调和函数

Based on the three dimensional surface data of the face and skull, methods of geometric analysis on them are presented through these parameters with their numerical calculating methods under the landmarks of the anthropologist and anthropometrist are proposed,which is the foundation of the statistical model that will be used in the field; 4Geometric analysis methods that based on the approximation coefficients of spherical harmonic functions are proposed.

以三维面皮和颅骨数据为基础，分别给出了基于公安法医学标志点和人体测量学标志点的几何分析主要途径和方法，确定了几何分析的主要参量，并给出了相关的计算方法，为建立颅面统计模型奠定基础； 4基于球面调和函数的颅面形态学的几何分析方法。

This paper presents the boundary value problem solved as the biharmonic equation of upper plane and strip domain with Green functional method and it deals with the numerical solution of bihormonic equation, using the Matlab program calculation to realize the visualization of numeric solution.

用Green函数法求解了区域为上半平面和带形区域的双调和泊松方程的边值问题；以及探讨了双调和方程的数值解，并用Matlab编程计算实现了双调和方程数值解的可视化。

In this paper we study the method of interpolation by radial basis functions in H~k(k ≥ 1) and give some error estimates. By means of such interpolation with a special kind of radial basis function, we construct a basis in H~k(k ≥ 1). Combined with the Galerkin method, this theory can be applied to solve boundary value problems for elliptic partial differential equations (such as the third boundary value problem for Poisson equation and the corresponding problem for the biharmonic equation), and some numerical experiments are also given.

本文从求解偏微分方程的角度出发，在被逼近函数u属于一般的Sobolev空间H~k(k≥1)的情形，引入了一种径向基函数插值方法，并建立了相应的误差估计；再利用这种插值性质，从一类特殊径向基函数出发构造Sobolev空间的一组基，针对Poisson方程第三类边值问题和重调和方程类似边值问题，为用无网格算法求解偏微分方程边值问题建立了相应的理论，并通过算例来验证了这一算法。

And the radius of the convergence set of the differentiated series is the same as for the orginal series.

定理5 如果幂级数的收敛半径为则其调和函数在内可导，且有逐项求导公式

In this paper, harmonic functions were used in defining the potential field.

应用调和函数定义了势场，选取合适的边界条件，解决船舶航行的路径规划问题。

"S.Axler",Harmonic Function Theory 2nd ed.

"，"调和函数理论第2版

In recent years, Feng Kang has advanced a more natural and direct redu-ction, i. e. the reduction via Green's formula and Green's function.

近年来冯康又提出一种更自然而直接的归化，即从Green公式及Green函数出发将微分方程边值问题化为边界上的含有广义函数意义下发散积分有限部分的奇异积分方程，这种归化在各种边界归化中占有特殊地位，被称为正则边界归化，本文将这一理论应用于重调和椭圆边值问题，研究了其正则归化的性质，并通过利用Green函数、Fourier分析及复变函数论方法等不同途径求出了在上半平面、单位圆内部、单位圆外部三种区域的Poisson积分公式及正则积分方程，其离散化可用于实际计算。

A 3-D face model is constructed by the imaging theory and spherical harmonics, combining with the principle of 3-D projection and PCA.

根据3-D数据的优势，利用图像成像原理和球面调和函数理论，结合3-D投影原理和PCA技术建立了一个3-D人脸模型。

By the expansion of a function in terms of spherical harmonics, solutions in form of separation about the asymmetry disturbance were studied. At last, by the theory of stability of differential equations, under the no limitation or less than 1 to the symmetry stationary radius, the asymmetry stability of stable and unstable conditions were obtained.

然后利用球调和函数展开，研究了有关扰动的偏微分方程的变量分离形式的解，最后根据微分方程稳定性定理，在肿瘤静止半径没有限制或肿瘤静止半径小于1的情况下，得到了非对称扰动的稳定和不稳定的条件。

To realize the realistic rendering of the scene in applying environment mapping to VR systems, an approach to fast calculating diffuse environment maps was proposed starting from the analysis of spherical harmonics.

为了把环境贴图应用于VR系统中，实现场景对象的真实感绘制，首先从分析球面调和函数入手，提出了漫反射环境纹理图的快速计算方法。

Results showed that 0.01g/mL extract of Oxalis corniculata L. had obvious antimicrobial effect on S. aureus, but all extract of Oxalis corniculata L has not any antimicrobial effect on E.

结果表明，0.01g/ml的提取物对摇床培养的金黄色葡萄球菌生长具有明显的抑制作用；但各种浓度的提取物对摇床培养的大肠杆菌生长均无抑制作用。

This will be totally different from Zaro's texture mod, it will be desaturated, dirty etc.

这会和Zaro的那个材质补丁有很大不同，这边是更多的不饱和色，整个都灰扑扑的。