测度

Let m be a finite μ-invariant measure of Q-matrix, as Q is totally stable, consists of a single absorbing state or single-instantaneous state, we prove that exists Q-processes P , in which m be a μ-invariant measure of P , and construct the Q-processes P .

第四章致力于μ不变测度的研究，设m是Q的有限μ不变测度，在Q是全稳定、单瞬时不可和准保守拟Q-矩阵，以及含有吸收态的情形，分别证明了存在Q过程P，使m是P的μ不变测度，并且都具体构造出Q过程。

Firstly, we consider the problem of the representation formula for the transition probability density of a right process under Girsanov transform. Secondly,we consider the problem of the MEMMs (minimal entropy martingale measure)for Markov switching Levy processes, and justify that the minimal entropy martingale measure is obtained by some regime switching Esscher transform. At last,we consider the problem of the invariant measure and the ergodic property of recurrent right processes. This paper includes 5 chapters.

首先，我们考虑了满足一定条件的右过程在Girsanov变换下的转移概率密度的表达式问题；其次，我们考虑了由Markov调制的Lévy过程的最小相对熵鞅测度的问题，证明了其最小相对熵鞅测度是某个状态转换Esscher变换；最后，我们考虑了右过程的不变测度及其遍历性的问题。

In Chapter 4we make more study about the properties of the multifractal Hausdorff mea-sure.The existence of finite measure subsets and the Frostman theorem areproved in this chapter.

第四章我们进一步讨论重分形Hausdorff测度的性质，证明了关于重分形Hausdorff测度的有限测度子集的存在性及Frostman定理。

In the fifth chapter,we mainly discuss the relationship of the metric space induced by a Loeb measure space and the metric space induced by a internal finitely additive measure space.

在第五章，我们重点讨论了由有限测度空间导出的度量空间和由内有限可加测度空间导出的度量空间的关系，然后在此关系的基础上研究了由内有限可加测度空间导出的度量空间的性质。

We prove that in every induced locally compact 〓fuzzy topological groupthere exist at least one regular fuzzy set-valued Haar measure.We define the fuzzymeasurable group and prove that the fuzzy set-valued Haar measure in a fuzzymeasurable group is essentially unique.We also construct the fuzzy topologicalstructure on a group by the fuzzy set-valued measure such that make it a fuzzytopological group.

证明了诱导局部紧〓模糊拓扑群上存在模糊集值Haar测度，定义了模糊可测群，通过对模糊可测群的研究证明了诱导局部紧〓模糊拓扑群上的模糊集值Haar测度本质上的唯一性，并利用模糊可测群上的模糊集值测度构造了群上的模糊拓扑结构使之成为模糊拓扑群。

First, a new similarity measure, named as arithmetic harmonic mean divergence measure, is induced.

首先，提出了新的相似性测度，算术调和均值距离测度AHM，图像通过最大化AHM全局配准，然后将图像划分成均匀子块，采用AHM测度刚性配准各对应子块，选取对应子块的中心作为标记点对，最后，用三次均匀B样条实现图像的弹性配准。

On the other hand, when we are proving the upper bounds of the Hausdorff measure functions for the level sets, although we get the answer to the cases for the symmetric processes, we had to consider multi-integral of highly dimensional functions, the proof is quite complex. Therefore, we suggest that the proof for the corresponding problem needs some new ways.

另一方面，我们在证明严格稳定过程的水平集的Hausdorff测度函数的上界时，虽然我们得到了对称的多参数稳定过程的水平集的Hausdorff测度函数上界，但是我们的方法必须考虑多重的高维积分，其证明过程也已经相当复杂，因此，我们认为，为了得到一般的严格稳定过程的水平集的Hausdorff测度函数上界，还需要引入其它新的研究方法。

For example,basic tools from calculus such as Fermat theorem,Rolle theorem and the intermediate value theorem may not necessarily hold and it is difficult to find a universal program for simulation in a model with various timescales,which attract attention of great deal researchers.In this PhD thesis,we first consider classification schemes for positive solutions of the first and second order dynamic systems.

在探讨测度链上的动力方程的动力学行为时人们所熟悉的基本工具诸如Fermat定理，Rolle定理以及介值定理等不再成立，同时很难找到适应不同测度链的模拟程序，这些在给测度链理论研究带来诸多困难的同时，也更引起了广大学者的兴趣。

Then,we discuss the question that when there is a q-process P,such thatπwhich is theμ-invariant measure orμ-subinvariant measure of a given q-pair,is theμ-invariant measure of P...

第三章分别分了三种情形，对给了保守或π-非保守q-对的μ-不变测度或μ-次不变测度π，何时存在q-过程P，使得π是P的μ-不变测度的问题进行了讨论研究。

The Hausdorff measure for the level sets of multi-parameter symmetric stable processes has been obtained and the inner connection of the measures and local times for it has been demonstrated, which describes the fractal nature for the level sets of this kind of processes. As we know, it is always one of the focuses in the field of random fractal. The reasult of the existence and continuity of the local time for multi-parameter stochastic processes with stable components has been got in the project. Then we gained the measure for the random fractal sets such as the range and graph of this kind of processes. Furthermore, a progress besides the mentioned above in the project is that we have resolved the multifractal analysis of the sample paths for Brownian sheet, that is the multifractal decomposition of white noise in high dimension.

解决了多指标对称稳定过程水平集的Hausdorff测度问题，同时给出N指标d维稳定过程水平集的Hausdorff测度下界的最佳估计，它们揭示了其水平集的测度与局部时之间的内在关系，刻划了多指标对称稳定过程水平集的分形特征，表明了研究此类问题的积极意义，该问题是随机分形研究领域的国内外学者所关注的热点问题之一；解决了多指标稳定分量过程的局部时存在性与连续性问题，进而解决了由多指标稳定分量过程产生的随机分形集--象集、图集的测度问题；此外，本研究项目还解决了布朗单样本轨道的重分形分析，即高维白噪音的重分形分析问题。

Singer Leona Lewis and former Led Zeppelin guitarist Jimmy Page emerged as the bus transformed into a grass-covered carnival float, and the pair combined for a rendition of "Whole Lotta Love".

歌手leona刘易斯和前率领的飞艇的吉他手吉米页出现巴士转化为基层所涵盖的嘉年华花车，和一双合并为一移交&整个lotta爱&。

This is Kate, and that's Erin.

这是凯特，那个是爱朗。