四元数矢丛的辛Pontryagin示性式和它的积分公式

设ρ（Ｍ）是微分流形Ｍ上的左四元数矢丛，文中给出ρ（Ｍ）的辛Ｐｏｎｔｒｙａｇｉｎ示性类和辛Ｐｏｎｔｒｙａｇｉｎ示性式的定义，并给出说明它们之间的联系的积分公式。     

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。     

等变上同调与ech超上同调的同构定理

首先证明了任意一个等变微分流形都存在等变良好开覆盖,且等变良好开覆盖集所组成的集合在全部开覆盖组成的集合中共尾.在此基础上,证明了等变上同调与ech超上同调的同构.此定理可应用于实代数簇的Deligne上同调研究.     

定积分公式及其应用

列举定积分计算的一些常用公式,实例说明它们在一此特殊函数求定积分计算方面的应用.     

妙用公式求积分

利用一些特殊的公式和技巧计算定积分,可以简化一些定积分的计算。     

含有三角函数的一个积分公式

定理:设f(x)在La,a十ZT〕上可积(T>0)甲(x)在〔a,a十T〕上可积,且满足条件:了(Za+一{f(t)己‘+2,一)一,(二)+,(二),贝。有{f(二)‘:- O}T{叫‘)dr{:‘T,“’“’证明:J了(·)‘一{:‘’,(·)‘·十{口+2个f(x)‘:.在右端第二个积分中,令‘=2a+ZT一‘,‘任〔a,a+TJ一则当:二a+T时,t=a+T;二二a+2少时,t=a;又由条件:f(2。+ZT一t)=一f(t)+p(t),因此{:‘’r了(·)‘一{:‘r,‘·,‘二 实用中验证函数满足条件f(Za+ZT一劝二一f(x)+中(x)井不困难,因为我们总可取p(x)二f(Za+ZT一:)+f仁).问题在于甲(:)是否容易积分.而当甲(:)为零,常数或…     

陈氏示性式的积分公式

陈国才去年在[1]中给出陈氏示性式的一个积分公式,他所用的方法与吴光磊[2]中的方法类似,都是把问题转化到复Grassmann流行中去解决。本文中将给出一种在流形上直接进行计算的方法,不必通过Grassmann流形,从而简化了陈国才的证明。最后我们将给出吴光磊的积分公式的一个简单的证明。     

复Grassmann流形上陈氏示性式的积分公式──谨以此文纪念我们的老师吴光

在本文中给出了复Ｇｒａｓｓｍａｎｎ流形上陈氏示性式Ｃ＿ｋ（Ω）的积分公式如下：     

Equivariant Novikov Inequalities

We establish an equivariant generalization of the Novikov inequalities which allows us to estimate the topology of the set of critical points of a closed basic invariant form by means of twisted equivariant cohomology of the manifold. We apply these inequalities to study cohomology of the fixed points set of a symplectic torus action. We show that in this case our inequalities are perfect, i.e. they are in fact equalities.     

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and let A be a Lie algebroid on M equipped with a compatible infinitesimal G-action. Using these data, we construct an equivariant cohomology of A and prove a related localization formula for the case of compact G. By way of application, we prove an analog of the Bott formula.     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

For an algebra with an action of a Hopf algebra we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for . We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg's theorem relating equivariant K-theory to ordinary K-theory of the C^*-algebra crossed product, and characterize equivariant vector bundles on quantum homogeneous spaces.     

Equivariant Extensions of Categorical Groups

If is a group, then the category of -graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from . In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreiers theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory).Mathematics Subject Classifications (2000) 18D10, 18B40, 20J05, 20J06.Partially supported by MTM2004-01060.     

数值积分公式中间点的渐近性质及其应用

主要研究了三类数值积分公式的中间点的渐近性质,得到了更一般性的结果.基于中间点的渐近性质,获得了数值积分的校正公式及其条件误差估计.数值例子显示了校正公式的精度明显高于对应的计算公式.     

Intersections and Translative Integral Formulas for Boundaries of Convex Bodies

Let K, L ⊂ ℝⁿ be two convex bodies with non–empty interiors and with boundaries ∂K, ∂L, and let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t ↦ χ(∂K ∩ (∂L + t)) and t ↦ χ(∂K ∩ (L + t)), t ∈ ℝⁿ, with respect to an n–dimensional Haar measure of ℝⁿ can be expressed in terms of certain mixed volumes of K and L. In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let K_r, L_r denote the outer parallel bodies of K, L at distance r ≥ 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of ∂K_r ∩ ∂L_r and ∂K_r ∩ L_r, respectively, is independent of r ≥ 0 if K° ∩ L° ≠ ∅︁ and if ∂K and ∂L intersect almost transversally.As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey .     

Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

Integral Multilinear Forms on C(K, X) Spaces

We use polymeasures to characterize when a multilinear form defined on a product of C(K, X) spaces is integral.