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.     

增长曲线模型中UMRE估计的存在性

对于设计矩阵不满秩，协方差阵任意或具有均匀结构或序列结构的正态增长曲线模型，本文讨论参数矩阵的一致最小风险同变（UMng）估计的存在性．在仿射变换群GI和转移交换群、二次损失和矩阵损失下本文分别获得存在回归系数矩阵的线性可估函数矩阵的UMRE估计的充要条件，推广了由[21]给出的在设计矩阵满秩下估计回归系数矩阵的结果．本文还首次证明了在群G1和二次损失下不存在协方差阵V和trV的UMRE估计．     

一种对称损失函数下正态总体刻度参数的估计

本文研究正态分布中刻度参数在损失函数L(σ，δ)=[(σ-δ)^2]/σδ下的最小风险同变估计及Bayes估计，并讨论(cT(x) d)^1/2形式估计的可容许性与不可容许性，我们发现在这种损失下σ的极大似然估计是不可容许的．     

Equivariant Sternberg–Chen Theorem

Smooth (C ) vector fields with linear symmetries and anti-symmetries are considered. We prove that provided symmetry group is compact then a smooth conjugacy in Sternberg–Chen Theorem can be chosen symmetric.     

Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems

We consider a special class of monotone dynamical systems and show that in this special class the stable and unstable manifolds of two hyperbolic periodic orbits always intersect transversally. The proof is based on the existence of a family of positively invariant nested cones.This paper is dedicated to Jack Hale on the occasion of his 60th birthday.     

EQUIVARIANT SELF EQUIVALENCES OF PRINCIPAL FIBRE BUNDLES

Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known.     

Lie Symmetries and Weak Transversality

We discuss the notion of weak transversality and clarify its role in the context of Lie group theory with several examples. In particular, we present some new results concerning models of fluid dynamics.     

Equivariant RO(G)-graded bordism theories

Let G be a finite group. The RO(G)-graded bordism theories of Pulikowski [7] and Kosniowski [3] are studied. Representing equivariant Thom spectra are constructed, and the relevant transversality results proved. New methods for splitting away from the order of G are described, and behavior in the presence of a gap hypothesis is examined.     

-actions with

Suppose that is a smooth -action on a closed smooth -dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each connected component of the fixed point set vanish in positive dimension. This paper shows that if 2^k\dim F$"> and each -dimensional part possesses the linear independence property, then bounds equivariantly, and in particular, is the best possible upper bound of if is nonbounding.

     

Representation Rings of Lie Superalgebras

Given a Lie superalgebra , we introduce several variants of the representation ring, built as subrings and quotients of the ring of virtual -supermodules, up to (even) isomorphisms. In particular, we consider the ideal of virtual -supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring on which the parity reversal operator takes the class of a virtual -supermodule to its negative. We also construct representation groups built from ungraded -modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring , including all degree shifts, is then a -graded ring in the complex case and a -graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings , and . We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring . In the real case, we obtain the expected 24-term version, as well as a surprising six-term version, of this periodic exact sequence. (Received: October 2004)