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 ...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.展开更多
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 Hermit...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.展开更多
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) peri...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.展开更多
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...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.展开更多
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 e...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.展开更多
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 pol...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.展开更多
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 communit...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.展开更多
Calculation of critical depth in open channels or closed conduits is a prerequisite for efficient hydraulic design,operation,and maintenance of irrigation channels and drainage ditches.Determination of critical depth ...Calculation of critical depth in open channels or closed conduits is a prerequisite for efficient hydraulic design,operation,and maintenance of irrigation channels and drainage ditches.Determination of critical depth in the trapezoidal cross section is of particular significance as it is one of the most widely used channel sections throughout the world,while no closedform analytical solutions exist.Based on the novel combined iteration-curve-fitting method,the existing equations were unified in the same function model,and two new equations were proposed for directly calculating critical depth in trapezoidal open channels.The maximum absolute relative errors of the two proposed equations are 0.00494%and 0.165%,respectively,in wide application ranges.Comparison and evaluation of the proposed and existing equations for calculating critical depth in trapezoidal open channels were also presented.The introduction and application of the novel method could make the process of function model establishment much more efficient,which provides more insights into the hydraulic calculations of channels and ditches.Moreover,this paper provides reference for the problems related to the empirical equations of high-degree polynomial equations.展开更多
基金Supported by the Nationat Natural Science Foun-dation of China(19531070)
文摘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.
文摘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.
文摘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.
文摘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.
基金King Fahd University of Petroleum and Minerals (KFUPM) for their support under research grant RG1502the material support and encouragements of the Saudi Center for Theoretical Physics (SCTP)
文摘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.
文摘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.
文摘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.
基金supported by Shanghai Agricultural Science and Technology Innovation Program,China(Industrial Upgrading Program)(Grant No.I2023004 and Grant No.I2023008).
文摘Calculation of critical depth in open channels or closed conduits is a prerequisite for efficient hydraulic design,operation,and maintenance of irrigation channels and drainage ditches.Determination of critical depth in the trapezoidal cross section is of particular significance as it is one of the most widely used channel sections throughout the world,while no closedform analytical solutions exist.Based on the novel combined iteration-curve-fitting method,the existing equations were unified in the same function model,and two new equations were proposed for directly calculating critical depth in trapezoidal open channels.The maximum absolute relative errors of the two proposed equations are 0.00494%and 0.165%,respectively,in wide application ranges.Comparison and evaluation of the proposed and existing equations for calculating critical depth in trapezoidal open channels were also presented.The introduction and application of the novel method could make the process of function model establishment much more efficient,which provides more insights into the hydraulic calculations of channels and ditches.Moreover,this paper provides reference for the problems related to the empirical equations of high-degree polynomial equations.
