are present to help managers to take appropriate measures and select the forecasting method that yields the lowest bullwhip effect.

Keywords: Supply chain, Bullwhip effect, Demand forecasting, Price fluctuations, Stochastic disturbance

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #13, pp. 127-149

Title of the Paper: **The Solution of a Parabolic Equation with Unbounded Flux Term **

Authors: **Huashui Zhan, Zheng Zhou, Xin Si **

Abstract:

Keywords: Cauchy problem, degenerate parabolic equation, existence, unbounded flux term

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #12, pp. 116-126

Title of the Paper: **Strong and Weak Augmentability in Calculus of Variations **

Authors: **Javier F. Rosenblueth **

Abstract: In this paper we derive sufficient conditions for local optimality, for the Lagrange problem in the calculus of variations involving mixed equality constraints, by means of the notion of augmented Lagrangians. It is well-known that the standard necessary conditions for that problem can be easily obtained under the assumption of augmentability, instead of the usual one of normality. On the other hand, as we show in this paper, the standard sufficient conditions for a strong (weak) minimum imply strong (weak) augmentability. Since this kind of augmentability implies that the extremal under consideration is a local solution, the results given provide an alternative approach to the classical theory of sufficient conditions.

Keywords: Augmentability, Lagrange problem, calculus of variations, equality constraints, sufficient conditions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #11, pp. 106-115

Title of the Paper: **The Influence of Conjugacy Class Sizes on the Structure of Finite Groups **

Authors: **Qingjun Kong **

Abstract: Let G be a finite p-solvable group and let G* be the set of p′-elements of primary and biprimary orders of G. We show that when the conjugacy class sizes of G* are {1, m,p^a,mp^a} with (m,p) = 1, then the p-complements of G are nilpotent and m is a prime power.

Keywords: Conjugacy class sizes, Nilpotent groups, Solvable groups, Sylow p-subgroup, Finite groups

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #10, pp. 97-105

Title of the Paper: **Existence and Nonexistence of Positive Solutions for a Singular Higher-Order Nonlinear Fractional Differential Equation **

Authors: **Kaihong Zhao, Ping Gong **

Abstract: In this paper, we establish some sufficient conditions for the existence and nonexstence of positive solutions to a general class of higher-order nonlinear fractional differential equation. The results are established by converting the problem into an equivalent integral equation and applying Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type. As applications, some examples are also provided to illustrate our main results.

Keywords: Fractional differential equations, Positive solutions, Boundary value problems, Fixed point theorem

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #9, pp. 87-96

Title of the Paper: **The Construction of A^3-Code from Singular Pseudo-Symplectic Geometry Over Finite Fields **

Authors: **Liu Yanqin, Gao You, Zhang Dake **

Abstract: A construction of A^3-code from singular pseudo-symplectic geometry over finite fields is presented. Under the assumption that the encoding rules of the transmitter, the receiver and the arbiter are chosen according to a uniform probability distribution, the parameters and the probabilities of success for different types of deceptions are computed.

Keywords: A^3-codes, Singular Pseudo-symplectic Geometry , Finite fields

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #8, pp. 77-86

Title of the Paper: **Feature Combination Methods Based on Graphs **

Authors: **Zhenfeng Zhu, Liya Fan **

Abstract: In pattern recognition, graph-based feature combination techniques attract many researchers to study this field. In this paper, we construct a unified framework based on graphs (GF), and derive that FDA, PCA, LPP, DLPP, MFA and MMC are special cases of GF, and then three new algorithms are proposed for GF, which are regularized GF (RGF), GF based on null space (NGF) and GF based on singular value decomposition (GF/SVD). Experiments are made on AVIRIS remote sensing image to illustrate the efficient and effective of our algorithms. The results show that the effects of proposed algorithms are very significantly.

Keywords: Framework based on graphs (GF), regularized GF, GF based on null space, GF based on singular value decomposition

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #7, pp. 67-76

Title of the Paper: **Well-Posedness of l-Set Optimization Problem Under Variable Order Structure **

Authors: **Dening Qu, Caozong Cheng **

Abstract: The concepts of well-posedness of l-set optimization problem under variable order structure are introduced, the metric characterizations and sufficient criteria of well-posedness of a l-set optimization problem are proposed, and the equivalent relations between the well-posedness of l-set optimization problem and that of a scalarization minimization problem are established. Finally, by discussing the lower semi-continuity and convexity of gap functions of l-set convex optimization problems, their well-posedness of are investigated.

Keywords: Well-posedness, l-set optimization, gap function, variable order structure

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #6, pp. 57-66

Title of the Paper: **Periodic Solutions for Three-Species Diffusive Systems with Beddington-Deangelis and Holling-Type Iii Schemes **

Authors: **Qiang-Jun Xie, Ze-Rong He, Zhe-Yong Qiu, Shu-Ping Wang **

Abstract: This paper is concerned with the three-species diffusive systems in a periodic environment, which arises in a one-prey and two-competing-predator population model with Beddington-Deangelis and Holling-type III schemes. By using eigenvalue analysis, bifurcation theories and Schauder estimates, the existence of positive periodic solutions of the single prey species system, two-species predator-prey systems with different functional responses and three-species periodic diffusive systems are investigated. The necessary and sufficient conditions are described by the principal eigenvalue of the periodic parabolic operators. Furthermore, the alternative sufficient conditions characterised by the integral form of the parameters of the systems are more convenient to the biological explanation.

Keywords: Positive periodic solutions, principal eigenvalue, Schauder estimates, global bifurcation, decoupling

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #5, pp. 47-56

Title of the Paper: **Probability Density Function of the Time to Kill for Heterogeneous Target **

Authors: **Jianjun Li, Liwei Liu, Tao Jiang **

Abstract: In this paper we propose the concepts about homogeneous and heterogeneous target. The homogeneous target is defined as one that hitting the different position of a target causes the same effect, whereas the heterogeneous target is allowed to differ. The large body of work on stochastic duels studied the homogeneous target. But, in many cases, it is unrealistic to assume that a target is homogeneous. We consider the heterogeneous target in this study, and obtain the probability density function of the time to kill. The general solution is obtained in quadrature form and specific solutions are derived for some particular cases. It is illustrated by an example with particular heterogeneous target.

Keywords: Stochastic duel, Heterogeneous target, Probability density function, Renewal equation, Military operations research, Applied probability

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #4, pp. 36-46

Title of the Paper: **Optimal Investment Problem for an Insurer and a Reinsurer Under the Proportional Reinsurance Model **

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

Abstract: This paper focuses on the optimal investment problem for an insurer and a reinsurer. The insurer’s and reinsurer’s surplus processes are both approximated by a Brownian motion with drift and the insurer can purchase proportional reinsurance from the reinsurer. In addition, both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset. We first study the optimization problem of minimizing the ruin probability for the insurer. Then according to the optimal reinsurance proportion chosen by the insurer, we study two optimal investment problems for the reinsurer: the problem of maximizing the exponential utility and the problem of minimizing the ruin probability. By solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations, wederive optimal strategies for both the insurer and the reinsurer explicitly. Furthermore, we find that the reinsurer’s optimal strategies under the two cases are equivalent for some special parameters. Finally, numerical simulations are presented to illustrate the effects of model parameters on the optimal strategies.

Keywords: Proportional reinsurance, Optimal investment, For a reinsurer, Hamilton-Jacobi-Bellman (HJB) equation, Exponential utility maximization, Ruin probability minimization

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #3, pp. 20-35

Title of the Paper: **Positive Periodic Solutions in Shifts δ± for a Class of Dynamic Equations with Feedback Control on Time Scales **

Authors: **Meng Hu, Pingli Xie **

Abstract:

Keywords: Periodic solution, Functional dynamic equation, feedback control, Shift operator, Time scale

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #2, pp. 10-19

Title of the Paper: **The Linear K-Arboricity of the Mycielski Graph M(Kn) **

Authors: **Shengjie He, Liancui Zuo **

Abstract: A linear k-forest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear k-arboricity Of G, denoted by lak(G), is the minimum number of linear k-forests needed to partition the edge set E(G) of G. In case that the lengths of paths are not restricted, we then have the linear arboricity of G, denoted by la(G). In this paper, the exact values of the linear 3-arboricity and the linear arboricity of the Mycielski graph M(Kn), and the linear k-arboricity of the Mycielski graph M(Kn) when n is even and k ≥ 5, are obtained.

Keywords: Linear k-forest, linear k-arboricity, Mycielski graph, bipartite difference

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 14, 2015, Art. #1, pp. 1-9