Showing papers on "Equivariant map published in 1990"
TL;DR: In this article, the simplical depth of a point is defined as the probability that the point is contained inside a random simplex whose vertices are $p + 1$ independent observations from the distribution of the point.
Abstract: For a distribution $F$ on $\mathbb{R}^p$ and a point $x$ in $\mathbb{R}^p$, the simplical depth $D(x)$ is introduced, which is the probability that the point $x$ is contained inside a random simplex whose vertices are $p + 1$ independent observations from $F$. Mathematically and heuristically it is argued that $D(x)$ indeed can be viewed as a measure of depth of the point $x$ with respect to $F$. An empirical version of $D(\cdot)$ gives rise to a natural ordering of the data points from the center outward. The ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and $L$-statistics. This generalized sample median and $L$-statistics are affine equivariant.
828 citations
TL;DR: In this paper, the dynamics and bifurcation theory of equivariant dynamical systems near relative equilibria, that is, group orbits invariant under the flow of an EIF, is discussed.
Abstract: This paper discusses the dynamics and bifurcation theory of equivariant dynamical systemsnear relative equilibria, that is, group orbits invariant under the flow of an equivariant vector field. The theory developed here applies, in particular, to secondary steady-state bifurcations from invariant equilibria. Let $\Gamma $ be a compact group of symmetries of $R^n $ and let $x_0 $ be in $R^n $. Suppose that f is a smooth $\Gamma $-equivariant vector field and $\Sigma $ the isotropy group of $x_0 $. It is shown that there exists a $\Sigma $-equivariant vector field $f_N $, defined on the space normal to X at $x_0 $, and that the local asymptotic dynamics of f are closely related to the local asymptotic dynamics of $f_N $. Next those bifurcations of X are studied which occur when an eigenvalue of $(df_N )_x $ crosses the imaginary axis. Properties of the vector field $f_N $ imply that branches of equilibria and periodic orbits of $f_N $ correspond to trajectories of f which are dense in tori. Field [Equivaria...
211 citations
TL;DR: In this paper, the moment map on the cotangent bundle is used to define a finite cristallographic reflection group Wx, which generalizes the little Weyl group of a symmetric space.
Abstract: Summary. Let G be a connected, reductive group defined over an algebraically closed field of characteristic zero. We assign to any G-variety X a finite cristallographic reflection group Wx by means of the moment map on the cotangent bundle. This generalizes the "little Weyl group" of a symmetric space. The Weyl group Wx is related to the equivariant compactification theory of X. We determine the closure of the image of the moment map and the generic isotropy group of the action of G on the cotangent bundle. As a byproduct we determine the ideal of elements of 11(g) which act trivially on X as a differential operator.
132 citations
TL;DR: In this paper, the cohomology groups and characteristic classes are computed in terms of these filtrations, and problems of linear algebra which arise from them are discussed, and the results are discussed.
Abstract: Equivariant bundles on toral varieties are described in terms of filtrations which arise canonically in the fiber over a fixed point. The cohomology groups and characteristic classes are computed in terms of these filtrations, and problems of linear algebra which arise from them are discussed. Bibliography: 20 titles.
103 citations
TL;DR: In this paper, the authors studied the S -equivariant de Rham cohomology of infinite dimensional S -manifolds and showed that the A-polynomial of X is an equivariant characteristic class of the normal bundle to X, considered as the space of constant loops.
Abstract: In this paper we study the S -equivariant de Rham cohomology of infinite dimensional S -manifolds. Our main example is the free loop space LX where X is a finite dimensional manifold with the circle acting by rotating loops. We construct a new form of equivariant cohomology h* which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces LX. This gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of Witten [16], Atiyah [2], and Bismut [5]. In particular we show, following Atiyah in [2], that the A-polynomial of X arises as an equivariant characteristic class, in the theory h*, of the normal bundle to X, considered as the space of constant loops, in LX .
39 citations
38 citations
TL;DR: Chudnovsky et al. as mentioned in this paper considered the index of the Dirac operator on the unicersal elliptic genus p(M) and showed that the index is a modular form of weight k for a congruence subgroup of SL(2, Z), which depends on the level of the representation of Spin(d).
37 citations
01 Mar 1990
TL;DR: In this article, it was shown that any subgroup of infinite index in the fundamental group of a hyperbolic 3-manifold M is geometrically finite.
Abstract: If G is a discrete subgroup of PSL (2; C) representing a fibred 3manifold and H the subgroup of G corresponding to the fibre, we show that any finitely generated subgroup of infinite index in H is geometrically finite. We will prove that if K is a finitely generated subgroup of infinite index in the fundamental group H of the fibre of a hyperbolic 3-manifold M of finite volume, then K is geometrically finite. Work of Bonahon [Bo] and Thurston [Th2] implies that the hypothesis that M be a bundle can be weakened. The result one obtains is that if K is a finitely generated subgroup of infinite index in H, and if H is a geometrically infinite surface group contained in the fundamental group of a geometrically finite hyperbolic 3-manifold M, then K is geometrically finite. For it follows from [Bo] and [Th2] that M is finitely covered by M' which is a bundle with fibre whose fundamental group is commensurable with H. The proof of our result depends on a theorem of Cannon and Thurston [C-T], which we will now describe. Let G be a discrete subgroup of ISO(H3) so that M = H3/G is a fibre bundle over the circle and let H be the subgroup of G corresponding to the fibre F. Let S be a hyperbolic 2-manifold homeomorphic to the fibre F and let h: S -, M denote the homeomorphism into M. We will identify 7I (S) with a subgroup H' of ISO (H2) and thus h* is an isomorphism of H' onto H. The map h: S -, M lifts to a map h: H2 -, H3 and we consider the compactifications B2 = H2 U SL and B3= H3 U S2 . TIhe main theorem of [C-T] asserts that when S is closed there is an equivariant extension j: B2 -, B3 of h. The case when S is not closed but has finite area has been considered by Fenley [F]. We will denote by joo the restriction of j to S. Then j., is a space-filling curve in the 2-sphere S2. With the above notations, our result follows. Received by the editors August 4, 1989. 1980 Mathematics Subject C lassification (1985 Revision). Primary 30F40, 55AO5.
32 citations
Book•
05 Oct 1990
TL;DR: Theorems in equivariant surgery related to periodicity theorems and products and periodicity for surgery up to pseudoequivalence are studied.
Abstract: Summary: Background material and basic results.- to equivariant surgery.- Relations between equivariant surgery theories.- Periodicity theorems in equivariant surgery.- Twisted product formulas for surgery with coefficients.- Products and periodicity for surgery up to pseudoequivalence.
30 citations
TL;DR: In this paper, the authors obtained the best equivariant estimator of μ under the loss function L(μd)= (μ−d)′(μ −d μ′μ)
28 citations
01 Apr 1990
TL;DR: In this article, an equivariant version of the Cheeger-Chou index theorem on spaces with conelike singularities was proposed. But it is not shown that equivariance of q functions near the origin can be obtained by a modification of the Bismut-Cheeger index.
Abstract: We prove the regulairty of equivariant eta functions near the origin. We also propose an equivariant version of the Cheeger-Chou index theorem on spaces with conelike singularities. 0. INTRODUCTION Let M be an odd-dimensional compact Riemannian spin manifold with a fixed spin structure, D the Dirac operator on M. The q function associated to D is defined by [1] (0.1) r(s,D)= (sign)A dimF(E.) where A runs over the nonzero eigenvalues of D and J7(E.) is the eigenspace of A. If T: M -, M is an isometry preserving the orientation and spin structure and d TD = D d T, where d T: 1(S(M)) -, F(S(M)) is the lift of d T: 1(TM) -JT(TM) to the spinors, then one can also define the equivariant q function [1, 5] by (0.2) 11T(S , D) = E(signA) Tr dIT(E,) In the first two sections we prove some basic properties of equivariant q functions and in ?3, we point out that a slight modification of Bismut-Cheeger [3] yields an equivariant index theorem for spaces with conelike singularities. Received by the editors April 28, 1989 and, in revised form, June 30, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 58G 10, 58G25.
TL;DR: In this article, the bifurcation of small periodic solutions at a non-semi-simple 1:1-resonance in equivariant conservative systems was studied.
Abstract: We study the bifurcation of small periodic solutions at a non-semi-simple 1:1-resonance in equivariant conservative or equivariant time-reversible systems. By using an equivariant Liapunov-Schmidt method and restricting to solutions with an appropriate isotropy, we reduce the problem to a scalar bifurcation equation. The analysis of this equation shows a bifurcation behaviour similar to that found for the Hamiltonian Hopf bifurcation.
Journal Article•
TL;DR: In this article, the authors investigated the equivariant embedding of metrizable G-spaces from the point of view of the theory of retracts, and showed that if one of the spaces X and G has countable weight and X is a G-A(N)R for metrizability, then X/G is an A(n)R this article.
Abstract: In this paper there is an investigation, for the case of a compact group G, of the orbit space X/G of a given G-space X, from the point of view of the theory of retracts. A particular case of the main result asserts that if one of the spaces X and G has countable weight and X is a G-A(N)R for metrizable spaces, then X/G is an A(N)R for metrizable spaces. New results about the equivariant embedding of metrizable G-spaces are also obtained. Bibliography: 28 titles.
TL;DR: In this paper, an extension of the equivariant bifurcation lemma to the case of nonlinear Lie point symmetries is presented. And the role of the fixed subspaces under the symmetry subgroups is played by some well specified flow-invariant manifolds.
Abstract: The author provides an extension of the equivariant bifurcation lemma to the case of nonlinear Lie point symmetries. In this extension, the role of the fixed subspaces under the symmetry subgroups is played by some well specified flow-invariant manifolds. Some typical examples are also considered.
Abstract: In both equivariant bifurcation theory [GSS, especially Chapter XIII] and physical theories of spontaneous symmetry breaking (for example, the Higgs-Landau theory [M]), there is the problem of determining the symmetries, stabilities and branching patterns for solutions of equations equivariant under a compact Lie group G. Very few general results and techniques are known for the analysis of this problem, though versions of a Maximum Isotropy Subgroup Conjecture have been conjectured, to the effect that generically all solution branches have maximal isotropy (see for example [G, M]). General results of this type are of particular interest for applications on account of the inherent complexity of the structure of isotropy subgroups, invariants and equivariants for ^-representations. In this note, we announce several new results for the general study of the symmetries and branching patterns for a large class of G-equivariant bifurcation problems. In particular, we give new counterexamples to the Maximal Isotropy Subgroup Conjecture and present examples where one can get precise information on the branching patterns. Our methods also show that one can get quite detailed information on these problems without full knowledge of the G-equivariants. To simplify our exposition, we assume G finite. Let F be a finite dimensional real Hubert space and G —• 0{V) be an absolutely irreducible representation of the finite group G. Let G act on V xR by g • (x, A) = {g • x, A) and let 8? = CTM(V x R, V) be the space of smooth G-equivariant maps of V x R to V. Give S? the C°°-topology; subsets of 8? are given the induced topology. Each ƒ e S? defines a one-parameter family (/Â)A€R °f e Q u i v a r i a n t vector fields on V. We have /(O, X) = 0, A € R. These are the trivial zeros of ƒ . We study zeros of ƒ bifurcating off the trivial zeros. Now D{f(0, A) = of(X)Idv ,
TL;DR: In this article, the question whether new anomalies appear, connected with the finite dimensional part of the gauge group isomorphic to the structure group of the theory, is investigated in a systematical way.
Journal Article•
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/conditions) are defined, i.e., toute utilisation commerciale ou impression systématique is constitutive d'une infraction pénale.
Abstract: L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: In this article, it was shown that for any Grothendieck topos 8 there exists a space X and a collection P of paths in X so that 6 is equivalent to the category of sheaves on X which are constant along the paths in P.
TL;DR: In this article, the Wishart matrix S [S ∽ Wp(n, Σ) and an independent multinomial vector X [X ∽ Np (μ, ǫ) (Ω, Ω) were combined to obtain the best multiple of S and the Stein-type truncated estimators.
Abstract: Given a Wishart matrix S [S ∽ Wp(n, Σ)] and an independent multinomial vector X [X ∽ Np (μ, Σ)], equivariant estimators of Σ are proposed. These estimators dominate the best multiple of S and the Stein-type truncated estimators.
01 Jan 1990
TL;DR: In this article, it was shown that the equivariant L-groups are equal for the smooth and locally linear PL-category simultaneously, and that the exact orbit sequence of the R-Rothenberg sequence does not always split in a way analogous to the splitting of the Equivariant K-theory.
Abstract: Equivariant algebraic K-theory decomposes as a sum of ordinary algebraic Ktheory of group rings (see [4, 5, 7, 12]). This follows for K t because the determinant of a upper triangular matrix is the product of the entries on the diagonal. For equivariant L-theory the situation is more complicated. We do obtain a set of exact orbit sequences similar to the neighbouring family sequences obtained by Connor and Floyd in [2]. But these exact sequences do not always split in a way analogous to the splitting of equivariant K-theory. There are easy counterexamples for G=Z/2 . However, if the transformation group has odd order, then the equivariant L-groups do in fact decompose in the expected fashion, cf. Theorem 2.11 below. We work in this paper in the smooth and locally linear PL-category simultaneously. A consequence of the existence of the exact orbit sequence is that the equivariant L-groups are equal for these two manifold categories. The paper is founded upon the definition of equivariant L-theory given in [9]. We refer the reader to that paper for the somewhat cumbersome definitions. A reference (I.?. ?) always refers to the first part [9]. It is our hope that the present definitions of equivariant L-groups and the calculational techniques presented here will make further calculations possible. It seems to us to be of some interest to evaluate equivariant L-groups for some of the standard 2-groups, for example, and to determine the equivariant Rothenberg sequence.
TL;DR: In this paper, a statistical model for data which takes values in R d and have elliptically distributed errors, and affine equivariant estimators μ and μ of a mean vector in R n and a d × d scatter matrix, expressions are given for the covarances of the estimators in terms of their expectations and some unknown constants that depend on the model and the estimator.
TL;DR: In this paper, the authors investigated the properties of a version of the general equilibrium model where symmetry considerations are explicitely taken into account, called the equivariant model, which is shown to be a natural framework to study sunspot equilibria.
TL;DR: It is shown that C(n, m) is generated by traces of products of the corresponding generic matrices and, as such, coincides with the center of the trace ring of m generic n by n matrices R (n,m) which is also the ring of equivariant maps from Xm, n to Mn(ℂ).
Abstract: Consider the vector space of m -tuples of n by n matrices . The linear group GL n (C) acts on X m, n by simultaneous conjugation. The corresponding ring of polynomial invariants will be denoted by C ( n, m ) and is called the ring of matrix invariants of m -tuples of n by n matrices. C. Procesi has shown in [8] that C ( n, m ) is generated by traces of products of the corresponding generic matrices and, as such, coincides with the center of the trace ring of m generic n by n matrices R ( n, m ) which is also the ring of equivariant maps from X m, n to M n (ℂ).