积分算子

In chapter 4 by means of the decomposition technique of Bojanic and Cheng together with some analysis methods we study rates of convergence of sequence of integral Baskakov operators for a class absolutely continuous functions.

第四章主要讨论了积分型Baskakov算子对绝对连续函数的逼近阶估计。在这章中主要利用经典的Bojanic-Cheng分解方法，结合分析技术，讨论了积分型Baskakov算子对绝对连续函数的点态收敛阶。

From our results, it can be known that the weakly asymptotic order of the information-based algorithm whose recover function is the classical Bernstein interpolation sequence is below that of the corresponding minimal information radius whose permissible information operator is function value.

通过我们的结果可以知道，以经典的Bernstein多项式算子列作为计算恢复函数的信息基算法，其在Wiener空间(或1-重积分Wiener空间)下的的平均误差的弱渐近阶均低于相应的以函数值计算为可允许信息算子的最小平均信息半径的弱渐近阶。

In the third chapter, we obtain the average errors(asymptotic order or weakly asymptotic order) of the Lagrange interpolation sequence, the Hermite-Fejer interpolation sequence and the Hermite interpolation sequence based on the Chebyshev nodes on the 1-fold integrated Wiener space.

第三章分别给出了基于第一类Chebyshev多项式零点的Lagrange插值算子列、Hermite-Fejer插值算子列和Hermite插值算子列在1-重积分Wiener空间下的平均误差的值。

In order to solve the problem,We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential manifold. The travel time are expressed as polynomials of the horizontal offset between the two points, and the single-square-root operator in frequency-wavenumber domain are expressed as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddle-point method.

针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量(地面点到地下相点的水平距离)的多项式，将频率波数域单平方根算子表示成波数的多项式，运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。

In order to solve the problem, We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddlepoint method.

针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量（地面点到地下相点的水平距离）的多项式，将频率波数域单平方根算子表示成波数的多项式，运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。

Further, with the help of Riccati equations, an infinite number of conservation laws for the solton hierarchy are deduced. For the sake of simplicity, taking the general TD hierarchy as an illustrative example, we prove that its 2×2 Lenard pair of operators forms a Hamiltonian pair. Thus the isospectral evolution TD hierarchy is the general Hamiltonian system and possesses the Bi-Hamiltonian structures and Multi-Hamiltonian structures. By using the method of derivation of functional under some constraint condition, a complete one-to-one correspondence between the Hamiltonian functions of the hierarchy and its conservation density functions can be built. These results can also be applied to the isospectral evolution soliton hierarchy of this paper. Finally, there's a gauge transformation between the spectral problem of this paper and the AKNS system. Moreover, the potentials in these spectral problems satisfy the general Miura transformation, the corresponding relationship between the two soliton hierarchies is also given.

进一步本文还通过特征函数的组合关系所满足的Riccati方程，得到了该等谱方程族的无穷多个守恒律；为简便起见，本文以广义TD族为例，由它的2×2 Lenard算子对的性质证明了此算子对为Hamilton算子对，这说明广义TD族是广义Hamilton系统且具有Bi-Hamilton结构和Multi-Hamilton结构；进而利用它的依赖于谱参数的一般守恒密度的积分在约束条件下求泛函导数的方法，得到了广义TD族的Hamilton函数与守恒密度之间的对应关系，这些性质对于由本文提出的2×2谱问题所导出的等谱孤子族仍成立；另外此谱问题与AKNS系统存在着规范变换，位势之间有广义Miura变换，而孤子方程之间也满足一定的等价关系。

In this paper, we will introduce the weighted modulus of smoothness and the weighted K-functional to discuss the direct and the converse theorems, the local saturation results on the simultaneous approximation on an infinite interval with Jacobi-weight by the integral operator, the lin...

本文利用带权光滑模与带权K--泛函讨论定义在无穷区间上的积分型算子Gamma算子线性组合对空间L_∞(0，∞)中函数的带Jacobi权同时逼近的正逆定理以及局部饱和结果，还讨论了Gamma算子的强逆不等式。

In chapter 1, as a preparative knowledge, the main results about symmetry theoryand Wus method are introduced, in addition some fundamental definitions, such as trans-formation group, infinitesimal operator, prolongation, first integral, infinitesimal criterionfbr invariance of an ODE, characteristic set are recalled.

第一章中作为预备知识简单介绍了对称理论和吴方法，并给出了一些基本概念和理论，如变换群、无穷小算子、算子延拓、首次积分、微分方程在对称下不变的判别准则和微分特征列集算法。

It is shown that an exponentially bounded integrated bisemigroups can be considered as the integration of a strongly continuous bisemigroup of bounded linear operators in some subspace of a Banach space with stronger norm topology,and also as the restriction of a strongly continuous bisemigroup of bounded linear operators in a bigger space with weaker norm topology.

证明了Banach空间x上的指数有界积分双半群可以作为x的某个子空间上具有较强范数拓扑下的有界线性算子强连续双半群的积分，同时也可作为较大空间上具有较弱范数拓扑下的有界线性算子强连续双半群积分的限制。

Our conclusion is : For general transport equation and some special transport equations (slab geometry, spherical symmetry, pillar symmetry) we can evaluate some relevant physical quantities and geometrical properties from the exiting photon flux. The method we will use is to transform the transport equation into integral equation by integrating it along its characteristics, then we prove the nonlinear operator K is monotone and continuous with respect to its arguments and obtain the existence of K~-1. Consequently, we can evaluate some relevant physical quantities and geometrical property of the medium.

我们得到的结论是：无论对于一般的迁移方程，还是当介质具有某种对称性时(板对称、球对称、柱对称)的迁移方程，我们都可以根据泄漏的光子流估计介质各种相关物理量和几何特性，所用方法都是将迁移方程通过沿特征线积分转化为积分方程的形式，然后证明非线性算子K关于自变量的单调性和连续性，从而得到逆算子K~(-1)存在，进而达到估计介质各种相关物理量和几何特性的目的。

By the time of its fall, most of the prisoners were writers who had written against the corruptions of the government.

到它被攻陷的时候，里面多数的犯人是写了反对政府贪污文章的作家。

The most obvious variation to ovum morphological character was that the color was changed from light green to sepiaceous in embryonic development, and all the ovums were almost hatched after 96h.

在胚胎发育过程中卵的形态特征最明显的变化是颜色从淡绿到深褐色，卵在发育96h后卵基本全部孵化。