...x in the category of $\mathbf{Z}$-modules over $\mathcal{O}_G$. Similarly, equivariant local cohomology can be described using modules over a category depending o
For $RO(G)$-graded equivariant ordinary cohomology one has to replace the orbit category $\mathcal{O}_G$ w
3 KB (473 words) - 18:52, 28 October 2016
$#C+1 = 30 : ~/encyclopedia/old_files/data/E036/E.0306090 Equivariant cohomology
...ogy]] that takes the action of some group into account. More precisely, an equivariant cohomology in the category of $ G $-
5 KB (772 words) - 11:12, 20 January 2021
An ordinary [[Equivariant cohomology|equivariant cohomology]] for a finite group $ G $,
cf., e.g., [[Equivariant cohomology|Equivariant cohomology]]). Important examples of coefficient systems are the [[Homotopy
5 KB (732 words) - 08:45, 26 March 2023
An [[Equivariant estimator|equivariant estimator]] for the shift parameter with respect to a group of real shifts,
if one uses an equivariant estimator $ \widehat \theta = \widehat \theta ( X) $
4 KB (549 words) - 19:05, 17 January 2024
$#C+1 = 21 : ~/encyclopedia/old_files/data/E036/E.0306100 Equivariant estimator
is an equivariant estimator, or that it preserves the structure of the problem of statistical
3 KB (368 words) - 19:37, 5 June 2020
A statistical decision problem is said to be $ G $-equivariant under a group $ G $
etc.) is $ G $-invariant or $ G $-equivariant. Under these conditions, the decision rule $ \delta : \omega \rightarrow
5 KB (735 words) - 07:44, 13 May 2022
...omments to) [[Morse theory|Morse theory]] and [[#References|[a1]]]) and to equivariant Morse functions (cf. [[#References|[a2]]] and [[#References|[a3]]]).
...TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Wasserman, "Equivariant differential topology" ''Topology'' , '''8''' (1969) pp. 127–150</TD><
3 KB (502 words) - 08:01, 6 June 2020
is an equivariant mapping. If $ G $
is an equivariant polynomial mapping from $ X $
3 KB (511 words) - 13:16, 7 April 2023
...uivariant" and also what is meant by "E1 and E2 are equivalent" , as $G$-equivariant vector bundles.
...m }$. The equivalence class of this representation depends only on the $G$-equivariant holomorphism class of $E$. If $M$ is a homogeneous $G$-space, this correspo
10 KB (1,550 words) - 16:55, 1 July 2020
equivariant (i.e. is a homomorphism of $ \Gamma $ -
equivariant homomorphisms is an equivalence of these categories.
4 KB (534 words) - 11:05, 17 December 2019
...References|[a2]]], Sect. II.5, and [[#References|[a3]]], Sect. 5.2. For an equivariant version, see [[#References|[a4]]], p. 300ff.
...valign="top">[a4]</td> <td valign="top"> J.P.C. Greenlees, J.P. May, "Equivariant stable homotopy theory" I.M. James (ed.) , ''Handbook of Algebraic Topolog
3 KB (432 words) - 16:46, 1 July 2020
====Equivariant Morse lemma====
...he change of variables $\varphi$ can be taken $G$-invariant. An analogous "equivariant Morse lemma" is true in the real-analytic and the differentiable context. C
5 KB (792 words) - 09:32, 28 June 2014
...^ { h G } = \operatorname { Map } _ { G } ( E _ { G } , X )$, the space of equivariant mappings from the contractible space $E _ { G }$ on which $G$ acts freely t
3 KB (522 words) - 16:57, 1 July 2020
In this case one considers equivariant decision rules $ \delta : X \rightarrow D $(
in the class of equivariant decision rules for every invariant loss function satisfying the following c
5 KB (653 words) - 20:28, 17 January 2024
...the identity mapping $S \rightarrow S$ (cf. also [[Equivariant cohomology|Equivariant cohomology]]), is a [[Homeomorphism|homeomorphism]] of $G \times_{ G _ { x
iv) there is an equivariant mapping $\pi : G ( S ) \rightarrow G ( x )$ that is the identity on $G ( x
11 KB (1,819 words) - 15:30, 1 July 2020
...p of order $r$ in dimension $m$. Several methods for finding $G_m ^ { r }$-equivariant mappings in the $C ^ { \infty }$-case are collected in [[#References|[a1]]]
4 KB (674 words) - 17:02, 1 July 2020
In particular, a morphism is called equivariant if $ \theta _ {b} = \theta $
an equivariant morphism is called a $ G $-
6 KB (847 words) - 20:45, 12 January 2024
...></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.M. Møller, "On equivariant function spaces" ''Pacific J. Math.'' , '''142''' (1990) pp. 103–119</
2 KB (333 words) - 19:38, 5 June 2020
...s for $G/P$: singular (or de Rham) cohomology, the Chow ring, K-theory, or equivariant or quantum versions of these theories. For each, the Schubert cycles form a
3 KB (371 words) - 18:45, 30 March 2012
...ry of simplices; covering spaces, and in particular Cayley graphs; and the equivariant theory. It is also essential for the [[Closed category|closed category]] st
...ry in the sense of D. Quillen, but stronger results in some areas, such as equivariant theory [[#References|[a4]]], can be obtained by constructing the appropriat
13 KB (1,937 words) - 13:10, 24 December 2020
equivariant mapping $ S: M _ {k} \rightarrow \mathop{\rm GL} ^ {k} ( n)/G $,
equivariant $ 1 $-
17 KB (2,677 words) - 19:40, 5 June 2020
...h take into account additional structures in certain bundles, for instance equivariant $ K $-theory.
In equivariant $ K $-theory one considers spaces $ X $
17 KB (2,459 words) - 07:32, 26 February 2022
===The equivariant case.===
This applies to functions which are equivariant under the action of a Lie group. Cf. [[#References|[a1]]]. There are applic
21 KB (3,095 words) - 08:01, 6 June 2020
are as in the last section of the main article above (e.g., for
equivariant $ K $-
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hodgkin, "The
equivariant Künneth theorem in <img align="absmiddle" border="0" src="https://www.ency
16 KB (2,248 words) - 22:15, 5 June 2020
equivariant mapping of the manifold of $ k $-
5 KB (766 words) - 22:14, 5 June 2020
...ign="top">[a6]</TD> <TD valign="top"> R.T. Živaljević, "User's guide to equivariant methods in combinatorics" ''Publ. Inst. Math. Belgrade'' , '''59 (73)'''
5 KB (674 words) - 19:43, 5 June 2020
...so [[Cohomology of groups|Cohomology of groups]]; [[Equivariant cohomology|Equivariant cohomology]].
16 KB (2,386 words) - 16:47, 20 January 2024
...rther progress was made until 1983, when Wu-yi Hsiang, in the framework of equivariant differential geometry, constructed infinitely many mutually non-congruent m
...rther results related to the spherical Bernstein problem, by investigating equivariant systems of the above type, as well as additional isoparametric foliations o
10 KB (1,518 words) - 21:22, 14 January 2021
...considered. In [[Linear regression|linear regression]] analysis, the first equivariant high-breakdown estimator was the least median of squares [[#References|[a5]
5 KB (759 words) - 08:12, 6 June 2020
...D valign="top">[a5]</TD> <TD valign="top"> D. Quillen, "The spectrum of an equivariant cohomology ring I–II" ''Ann. of Math.'' , '''94''' (1971) pp. 549–602 {
6 KB (868 words) - 22:16, 5 February 2021
...ven a reformulation of Alperin's conjecture which is expressed in terms of equivariant [[K-theory| $ K $-theory]]. E.C. Dade has continued the pattern of predict
...} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J. Thévenaz, "Equivariant $K$-theory and Alperin's conjecture" ''J. Pure Appl. Algebra'' , '''85''' (
8 KB (1,167 words) - 06:46, 26 March 2023
...at do not disrupt the symmetry of the problem (see [[Equivariant estimator|Equivariant estimator]]).
17 KB (2,334 words) - 19:44, 7 January 2024
equivariant moment mapping
7 KB (1,006 words) - 08:24, 6 June 2020
<TR><TD valign="top">[7]</TD> <TD valign="top"> D.B. Zagier, "Equivariant Pontryagin classes and applications to orbit spaces" , Springer (1972)</TD
7 KB (1,066 words) - 06:29, 26 March 2023
...angular momentum, hence its name. The momentum mapping is infinitesimally equivariant for the $\frak g$-actions if $j$ is a homomorphism of Lie algebras. It is a
6 KB (935 words) - 10:58, 2 July 2020
for some smooth $G$-equivariant map $\mu:M\to \mathfrak{g}^*$. Here $\mu^*:\mathfrak{g}\to C^\infty(M)$ den
6 KB (1,109 words) - 13:50, 12 December 2013
the equivariant index theorem for a single elliptic operator equivariant with respect to a compact group $G$ (Atiyah–Segal, [[#References|[a5]]]);
===Equivariant index theorem.===
35 KB (5,243 words) - 17:45, 1 July 2020
equivariant mapping of the principal $ \mathop{\rm GL} ^ {k} ( n) $-
9 KB (1,337 words) - 22:16, 5 June 2020
The existence of an equivariant chain mapping
8 KB (1,093 words) - 08:21, 13 January 2024
is equivariant under the action of $ G $
8 KB (1,274 words) - 22:13, 5 June 2020
...">[32]</TD> <TD valign="top"> N. Berline, M. Vergne, "A computation of the equivariant index of the Dirac operator" ''Bull. Soc. Math. France'' , '''113''' (1985)
41 KB (5,908 words) - 22:12, 5 June 2020
== Equivariant partition models ==
is said to be equivariant under the induced action of
36 KB (5,856 words) - 18:32, 17 April 2024
...e Conley index has been refined and extended in several directions. For an equivariant version together with a product structure on the cohomology level, see [[#R
9 KB (1,428 words) - 16:58, 1 July 2020
equivariant $ 1 $-
11 KB (1,659 words) - 08:01, 21 January 2024
The Dixmier mapping is an equivariant mapping with respect to the adjoint [[Algebraic group|algebraic group]] $G$
11 KB (1,723 words) - 17:02, 1 July 2020
equivariant mappings $ S: M _ {k} \rightarrow W $.
11 KB (1,705 words) - 14:55, 7 June 2020
with an $\operatorname{SL} ( 2 , \mathbf{Q} )$-equivariant topology that is the given one on $H$ and makes $\mathbf{P} ^ { 1 } ( \math
10 KB (1,509 words) - 16:57, 1 July 2020
...lds outside, possibly, a set of singular points of measure zero. Later the equivariant Plateau problem was solved by W.Y. Hsiang and Lawson [[#References|[16]]].
...line of research on the multi-dimensional Plateau problem, namely that of equivariant multi-dimensional Plateau problems. Among the globally-minimal surfaces, on
29 KB (4,259 words) - 11:43, 3 March 2022
...v developed a solution to this problem. Kasparov and others have used the equivariant version of Kasparov $K$-theory to prove the strong Novikov conjecture on h
15 KB (2,316 words) - 21:05, 20 April 2012
...are the cross-sections of this bundle, or, equivalently, the $ \Gamma $-equivariant cross-sections of the trivial bundle $ X \times V \rightarrow X $ .
17 KB (2,502 words) - 06:16, 12 July 2022