Connections with prescribed first Pontrjagin form

Let be a principal bundle over a manifold of dimension . If , then we prove that every differential 4-form representing the first Pontrjagin class of is the Pontrjagin form of some connection on .     

Pontrjagin空间上算子代数理想的结构

对Π_k空间上一般对称算子代数,给出了对称理想的结构的两个结果．(1)令A是Π_k空间上一般对称算子代数．若M_1∩M_2≠{0},则存在对■~((k))不变的子空间V∈~(k)H~(k),满足M_1∩M_2=F(V) J,这里J=(■),T属于k×k矩阵代数,V=(R){VXX│X∈D},R和R⊥是对*-算子代数A_p~(k)不变的．(2)令A是Π_k空间上一般对称算子代数．设△=M_1∩M_2≠{0}．则M_2：△ U(Q),其中U(Q)是下列元的集(■),这里B∈A_p,q_i是算子代数U到R~⊥的线性映射,并满足条件：q(A B)=Aq(B),A,B∈A_p．     

Pontriagin空间上算子的交换性

证明了:当Pontrjagin空间上的正规算子A没有零性不变子空间时,Putnam- Fuglede定理成立;当正规算子A有零性不变子空问时,通过构造反例说明此时Putnam- Fuglede定理不成立,并对Π1空间上算子相关的交换性条件进行了讨论,得到了Π1空间上算子代数的二次交换定理．     

Pontrjagin空间上算子代数理想的对称性

本文证明了非退化的JC*-代数和非退化的JVN-代数的理想必是对称的.对退化的JC*-代数给出两种对称的理想和两种非对称的理想,说明了π1空间上JVN-代数除了第三类代数外无非对称理想,还构造两个关于JC*-代数理想对称性的例子.     

Pontrjagin空间上算子集合的两个稠密性定理

Ｋａｐｌａｎｓｋｙ稠密性定理￣［１］是ｖｏｎＮｅｕｍａｎｎ代数和Ｃ￣＊代数理论中一个基本而重要的定理。算子代数中许多深刻的结果都是以此为工具导出的。要在不定度规空间上探讨算子代数的性质，人们自然会关心在这类空间上是否存在同一类型的结果。本文的主要目的就是在Ｐｏｎｔｒｊａｇｉｎ空间上给出一个相应的稠密性定理。同时，我们还将给出关于完全正则自共轭算子的另一个稠密性的结果。     

Grassmann流形上Pontrjagin示性式的积分式公式

命Ｇ_n+m,n是一个Ｇｒａｓｓｍａｎｎ流形，Ｚ_2k＝是Ｇ_n+m,n的一个Ｓｃｈｕｂｅｒｔ流形。命Ｅ是Ｇ_n+m,n上的规范矢丛，ＥＣ是Ｅ的变化，Ｅ是ＥＣ的相配酉（ｎ－２ｋ）－标架丛，并且Ｅ′是ＥＣ的一个矢丛，本文证明了下列积分公式：其中ｃ_4k是Ｇ_n+m,n的４ｋ链，Ｐ_k（Ω）是Ｇ_n+m,n的第ｋ个Ｐｏｎｔｒｊａｇｉｎ示性式，是定义在Ｅ和Ｅ′上的（４ｋ－１）－形式。     

Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件

本文研究Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件,得到定理:设C0(U),C1(U,L,R,D,V),C2a(U),C2b(U,R),C3a(U),C3b(U,R)分别是Ⅱk空间上第0,Ⅰ,Ⅱa,Ⅱb,Ⅲa和Ⅲb类的算子代数,则(1)C0(U),C2a(U)或C3a(U)为一致闭(弱闭)的等价条件是U是Hibert空间G上的C*-代数(W*-代数;(2)C1(U,L,R,D,V)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间,V是闭算子,L对称闭的;(3)C2b(U,R)或C3b(U,R)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间．     

Syzygies of abelian varieties

We prove a conjecture of R. Lazarsfeld on the syzygies (of the homogeneous ideal) of abelian varieties embedded in projective space by multiples of an ample line bundle. Specifically, we prove that if is an ample line on an abelian variety, then satisfies the property as soon as . The proof uses a criterion for the global generation of vector bundles on abelian varieties (generalizing the classical one for line bundles) and a criterion for the surjectivity of multiplication maps of global sections of two vector bundles in terms of the vanishing of the cohomology of certain twists of their Pontrjagin product.

     

PrincipalG-bundles on nodal curves

LetG be a connected semisimple affine algebraic group defined over C. We study the relation between stable, semistable G-bundles on a nodal curveY and representations of the fundamental group ofY. This study is done by extending the notion of (generalized) parabolic vector bundles to principal G-bundles on the desingularizationC ofY and using the correspondence between them and principal G-bundles onY. We give an isomorphism of the stack of generalized parabolic bundles onC with a quotient stack associated to loop groups. We show that if G is simple and simply connected then the Picard group of the stack of principal G-bundles onY is isomorphic to ⊕_m Z,m being the number of components ofY.     

A Generalisation of Nagata's Theorem on Ruled Surfaces

We prove a generalisation of a theorem of Nagata on ruled surface to the case of the fiber bundle E/P X, associated to a principal G-bundle E. Using this we prove boundedness for the isomorphism classes of semi-stable G-bundles in all characteristics.     

Curvature identities derived from the integral formula for the first Pontrjagin number

We give an integral formula for the first Pontrjagin number of a compact almost Hermitian surface and derive curvature identities from the integral formula based on the fundamental fact that the first Pontrjagin number in the deRham cohomology group is a topological invariant. Further, we provide some applications of the identities.     

A note on quadratic forms

We extend an interesting theorem of Yuan [12] for two quadratic forms to three matrices. Let C ₁, C ₂, C ₃ be three symmetric matrices in ℜ ^n×n , if max{x ^T C ₁ x,x ^T C ₂ x,x ^T C ₃ x}≥0 for all x∈ℜ ⁿ , it is proved that there exist t _i ≥0 (i=1,2,3) such that ∑ _i=1 ³ t _i =1 and ∑ _i=1 ³ t _i C _i has at most one negative eigenvalue. Received February 18, 1997 / Revised version received October 1, 1997? Published online June 11, 1999     

A note on biquadratic forms

It is shown that semidefinite quadratic forms in two by n variables are sums of squares of bilinear forms.     

A note on Siegel modular forms

This paper gives a new identification for Siegel modular forms with respect to any congruence subgroup by investigating the properties of their Fourier-Jacobi expansions, and verifies a comparison theorem for the dimensions of the spaces Skn (Γn) and J0k, 1 (Γn) with small weight k. These results can be used to estimate the dimension of the space of modular forms.     

A note on tight extensive game forms

Associated with each extensive game form are the α- and β-effectivity functions,E _α andE _β. These functions characterize the effective capabilities which coalitions command under pessimistic and optimistic cooperative behavior respectively. A game form is said to be tight whenE _α ≡E _β, in which case pessimistic, optimistic, and intermediate types of behavior all support the same effective power. Dalkey's condition for Nash consistency, essentially perfect information, is found to be a necessary and sufficient condition for a game form to be tight.     

General forms of operator algebras on Pontrjagin space with neutral invariant subspaces

In this paper, we classify the general symmetric operator algebras on Pontrjagin space into six classes by using of symmetric ideals, and give the general forms for all the general symmetric operator algebras in each class.     

A note on harmonic forms on complete manifolds

In this note, we will prove that under certain conditions, the space of polynomial growth harmonic functions and harmonic forms with a fixed growth rate on manifolds which are asymptotically nonnegatively curved is finite dimensional. This is a partial generalization of the works of Li and Colding-Minicozzi. We will also give an explicit estimate for the dimension in case the manifold is a complete surface of finite total curvature. This is a generalization to harmonic forms of the work of Li and the author.

     

A note on principal parts on projective space and linear representations

Let be a closed subgroup of a linear algebraic group defined over a field of characteristic zero. There is an equivalence of categories between the category of linear finite-dimensional representations of , and the category of finite rank -homogeneous vector bundles on . In this paper we will study this correspondence for the sheaves of principal parts on projective space, and we describe the representation corresponding to the principal parts of a line bundle on projective space.

     

A note on moment inequality for quadratic forms

Moment inequality for quadratic forms of random vectors is of particular interest in covariance matrix testing and estimation problems. In this paper, we prove a Rosenthal-type inequality, which exhibits new features and certain improvement beyond the unstructured Rosenthal inequality of quadratic forms when dimension of the vectors increases without bound. Applications to test the block diagonal structures and detect the sparsity in the high-dimensional covariance matrix are presented.