辛平延的换位子之积

令F为特征不为2的域,V表F上2n维向量空间,Sp2n（V）表V上的辛群,对任一σ∈SP2n（V）,记resσ＝dim（σ－1）V．本文证明了当|F|＞9时,SP2n（V）中每一元素σ可表成不超过＋3个辛平延的换位之积。     

局部环上辛群生成定理

讨论了局部环上辛群的生成系 ,在探讨生成长度时 ,引入变换亏数的概念 ,用亏数和剩余数来刻划变换的生成长度     

射影辛群PSp(2n,F)中几类极大子群

本文定出了PS_p(2n,F)(n≥2)中含T-子群的全部极大子群(T-子群是指由某一方向上全部辛平延组成的子群).当F≠F₂时,这些极大子群的所有可能的类型为:PS_p(2n,F)所作用的空间V的全迷向子空间或维数小于n的非退化子空间的定驻子群;以及将V分成一些同维数的子空间的正交和时,这些子空间组成的集合的定驻子群(F=F₂的情形将在另一篇文章中讨论)。     

S_p(2n,F_2)中含长根子群的极大子群系

在[1]中,我们确定了当F≠F_2时射影辛群PS_p(2n,F)中含T-子群的全部极大子群。这里,T-子群是指任一方向上全体辛平延组成的群,也就是长根子群。本文将定出S_p(2n,F_2)中含长根子群的全部极大子群,从而完成对射影辛群中含长根子群的极大子群的分类。本文的结果是:     

关于同调维数若干命题的讨论

在[1]中,作者讨论了有限整体维数的半局部环上有限生成非零模的同调维数和余维数的和的性质;定义了模N的次全维数(pro-total dimension)ptd_sN、次整体维数(pro-glo-bal dimension) pgd_sN,以及半局部环S本身的全维数 (total dimennsion)tdS;紧接着在§4讨论了上述各种维数之间的关系,然后在§6应用前面的结果去研究tdS=2的半局部环的结构。但我们发现[1]文中的某些结论不成立,本文试图举出[1]文中某些命题不成     

弱半局部环的同调性质

环R称为弱半局部环,如果R/J(R)是Von Neumann正则环.给出了一个交换环是弱半局部环的充分且必要条件;还讨论了交换凝聚弱半局部环及其模的同调维数.     

右有限连通的正则半局部环

设Ｒ是一个右有限连通的正则半局部环。Ｒ是一个整环上的全矩阵环的充分必要条件被给出。同时，讨论了不同调维数时，Ｒ的结构。     

局部环上正交群生成定理

本文研究了局部环上正交空间中正交变换用对称表出的问题。讨论了表出长度与剩余数和亏数的关系,给出了表出长度时用剩余数和亏数所限定的界限。     

关于同调维数的一些注记

本文推广了环的R-序列的概念,引进了相伴R-序列,讨论了Jacobson根具有AR-性质的Noetherian环的同调维数及其结构性质,推广了交换局部环和半局部环的一些同调性质。     

Gr-凝聚环的小有限分次投射维数gr.fp.dimR

文 [1],[2 ]分别研究了Gr NoetherGr 局部 (半局部 )环的同调维数 ,本文主要进一步讨论Gr 凝聚Gr 半局部环的同调性质 .在§ 1中 ,主要刻画交换Gr 凝聚Gr 半局部环R的分次弱整体维数gr.gl.w .dimR ;在§ 2中 ,定义了分次环R的小有限分次投射维数gr.fp .dimR .刻画了gr.fp .dimR =gr .gl.w .dimR的Gr 凝聚环 .由于Gr Noether环是Gr 凝聚的 ,因而本文所得的结果对于Gr Noether环是自然成立的 .同时 ,本文所得的结果 ,也可视为文 [4 ]关于一般交换凝聚环相应结论的推广 .     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

Four-Dimensional Simply Connected Symplectic Symmetric Spaces

A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, ) belonging to this class is diffeomorphic to Rⁿwith the property that every tensor field on M invariant by the transvection group is constant; in particular, is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rⁿwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.     

Critical Landscape Topology for Optimization on the Symplectic Group

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.     

局部环上典型群的生成问题

回顾了域上典型群的生成问题所得成果,对局部环上典型群生成问题的研究,作了一个展示,并按展示的范式,给出了辛群生成定理的证明.     

一类条件数为常数的随机辛阵的性质

对A.Bunse-Gerstner和V.Mehrmann使用的一种机辛阵的性质进行了研究。证明了1）其可以通过正交相似变换化为一种特殊的Schur标准型；2）其条件数为一常数；（3）该常数约为2．618。     

Symplectic supercuspidal representations and related problems

In this paper, we summarize the basic structures and properties of irreducible symplectic supercuspidal representations of GLn(F) over a p-adic local field F with characteristic zero, and explore possible topics for further investigation.     

On subnormal subgroups of linear groups

We describe the subnormal subgroups of 2-dimensional linear groups over local and full rings in which 2 is invertible, as well as the subnormal subgroups of symplectic groups over local rings in which 2 is invertible.     

Deformation Quantization of Symplectic Fibrations

A symplectic fibration is a fibre bundle in the symplectic category (a bundle of symplectic fibres over a symplectic base with a symplectic structure group). We find the relation between the deformation quantization of the base and the fibre, and that of the total space. We consider Fedosov's construction of deformation quantization. We generalize the Fedosov construction to the quantization with values in a bundle of algebras. We find that the characteristic class of deformation of a symplectic fibration is the weak coupling form of Guillemin, Lerman, and Sternberg. We also prove that the classical moment map could be quantized if there exists an equivariant connection.     

Reissner板弯曲的辛求解体系

基于Reissner板弯曲问题的Hellinger-Reissner变分原理，通过引入对偶变量，导出Reissner板弯曲的Hamilton对偶方程组．从而将该问题导人到哈密顿体系，实现从欧几里德空间向辛几何空间．拉格朗日体系向哈密顿体系的过渡．于是在由原变量及其对偶变量组成的辛几何空间内，许多有效的数学物理方法如分离变量法和本征函数向量展开法等均可直接应用于Reissner板弯曲问题的求解．这里详细求解出Hamilton算子矩阵零本征值的所有本征解及其约当型本征解，给出其具体的物理意义．形成了零本征值本征向量之间的共轭辛正交关系．可以看到，这些零本征值的本征解是Saint—Venant问题所有的基本解，这些解可以张成一个完备的零本征值辛子空间．而非零本征值的本征解是圣维南原理所覆盖的部分．新方法突破了传统半逆解法的限制，有广阔的应用前景。