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1.
There has been much work (cf. [1], [4], [7]-[12]and references of [8]) on the stabilization of hyperbolic systems. Most of the papers, with the exception of [4] and[12], are concerned only with feedbacklaws such that the resulting systems are dissipative. As is pointed out at the beginning of [7], the dissipative property demands  相似文献   

2.
A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once.In this paper,we study 1-planar graph joins.We prove that the join G + H is 1-planar if and only if the pair [G,H] is subgraph-majorized by one of pairs [C3 ∪ C3,C3],[C4,C4],[C4,C3],[K2,1,1,P3] in the case when both elements of the graph join have at least three vertices.If one element has at most two vertices,then we give several necessary/sufficient conditions for the bigger element.  相似文献   

3.
The degree of approximation to a function f(x)∈C[-1,1] by (U, λ) means and f(x) ∈ Lpw by (Jr) means are discussed, some results in the literatures [1],[2],[3] have been improved.  相似文献   

4.
Recently there has been a considerable amount of work on the existence of Tperiodic solutions for Hamiltonian systems with singular potentials, (see [1]—[7],[10], [11], [13], [14]). In this paper we will study the existence of T-periodic solutions for nonconservative second-order dynamical systems  相似文献   

5.
1 IntroductionThe solution of the following stochastic differential equationis called the geometric Brownian motion, where a(t), b(t) are deterministic functions of ti ac isa Brownian motion, and ddt is the It6 integral. This equation has been successfully applied tothe financial problems such as modeling the prices of stocks, sc.e for example, [81, [7], [14], [13],[19], [21]. When the initial condition is given, i.e. xo = x, the solution isIt is known that in the financial market, it is als…  相似文献   

6.
Let L^2([0, 1], x) be the space of the real valued, measurable, square summable functions on [0, 1] with weight x, and let n be the subspace of L2([0, 1], x) defined by a linear combination of Jo(μkX), where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2([0, 1], x), let E(f, n) be the error of the best L2([0, 1], x), i.e., approximation of f by elements of n. The shift operator off at point x ∈[0, 1] with step t ∈[0, 1] is defined by T(t)f(x)=1/π∫0^π f(√x^2 +t^2-2xtcosO)dθ The differences (I- T(t))^r/2f = ∑j=0^∞(-1)^j(j^r/2)T^j(t)f of order r ∈ (0, ∞) and the L^2([0, 1],x)- modulus of continuity ωr(f,τ) = sup{||(I- T(t))^r/2f||:0≤ t ≤τ] of order r are defined in the standard way, where T^0(t) = I is the identity operator. In this paper, we establish the sharp Jackson inequality between E(f, n) and ωr(f, τ) for some cases of r and τ. More precisely, we will find the smallest constant n(τ, r) which depends only on n, r, and % such that the inequality E(f, n)≤ n(τ, r)ωr(f, τ) is valid.  相似文献   

7.
Let f(x)∈C[-1, 1], and ∏_n be the class of algebraic poly-nomials of degree n. It is well known that algebraic polynomialapproximation can be improved near the end points of the interval[-1, 1]. This fact was first observed by A. F. Timan, and laterJ. Brudnyi proved that for any positive integer k, there is aρ_n(x)∈∏_n such that  相似文献   

8.
1 IntroductionConsider the functional differential equations (FDE's) with finite delay ofthe form:where x,(0) = x(t 0) for --r 5 0 5 0 with some r 2 0 and f: [T, co] x C RI with C = C([--r, 0], R') the space of continuous functions mapping [--r, 0]into l-dimensional real space RI. For any yi E C we define the norm of yi as:where l' I is any norm in RI.Inspired by the ideas in [l--3], we develop a new technique in this work tostudy the stability of FDE's, where the components xl) xZt…  相似文献   

9.
1. IntroductionWe deal with the Duffing equationi g(x) ~ p(t), (1.1)where g(x), p(t) E C(R, R) and p(t) is periodic, whose least period is 27. The multiplicityof periodic solutions of Equation (l.1) has been widely studied since the 50s. In [1], T.Ding studied the multiplicity of periodic solutions of Equation (1.1) under the followingconditions.(gi) Let g(x) E C'(R, R), and let K be a positive constant, such that(gi) There exist two constallts Ac > 0 and MO > 0 such thatg(x) w~ > AO…  相似文献   

10.
In this paper the concept of positive definite bilinear matrix moment functional. acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b], is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Kiemann -Lebesgue matrix property and the bessel-parseval matrix inequality are proved. The concept of total set vjith respect to a positive definite matrix functional is introduced , and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional, is established.  相似文献   

11.
成礼智 《计算数学》1999,21(4):451-462
1.引言考虑线性方程组TNx=b(1.1)其中TN=(ti,j)是NxN对称正定(SPD)Toeplitz矩阵,即ti,j=t|i-j|(i,j=0,1,...,N-1)且TN的所有特征值均为正数,并表为TN:=T(t。,ti,...,tN-1).如果我们用预条件子共轭梯度法(PCG)求解方程组(1.1),最关健的任务是构造出高效的预条件子.而预条件子最自然的选择似乎其逆矩阵易求且构成矩阵TN的某种最优逼近.由于循环矩阵CN的逆矩阵CR'仍为循环矩阵,因此CN和CH'与向量的乘积可通is速Fourier…  相似文献   

12.
We study the numerical solution of a block system T m,n x=b by preconditioned conjugate gradient methods where T m,n is an m×m block Toeplitz matrix with n×n Toeplitz blocks. These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose new preconditioners for block systems based on circulant preconditioners. From level-1 circulant preconditioner we construct our first preconditioner q 1(T m,n ) which is the sum of a block Toeplitz matrix with Toeplitz blocks and a sparse matrix with Toeplitz blocks. By setting selected entries of the inverse of level-2 circulant preconditioner to zero, we get our preconditioner q 2(T m,n ) which is a (band) block Toeplitz matrix with (band) Toeplitz blocks. Numerical results show that our preconditioners are more efficient than circulant preconditioners.  相似文献   

13.
In this paper we present an algorithm for the construction of the superoptimal circulant preconditioner for a two-level Toeplitz linear system. The algorithm is fast, in the sense that it operates in FFT time. Numerical results are given to assess its performance when applied to the solution of two-level Toeplitz systems by the conjugate gradient method, compared with the Strang and optimal circulant preconditioners.  相似文献   

14.
Both theoretical analysis and numerical experiments in the literature have shown that the Tyrtyshnikov circulant superoptimal preconditioner for Toeplitz systems can speed up the convergence of iterative methods without amplifying the noise of the data. Here we study a family of Tyrtyshnikov‐based preconditioners for discrete ill‐posed Toeplitz systems with differentiable generating functions. In particular, we show that the distribution of the eigenvalues of these preconditioners has good regularization features, since the smallest eigenvalues stay well separated from zero. Some numerical results confirm the regularization effectiveness of this family of preconditioners. Copyright © 2009 John Wiley & Sons, Ltd.  相似文献   

15.
In the general case of multilevel Toeplitz matrices, we recently proved that any multilevel circulant preconditioner is not superlinear (a cluster it may provide cannot be proper). The proof was based on the concept of quasi-equimodular matrices, although this concept does not apply, for example, to the sine-transform matrices. In this paper, with a new concept of partially equimodular matrices, we cover all trigonometric matrix algebras widely used in the literature. We propose a technique for proving the non-superlinearity of certain frequently used preconditioners for some representative sample multilevel matrices. At the same time, we show that these preconditioners are, in a certain sense, the best among the sublinear preconditioners (with only a general cluster) for multilevel Toeplitz matrices.

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16.
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.  相似文献   

17.
Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis. Received December 19, 1995 / Revised version received September 15, 1997  相似文献   

18.
For large-scale image deconvolution problems, the iterative regularization methods can be favorable alternatives to the direct methods. We analyze preconditioners for regularizing gradient-type iterations applied to problems with 2D band Toeplitz coefficient matrix. For problems having separable and positive definite matrices, the fit preconditioner we have introduced in a previous paper has been shown to be effective in conjunction with CG. The cost of this preconditioner is of O(n2) operations per iteration, where n2 is the pixels number of the image, whereas the cost of the circulant preconditioners commonly used for this type of problems is of O(n2 log n) operations per iteration. In this paper the extension of the fit preconditioner to more general cases is proposed: namely the nonseparable positive definite case and the symmetric indefinite case. The major difficulty encountered in this extension concerns the factorization phase, where a further approximation is required. Three approximate factorizations are proposed. The preconditioners thus obtained have still a cost of O(n2) operations per iteration. A numerical experimentation shows that the fit preconditioners are competitive with the regularizing Chan preconditioner, both in the regularizing efficiency and the computational cost. AMS subject classification (2000) 65F10, 65F22.Received October 2003. Accepted December 2004. Communicated by Lars Eldén.  相似文献   

19.
20.
Boundary value methods (BVMs) for ordinary differential equations require the solution of non‐symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block‐circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is Ak1,k2‐stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block‐circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.  相似文献   

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