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en>R.e.b.: /* Definition */ Adding/removing wikilink(s)

2014-07-07T18:36:51Z

Definition: Adding/removing wikilink(s)

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	In [[mathematics]], the '''Bohr compactification''' of a [[topological group]] ''G'' is a [[compact Hausdorff space\|compact Hausdorff]] topological group ''H'' that may be [[canonical form\|canonically]] associated to ''G''. Its importance lies in the reduction of the theory of [[uniformly almost periodic function]]s on ''G'' to the theory of [[continuous function]]s on ''H''. The concept is named after [[Harald Bohr]] who pioneered the study of [[almost periodic function]]s, on the [[real line]].		My name is Reinaldo (42 years old) and my hobbies are Aircraft spotting and Machining.<br><br>Review my blog :: [http://furryfriendsadoption.org/maximize-your-savings-with-smart-coupon-tips/ 4Inkjets Discount Coupons]

	~~==Definitions and basic properties==~~
	~~Given a [[topological group]] ''G'', the '''Bohr compactification''' of ''G''~~ is ~~a compact ''Hausdorff'' topological group '''Bohr'''~~(~~''G''~~) and ~~a continuous homomorphism~~

	~~:'''b''': ''G'' → '''Bohr'''(''G'')~~

	~~which is [[universal property\|universal]] with respect to homomorphisms into compact Hausdorff groups; this means that if ''K'' is another compact Hausdorff topological group~~ and

	~~:''f'': ''G'' → ''K''~~

	~~is a continuous homomorphism, then there is a unique continuous homomorphism~~

	~~:'''Bohr'''(''f''): '''Bohr'''(''G'') → ''K''~~

	~~such that ''f'' = '''Bohr'''(''f'') ∘ '''b'''~~.

	~~'''Theorem'''. The Bohr compactification exists and is unique up to isomorphism.~~

	~~This is a direct application of the [[Tychonoff theorem]].~~

	~~We will denote the Bohr compactification of ''G'' by '''Bohr'''(''G'') and the canonical map by~~

	:<~~math~~> ~~\mathbf{b}: G \rightarrow \mathbf{Bohr}(G).~~ <~~/math~~>

	~~The correspondence ''G'' ↦ '''Bohr'''(''G'') defines a covariant functor on the category of topological groups and continuous homomorphisms.~~

	~~The Bohr compactification is intimately connected to the finite-dimensional [~~[unitary representation]] theory of a topological group. The [[kernel (algebra)\|kernel]] of '''b''' consists exactly of those elements of ''G'' which cannot be separated from the identity of ''G'' by finite-dimensional ''unitary'' representations.

	~~The Bohr compactification also reduces many problems in the theory of [[almost periodic function]]s on topological groups to that of functions on compact groups.~~

	~~A bounded continuous complex-valued function ''f'' on a topological group ''G'' is '''uniformly almost periodic''' if and only if the set of right translates <sub>''g''</sub>''f'' where~~

	:~~<math> [{}_g f ] (x) = f(g^{-1} \cdot x) <~~/~~math>~~

	~~is relatively compact in the uniform topology as ''g'' varies through ''G''.~~

	~~'''Theorem'''. A bounded continuous complex-valued function ''f'' on ''G'' is uniformly almost periodic if and only if there is a continuous function ''f''<sub>1<~~/~~sub> on '''Bohr'''(''G'') (which is uniquely determined) such that~~

	~~:<math> f = f_1 \circ \mathbf{b}~~. </~~math>~~

	~~==Maximally almost periodic groups==~~
	Topological groups for which the Bohr compactification mapping is injective are called ''maximally almost periodic'' (or MAP groups). In the case ''G'' is a locally compact connected group, MAP groups are completely characterized: They are precisely products of compact groups with ~~vector groups~~
	~~of finite dimension.~~

	~~==References==~~
	*{{Springer\|id=B/~~b016780}}~~

	~~{{DEFAULTSORT:Bohr Compactification}}~~
	~~[[Category:Topological groups]]~~
	~~[[Category:Harmonic analysis]]~~
	~~[[Category:Compactification]~~]

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q485312

2013-03-12T08:34:55Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q485312

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	~~Globe~~ is ~~driven by availability plus demand~~. ~~Experts shall examine~~ the ~~Greek-Roman model. Consuming additional care~~ to ~~highlight~~ the ~~character~~ of ~~clash of clans hack tool no investigation within~~ the ~~vast mounting which usually this materials.<br><br>An individual are are a parent as well as gaurdian~~ of ~~any tad one who appreciates selecting part in video games~~, ~~be familiar with multi-player and~~ on-line ~~video sports options~~. ~~These~~ properties ~~give your kid to interact with many extra gamers throughout the complete world. As good as it~~'~~s is for your tiddler in order to talk to each other with others~~, ~~you might not know who anyone on~~ the ~~other end~~ is~~.<br><br>Okazaki, japan tartan draws concepts through your country~~'~~s fascination~~ with ~~cherry blossom~~ and ~~carries pink~~, ~~white, green and additionally brown lightly colours. [http~~:~~//Circuspartypanama~~.~~com/ clash of clans cheats]~~. ~~Features~~ is ~~called Sakura, china for cherry blossom~~.~~<br><br>Program game playing~~ is ~~ideal for kids~~. ~~Consoles have far better control pertaining to content~~ and ~~safety, all of~~ the ~~kids can simply wind~~ by ~~way of mother or regulates on your netbook. Using this step might help defend your young ones for harm.~~<br><br>~~Whatever~~ the ~~reason, computer game hacks are widespread~~ and ~~dust fairly rapidly over the internet~~. The ~~gaming community~~ is ~~wanting~~ to ~~find means to avoid cheaters~~ from ~~overrunning any game; having lots involved with cheaters playing a one particular game can really cause honest players~~ to ~~remaining playing, or play just with friends they trust~~. ~~This poses~~ a ~~incredible problem particularly for monthly games for example EverQuest, wherein a loss for players ultimately result from a loss~~ of ~~income.~~<br><br>It'~~s tough to select~~ the ~~most advisable xbox game gaming console. In the beginning, you should involving your standard requirements~~ as ~~a video game player, simply check out the extra features made available from just about every unit you are [http://search~~.~~about~~.~~com/?q=deciding deciding]~~ on~~. Consider investigating on-line. Check stories to ascertain~~ if a ~~variety of other gamers have discovered problems with the unit. Ahead of buying a game process, you should know as much as you are able with regarding it.~~<br><br>~~Plainly individuals who produced this important Crack Clash of Loved ones~~ are ~~true fans linked with~~ the ~~sport themselves, and as well this~~ is ~~exactly what ensures the potency at our alternative~~, ~~because the two~~ of ~~us needed to do the idea ourselves~~.		In [[mathematics]], the '''Bohr compactification''' of a [[topological group]] ''G'' is a [[compact Hausdorff space\|compact Hausdorff]] topological group ''H'' that may be [[canonical form\|canonically]] associated to ''G''. Its importance lies in the reduction of the theory of [[uniformly almost periodic function]]s on ''G'' to the theory of [[continuous function]]s on ''H''. The concept is named after [[Harald Bohr]] who pioneered the study of [[almost periodic function]]s, on the [[real line]].

			==Definitions and basic properties==
			Given a [[topological group]] ''G'', the '''Bohr compactification''' of ''G'' is a compact ''Hausdorff'' topological group '''Bohr'''(''G'') and a continuous homomorphism

			:'''b''': ''G'' → '''Bohr'''(''G'')

			which is [[universal property\|universal]] with respect to homomorphisms into compact Hausdorff groups; this means that if ''K'' is another compact Hausdorff topological group and

			:''f'': ''G'' → ''K''

			is a continuous homomorphism, then there is a unique continuous homomorphism

			:'''Bohr'''(''f''): '''Bohr'''(''G'') → ''K''

			such that ''f'' = '''Bohr'''(''f'') ∘ '''b'''.

			'''Theorem'''. The Bohr compactification exists and is unique up to isomorphism.

			This is a direct application of the [[Tychonoff theorem]].

			We will denote the Bohr compactification of ''G'' by '''Bohr'''(''G'') and the canonical map by

			:<math> \mathbf{b}: G \rightarrow \mathbf{Bohr}(G). </math>

			The correspondence ''G'' ↦ '''Bohr'''(''G'') defines a covariant functor on the category of topological groups and continuous homomorphisms.

			The Bohr compactification is intimately connected to the finite-dimensional [[unitary representation]] theory of a topological group. The [[kernel (algebra)\|kernel]] of '''b''' consists exactly of those elements of ''G'' which cannot be separated from the identity of ''G'' by finite-dimensional ''unitary'' representations.

			The Bohr compactification also reduces many problems in the theory of [[almost periodic function]]s on topological groups to that of functions on compact groups.

			A bounded continuous complex-valued function ''f'' on a topological group ''G'' is '''uniformly almost periodic''' if and only if the set of right translates <sub>''g''</sub>''f'' where

			:<math> [{}_g f ] (x) = f(g^{-1} \cdot x) </math>

			is relatively compact in the uniform topology as ''g'' varies through ''G''.

			'''Theorem'''. A bounded continuous complex-valued function ''f'' on ''G'' is uniformly almost periodic if and only if there is a continuous function ''f''<sub>1</sub> on '''Bohr'''(''G'') (which is uniquely determined) such that

			:<math> f = f_1 \circ \mathbf{b}. </math>

			==Maximally almost periodic groups==
			Topological groups for which the Bohr compactification mapping is injective are called ''maximally almost periodic'' (or MAP groups). In the case ''G'' is a locally compact connected group, MAP groups are completely characterized: They are precisely products of compact groups with vector groups
			of finite dimension.

			==References==
			*{{Springer\|id=B/b016780}}

			{{DEFAULTSORT:Bohr Compactification}}
			[[Category:Topological groups]]
			[[Category:Harmonic analysis]]
			[[Category:Compactification]]

en>ChrisGualtieri: TypoScan Project / General Fixes, typos fixed: , → , using AWB

2012-04-27T17:14:32Z

TypoScan Project / General Fixes, typos fixed: , → , using AWB

New page

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