给定能量的N体型问题多个几何上不同的周期轨道的存在性

本文利用等变的Ｌｊｕｓｔｅｒｎｉｋ－Ｓｃｈｎｉｒｅｌｍａｎｎ理论证明了平面上的一类给定能量的Ｎ体型问题至少存在２·（Ｎ—２）·２Ｎ－３个几何上不同的非碰撞周期轨道．     

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.     

A Cl closing lemma for equivariant endomorphisms under actions of finite groups

A C^l closing lemma is proved for equivariant endomorphisms under actions of finite groups. Our result shows that, for such an endomorphism, a nonwandering orbit together with its symmetric conjugates can be closed up under a Cl-small equivariant perturbation Project partially supported by the National Natural Science Foundation of China (Grant No. 19531070).     

含两组状态变量等变分歧问题开折的唯一性和稳定性

基于奇点理论中光滑映射芽的接触等价,讨论多参数等变分歧问题关于接触等价的开折的唯一性和稳定性.将这种等变分歧问题的状态变量分为两组,其中一组的诸状态变量可以独立地变化,而属于另一组的诸状态变量在变化过程中依赖于前一组中的诸状态变量.得出了在接触等价下,满足一定条件的等变分歧问题的万有开折是唯一的,并且给出了一定条件下等变分歧问题开折稳定的一个充要条件.     

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.     

Bifurcation of limit cycles near equivariant compound cycles

In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.     

Multiple geometrically distinct closed noncollision orbits of fixed energy for N-body type problems with strong force potentials

Using the equivariant Ljusternik-Schnirelmann theory, we prove that there are at least geometrically distinct noncollision orbits with prescribed energy for a class of planar N-body type problems with strong force potentials.

     

含两组状态变量的等变分歧问题在左右等价群下的开折

基于奇点理论中光滑映射芽的左右等价关系, 讨论多参数等变分歧问题关于左右等价的开折．将这种等变分歧问题的状态变量分为两组,其中属于同一组的诸状态变量可以独立地变化,而属于另一组的诸状态变量在变化过程中依赖于前一组中的诸状态变量．应用光滑映射芽开折理论中的相关方法和技巧,得到了一个含两组状态变量的多参数等变分歧问题的开折是通用开折的充要条件．     

On the Least Number of Fixed Points of an Equivariant Map

The problem on the least number of fixed points of an equivariant map of a compact polyhedron on which a finite group acts is considered. For such a map, the least number of fixed points and the least number of fixed orbits are estimated in terms of invariants of the type of Nielsen numbers. The estimates obtained are sharp. The results are similar to those of P. Wong, but their assumptions are essentially weaker. Some notations are refined. The proofs are constructive.     

Spaces of Maps into Classifying Spaces for Equivariant Crossed Complexes, II: The General Topological Group Case

The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces.     

含两组状态变量且参数具有对称性的等变分歧问题及其开折的稳定性

基于奇点理论中光滑映射芽的接触等价关系,讨论含两组状态变量且分歧参数带有对称性的等变分歧问题及其开折的稳定性,得到了一些基本结果,并且用横截性条件刻划了等变分歧问题的稳定性．     

Equivariant resolution of points of indeterminacy

We prove an equivariant form of Hironaka's theorem on elimination of points of indeterminacy. Our argument uses canonical resolution of singularities and an extended version of Sumihiro's equivariant Chow lemma.

     

Estimation of the entropy of a multivariate normal distribution

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.     

The Universal Functorial Equivariant Lefschetz Invariant

We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K₀ of the category of “ ϕ -endomorphisms of finitely generated free RΠ(G, X)-modules”. We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds. Received: February 2006     

Shape Morphisms to Transitive G-Spaces

The following problem plays an important role in shape theory: find conditions that guarantee that a shape morphism F:X Y of a topological space X to a topological space Y is generated by a continuous mapping f:X Y. In the present paper, we study this problem in equivariant shape theory and give a solution for shape-equivariant morphisms to transitive G-spaces, where G is a compact group with countable base. As a corollary, we prove a sufficient condition for equivariant shapes of a G-space X to be equal to the group G itself. We also prove some statements concerning equivariant bundles that play the key role in the proof of the main results and are of interest on their own.     

The involution on the equivariant Whitehead group

For a finite group G we define an involution on the equivariant Whitehead group given by reversing the direction of an equivariant h-cobordism. It turns out that the involution is not compatible with the splitting of the equivariant Whitehead group into a direct sum of algebraic Whitehead groups, certain correction terms involving the transfer maps of the normal sphere bundles of the various fixed point sets come in. However, if the group has odd order, these transfer maps all vanish. We prove a duality formula for a G-homotopy equivalence (f f): (M; M) (N, N) relating the equivariant Whitehead torsion of f and (f,f).     

Comparing different versions of tiling cohomology

We establish direct isomorphisms between different versions of tiling cohomology. The first version is the direct limit of the cohomologies of the approximants in the Anderson-Putnam-Gähler system, the second is the recently introduced PV-cohomology of Savinien and Bellissard and the third is pattern equivariant cohomology. For the last two versions one can define weak cohomology groups. We show that the isomorphisms extend to the weak versions. This leads to an alternative formulation of the pattern equivariant mixed quotient group which describes deformations of the tiling modulo topological conjugacy.     

Equivariant Novikov conjecture for groups acting on Euclidean buildings

We prove the equivariant Novikov conjecture for groups acting on Euclidean buildings by using an equivariant Hilsum-Skandalis method. We also obtain an equivariant version of the Connes-Gromov-Moscovici theorem for almost flat -algebra bundles.

     

Invariants de Maslov équivariants

We construct two cohomological invariants associated to pairs of Lagrangian sub-bundles of a symplectic bundle on a compact manifold upon which a compact Lie group is acting. The first invariant, which we call the classical equivariant Maslov H-invariant, provides an obstruction to Lagrangian transversality and lives in the Borel cohomology. The second invariant, which we call the equivariant Maslov U-invariant, generalises the author's results in K-Theory 13 (1998), 347–361 to the equivariant context and provides a necessary and sufficient condition for equivariant Lagrangian transversality, up to homotopic stability, and lives in the U-theory (intermediate between the real complex K-theories). As an application, we show that two Lagrangian sub-bundles of a symplectic bundle on a homogeneous space are always stably transverse.