Rate of Convergence of Chernoff Approximations of Operator $$C_0$$ -Semigroups

Mathematical Notes -     

迁移方程的扩散近似的收敛性

导出了迁移方程的扩散近似方程．说明了它的离散纵标方法在区间内和边界上都有扩散极限，它的解关于一致地收敛于迁移方程的解．其收敛性的证明是依据其渐近扩散展开式，在边界层上得到的误差估计逼近其离散纵标方法的解．     

算子级数的序列赋值收敛性

对于巴拿赫空间上的经典矢量值序列空间,引入了一类重要的子集,它包含了该序列空间中的全部金有界集.利用该子集,得到了一个算子级数序列赋值收敛定理.完全去掉了通常对算予的线性约束.它不仅在理论上很重要,而且也增加了应用的可能性.     

Diophantine Approximations and Integer Points of Cones

The purpose of this note is to present a relation between directed best approximations of a rational vector and the elements of the minimal Hilbert basis of certain rational pointed cones. Furthermore, we show that for a special class of these cones the integer Carathéodory property holds true. Received May 6, 1998 RID="*" ID="*" Supported by a "Leibniz Preis" of the German Science Foundation (DFG) awarded to M. Gr?tschel. RID="†" ID="†" Supported by a "Gerhard-Hess-Forschungsf?rderpreis" of the German Science Foundation (DFG).     

Subseries Convergence and Unconditionally Convergence of Compact Operator Series

Let(E,r)bealocallyconvexHausdorfftopologicalvectorspace.AseriesZxiinEissaidtobesubseriesr-convergent(-s.c.),ifforeachstrictlyincreasingsequence{n,}inN,theseriesZx"'isT-convergent.TheseriesZx,issaidtobeunconditionallyr-convergent(r-u.c.),ifforeachpermutationffofN,theseriesZx,<,,ls.,r-convergence.rheseriesZx,issaidtobesubseriesunconditional1yT-convergent(r-s.u.c.)ifforeachstrictlyincreasingseqtience{n,}inN,theseriesZx.,isunconditionallyr-convergent.If(E,r)isasequentiallycompletespace,thenT…     

Non-simple Turning Points and Cusps

Stimulated by a problem in the theory of thermal ignition, weprove some results on non-simple turning points and correspondingcusps for two parameter-dependent non-linear problems. Theseresults provide a theoretical basis for the numerical computationof non-simple turning points or cusps. The numerical procedureis illustrated for the thermal ignition problem.     

Continuous Approximations of Multivalued Mappings and Fixed Points

In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset X of the Hilbert space H which satisfies the tangential condition , where T _X (x) is the cone tangent to the set X at a point x. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 212–222.Original Russian Text Copyright © 2005 by B. D. Gel’man.     

广义算子半群的遍历及概率逼近定理

首先给出了广义算子半群的Abel-遍历和Cesaro-遍历的定义,对两种遍历的性质进行了刻画,研究了两种遍历的等价条件.其次,利用Pettis积分、算子值数学期望及广义连续修正模等工具给出广义算子半群的概率逼近表达式.     

无点滤子收敛

我们在ｌｏｃａｌｅ上定义了一种新的滤子收敛概念．对空间式ｌｏｃａｌｅ,这种收敛与拓扑收敛等价,并且适应于任意ｌｏｃａｌｅ．作为应用,我们给出了ｌｏｃａｌｅ紧的程度的刻划和 Ｃａｕｃｈｙ完备性的描述．     

On the Superlinear Convergence of the Successive Approximations Method

The Ostrowski theorem is a classical result which ensures the attraction of all the successive approximations x _k+1 = G(x _k ) near a fixed point x*. Different conditions [ultimately on the magnitude of G(x*)] provide lower bounds for the convergence order of the process as a whole. In this paper, we consider only one such sequence and we characterize its high convergence orders in terms of some spectral elements of G(x*); we obtain that the set of trajectories with high convergence orders is restricted to some affine subspaces, regardless of the nonlinearity of G. We analyze also the stability of the successive approximations under perturbation assumptions.     

The Rate of Convergence of Finite-Difference Approximations for Bellman Equations with Lipschitz Coefficients

We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order h^1/2 for certain types of finite-difference schemes are obtained.     

A New Method for Approximations in Probability and Operator Theories

The aim of this work is to introduce a new approach in normal approximations, approximations of semigroups of operators, and approximations by accompanying laws. We describe the method and provide several examples showing how the method works. Loosely speaking, the approach is based on multiplicative representations of differences to estimate. The method considerably reduces the technical part of proofs compared to the traditional approaches. In probability theory, it nicely applies to the nonidentically distributed case; another advantage of the method is its extendability to the case of dependent random variables.     

Indefinite Eigenvalue Problems with Several Singular Points and Turning Points

We consider a general class of eigenvalue problems with two-point boundary conditions on a finite interval generated by a differential equation with an indefinite weight function which has several zeros and/or poles. As a basic result we derive asymptotic estimates for a special fundamental system of solutions of the corresponding differential equation and determine the asymptotic distribution of the eigenvalues. Finally we prove the uniform convergence of eigenfunction expansions for some class of functions f.     

广义W-Bernstein-Kantorovich算子的构造、收敛性及其渐近表示

运用Kantorovich型算子的构造方法,将一种新的基本Kantorovich算子与广义W-Bern-stein算子复合得到一种新的广义W-Bernstein-Kantorovich算子,证明了它的收敛性并给出了它的Mamedov渐近表达式．     

拟增算子的多个不动点

该文应用变分方法和 Banach 空间的常微分方程理论证明了关于拟增算子的一个三不动点定理.     

Convergence Analysis for Pseudospectral Multidomain Approximations of Linear Advection Equations

Legendre and Chebyshev collocation schemes are proposed forthe numerical approximation of first order linear hyperbolicequations, by a domain decomposition procedure. Spectral convergenceestimates are provided both for Legendre and Chebyshev Gauss-Lobattonodes.     

Global Convergence of a Nonlinear Programming Method Using Convex Approximations

The method of moving asymptotes (MMA) and its globally convergent extension SCP (sequential convex programming) are known to work well for certain problems arising in structural optimization. In this paper, the methods are extended for a general mathematical programming framework and a new scheme to update certain penalty parameters is defined, which leads to a considerable improvement in the performance. Properties of the approximation functions are outlined in detail. All convergence results of the traditional methods are preserved.     

渐近伪压缩型映象不动点的迭代逼近

在赋范空间中引入有限族渐近伪压缩型映象,在较弱条件下,在赋范空间中建立了有限族渐近伪压缩型映象不动点的带误差的迭代算法的一个强收敛定理.也给出几个例子说明结果的有效性与广泛性,从而改进和推广了Rafiq和其他人的结果.