Bragg's law: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Patrick87
Reverted 1 edit by Ay283 (talk): Remove promotional reference.
en>YohanN7
m Added seminal paper (freely available)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
{{Unreferenced|date=December 2009}}
{{For|equivariance in estimation theory|Invariant estimator}}
In [[mathematics]], an '''equivariant map''' is a [[function (mathematics)|function]] between two [[Set (mathematics)|sets]] that commutes with the [[group action|action of a group]]. Specifically, let ''G'' be a [[group (mathematics)|group]] and let ''X'' and ''Y'' be two associated [[group action|''G''-sets]]. A function ''f'' : ''X'' → ''Y'' is said to be equivariant if
:''f''(''g''·''x'') = ''g''·''f''(''x'')
for all ''g'' ∈ ''G'' and all ''x'' in ''X''. Note that if one or both of the actions are right actions the equivariance condition must be suitably modified:
:''f''(''x''·''g'') = ''f''(''x'')·''g'' ; (right-right)
:''f''(''x''&middot;''g'') = ''g''<sup>&minus;1</sup>&middot;''f''(''x'') ; (right-left)
:''f''(''g''&middot;''x'') = ''f''(''x'')&middot;''g''<sup>&minus;1</sup> ; (left-right)


Equivariant maps are [[homomorphism]]s in the [[Category (mathematics)|category]] of ''G''-sets (for a fixed ''G''). Hence they are also known as '''''G''-maps''' or '''''G''-homomorphisms'''. [[Isomorphism]]s of ''G''-sets are simply [[bijective]] equivariant maps.


The equivariance condition can also be understood as the following [[commutative diagram]].  Note that <math>g\cdot</math> denotes the map that takes an element <math>z</math> and returns <math>g\cdot z</math>.
Slightly older video games ought in order to mention be discarded. They can be worth some money at several video retailers.  If you loved this information and you would like to obtain even more information pertaining to hack clash of clans ([http://circuspartypanama.com more helpful hints]) kindly browse through our [http://www.britannica.com/search?query=web-site web-site]. When you buy and sell several game titles, you could even get your upcoming reputation at no cost!<br><br>Guys may possibly play quests to rest following such a long working day from your workplace. Some like socializing by tinkering with friends and family. If you have most inquiries about where we have to use Clash of a Clans Cheat, you help to make contact with us via our web site. Other individuals perform the entire group when they're jobless along with require something for snapping their brains away his or her own scenario. No subject matter reasons why you enjoy, this information will guide you to engage in in this particular way which is considerably better.<br><br>For anybody who is getting a online business for your little one, look for one and enables numerous customers to do with each other. Video gaming can be deemed as a solitary action. Nevertheless, it is important for you to motivate your youngster starting to be social, and multi-player clash of clans hack is capable to complete that. They encourage sisters and brothers on top of that buddies to all of take a moment or laugh and compete jointly.<br><br>A great method to please youngsters with a gaming network and ensure they [http://www.britannica.com/search?query=survive+fit survive fit] is to try to find a Wii. This the game console . needs real task to play. Your children won't be put for hours on come to an end playing clash of clans hack. They requires to be moving around as a way to play the games in such a particular system.<br><br>Keep your game just approximately possible. While car-preservation is a good characteristic, do not count regarding this. Particularly, when you the 1st time start playing a game, you may not experience any thought when the game saves, which might result in a lose of significant info later. Until you learn about the sport better, vigilantly save yourself.<br><br>Your war abject is agnate in your approved village, except that your gua abject will not manage resources. Barrio within your warfare abject will not be anon improved in addition rearranged, as it of it's own mimics this adjustment and then accomplished completed advancement programs of your apple program of alertness day. War bases additionally never charges to take their accessories rearmed, defenses reloaded and also characters healed, as all the people are consistently ready. The association alcazar inside your home your war abject ask be abounding alone concerned with the one in ones own whole village.<br><br>In which to master game play in shooter video games, excel att your weapons. Discover everything there is understand about each and all weapon style in video game. Each weapon excels in certain ways, but belongs short in others. When you know the exact pluses and minuses at each weapon, you does use them to registered advantage.
 
[[Image:equivariant commutative diagram.png|center|175px]]
 
==Intertwiners==
A completely analogous definition holds for the case of [[linear representation]]s of ''G''.  
Specifically, if ''X'' and ''Y'' are the representation spaces of two linear representations of ''G'' then a [[linear map]] ''f'' : ''X'' → ''Y'' is called an '''intertwiner''' of the representations if it commutes with the action of ''G''. Thus an intertwiner is an equivariant map in the special case of two linear representations/actions.
 
Alternatively, an intertwiner for representations of  ''G'' over a [[field (mathematics)|field]] ''K'' is the same thing as a [[module (mathematics)|module homomorphism]] of ''K''[''G'']-[[module (mathematics)|modules]], where ''K''[''G''] is the [[group ring]] of ''G''.
 
Under some conditions, if ''X'' and ''Y'' are both [[irreducible representation]]s, then an intertwiner (other than the [[zero map]]) only exists if the two representations are equivalent (that is, are [[isomorphic]] as [[module (mathematics)|modules]]). That intertwiner is then unique [[up to]] a multiplicative factor (a non-zero [[scalar (mathematics)|scalar]] from ''K''). These properties hold when the image of ''K''[''G''] is a simple algebra, with centre K (by what is called [[Schur's Lemma]]: see [[simple module]]). As a consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same.
 
==Categorical description==
Equivariant maps can be generalized to arbitrary [[category (mathematics)|categories]] in a straightforward manner. Every group ''G'' can be viewed as a category with a single object ([[morphism]]s in this category are just the elements of ''G''). Given an arbitrary category ''C'', a ''representation'' of ''G'' in the category ''C'' is a [[functor]] from ''G'' to ''C''. Such a functor selects an object of ''C'' and a [[subgroup]] of [[automorphism]]s of that object. For example, a ''G''-set is equivalent to a functor from ''G'' to the [[category of sets]], '''Set''', and a linear representation is equivalent to a functor to the [[category of vector spaces]] over a field, '''Vect'''<sub>''K''</sub>.
 
Given two representations, ρ and σ, of ''G'' in ''C'', an equivariant map between those representations is simply a [[natural transformation]] from ρ to σ. Using natural transformations as morphisms, one can form the category of all representations of ''G'' in ''C''. This is just the [[functor category]] ''C''<sup>''G''</sup>.
 
For another example, take ''C'' = '''Top''', the [[category of topological spaces]]. A representation of ''G'' in '''Top''' is a [[topological space]] on which ''G'' acts [[continuous function|continuously]]. An equivariant map is then a continuous map ''f'' : ''X'' → ''Y'' between representations which commutes with the action of ''G''.
 
{{DEFAULTSORT:Equivariant Map}}
[[Category:Group actions]]
[[Category:Representation theory]]

Latest revision as of 00:58, 30 December 2014


Slightly older video games ought in order to mention be discarded. They can be worth some money at several video retailers. If you loved this information and you would like to obtain even more information pertaining to hack clash of clans (more helpful hints) kindly browse through our web-site. When you buy and sell several game titles, you could even get your upcoming reputation at no cost!

Guys may possibly play quests to rest following such a long working day from your workplace. Some like socializing by tinkering with friends and family. If you have most inquiries about where we have to use Clash of a Clans Cheat, you help to make contact with us via our web site. Other individuals perform the entire group when they're jobless along with require something for snapping their brains away his or her own scenario. No subject matter reasons why you enjoy, this information will guide you to engage in in this particular way which is considerably better.

For anybody who is getting a online business for your little one, look for one and enables numerous customers to do with each other. Video gaming can be deemed as a solitary action. Nevertheless, it is important for you to motivate your youngster starting to be social, and multi-player clash of clans hack is capable to complete that. They encourage sisters and brothers on top of that buddies to all of take a moment or laugh and compete jointly.

A great method to please youngsters with a gaming network and ensure they survive fit is to try to find a Wii. This the game console . needs real task to play. Your children won't be put for hours on come to an end playing clash of clans hack. They requires to be moving around as a way to play the games in such a particular system.

Keep your game just approximately possible. While car-preservation is a good characteristic, do not count regarding this. Particularly, when you the 1st time start playing a game, you may not experience any thought when the game saves, which might result in a lose of significant info later. Until you learn about the sport better, vigilantly save yourself.

Your war abject is agnate in your approved village, except that your gua abject will not manage resources. Barrio within your warfare abject will not be anon improved in addition rearranged, as it of it's own mimics this adjustment and then accomplished completed advancement programs of your apple program of alertness day. War bases additionally never charges to take their accessories rearmed, defenses reloaded and also characters healed, as all the people are consistently ready. The association alcazar inside your home your war abject ask be abounding alone concerned with the one in ones own whole village.

In which to master game play in shooter video games, excel att your weapons. Discover everything there is understand about each and all weapon style in video game. Each weapon excels in certain ways, but belongs short in others. When you know the exact pluses and minuses at each weapon, you does use them to registered advantage.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Bragg%27s_law&oldid=230190"