On a fast direct elliptic solver by a modified Fourier method

We describe high order numerical algorithms for the solution of second order elliptic equations in rectangular domains. These algorithms are based on the Fourier method in combination with a subtraction procedure. The singularities at the corner points, arising due to non-smoothness of the boundaries, are treated explicitly using properly constructed singular corner functions. The present algorithm is a generalization of the Fast Poisson Solver developed in our previous paper. This revised version was published online in August 2006 with corrections to the Cover Date.     

海浪谱的光学探测及信息处理

建立了一套光学——计算机混合处理系统用以分析海浪航空底片,并得到了海浪方向谱和能谱。实验结果与现有的海浪理论符合较好,同时显示了该方法有可能发展成一种实用的实时海浪探测方法。     

苯对N,N-二甲基甲酰胺化合物~1H NMR的影响研究

本文研究了N,N-二甲基甲酰胺(DMF)在四氯化碳和苯混合溶剂中的~1H NMR。观察到N,N-二甲基甲酰胺(DMF)分子中两个甲基和混合溶剂中苯的共振吸收峰的化学位移随苯的摩尔分数增加而逐渐移向高场,而且两个甲基共振吸收峰向高场移动的程度不同,α甲基较β甲基为甚。DMF分子中两个甲基和混合溶剂中苯的化学位移可以用线性方程表示(δ=A+Bx),这里x是混合溶剂中苯的摩尔分数。相关系数R接近于-1。     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

用频域法实现圆环形目标物的计数

研究了用频域法实现圆环形目标物计数的方法。对空域中含有圆环形目标物的二值图像，经傅里叶变换后在频域内进行滤波，再经傅里叶反变换后取一阈值进行处理，则对应于圆环形目标物即变为一个实心体，然后对实心体进行计数即可实现对圆环形目标物的计数。对该方法进行了数学描述，并分别用模拟图像和实际图像进行了实验。结果表明，该方法消除噪声的效果好，对部分封闭圆的检测同样有效，且可用于目标物有粘连的情况。该方法对圆环形目标物的尺寸相差不大的情况更为有效。     

Simplified analytic solutions and a novel fast algorithm for Yb-doped double-clad fiber lasers

In this paper, continuous wave Yb³⁺-doped double-clad fiber lasers (DCFLs) with linear-cavity are investigated theoretically and numerically using the rate equations. Under the steady state conditions, the simplified analytic solutions of Yb³⁺-doped DCFLs under considering the scattering loss are deduced in the strongly pump condition. Compared with the known analytic solutions in published literatures, our analytic solutions are more accurate, especially, at higher reflectivity of output mirror. In addition, a fast and stable algorithm based on the Newton-Raphson method is proposed to simulate numerically Yb³⁺-doped DCFLs. The results by simplified analytic solutions are in good agreement with those by the numerical simulation. Moreover, we have performed the optimization of an Yb³⁺-doped DCFL using the simplified analytic solutions and the numerical simulations, respectively.     

聚变等离子体中的快离子压强

本文运用Fokker-Planck方程的慢化近似,考虑存在多种离子成份(包括杂质),假定它们具有共同的温度,我们得到了聚变产生的快离子压强的简单封闭形式表示式。表明在典型的工作温度下(～60keV),D-~3He等离子体中聚变产生的快离子压强约为本底热压强的20％,这与D-T等离子体工作在20keV时的比值几乎相同。因此,D-~3He和D-T在它们相应的预期工作温度下,它们各自的快离子压强对总压强的影响是类似的,然而在更高的温度下,这个比值将变得更大。     

Sparse generalized Fourier transforms

Block-diagonalization of sparse equivariant discretization matrices is studied. Such matrices typically arise when partial differential equations that evolve in symmetric geometries are discretized via the finite element method or via finite differences. By considering sparse equivariant matrices as equivariant graphs, we identify a condition for when block-diagonalization via a sparse variant of a generalized Fourier transform (GFT) becomes particularly simple and fast. Characterizations for finite element triangulations of a symmetric domain are given, and formulas for assembling the block-diagonalized matrix directly are presented. It is emphasized that the GFT preserves symmetric (Hermitian) properties of an equivariant matrix. By simulating the heat equation at the surface of a sphere discretized by an icosahedral grid, it is demonstrated that the block-diagonalization is beneficial. The gain is significant for a direct method, and modest for an iterative method. A comparison with a block-diagonalization approach based upon the continuous formulation is made. It is found that the sparse GFT method is an appropriate way to discretize the resulting continuous subsystems, since the spectrum and the symmetry are preserved. AMS subject classification (2000)  43A30, 65T99, 20B25     

Optimal reconstruction of the solution of the wave equation from inaccurate initial data

In the present paper, we consider the problem of the optimal reconstruction of the solution of the wave equation from the approximate values of the Fourier coefficients of the function specifying the initial form of the string. For an operator defined on the weight space of vectors from l ₂, we present the solution of the more general problem of reconstruction from the approximate values of the coordinates of these vectors.     

On multiple integrals of special form

Multiple integrals generalizing the iterated kernels of integral operators are expressed as single integrals in the case of a special representation of the kernel (this is our theorem). Besides integral equations, Markov processes involve these integrals as well. As a consequence of the theorem, we obtain transition probability densities of certain Markov processes. As an illustration, we consider nine examples.