The Center and the Fine Focus for a Class of Quartic Polynomial Poincare Equations

We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fine focus at the origin for this class of equations is at most 6. Key words quartic polynomial Poincare equation - center - fine focus - order CLC number O 175. 12 Foundation item: Supported by the National Natural Science Foundation of China (19531070)Biography: TIAN De-sheng (1966-), male, Ph. D candidate, research direction: qualitative theory of differential equation.

HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

The Existence of Periodic Solutions of a Class of <i>n</i>-Degree Polynomial Differential Equations*

This paper deals with a class of n-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (n is an odd number) or two (n is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.

Solve the Polynomial Functions Conditional Extreme by Applying the Groebner Basis Method

In this paper, an algebraic method which is based on the groebner bases theory is proposed to solve the polynomial functions conditional extreme. Firstly, we describe how to solve conditional extreme value problems by establishing Lagrange functions and calculating the differential equations derived from the Lagrange functions. Then, by solving the single variable polynomials in the groebner basis, the solution of polynomial equations could be derived successively. We overcome the high number of variables and constraints in the extreme value problem. Finally, this paper illustrates the calculation process of this method through the general procedures and examples in solving questions of conditional extremum of polynomial function.

A Fast Method to Compute the Inertia of Bezout Matrix and Its Application

In this paper, we present a fast and fraction free procedure for computing the inertia of Bezout matrix and we can determine the numbers of different real roots and different pairs of conjugate complex roots of a polynomial equation with integer coefficients quickly based on this result.

Solving sparse non-negative tensor equations： algorithms and applications

We study iterative methods for solving a set of sparse non-negative tensor equations （multivariate polynomial systems） arising from data mining applications such as information retrieval by query search and community discovery in multi-dimensional networks. By making use of sparse and non-negative tensor structure, we develop Jacobi and Gauss-Seidel methods for solving tensor equations. The multiplication of tensors with vectors are required at each iteration of these iterative methods, the cost per iteration depends on the number of non-zeros in the sparse tensors. We show linear convergence of the Jacobi and Gauss-Seidel methods under suitable conditions, and therefore, the set of sparse non-negative tensor equations can be solved very efficiently. Experimental results on information retrieval by query search and community discovery in multi-dimensional networks are presented to illustrate the application of tensor equations and the effectiveness of the proposed methods.

Solution of Spin and Pseudo-Spin Symmetric Dirac Equation in (1＋1) Space-Time Using Tridiagonal Representation Approach

The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus(and minus) the vector potential.We also include pseudo-scalar potentials in the interaction.The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis,which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric.This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction.We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.