To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the s...To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.展开更多
Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be a...Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.展开更多
We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation in...We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate.And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well.Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function.The linearly dependent relation between two eigenfunctions is also studied.展开更多
In this paper, we use the representation of the solutions of the focusing nonlinear Schrodinger equation we have constructed recently, in terms of wronskians;when we perform a special passage to the limit, we get quas...In this paper, we use the representation of the solutions of the focusing nonlinear Schrodinger equation we have constructed recently, in terms of wronskians;when we perform a special passage to the limit, we get quasi-rational solutions expressed as a ratio of two determinants. We have already construct breathers of orders N = 4, 5, 6 in preceding works;we give here the breather of order seven.展开更多
The computational problems of two special determinants are investigated. Those determinants appear in the construction of the function-valued Pade-type approximation for computing Fredholm integral equation of the sec...The computational problems of two special determinants are investigated. Those determinants appear in the construction of the function-valued Pade-type approximation for computing Fredholm integral equation of the second kind. The main tool to be used in this paper is the well-known Schur complement theorem.展开更多
Estimations of parametric functions under a system of linear regression equations with correlated errors across equations involve many complicated operations of matrices and their generalized inverses. In the past sev...Estimations of parametric functions under a system of linear regression equations with correlated errors across equations involve many complicated operations of matrices and their generalized inverses. In the past several years, a useful tool -- the matrix rank method was utilized to simplify various complicated operations of matrices and their generalized inverses. In this paper, we use the matrix rank method to derive a variety of new algebraic and statistical properties for the best linear unbiased estimators （BLUEs） of parametric functions under the system. In particular, we give the necessary and sufficient conditions for some equalities, additive and block decompositions of BLUEs of parametric functions under the system to hold.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 10271074)
文摘To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.
基金Project supported by the National Natural Science Foundation of China(Grant No.10271074)
文摘Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.
基金Project supported by the National Natural Science Foundation of China(Grant No.11975196)partially by SIP,Instituto Politecnico Nacional(IPN),Mexico(Grant No.20210414)。
文摘We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate.And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well.Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function.The linearly dependent relation between two eigenfunctions is also studied.
文摘In this paper, we use the representation of the solutions of the focusing nonlinear Schrodinger equation we have constructed recently, in terms of wronskians;when we perform a special passage to the limit, we get quasi-rational solutions expressed as a ratio of two determinants. We have already construct breathers of orders N = 4, 5, 6 in preceding works;we give here the breather of order seven.
基金The work is supported by the National Natural Science Foundation of China （10271074）by the Special Funds for Major Specialities of Shanghai Education Committee.
文摘The computational problems of two special determinants are investigated. Those determinants appear in the construction of the function-valued Pade-type approximation for computing Fredholm integral equation of the second kind. The main tool to be used in this paper is the well-known Schur complement theorem.
基金Supported by National Natural Science Foundation of China (Grant No. 70871073)
文摘Estimations of parametric functions under a system of linear regression equations with correlated errors across equations involve many complicated operations of matrices and their generalized inverses. In the past several years, a useful tool -- the matrix rank method was utilized to simplify various complicated operations of matrices and their generalized inverses. In this paper, we use the matrix rank method to derive a variety of new algebraic and statistical properties for the best linear unbiased estimators （BLUEs） of parametric functions under the system. In particular, we give the necessary and sufficient conditions for some equalities, additive and block decompositions of BLUEs of parametric functions under the system to hold.
