辛群胚与泊松群胚作用的充要条件

研究了辛群胚与泊松群胚的作用.利用李群胚作用及相关性质,得到了李群胚作用成为辛群胚和泊松群胚作用的充要条件,推广了辛群胚和泊松群胚的性质,为辛群胚与泊松群胚理论的进一步研究起到了推动作用.     

关于李群胚和泊松作用的讨论

定义了纤维丛的相配群胚的概念,从作用的角度研究了李群胚与主丛的关系;给出了一个泊松群胚在泊松流形上的作用是泊松作用的充要条件;文末得到了一些关于泊松流形上Casimir函数的结果.     

关于群胚作用和李双代数胚态射

在这篇文章中，我们讨论了李双代数胚之间的态射，得到了一些李双代数胚之间态射的性质．研究了泊松群胚在泊松流形上的泊松作用，以及这个泊松作用与被作用流形的切李双代数胚到作用泊松群胚的切李双代数胚之间的态射的关系，得到了一些有用的结论。     

G匚W上的泊松群胚结构

从泊松作用的角度考察了群胚上的半直积结构,定义了泊松群胚对泊松群胚的泊松作用,讨论了其性质,并证明了两个泊松群胚的半直积仍是泊松群胚,从而对群胚的半直积结构有了更多的认识.     

关于辛群胚的拉格朗日双截面

本文研究了辛群胚上的拉格朗日双截面.利用李群胚和辛群胚的相关性质, 得到了李群胚和辛群胚的李代数胚的截面空间中向量场X的指数映射能成为拉格朗日双截面的充分必要条件,并给出了辛群胚之间的一种同构,推广了拉格朗日双截面在群胚理论中的应用.     

群胚上的Dirac结构及泊松约化

本文详细讨论了李双代数胚中的Dirac结构、群胚上的Dirac结构。利用Dirac结构的特征对的概念,给出了作用不变Dirac结构,拉回Dirac结构等概念的新的刻画。最后利用Dirac结构的有关性质,讨论了泊松齐性空间和泊松群胚作用的约化。     

李群的余切丛上的两种辛群胚结构

本文研究了李群的余切丛上的辛群胚结构.利用李代数胚的对偶丛上有自然诱导的泊松结构,.构造出了同一余切丛上的不同的辛群胚结构,推广了辛群胚的性质.     

复合泊松过程的可加性

对复合泊松分布可加性的研究在许多的文献中都可以看到,本文首先应用特征函数的方法证明了复合泊松分布的可加性.以此为基础,结合对随机过程相关性质的讨论,证明了复合泊松过程也具有与复合泊松分布可加性相似的,某种意义上的可加性性质.     

双复合泊松风险模型下的投资问题

在经典风险模型基础上,研究了保险公司保费收入和索赔均服从复合泊松过程的双复合泊松风险模型,针对最优投资策略和求解破产时刻惩罚金期望折现函数的问题,利用重期望公式和马氏性得到期望折现函数满足的带边界条件的二阶积分微分方程,通过高效的Sinc数值方法求出折现函数的近似数值解,从而由图像分析破产概率变化的趋势.     

一类薛定谔-泊松方程解的存在性

本文研究一类具有变号权的薛定谔-泊松方程{-△u+u+k(x)φu=a(x)|u|p-1u,x∈R3,-△φ=k(x)u2,x∈R3解的存在性,其中3≤p＜5,a(x)为一连续的变号权且lim|x|→∞=a∞＜0,k(x)连续且k(x)∈L2(R3).我们将证明该方程至少存在一个非平凡的解.     

关于Poisson群胚的结构

令（ΓP,α,β）是Poisson群胚．如果它的每个α-纤维与β-纤维至多交于一点,则Γ在任一点x的特征分布有直和分解△（x）=△α（x）+△β（x）,其中△α（x）Txα－1（u）,△β（x）Txβ-1（u）且它们都是△（x）的辛子空间．由此得到辛叶Sx的辛子流形S和S,使在映射α之下,S辛微分同胚于P中辛叶Su,在映射β之下,S反辛微分同胚于P中辛叶Sv（定理4和5）．对于一般的Poisson群胚,也可得到类似的S和S,它们差一局部辛微分同胚是唯一确定的（定理6）．把以上结果用于辛群胚,还可得到一些更具体的性质（定理7及其推论）．     

On the Action of a Groupoid on a Manifold

In this paper,some properties of reduction for symplectic F-spaces are discussed.The properties of stable subgroups are discussed.We find that the symplectic action of a symplectic groupoid on a symplectic manifold can induce a symplectic map between reduced symplectic manifolds.This symplectic action can be characterized by the action of its induced symplectic groupoid on a symplectic manifold.Lastly,we shall discuss Poisson reduction and give a Poisson reduction theorem.     

Groupoid的诱导表示

设Ｇ为第二可数局部紧有Ｈａａｒ系的Ｇｒｏｕｐｏｉｄ， Ｈ为子Ｇｒｏｕｐｏｉｄ闭于Ｇ，则可得Ｇｒｏｕｐｏｉｄ　Ｈ＼Ｇ２，我们证明了Ｃ＊（Ｈ）与Ｃ＊（Ｈ＼Ｇ２）是Ｍｏｒｉｔａ等价的，从而回答了［１］中的问题．利用此非本原双模及定义Ｃ＊（Ｇ）到Ｍ（Ｃ＊（Ｈ＼Ｇ２））的映射，得到了由Ｃ＊（Ｈ）到Ｃ＊（Ｇ）的诱导表示．特别在群丛情形，我们定义了Ｃ＊（Ｈ）→Ｍ（Ｃ＊（Ｇ））的映射，并具体得到了诱导表示的积分形式的表达式．     

A Poisson manifold of strong compact type

We construct a corank one Poisson manifold which is of strong compact type, i.e., the associated Lie algebroid structure on its cotangent bundle is integrable, and the source 1-connected (symplectic) integration is compact. The construction relies on the geometry of the moduli space of marked K3 surfaces.     

Bisections of graphs

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions and conjectures of Bollobás and Scott, we study maximum bisections of graphs. First, we extend the classical Edwards bound on maximum cuts to bisections. A simple corollary of our result implies that every graph on n vertices and m   edges with no isolated vertices, and maximum degree at most n/3+1$n/3+1$, admits a bisection of size at least m/2+n/6$m/2+n/6$. Then using the tools that we developed to extend Edwards?s bound, we prove a judicious bisection result which states that graphs with large minimum degree have a bisection in which both parts span relatively few edges. A special case of this general theorem answers a conjecture of Bollobás and Scott, and shows that every graph on n vertices and m   edges of minimum degree at least 2 admits a bisection in which the number of edges in each part is at most (1/3+o(1))m$(1/3+o(1))m$. We also present several other results on bisections of graphs.     

On the existence of global bisections of Lie groupoids

We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed point is also discussed. We give some interesting applications of these results.     

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.     

Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces

We obtain the full classification of coisotropic and polar isometric actions of compact Lie groups on irreducible Hermitian symmetric spaces.