Smale conjecture - Revision history

en>Wavelength: inserting 1 hyphen: —> "Low-dimensional"—User talk:Wavelength#Hyphenation [to Archive 6]

2014-04-16T01:43:07Z

inserting 1 hyphen: —> "Low-dimensional"—User talk:Wavelength#Hyphenation [to Archive 6]

← Older revision		Revision as of 03:43, 16 April 2014
Line 1:		Line 1:
	In mathematics, the '''Coxeter complex''', named after [[H. S. M. Coxeter]], is a geometrical structure (a [[simplicial complex]]) associated to a [[Coxeter group]]. Coxeter complexes are the basic objects that allow the construction of [[building (mathematics)\|buildings]]; they form the apartments of a building.		I like Sewing. <br>I also to learn Portuguese in my free time.<br><br>My page: [http://www.gwu-aqe.org/ gw timeshare services]

	~~==Construction==~~
	~~===The canonical linear representation===~~
	The first ingredient in the construction of the Coxeter complex associated to a Coxeter group ''W'' is a certain [[representation (mathematics)\|representation]] of ''W'', called the canonical representation of ''W''.

	Let <math>(W,S)</math> be a [[Coxeter group\|Coxeter system]] associated to ''W'', with [[Coxeter group#Coxeter matrix\|Coxeter matrix]] <math> M = (m(s,t))_{s,t \in S}</math>. The canonical representation is given by a vector space ''V'' with basis of formal symbols <math> (e_s)_{s \in S}</math>, which is equipped with the symmetric bilinear form <math> B(e_s,e_t) = - \cos \left ( \frac{\pi}{m(s,t)} \right )</math>. The action of ''W'' on this vector space ''V'' is then given by <math> s(v) = v - 2 \frac{B(e_s,v)}{B(e_s,e_s)}e_s</math>, as motivated by the expression for reflections in [[root system]]s.

	This representation has several foundational properties in the theory of Coxeter groups; for instance, the bilinear form ''B'' is positive definite if and only if ''W'' is finite. It is (always) a [[faithful representation]] of ''W''.

	~~===Chambers and the Tits cone===~~
	One can think of this representation as expressing ''W'' as some sort of reflection group, with the caveat that ''B'' might not be positive definite. It becomes important then to distinguish the representation ''V'' with its dual ''V''<sup></sup>. The vectors <math>e_s</math> lie in ''V'', and have corresponding dual vectors <math>e_s^\vee</math> in ''V''<sup></sup>, given by:

	~~: <math> \langle e_s^\vee, v \rangle = 2 \frac{B(e_s,v)}{B(e_s,e_s)},</math>~~

	~~where the angled brackets indicate the natural pairing of a dual vector in ''V''<sup>*</sup> with a vector of ''V'', and ''B'' is the bilinear form as above.~~

	~~Now ''V'' acts on ''V''<sup>*</sup>, and the action satisfies the formula~~

	~~: <math> s(f) = f - \langle f, e_s \rangle e_s^\vee, \, </math>~~

	for <math> s \in S</math> and any ''f'' in ''V''<sup></sup>. This expresses ''s'' as a reflection in the hyperplane <math> H_s = \{ f \in V^: \langle f, e_s \rangle = 0 \}</math>. One has the fundamental chamber <math> \mathcal{C} = \{ f \in V^* : \langle f, e_s \rangle > 0 \ \forall s \in S \}</math>, this has faces the so-called walls, <math>H_s</math>. The other chambers can be obtained from <math>\mathcal{C}</math> by translation: they are the <math> w\mathcal{C} </math> for <math> w \in W</math>.

	Given a fundamental chamber <math> \mathcal{C}</math>, the '''Tits cone''' is defined to be <math> X = \bigcup_{w \in W} w \overline{\mathcal{C}}</math>. This need not be the whole of ''V''<sup>*</sup>. Of major importance is the fact that the Tits cone ''X'' is convex. The action of ''W'' on the Tits cone ''X'' has [[fundamental domain]] the fundamental chamber <math> \mathcal{C}</math>.

	~~===The Coxeter complex===~~
	Once one has defined the Tits cone ''X'', the Coxeter complex <math>\Sigma(W,S)</math> of ''W'' with respect to ''S'' can be defined as the quotient of ''X'', with the origin removed, under multiplication by the positive reals, <math> \Sigma(W,S) = (X \setminus \{ 0 \}) / \mathbb{R}_{>0}^\times</math>.

	~~==Examples==~~
	~~===Finite dihedral groups===~~
	~~The dihedral groups <math>D_n</math> (of order 2''n'') are Coxeter groups, of corresponding type <math> \mathrm{~~I~~}_2(n)</math>~~. ~~These have the presentation~~ <~~math~~> ~~\left \langle s, t \, \left \| \, s^2, t^2, (st)^n \right \rangle \right . </math>.~~

	~~The canonical linear representation of <math> \mathrm{~~I~~}_2(n)</math> is the usual reflection representation of the dihedral group, as acting on a ''n''-gon in the plane (so <math> V = \mathbb{R}^2</math>~~ in ~~this case)~~. ~~For instance, in the case ''n'' = 3, we get the Coxeter group of type~~ <~~math~~> ~~\mathrm{I}_2(3) = \mathrm{A}_2~~<~~/math~~>, acting on an equilateral triangle in the plane. Each reflection ''s'' has an associated hyperplane ''H''<sub>''s''</sub> in the dual vector space (which can be canonically identified with the vector space itself using the bilinear form ''B'', which is an inner product in this case as remarked above), these are the walls. They cut out chambers, as seen below:

	[~~[File~~:~~Spherical dihedral complex.svg \| 300px]]~~

	~~The Coxeter complex is then the corresponding 2''n''-gon, as in the previous image. This is a simplicial complex of dimension 1, and it can be colored by cotype.~~

	~~===The infinite dihedral group===~~
	~~Another motivating example is the [[infinite dihedral group]] <math> D_{\infty} <~~/~~math>. This can be seen as the group of symmetries of the real line that preserves the set of points with integer coordinates; it is generated by the reflections in <math> x= 0 <~~/~~math> and <math> x = 1 </math>. This group has the Coxeter presentation <math> \left \langle s, t \, \left \| \, s^2, t^2 \right \rangle \right . </math>.~~

	~~In this case, it is no longer possible to identify ''V'' with the dual space ''V''<sup></sup>, as ''B'' is not positive definite~~. ~~It is then better to work solely with ''V''<sup></sup>, which is where the hyperplanes are defined. This then gives the following picture:~~

	~~[[Image:Affine dihedral complex.svg\|300px]]~~

	In this case, the Tits cone is not the whole plane, but only the upper half plane. Quotienting out by the positive reals then yields another copy of the real line, with marked points at the integers. This is the Coxeter complex of the infinite dihedral group.

	~~==Alternative construction of the Coxeter complex==~~
	Another description of the Coxeter complex uses standard cosets of the Coxeter group ''W''. A standard coset is a coset of the form <math> w W_J </math>, where <math> W_J = \langle J \rangle</math> for some subset ''J'' of ''S''. For instance, <math> W_S = W </math> and <math> W_\emptyset = \{1\}</math>.

	The Coxeter complex <math> \Sigma(W,S) </math> is then the [[poset]] of standard cosets, ordered by reverse inclusion. This has a canonical structure of a simplicial complex, as do all posets that satisfy:
	*Any two elements have a greatest lower bound.
	*The poset of elements less than or equal to any given element is isomorphic to the poset of subsets of <math> \{1, 2, \ldots, n \}</math> for some integer ''n''.

	~~==Properties==~~
	~~The Coxeter complex associated to <math>(W,S)</math> has dimension <math>\|S\|~~-~~1</math>~~. ~~It is homeomorphic to a <math>(\|S\|-1)<~~/~~math>-sphere if ''W'' is finite and is [[contractible space\|contractible]] if ''W'' is infinite.~~

	~~==See also==~~
	*[[Building (mathematics)\|Buildings]]
	*[[Weyl group]]
	*[[Root system]]

	~~==References==~~
	* Peter Abramenko and Kenneth S. Brown, ''Buildings, Theory and Applications''. Springer, 2008.

	~~[[Category:Group theory]]~~
	~~[[Category:Coxeter groups\| ]]~~
	~~[[Category:Algebraic combinatorics]]~~
	~~[[Category:Geometric group theory]]~~
	~~[[Category:Mathematical structures]~~]

en>Michael Hardy: The omission of the man's name is lunacy.

2013-06-24T07:00:21Z

The omission of the man's name is lunacy.

← Older revision		Revision as of 09:00, 24 June 2013
Line 1:		Line 1:
	~~Temple Run: The premise of this game~~ is ~~very uncomplicated: as an archaeologist trying~~ to ~~analysis an ancient temple, you~~ are ~~supposed~~ to ~~run and save oneself from~~ a group of ~~natives who are attempting to kill you~~. ~~Even so~~, ~~there are some aesthetic differences~~ to ~~the product~~'s ~~overall design~~. ~~If you invest time speaking on the telephone~~ with ~~close friends and playing on~~ the ~~net games~~, ~~when they go to study?~~ The ~~frequent gaming platform makes it possible~~ for ~~persons~~ of a group to ~~compete~~ with ~~absolute strangers as well. 3~~.<br><br>~~Men~~ have ~~seen~~ a ~~lot~~ of ~~technology~~ and ~~enjoyed~~ the ~~privilege of being~~ in the ~~most up~~-to~~-date trend via numerous a gadgets that have come our way~~ in the ~~previous couple~~ of ~~years~~. ~~Embedded in~~ the ~~BlackBerry 8830~~ is ~~16 MB RAM which gives assistance to substantial quantity~~ of ~~applications like games and net and so forth~~. The ~~time~~ has ~~come~~ to ~~reduce~~ the ~~cord and set your advertising-mix totally [http:~~//~~cleversplitter~~.~~com~~/~~finding-systems-throughout-farm-heroes-saga-cheat-tool-free-download~~/ ~~farm heroes saga cheat tool free download] to keep up with your active customers~~. ~~Also~~, ~~there~~ is a ~~separate camera for~~ the ~~video calling and it~~ has ~~diverse video features~~. ~~If you have ever spent countless hours trying to figure~~ out ~~all~~ the ~~intricacies of a brand new game you~~'~~ve just purchased then~~ a ~~suitable strategy guide~~ can be ~~your finest pal~~.<br><br>~~Now~~ that the ~~iPhone~~ is ~~readily available on two carriers~~ in the U.S. This is ~~a simple concept~~, but the ~~game has received more than 250 million downloads to date across all platforms~~, with the ~~Android version accounting~~ for ~~a massive quantity~~ of ~~these~~. ~~They have every thing that you can envision~~ and ~~at times~~, ~~applications which are even beyond your otherwise~~ standard ~~imagination~~. The ~~iconic 4-button pad~~ is ~~still there~~, as is ~~the d~~-~~pad~~, and ~~the dual manage sticks have been replaced with two round touch pads~~. ~~Practically all the i-mate mobile phones are sleekly designed that they automatically win the appreciation of the onlookers~~.		In mathematics, the '''Coxeter complex''', named after [[H. S. M. Coxeter]], is a geometrical structure (a [[simplicial complex]]) associated to a [[Coxeter group]]. Coxeter complexes are the basic objects that allow the construction of [[building (mathematics)\|buildings]]; they form the apartments of a building.

			==Construction==
			===The canonical linear representation===
			The first ingredient in the construction of the Coxeter complex associated to a Coxeter group ''W'' is a certain [[representation (mathematics)\|representation]] of ''W'', called the canonical representation of ''W''.

			Let <math>(W,S)</math> be a [[Coxeter group\|Coxeter system]] associated to ''W'', with [[Coxeter group#Coxeter matrix\|Coxeter matrix]] <math> M = (m(s,t))_{s,t \in S}</math>. The canonical representation is given by a vector space ''V'' with basis of formal symbols <math> (e_s)_{s \in S}</math>, which is equipped with the symmetric bilinear form <math> B(e_s,e_t) = - \cos \left ( \frac{\pi}{m(s,t)} \right )</math>. The action of ''W'' on this vector space ''V'' is then given by <math> s(v) = v - 2 \frac{B(e_s,v)}{B(e_s,e_s)}e_s</math>, as motivated by the expression for reflections in [[root system]]s.

			This representation has several foundational properties in the theory of Coxeter groups; for instance, the bilinear form ''B'' is positive definite if and only if ''W'' is finite. It is (always) a [[faithful representation]] of ''W''.

			===Chambers and the Tits cone===
			One can think of this representation as expressing ''W'' as some sort of reflection group, with the caveat that ''B'' might not be positive definite. It becomes important then to distinguish the representation ''V'' with its dual ''V''<sup></sup>. The vectors <math>e_s</math> lie in ''V'', and have corresponding dual vectors <math>e_s^\vee</math> in ''V''<sup></sup>, given by:

			: <math> \langle e_s^\vee, v \rangle = 2 \frac{B(e_s,v)}{B(e_s,e_s)},</math>

			where the angled brackets indicate the natural pairing of a dual vector in ''V''<sup>*</sup> with a vector of ''V'', and ''B'' is the bilinear form as above.

			Now ''V'' acts on ''V''<sup>*</sup>, and the action satisfies the formula

			: <math> s(f) = f - \langle f, e_s \rangle e_s^\vee, \, </math>

			for <math> s \in S</math> and any ''f'' in ''V''<sup></sup>. This expresses ''s'' as a reflection in the hyperplane <math> H_s = \{ f \in V^: \langle f, e_s \rangle = 0 \}</math>. One has the fundamental chamber <math> \mathcal{C} = \{ f \in V^* : \langle f, e_s \rangle > 0 \ \forall s \in S \}</math>, this has faces the so-called walls, <math>H_s</math>. The other chambers can be obtained from <math>\mathcal{C}</math> by translation: they are the <math> w\mathcal{C} </math> for <math> w \in W</math>.

			Given a fundamental chamber <math> \mathcal{C}</math>, the '''Tits cone''' is defined to be <math> X = \bigcup_{w \in W} w \overline{\mathcal{C}}</math>. This need not be the whole of ''V''<sup>*</sup>. Of major importance is the fact that the Tits cone ''X'' is convex. The action of ''W'' on the Tits cone ''X'' has [[fundamental domain]] the fundamental chamber <math> \mathcal{C}</math>.

			===The Coxeter complex===
			Once one has defined the Tits cone ''X'', the Coxeter complex <math>\Sigma(W,S)</math> of ''W'' with respect to ''S'' can be defined as the quotient of ''X'', with the origin removed, under multiplication by the positive reals, <math> \Sigma(W,S) = (X \setminus \{ 0 \}) / \mathbb{R}_{>0}^\times</math>.

			==Examples==
			===Finite dihedral groups===
			The dihedral groups <math>D_n</math> (of order 2''n'') are Coxeter groups, of corresponding type <math> \mathrm{I}_2(n)</math>. These have the presentation <math> \left \langle s, t \, \left \| \, s^2, t^2, (st)^n \right \rangle \right . </math>.

			The canonical linear representation of <math> \mathrm{I}_2(n)</math> is the usual reflection representation of the dihedral group, as acting on a ''n''-gon in the plane (so <math> V = \mathbb{R}^2</math> in this case). For instance, in the case ''n'' = 3, we get the Coxeter group of type <math> \mathrm{I}_2(3) = \mathrm{A}_2</math>, acting on an equilateral triangle in the plane. Each reflection ''s'' has an associated hyperplane ''H''<sub>''s''</sub> in the dual vector space (which can be canonically identified with the vector space itself using the bilinear form ''B'', which is an inner product in this case as remarked above), these are the walls. They cut out chambers, as seen below:

			[[File:Spherical dihedral complex.svg \| 300px]]

			The Coxeter complex is then the corresponding 2''n''-gon, as in the previous image. This is a simplicial complex of dimension 1, and it can be colored by cotype.

			===The infinite dihedral group===
			Another motivating example is the [[infinite dihedral group]] <math> D_{\infty} </math>. This can be seen as the group of symmetries of the real line that preserves the set of points with integer coordinates; it is generated by the reflections in <math> x= 0 </math> and <math> x = 1 </math>. This group has the Coxeter presentation <math> \left \langle s, t \, \left \| \, s^2, t^2 \right \rangle \right . </math>.

			In this case, it is no longer possible to identify ''V'' with the dual space ''V''<sup></sup>, as ''B'' is not positive definite. It is then better to work solely with ''V''<sup></sup>, which is where the hyperplanes are defined. This then gives the following picture:

			[[Image:Affine dihedral complex.svg\|300px]]

			In this case, the Tits cone is not the whole plane, but only the upper half plane. Quotienting out by the positive reals then yields another copy of the real line, with marked points at the integers. This is the Coxeter complex of the infinite dihedral group.

			==Alternative construction of the Coxeter complex==
			Another description of the Coxeter complex uses standard cosets of the Coxeter group ''W''. A standard coset is a coset of the form <math> w W_J </math>, where <math> W_J = \langle J \rangle</math> for some subset ''J'' of ''S''. For instance, <math> W_S = W </math> and <math> W_\emptyset = \{1\}</math>.

			The Coxeter complex <math> \Sigma(W,S) </math> is then the [[poset]] of standard cosets, ordered by reverse inclusion. This has a canonical structure of a simplicial complex, as do all posets that satisfy:
			*Any two elements have a greatest lower bound.
			*The poset of elements less than or equal to any given element is isomorphic to the poset of subsets of <math> \{1, 2, \ldots, n \}</math> for some integer ''n''.

			==Properties==
			The Coxeter complex associated to <math>(W,S)</math> has dimension <math>\|S\|-1</math>. It is homeomorphic to a <math>(\|S\|-1)</math>-sphere if ''W'' is finite and is [[contractible space\|contractible]] if ''W'' is infinite.

			==See also==
			*[[Building (mathematics)\|Buildings]]
			*[[Weyl group]]
			*[[Root system]]

			==References==
			* Peter Abramenko and Kenneth S. Brown, ''Buildings, Theory and Applications''. Springer, 2008.

			[[Category:Group theory]]
			[[Category:Coxeter groups\| ]]
			[[Category:Algebraic combinatorics]]
			[[Category:Geometric group theory]]
			[[Category:Mathematical structures]]

en>He to Hecuba: added Category:Smooth manifolds using HotCat

2012-02-18T12:42:17Z

added Category:Smooth manifolds using HotCat

New page

Temple Run: The premise of this game is very uncomplicated: as an archaeologist trying to analysis an ancient temple, you are supposed to run and save oneself from a group of natives who are attempting to kill you. Even so, there are some aesthetic differences to the product's overall design. If you invest time speaking on the telephone with close friends and playing on the net games, when they go to study? The frequent gaming platform makes it possible for persons of a group to compete with absolute strangers as well. 3.<br><br>Men have seen a lot of technology and enjoyed the privilege of being in the most up-to-date trend via numerous a gadgets that have come our way in the previous couple of years. Embedded in the BlackBerry 8830 is 16 MB RAM which gives assistance to substantial quantity of applications like games and net and so forth. The time has come to reduce the cord and set your advertising-mix totally [http://cleversplitter.com/finding-systems-throughout-farm-heroes-saga-cheat-tool-free-download/ farm heroes saga cheat tool free download] to keep up with your active customers. Also, there is a separate camera for the video calling and it has diverse video features. If you have ever spent countless hours trying to figure out all the intricacies of a brand new game you've just purchased then a suitable strategy guide can be your finest pal.<br><br>Now that the iPhone is readily available on two carriers in the U.S. This is a simple concept, but the game has received more than 250 million downloads to date across all platforms, with the Android version accounting for a massive quantity of these. They have every thing that you can envision and at times, applications which are even beyond your otherwise standard imagination. The iconic 4-button pad is still there, as is the d-pad, and the dual manage sticks have been replaced with two round touch pads. Practically all the i-mate mobile phones are sleekly designed that they automatically win the appreciation of the onlookers.