A solution to the reparametrization of Bézier curves by sine transformation of Bemstein basis is presented. The new effective reparametrization method is given through the following procedures： educing Sine Bems...A solution to the reparametrization of Bézier curves by sine transformation of Bemstein basis is presented. The new effective reparametrization method is given through the following procedures： educing Sine Bemstein-Bézier Class-SBBC function, defining SBBC curve and discussing the relation between SBBC and Bézier curve.展开更多
In this paper, we study an operator s which maps every n-by-n symmetric matrix A, to a matrix s(A_n) that minimizes || B_n-A_n || F over the set of all matrices B_n, that can be diagonalized by the sine transform. The...In this paper, we study an operator s which maps every n-by-n symmetric matrix A, to a matrix s(A_n) that minimizes || B_n-A_n || F over the set of all matrices B_n, that can be diagonalized by the sine transform. The matrix s(A_n), called the optimal sine transform preconditioner, is defined for any n-by-n symmetric matrices A_n. The cost of constructing s(A_n) is the same as that of optimal circulant preconditioner c(A_n) which is defined in [8], The s(A_n) has been proved in [6] to be a good preconditioner in solving symmetric Toeplitz systems with the preconditioned conjugate gradient (PCG) method. In this paper, we discuss the algebraic and geometric properties of the operator s, and compute its operator norms in Banach spaces of symmetric matrices. Some numerical tests and an application in image restoration are also given.展开更多
In this paper, a fast algorithm for the discrete sine transform(DST) of a Toeplitz matrix of order N is derived. Only O(N log N) + O(M) time is needed for the computation of M elements. The auxiliary storage requireme...In this paper, a fast algorithm for the discrete sine transform(DST) of a Toeplitz matrix of order N is derived. Only O(N log N) + O(M) time is needed for the computation of M elements. The auxiliary storage requirement is O(N). An application of the new fast algorithm is also discussed.展开更多
Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + ...Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + 1). It is known that if r = a satisfies homogeneous boundary conditions on all boundary lines ?in addition to non-homogeneous ones on the circular boundary , then an explicit expression of in terms of eigen-functions can be found through the classical method of separation of variables. But when the boundary?condition given on the circular boundary r = a is homogeneous, it is not possible to define a discrete set of eigen-functions. In this paper one shows that if the homogeneous condition in question is of the Dirichlet (or Neumann) type, then the logarithmic sine transform (or logarithmic cosine transform) defined by (or ) may be effective in solving the problem. The inverses of these transformations are expressed through the same kernels on or . Some properties of these transforms are also given in four theorems. An illustrative example, connected with the heat transfer in a two-part wedge domain, shows their effectiveness in getting exact solution. In the example in question the lateral boundaries are assumed to be non-conducting, which are expressed through Neumann type boundary conditions. The application of the method gives also the necessary condition for the solvability of the problem (the already known existence condition!). This kind of problems arise in various domain of applications such as electrostatics, magneto-statics, hydrostatics, heat transfer, mass transfer, acoustics, elasticity, etc.展开更多
In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the ...In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.展开更多
We consider complex-valued functions f ∈ L^1 （R^2＋）, where R＋ ：= [0,∞）, and prove sufficient conditions under which the double sine Fourier transform fss and the double cosine Fourier transform fcc belong to o...We consider complex-valued functions f ∈ L^1 （R^2＋）, where R＋ ：= [0,∞）, and prove sufficient conditions under which the double sine Fourier transform fss and the double cosine Fourier transform fcc belong to one of the two-dimensional Lipschitz classes Lip（a,β） for some 0 〈 α,β ≤ 1; or to one of the Zygmund classes Zyg（α,β） for some 0 〈 α,β ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f ∈ L^1 （R^2＋）.展开更多
Provides information on a study which discussed the properties of eigenvalues for the solutions of symmetric positive definite Toeplitz systems, skew circulant and sine transform based properties. Eigenvalues of vario...Provides information on a study which discussed the properties of eigenvalues for the solutions of symmetric positive definite Toeplitz systems, skew circulant and sine transform based properties. Eigenvalues of various preconditioners; Design of positive sine transform based preconditioners; Clustering property of the preconditioners; Numerical results.展开更多
Firstly, using the improved homogeneous balance method, an auto-Darboux transformation (ADT) for the Brusselator reaction diffusion model is found. Based on the ADT, several exact solutions are obtained which contain ...Firstly, using the improved homogeneous balance method, an auto-Darboux transformation (ADT) for the Brusselator reaction diffusion model is found. Based on the ADT, several exact solutions are obtained which contain some authors' results known. Secondly, by using a series of transformations, the model is reduced into a nonlinear reaction diffusion equation and then through using sine-cosine method, more exact solutions are found which contain soliton solutions.展开更多
基金Supported by the Science Research Foundation of Zhejiang Office of Education (20050718)
文摘A solution to the reparametrization of Bézier curves by sine transformation of Bemstein basis is presented. The new effective reparametrization method is given through the following procedures： educing Sine Bemstein-Bézier Class-SBBC function, defining SBBC curve and discussing the relation between SBBC and Bézier curve.
文摘In this paper, we study an operator s which maps every n-by-n symmetric matrix A, to a matrix s(A_n) that minimizes || B_n-A_n || F over the set of all matrices B_n, that can be diagonalized by the sine transform. The matrix s(A_n), called the optimal sine transform preconditioner, is defined for any n-by-n symmetric matrices A_n. The cost of constructing s(A_n) is the same as that of optimal circulant preconditioner c(A_n) which is defined in [8], The s(A_n) has been proved in [6] to be a good preconditioner in solving symmetric Toeplitz systems with the preconditioned conjugate gradient (PCG) method. In this paper, we discuss the algebraic and geometric properties of the operator s, and compute its operator norms in Banach spaces of symmetric matrices. Some numerical tests and an application in image restoration are also given.
基金This work is supported by National Natural Science Foundation of China No. 10271099.Received: Sep. 25, 2004.
文摘In this paper, a fast algorithm for the discrete sine transform(DST) of a Toeplitz matrix of order N is derived. Only O(N log N) + O(M) time is needed for the computation of M elements. The auxiliary storage requirement is O(N). An application of the new fast algorithm is also discussed.
文摘Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + 1). It is known that if r = a satisfies homogeneous boundary conditions on all boundary lines ?in addition to non-homogeneous ones on the circular boundary , then an explicit expression of in terms of eigen-functions can be found through the classical method of separation of variables. But when the boundary?condition given on the circular boundary r = a is homogeneous, it is not possible to define a discrete set of eigen-functions. In this paper one shows that if the homogeneous condition in question is of the Dirichlet (or Neumann) type, then the logarithmic sine transform (or logarithmic cosine transform) defined by (or ) may be effective in solving the problem. The inverses of these transformations are expressed through the same kernels on or . Some properties of these transforms are also given in four theorems. An illustrative example, connected with the heat transfer in a two-part wedge domain, shows their effectiveness in getting exact solution. In the example in question the lateral boundaries are assumed to be non-conducting, which are expressed through Neumann type boundary conditions. The application of the method gives also the necessary condition for the solvability of the problem (the already known existence condition!). This kind of problems arise in various domain of applications such as electrostatics, magneto-statics, hydrostatics, heat transfer, mass transfer, acoustics, elasticity, etc.
文摘In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.
基金Supported partially by the Program TMOP-4.2.2/08/1/2008-0008 of the Hungarian National Development Agency
文摘We consider complex-valued functions f ∈ L^1 （R^2＋）, where R＋ ：= [0,∞）, and prove sufficient conditions under which the double sine Fourier transform fss and the double cosine Fourier transform fcc belong to one of the two-dimensional Lipschitz classes Lip（a,β） for some 0 〈 α,β ≤ 1; or to one of the Zygmund classes Zyg（α,β） for some 0 〈 α,β ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f ∈ L^1 （R^2＋）.
基金This work is supported by Chinese Natural Science Foundation (No: 9601012 ).
文摘Provides information on a study which discussed the properties of eigenvalues for the solutions of symmetric positive definite Toeplitz systems, skew circulant and sine transform based properties. Eigenvalues of various preconditioners; Design of positive sine transform based preconditioners; Clustering property of the preconditioners; Numerical results.
基金国家自然科学基金，NKBRD of China，Doctor Foundation of Education Commission of China
文摘Firstly, using the improved homogeneous balance method, an auto-Darboux transformation (ADT) for the Brusselator reaction diffusion model is found. Based on the ADT, several exact solutions are obtained which contain some authors' results known. Secondly, by using a series of transformations, the model is reduced into a nonlinear reaction diffusion equation and then through using sine-cosine method, more exact solutions are found which contain soliton solutions.
