逼近定理

The paper applies algebraic geometry, computational geometry, approximation theory to study the following problems: the Nother type theory and the Riemann-Roch type theory of the piecewise algebraic curve; the number of real intersection points of piecewise algebraic curves; the real piecewise algebraic variety and the B-net resultant of polynomials.

本文应用代数几何，计算几何，函数逼近论等学科的基本理论，分别就分片代数曲线的Nother型与Riemann-Roch型定理；分片代数曲线的实交点数；实分片代数簇以及多项式的B-网结式进行研究。

Thirdly, for Bernstein-Durrmeyer-Bezier operators, we have given the approximation equivalent theorem in L_p(1 ≤ p≤∞).

三、我们给出了Bernstein-Durrmeyer-Bézier算子于L_p(1≤p≤∞)空间中逼近的等价定理。

Secondly, the approximation equivalent theorem for Bernstein-Kantorovich-Bezier operators have been studied by using of smooth modulus ω_φ_p in the space L_p(1 ≤ p≤∞).

二、利用光滑模ω_φ_p和与之相对应的K泛函研究了Bernstein-Kantorovich-Bézier算子于L_p(1≤p≤∞)空间中的逼近正、逆定理和等价定理。

A fast algorithm for calculating the inverse and Moore-Penrose inverse of the permutation factor circulant matrix is presented. Then, a fast algorithm for the product of the permutation factor circulant matrix is given. Finally, The least-square solutions of inverse problems and the optimal approximation problems for the permutation factor circulant matrix are discussed.

首先给出了一种求n阶置换因子循环矩阵的逆阵、自反g-逆、群逆、Moore-Penrose逆的快速算法，然后给出了计算两n阶置换因子循环矩阵之乘积阵的一种快速算法，最后给出了其反问题的最小二乘解的存在性定理及它的一般表示，并证明了逼近矩阵的存在唯一性且给出它的具体表达式。

With a constructed theorem, the author comes to a conclusion that this method, in a sense, equals to the algorithm of the continued fraction.

通过构造的一个定理对这个算法进行的分析讨论证实，闰周算法是一种很好的实数有理逼近算法，事实上，从某种意义上讲，它与连分数展开算法是等价的。

In this paper, an important property and a theorem have been given about the matrix Thiele's continued fraction which is used to approximate the function of e. Based on the property, the Thiele algorithm is improved. A numeral example is also provided.

文章利用代数知识给出了矩阵指数函数e的连分式逼近函数的一个重要性质和定理，并在此基础上对矩阵指数函数e的连分式算法进行改进，最后用数值例子来验证其可行性。

In Chapter 3, by using the technique of finite dimensional approximation of Seiberg-Wittentheory, the equivariant K-theory and some other methods, we study the problem of topologicalrestriction when there are smooth groups acting on Spin 4-manifolds such as the alternatinggroup A_5, cyclic group Z_6 and symmetric group S_3 with actions of odd type, consequently weimprove Furutas 10/8 theorem under the condition of group actions.

第三章运用Seiberg-Witten理论的有限维逼近技巧和等变K-理论等工具，研究了Spin 4-流形X上分别存在光滑交错群A_5、循环群Z_6和奇型三阶对称群S_3作用时，X的拓扑限制问题，在群作用下改进了Furuta的10／8定理。

The boundary value problem for first-order differential equations with a parameter refers to the finite solution condition of equation which includes unknown quantity and its first-order derivative that not only relies on interval value but also relies on the parameter of the equation,which is always settled by fixed point theory or successive approximation method.

含一个参数的一阶微分方程边值问题是指：含有未知量及它的一阶导数的方程的定解条件不仅依赖于在区间端点的取值，而且依赖于方程中的参数。通常它用不动点定理或逐步逼近法来解决。

The mechanism of the least squares approximation that MFNN with finite nodes carried out on a given discrete point set is made, which, for the first time, results in an important theorem.

分析了有限规模多层前向网络用于离散点集上的最小二乘逼近的机理，首次给出了一个有关的定理。

1Because some conditions can not be fulfilled,one can not applythe known results in stochastic approximation directly.

1由于有些条件不能满足，现有的随机逼近结果无法直接利用，我们巧妙地克服了这一困难，在较弱的条件(文中H.4.1—H.4.3)下，给出了这类算法的严格收敛性分析(定理4.3.2，定理4.3.3)。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。