Steiner tree problem - Revision history

2001:638:403:5050:224:1DFF:FEDE:83A3: problem -> case

2015-01-02T09:28:27Z

problem -> case

← Older revision		Revision as of 11:28, 2 January 2015
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	~~{{Refimprove\|date=March 2011}}~~		These folks call me Gabrielle. I am a [http://Data.gov.uk/data/search?q=trustworthy+cashier trustworthy cashier] and I'm preparing pretty good financially. As a girl what I definitely like is going to karaoke but I in no way made a dime in addition to it. My husband and I select to reside in Guam but I will have actually to move in a new year or two. See what is new on my blog here: http://circuspartypanama.com<br><br>Take a look at my blog post :: [http://circuspartypanama.com clash of clans hack tools]
	~~In [[group theory]], the '''growth rate''' of a [[group (mathematics)\|group]] with respect to a symmetric [[generating set of a group\|generating set]] describes the size of balls in the group~~. ~~Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length ''n''.~~

	~~==Definition==~~
	~~Suppose ''G'' is a finitely generated group; and ''T'' is a finite ''symmetric'' set of [[Generating set of a group\|generator]]s~~
	~~(symmetric means that if <math> x \in T </math> then <math> x^{-1} \in T </math>).~~
	~~Any element <math> x \in G </math> can be expressed as~~ a [~~[string (computer science)#Formal theory\|word]] in the ''T''-alphabet~~

	:~~<math> x = a_1 \cdot a_2 \cdots a_k \mbox{ where } a_i\in T. <~~/~~math>~~

	~~Let us consider the subset of all elements of ''G'' which can be presented by such a word of length ≤ ''n''~~

	~~:<math>B_n(G,T) = \{x\in G \| x = a_1 \cdot a_2 \cdots a_k \mbox{ where } a_i\in T \mbox{ and } k\le n\}.<~~/~~math>~~

	~~This set is just the [[Ball (mathematics)\|closed ball]] of radius ''n'' in the [[word metric]] ''d'' on ''G'' with respect to the generating set ''T'':~~

	~~:<math>B_n(G,T) = \{x\in G \| d(x, e)\le n\}~~.~~</math>~~

	~~More geometrically, <math>B_n(G,T)</math> is the set of vertices in the [[Cayley graph]] with respect to ''T'' which are within distance ''n'' of the identity~~.

	~~Given two nondecreasing positive functions ''a'' and ''b'' one can say that~~
	~~they are equivalent (<math>a\sim b<~~/~~math>) if there is a constant ''C'' such that~~

	~~:<math> a(n~~/ ~~C) \leq b(n) \leq a(Cn),\, </math>~~

	~~for example <math> p^n\sim~~ q~~^n </math> if <math> p,q>1 </math>.~~

	~~Then the growth rate of the group ''G'' can be defined as the corresponding [[equivalence class~~]~~] of the function~~
	~~:<math>\#(n)=\|B_n(G,T)\|, </math>~~
	~~where <math>\|B_n(G,T)\|</math> denotes the number of elements in the set <math>B_n(G,T)</math>.~~
	~~Although the function <math>\#(n)</math> depends on the set of generators ''T'' its rate of~~
	~~growth does not (see below)~~ and ~~therefore the rate of growth gives an invariant of a group.~~

	~~The word metric ''d'' and therefore sets <math>B_n(G,T)</math> depend~~
	on the generating set ''T''. However, any two such metrics are [[Lipschitz continuity#Lipschitz continuity in metric spaces\|''bilipschitz'']] [[equivalence class\|''equivalent'']] in the following sense: for finite symmetric generating sets ''E'', ''F'', there is a positive constant ''C'' ~~such that~~
	~~:<math> {1\over C} \ d_F(x,y) \leq d_{E}(x,y) \leq C \ d_F(x,y)~~. ~~</math>~~
	As ~~an immediate corollary of this inequality we get that the growth rate does not depend on the choice of generating set.~~

	~~==Polynomial and exponential growth==<!-- This section is linked from [[Hyperbolic geometry]] -->~~

	If

	~~:<math>\#(n)\le C(n^k+1)</math>~~

	~~for some <math>C,k<\infty</math> we say that ''G'' has~~ a ~~'''polynomial growth rate'''.~~
	~~The infimum <math>k_0</math> of such ''k'''s~~ is ~~called the '''order of polynomial growth'''.~~
	~~According~~ to ~~[[Gromov's theorem on groups of polynomial growth\|Gromov's theorem]],~~ a ~~group of polynomial growth is [[virtually nilpotent]], i.e.~~ it ~~has a [[nilpotent group\|nilpotent]] [[subgroup]] of finite [[Index of a subgroup\|index]]~~. ~~In particular, the order of polynomial growth <math>k_0</math> has~~ to ~~be a [[natural numbers\|natural number]] and~~ in ~~fact <math>\#(n)\sim n^{k_0}</math>.~~

	~~If <math>\#(n)\ge a^n</math> for some <math>a>1</math> we say that ''G'' has an '''[[exponential growth]] rate'''.~~
	~~Every [[finitely generated group\|finitely generated]] ''G'' has at most exponential growth, i.e. for some <math>b>1</math> we~~ have ~~<math>\#(n)\le b^n</math>.~~

	~~If <math>\#(n)</math> grows more slowly than any exponential function, ''G'' has~~ a ~~'''subexponential growth rate'''~~. ~~Any such group is [[amenable group\|amenable]].~~

	~~==Examples==~~

	* A [[free group]] with a finite rank ''k'' > 1 has an exponential growth rate.

	* A [[finite group]] has constant growth – polynomial growth of order 0 – and includes fundamental groups of manifolds whose universal cover is compact.

	* If ''M'' is a closed negatively curved [[Riemannian manifold]] then its [[fundamental group]] <math>\pi_1(M)</math> has exponential growth rate. [[John Milnor\|Milnor]] proved this using the fact that the [[word metric]] on ~~<math>\pi_1(M)<~~/~~math> is [[Glossary of Riemannian and metric geometry#Q\|quasi-isometric]] to the [[Covering map\|universal cover]] of ''M''.~~

	* '''Z'''<sup>''d''</~~sup> has a polynomial growth rate of order ''d''~~.

	* The [[discrete Heisenberg group]] ''H''<~~sub~~>3<~~/sub~~> ~~has~~ a polynomial growth rate of order 4. This fact is a special case of the general theorem of [[Hyman Bass\|Bass]] and [[Yves Guivarch\|Guivarch]] that is discussed in the article on [[Gromov's theorem on groups of polynomial growth\|Gromov's theorem]].

	* The [[lamplighter group]] has an exponential growth. <!-- This is a rare example of a solvable group with exponential growth. -->

	* The existence of groups with '''intermediate growth''', i.e. subexponential but not polynomial was open for many years. It was asked by [[John Milnor\|Milnor]] in 1968 and was finally answered in the positive by [[Rostislav Grigorchuk\|Grigorchuk]] in 1984. There are still open questions in this area and a complete picture of which orders of growth are possible and which are not is missing.

	* The [[triangle group]]s include 3 finite groups (the spherical ones, corresponding to sphere), 3 groups of quadratic growth (the Euclidean ones, corresponding to Euclidean plane), and infinitely many groups of exponential growth (the hyperbolic ones, corresponding to the hyperbolic plane).

	~~==See also==~~
	* [[Isoperimetric dimension#Consequences of isoperimetry\|Connections to isoperimetric inequalities]]

	~~==References==~~

	* [[John Milnor\|J. Milnor]], ''A note on curvature and fundamental group'', J. Differential Geometry '''2''' (1968), 1–7.
	* R. I. Grigorchuk, ''Degrees of growth of finitely generated groups and the theory of invariant means.'', Izv. Akad. Nauk SSSR Ser. Mat. '''48:~~5''' (1984), 939–985 (Russian)~~.

	~~==Further reading==~~
	*{{cite arxiv \|author=Rostislav Grigorchuk and [[Igor Pak]] \|title=Groups of ~~Intermediate Growth: an Introduction for Beginners \|year=2006 \|eprint=math.GR/0607384}}~~

	~~[[Category:Infinite group theory]]~~
	~~[[Category:Metric geometry]~~]

en>David Eppstein: see also Opaque forest problem

2014-01-20T02:47:05Z

see also Opaque forest problem

← Older revision		Revision as of 04:47, 20 January 2014
Line 1:		Line 1:
			{{Refimprove\|date=March 2011}}
			In [[group theory]], the '''growth rate''' of a [[group (mathematics)\|group]] with respect to a symmetric [[generating set of a group\|generating set]] describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length ''n''.

			==Definition==
			Suppose ''G'' is a finitely generated group; and ''T'' is a finite ''symmetric'' set of [[Generating set of a group\|generator]]s
			(symmetric means that if <math> x \in T </math> then <math> x^{-1} \in T </math>).
			Any element <math> x \in G </math> can be expressed as a [[string (computer science)#Formal theory\|word]] in the ''T''-alphabet

	~~Adrianne~~ is ~~what you most likely will call me but Which i don~~'~~t like when everyone use my full logo~~. ~~What I love doing~~ is in ~~fact~~ to [~~http~~://~~Mondediplo~~.~~com~~/~~spip~~.~~php?page~~=~~recherche&recherche~~=~~play~~+~~croquet play croquet~~] ~~and now I have~~ a ~~chance~~ to ~~take on new things~~. ~~The job I~~'~~ve been~~ [~~http:~~//~~Www~~.~~Google~~.~~com/search?q~~=~~occupying+designed&btnI~~=~~lucky occupying designed~~] ~~for years is~~ an ~~sequence clerk~~. ~~My husband~~ and ~~I chose to reside in Guam but I will gain to move in~~ a ~~year or two~~. ~~You does find my website here: http:~~//~~Prometeu~~.~~net~~/~~<br~~><br>~~Feel free~~ to ~~surf~~ to ~~my blog post~~ ... [~~http~~:~~//Prometeu~~.~~net~~/ ~~clash of clans cheats no survey~~]		:<math> x = a_1 \cdot a_2 \cdots a_k \mbox{ where } a_i\in T. </math>

			Let us consider the subset of all elements of ''G'' which can be presented by such a word of length ≤ ''n''

			:<math>B_n(G,T) = \{x\in G \| x = a_1 \cdot a_2 \cdots a_k \mbox{ where } a_i\in T \mbox{ and } k\le n\}.</math>

			This set is just the [[Ball (mathematics)\|closed ball]] of radius ''n'' in the [[word metric]] ''d'' on ''G'' with respect to the generating set ''T'':

			:<math>B_n(G,T) = \{x\in G \| d(x, e)\le n\}.</math>

			More geometrically, <math>B_n(G,T)</math> is the set of vertices in the [[Cayley graph]] with respect to ''T'' which are within distance ''n'' of the identity.

			Given two nondecreasing positive functions ''a'' and ''b'' one can say that
			they are equivalent (<math>a\sim b</math>) if there is a constant ''C'' such that

			:<math> a(n/ C) \leq b(n) \leq a(Cn),\, </math>

			for example <math> p^n\sim q^n </math> if <math> p,q>1 </math>.

			Then the growth rate of the group ''G'' can be defined as the corresponding [[equivalence class]] of the function
			:<math>\#(n)=\|B_n(G,T)\|, </math>
			where <math>\|B_n(G,T)\|</math> denotes the number of elements in the set <math>B_n(G,T)</math>.
			Although the function <math>\#(n)</math> depends on the set of generators ''T'' its rate of
			growth does not (see below) and therefore the rate of growth gives an invariant of a group.

			The word metric ''d'' and therefore sets <math>B_n(G,T)</math> depend
			on the generating set ''T''. However, any two such metrics are [[Lipschitz continuity#Lipschitz continuity in metric spaces\|''bilipschitz'']] [[equivalence class\|''equivalent'']] in the following sense: for finite symmetric generating sets ''E'', ''F'', there is a positive constant ''C'' such that
			:<math> {1\over C} \ d_F(x,y) \leq d_{E}(x,y) \leq C \ d_F(x,y). </math>
			As an immediate corollary of this inequality we get that the growth rate does not depend on the choice of generating set.

			==Polynomial and exponential growth==<!-- This section is linked from [[Hyperbolic geometry]] -->

			If

			:<math>\#(n)\le C(n^k+1)</math>

			for some <math>C,k<\infty</math> we say that ''G'' has a '''polynomial growth rate'''.
			The infimum <math>k_0</math> of such ''k'''s is called the '''order of polynomial growth'''.
			According to [[Gromov's theorem on groups of polynomial growth\|Gromov's theorem]], a group of polynomial growth is [[virtually nilpotent]], i.e. it has a [[nilpotent group\|nilpotent]] [[subgroup]] of finite [[Index of a subgroup\|index]]. In particular, the order of polynomial growth <math>k_0</math> has to be a [[natural numbers\|natural number]] and in fact <math>\#(n)\sim n^{k_0}</math>.

			If <math>\#(n)\ge a^n</math> for some <math>a>1</math> we say that ''G'' has an '''[[exponential growth]] rate'''.
			Every [[finitely generated group\|finitely generated]] ''G'' has at most exponential growth, i.e. for some <math>b>1</math> we have <math>\#(n)\le b^n</math>.

			If <math>\#(n)</math> grows more slowly than any exponential function, ''G'' has a '''subexponential growth rate'''. Any such group is [[amenable group\|amenable]].

			==Examples==

			* A [[free group]] with a finite rank ''k'' > 1 has an exponential growth rate.

			* A [[finite group]] has constant growth – polynomial growth of order 0 – and includes fundamental groups of manifolds whose universal cover is compact.

			* If ''M'' is a closed negatively curved [[Riemannian manifold]] then its [[fundamental group]] <math>\pi_1(M)</math> has exponential growth rate. [[John Milnor\|Milnor]] proved this using the fact that the [[word metric]] on <math>\pi_1(M)</math> is [[Glossary of Riemannian and metric geometry#Q\|quasi-isometric]] to the [[Covering map\|universal cover]] of ''M''.

			* '''Z'''<sup>''d''</sup> has a polynomial growth rate of order ''d''.

			* The [[discrete Heisenberg group]] ''H''<sub>3</sub> has a polynomial growth rate of order 4. This fact is a special case of the general theorem of [[Hyman Bass\|Bass]] and [[Yves Guivarch\|Guivarch]] that is discussed in the article on [[Gromov's theorem on groups of polynomial growth\|Gromov's theorem]].

			* The [[lamplighter group]] has an exponential growth. <!-- This is a rare example of a solvable group with exponential growth. -->

			* The existence of groups with '''intermediate growth''', i.e. subexponential but not polynomial was open for many years. It was asked by [[John Milnor\|Milnor]] in 1968 and was finally answered in the positive by [[Rostislav Grigorchuk\|Grigorchuk]] in 1984. There are still open questions in this area and a complete picture of which orders of growth are possible and which are not is missing.

			* The [[triangle group]]s include 3 finite groups (the spherical ones, corresponding to sphere), 3 groups of quadratic growth (the Euclidean ones, corresponding to Euclidean plane), and infinitely many groups of exponential growth (the hyperbolic ones, corresponding to the hyperbolic plane).

			==See also==
			* [[Isoperimetric dimension#Consequences of isoperimetry\|Connections to isoperimetric inequalities]]

			==References==

			* [[John Milnor\|J. Milnor]], ''A note on curvature and fundamental group'', J. Differential Geometry '''2''' (1968), 1–7.
			* R. I. Grigorchuk, ''Degrees of growth of finitely generated groups and the theory of invariant means.'', Izv. Akad. Nauk SSSR Ser. Mat. '''48:5''' (1984), 939–985 (Russian).

			==Further reading==
			*{{cite arxiv \|author=Rostislav Grigorchuk and [[Igor Pak]] \|title=Groups of Intermediate Growth: an Introduction for Beginners \|year=2006 \|eprint=math.GR/0607384}}

			[[Category:Infinite group theory]]
			[[Category:Metric geometry]]

128.197.164.27: /* Generalization of minimum spanning tree */

2012-08-02T01:18:24Z

Generalization of minimum spanning tree

New page

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