In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.
Using the second Green formula, the boundary problem of Laplace equation satisfied by potential function of static electric field is transformed to the problem of the boundary integral equation, and then a boundary in...Using the second Green formula, the boundary problem of Laplace equation satisfied by potential function of static electric field is transformed to the problem of the boundary integral equation, and then a boundary integral equation approach is established by partitioning boundary using linear boundary element.展开更多
The prolongation structure technique of Wahlquist and Estanbrook is improved and applied to a newequation proposed by Z.J.Qiao [J.Math.Phys.48 (2007) 082701].Two potentials and two pseudopotentials areobtained,from wh...The prolongation structure technique of Wahlquist and Estanbrook is improved and applied to a newequation proposed by Z.J.Qiao [J.Math.Phys.48 (2007) 082701].Two potentials and two pseudopotentials areobtained,from which a new type of inverse scattering problem,Lax equations,and infinite number of conservation lawsare obtained.展开更多
By introducing an image plane, the inverse heat conduction problem with free boundary is transformed into one with completely known boundaryt which is much simpler to handle.As a by-product, the classical Kirchhoff’s...By introducing an image plane, the inverse heat conduction problem with free boundary is transformed into one with completely known boundaryt which is much simpler to handle.As a by-product, the classical Kirchhoff’s transformation for accounting for variable conductivity is rederived and an invariance property of the inverse problem solution with respect to variable conductivity is indicated. Then a pair of complementary extremum principles are established on the image plane, providing a sound theoretical foundation for the Ritz’s method and finite element method (FEM).An example solved by FEM is also given.展开更多
Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9,...Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9, or b=c=7, then R b,c (n)》n 5/6+1/b+1/c for all sufficiently large n.展开更多
It is well-known that many Krylov solvers for linear systems,eigenvalue problems,andsingular value decomposition problems have very simple and elegant formulas for residual norms.Theseformulas not only allow us to fur...It is well-known that many Krylov solvers for linear systems,eigenvalue problems,andsingular value decomposition problems have very simple and elegant formulas for residual norms.Theseformulas not only allow us to further understand the methods theoretically but also can be usedas cheap stopping criteria without forming approximate solutions and residuals at each step beforeconvergence takes place.LSQR for large sparse linear least squares problems is based on the Lanczosbidiagonalization process and is a Krylov solver.However,there has not yet been an analogouslyelegant formula for residual norms.This paper derives such kind of formula.In addition,the authorgets some other properties of LSQR and its mathematically equivalent CGLS.展开更多
文摘In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.
文摘Using the second Green formula, the boundary problem of Laplace equation satisfied by potential function of static electric field is transformed to the problem of the boundary integral equation, and then a boundary integral equation approach is established by partitioning boundary using linear boundary element.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 11075055, 61021004, 10735030the Shanghai Leading Academic Discipline Project, China under Grant No. B412the Program for Changjiang Scholars and the Innovative Research Team in University of Ministry of Education of China under Grant No. IRT 0734
文摘The prolongation structure technique of Wahlquist and Estanbrook is improved and applied to a newequation proposed by Z.J.Qiao [J.Math.Phys.48 (2007) 082701].Two potentials and two pseudopotentials areobtained,from which a new type of inverse scattering problem,Lax equations,and infinite number of conservation lawsare obtained.
文摘By introducing an image plane, the inverse heat conduction problem with free boundary is transformed into one with completely known boundaryt which is much simpler to handle.As a by-product, the classical Kirchhoff’s transformation for accounting for variable conductivity is rederived and an invariance property of the inverse problem solution with respect to variable conductivity is indicated. Then a pair of complementary extremum principles are established on the image plane, providing a sound theoretical foundation for the Ritz’s method and finite element method (FEM).An example solved by FEM is also given.
文摘Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9, or b=c=7, then R b,c (n)》n 5/6+1/b+1/c for all sufficiently large n.
基金supported in part by the National Science Foundation of China under Grant No. 10771116the Doctoral Program of the Ministry of Education under Grant No. 20060003003
文摘It is well-known that many Krylov solvers for linear systems,eigenvalue problems,andsingular value decomposition problems have very simple and elegant formulas for residual norms.Theseformulas not only allow us to further understand the methods theoretically but also can be usedas cheap stopping criteria without forming approximate solutions and residuals at each step beforeconvergence takes place.LSQR for large sparse linear least squares problems is based on the Lanczosbidiagonalization process and is a Krylov solver.However,there has not yet been an analogouslyelegant formula for residual norms.This paper derives such kind of formula.In addition,the authorgets some other properties of LSQR and its mathematically equivalent CGLS.
