特殊函数

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。

By combining the theory of sectionally holomorphic function, the analytical extending method, Cauchy model integral, the analysis of the singularity of complex functions and Riemann boundary problem, analytic relations between the complex potentials were obtained.

研究压电材料在均匀热流作用下螺型位错与圆弧裂纹的相互作用，综合运用复变函数分区全纯理论、解析函数奇性主部分析方法、解析延拓原理、Cauchy型积分以及Riemann边值问题求解方法，获得各复势函数之间的解析关系，进一步得到了特殊情况下所讨论问题的封闭解，并求解出像力随温度梯度和位错位置变化的表达式。

By combining the theory of sectionally holomorphic function,the analytical extending method,Cauchy model integral,the analysis of the singularity of complex functions and Riemann boundary problem,analytic relations between the complex potentials were obtained.

综合运用复变函数分区全纯理论、解析函数奇性主部分析方法、解析延拓原理、Cauchy型积分以及Riemann边值问题求解方法，获得各复势函数之间的解析关系，进一步得到了特殊情况下所讨论问题的封闭解，并求解出像力随温度梯度和位错位置变化的表达式。

By combining the theory of sectionally holomorphic function, the analytical extending method, Cauchy model integral, the analysis of the singularity of complex functions and the method for Riemann boundary problem, analytic relations between the complex potentials are obtained.

综合运用复变函数分区全纯理论、解析函数奇性主部分析方法、解析延拓原理、Cauchy型积分以及Riemann边值问题求解力法，导出各复势函数之间的解析关系，进一步得到特殊情况下所讨论问题的封闭解，并解出像力随温度梯度和位错位置变化的表达式。

In addition,we show that the transformed conditional distribution function is of the same family as that of the original one as long as the moment generating function of the original distribution function exists,which is illustrated by two special cases:Normal distribution and Normal Inverse Gaussiandistribution.

此外，本文还通过两个特殊情况(正态分布和正态逆高斯分布)说明了：只要初始分布函数的母函数存在，转化后的条件分布函数保持原有的分布类型。

In this paper, the computation of original likelihood function is transformed into the optimal problem of its low bound through the conjugate representation of the convex function, and a additional parameter is introduced to control the smoothing degree of low bound, and so the performance of local optimal solution can be improved.

我们利用凸泛函的对偶性，将原似然函数的复杂计算问题转化为对其下界函数的优化问题，从而降低学习算法的复杂性；并通过引入一个参数来控制下界函数的光滑程度来改善局部最优解的性能；指出了传统的期望最大化算法(Expectation Maximization Algorithm)是变分学习算法的特殊情况。

We show the different properties between covariance risk allocation function and relative risk allocation function and prove that the commonly used risk budgeting model is a special form of risk allocation function.

我们证明并比较了在标准差风险度量下，协方差风险分配函数与相对风险分配函数性质上的差异，并证明风险预算也是一种特殊形式的风险分配函数。

Although the merit function introduced here is the sum of squares of overdetermined equations, by using the special structure of it, we successfully form the Newton equation by avoiding the product of Jacobi matrices and using only the Jacobi matrix of the function defining NCP.

对非线性互补问题，利用所构造价值函数具有的特殊非线性最小二乘结构，成功实现了具有一般意义的构造Newton方程时仅利用价值函数的一阶导数且无须计算相应残量函数Jacobi矩阵乘积的Newton法。

The thesis demonstrated the function dependency and the greatest lower bound ofλ-spiral function and proved the inclusion relation of the two functions.

对λ-螺旋形函数的从属性及星函数实部的精确下界给出了证明，并证明了两族函数的包含关系，给出了特殊情况的系数估计。

An integral operator I(subscript n+p) defined by Hadamard product is introduced. Making use of this operator, the subclass S(superscript * subscript n+p) of p(superscript -) valent analytic function is defined in the open unit disk, the inclusion relation of S(superscript * subscript n+p+1)S(superscript * subscript n+p) and the best dominant function of differential subordination q1 are obtained. Furthermore, some conclusions are made according to the special parameters A and B.

用Hadamard卷积定义线性算子I，并利用算子I定义在单位圆内的解析的p叶函数类S(上标*下标 n+p)，给出了此函数类的包含关系S(上标*下标 n+p+1)S和微分从属的最佳控制函数q1，并根据参数A，B取不同的特殊值得出了相应的推论。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。