Excess molar quantity - Revision history

5.15.19.69: /* Relation to activity coefficients */ text

2014-09-08T08:50:57Z

Relation to activity coefficients: text

← Older revision		Revision as of 10:50, 8 September 2014
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5.15.28.131: /* Examples */ rel

2014-03-03T23:07:59Z

Examples: rel

← Older revision		Revision as of 01:07, 4 March 2014
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	~~In [[mathematics]],~~ '~~''random groups''' are certain [[Group (mathematics)\|group]]~~s ~~obtained by a probabilistic construction~~. ~~They were introduced by [[Mikhail_Gromov_(mathematician)\|Misha Gromov]]~~ to ~~answer questions such as "What does a typical group look like?"~~		Footwear is each of essential items to top off one's outfit. There is definitely not to say about fashion trend the family leave aside shoes and boots. Nowadays, shoes not only protect your feet from cold or obstacle, they play more important role on fashion advised. For most people, both of them are necessary. If you are searching for one kind of shoes which can help you go in the fashion trend and also comfy to wear in winter, Ugg shoes are all the choice.<br><br>The ringing of the doorbell interrupts Henry's evening ritual: watching TV, taking in the ugg news of disasters and wars from all over the internet. Only a door-to-door salesman would develop the gall to bother mankind at that time period. And true to character, he -- she? -- isn't content to ring once. There it goes again.<br><br>Do fretting or constant the exact time when ugg boots become fasionable? Perhaps, till now, few people can deliver a clear take. But nobody can deny the simple fact that these boots are popular all around the globe. Although they look ugly, they get become a way icon. Most people like these boots, nonetheless all individuals can afford them. The real boots are seen as expensive. If you are facing any question, perhaps my article can offer some advices. Here I just want to tell some places or practical ways to discover a cheap overshoes. Of course, cheaper boots are not fake ones; you can some places where ugg boots with discounted prices are in existence.<br><br>DO Not alone settle for cheap boots. Yes, do they protect through the weather. But invest a little extra for better designs. Take note of especially the direction they fit your legs.<br><br>Henry thump-thump-thumps to your kitchen. Carolyn is filling the kettle with water -- so she's found it -- sporting that mom-smile, the one who makes him feel becoming ten year-old who hasn't done his homework. But suddenly that smile falls off her face. Henry can almost hear it shatter on the floor, like a glass ashtray. Hah, triumph! But what has he completed to earn the situation?<br><br>Though I have made these suggestions, yet each to her own style. You can mix and match different style to choose your gown. Keep experimenting and make your own style statement with Australian Ugg " booties ".<br><br>If you are you looking for more information in regards to [http://horizonafrica.com/img/ ugg boots outlet] look at the web-page.

	~~It so happens that~~, ~~once a precise definition is given~~, ~~random groups satisfy some properties with very high probability, whereas other properties fail with very high probability~~. For ~~instance~~, ~~very probably random groups~~ are ~~[[hyperbolic group]]s~~. ~~In this sense,~~ one ~~can say that "most groups are hyperbolic".~~

	~~==Definition==~~

	~~The definition~~ of ~~random groups depends on a probabilistic model on the set of possible groups. Various such probabilistic models yield different (but related) notions of random groups.~~

	~~Any group~~ can ~~by defined by a [[Presentation of a group\|group presentation]] involving generators and relations. For instance,~~ the ~~Abelian group <math>\mathbb{Z}\times\mathbb{Z}</math> has a presentation with two generators <math>a</math>~~ and ~~<math>b</math>~~, ~~and~~ the ~~relation~~ <~~math~~>~~ab=ba</math>, or equivalently~~ <~~math~~>~~aba^{-1}b^{-1}=1</math>.~~ The ~~main idea of random groups is to start with a fixed number of group generators <math>a_1,\,a_2,\,\ldots,\,a_m</math>, and imposing relations~~ of the ~~form <math>r_1=1~~,~~\,r_2=1,\,\ldots,\,r_k=1</math> where each <math>r_j</math> is a random word involving~~ the ~~letters <math>a_i</math>~~ and ~~their formal inverses <math>a_i^{-1}</math>~~. ~~To specify~~ a ~~model of random groups is~~ to ~~specify a precise way in which <math>m</math>, <math>k</math> and~~ the ~~random relations <math>r_j</math> are chosen~~.

	~~Once the random relations <math>r_k</math> have been chosen~~, ~~the resulting random group~~ <~~math~~>G<~~/math~~> ~~is defined in~~ the ~~standard way for group presentations, namely: <math>G</math> is the quotient of the [[free group]] <math>F_m</math> with generators <math>a_1~~,\,~~a_2,\,\ldots,\,a_m</math>, by~~ the ~~normal subgroup <math>R\subset F_m</math> generated by~~ the ~~relations <math>r_1\~~,~~r_2~~,~~\,\ldots,\,r_k</math>~~ seen as ~~elements of <math>F_m</math>:~~

	~~: <math>G=F_m/\langle r_1~~,~~\,r_2,\,\ldots,\,r_k \rangle~~.~~</math>~~

	~~==The few-relator model of random groups==~~

	~~The simplest model of random groups is the ''few-relator model''~~. ~~In this model~~, ~~a number of generators <math>m\geq 2</math> and a number of relations <math>k\geq 1</math>~~ are ~~fixed~~. ~~Fix an additional parameter~~ <~~math~~>~~\ell~~<~~/math~~> ~~(the length of the relations), which is typically taken very large~~.

	~~Then~~, the ~~model consists in choosing the relations <math>r_1\,r_2,\,\ldots,\,r_k</math> at random, uniformly and independently among all possible [[reduced word]]s~~ of ~~length at most <math>\ell</math> involving~~ the ~~letters~~ <~~math~~>~~a_i~~<~~/math~~> ~~and their formal inverses <math>a_i^{~~-~~1}</math>~~.

	~~This model~~ is ~~especially interesting when~~ the ~~relation length <math>\ell</math> tends to infinity:~~ with ~~probability tending to <math>1</math> as <math>\ell\to\infty</math> a random group in this model is [[hyperbolic group\|hyperbolic]] and satisfies other nice properties.~~

	~~==Further remarks==~~

	~~More refined models of random groups have been defined.~~

	~~For instance~~, in the '~~'density model'', the number of relations is allowed to grow with the length of the relations~~. ~~Then there is a sharp "phase transition" phenomenon: if the number of relations is larger than some threshold, the random group "collapses" (because the relations allow to show~~ that ~~any word is equal to any other), whereas below the threshold the resulting random group is infinite and hyperbolic~~.

	~~Constructions of random groups~~ can ~~also be twisted in specific ways to build group with particular properties~~. ~~For instance~~, ~~Gromov used this technique~~ to ~~build new groups that are counter-examples to an extension of~~ the ~~[[Baum-Connes conjecture]].~~

	~~== References ==~~

	* [[Mikhail Gromov (mathematician)\|Mikhail Gromov]]. ''Hyperbolic groups.'' Essays in group theory, ~~75–263, Math~~. ~~Sci. Res. Inst. Publ., 8, Springer, New York, 1987.~~

	* [[Mikhail Gromov (mathematician)\|Mikhail Gromov]]. "~~Random walk in random groups.~~" ~~''Geom~~. ~~Funct. Anal.'', vol. 13 (2003), 73–146.~~

	[~~[Category~~:~~Geometric group theory]]~~
	~~[[Category:Properties of groups]]~~
	~~[[Category:Combinatorics on words]~~]

82.137.11.19: /* External links */ ionic solutions mixtures

2014-01-04T20:11:55Z

External links: ionic solutions mixtures

← Older revision		Revision as of 22:11, 4 January 2014
Line 1:		Line 1:
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	~~my web page~~ - [~~http~~://~~www~~.~~tobler-verlag~~.ch/~~media~~/~~galerie~~/jp/~~shop~~/nb/ ~~ニューバランス店舗~~]		It so happens that, once a precise definition is given, random groups satisfy some properties with very high probability, whereas other properties fail with very high probability. For instance, very probably random groups are [[hyperbolic group]]s. In this sense, one can say that "most groups are hyperbolic".

			==Definition==

			The definition of random groups depends on a probabilistic model on the set of possible groups. Various such probabilistic models yield different (but related) notions of random groups.

			Any group can by defined by a [[Presentation of a group\|group presentation]] involving generators and relations. For instance, the Abelian group <math>\mathbb{Z}\times\mathbb{Z}</math> has a presentation with two generators <math>a</math> and <math>b</math>, and the relation <math>ab=ba</math>, or equivalently <math>aba^{-1}b^{-1}=1</math>. The main idea of random groups is to start with a fixed number of group generators <math>a_1,\,a_2,\,\ldots,\,a_m</math>, and imposing relations of the form <math>r_1=1,\,r_2=1,\,\ldots,\,r_k=1</math> where each <math>r_j</math> is a random word involving the letters <math>a_i</math> and their formal inverses <math>a_i^{-1}</math>. To specify a model of random groups is to specify a precise way in which <math>m</math>, <math>k</math> and the random relations <math>r_j</math> are chosen.

			Once the random relations <math>r_k</math> have been chosen, the resulting random group <math>G</math> is defined in the standard way for group presentations, namely: <math>G</math> is the quotient of the [[free group]] <math>F_m</math> with generators <math>a_1,\,a_2,\,\ldots,\,a_m</math>, by the normal subgroup <math>R\subset F_m</math> generated by the relations <math>r_1\,r_2,\,\ldots,\,r_k</math> seen as elements of <math>F_m</math>:

			: <math>G=F_m/\langle r_1,\,r_2,\,\ldots,\,r_k \rangle.</math>

			==The few-relator model of random groups==

			The simplest model of random groups is the ''few-relator model''. In this model, a number of generators <math>m\geq 2</math> and a number of relations <math>k\geq 1</math> are fixed. Fix an additional parameter <math>\ell</math> (the length of the relations), which is typically taken very large.

			Then, the model consists in choosing the relations <math>r_1\,r_2,\,\ldots,\,r_k</math> at random, uniformly and independently among all possible [[reduced word]]s of length at most <math>\ell</math> involving the letters <math>a_i</math> and their formal inverses <math>a_i^{-1}</math>.

			This model is especially interesting when the relation length <math>\ell</math> tends to infinity: with probability tending to <math>1</math> as <math>\ell\to\infty</math> a random group in this model is [[hyperbolic group\|hyperbolic]] and satisfies other nice properties.

			==Further remarks==

			More refined models of random groups have been defined.

			For instance, in the ''density model'', the number of relations is allowed to grow with the length of the relations. Then there is a sharp "phase transition" phenomenon: if the number of relations is larger than some threshold, the random group "collapses" (because the relations allow to show that any word is equal to any other), whereas below the threshold the resulting random group is infinite and hyperbolic.

			Constructions of random groups can also be twisted in specific ways to build group with particular properties. For instance, Gromov used this technique to build new groups that are counter-examples to an extension of the [[Baum-Connes conjecture]].

			== References ==

			* [[Mikhail Gromov (mathematician)\|Mikhail Gromov]]. ''Hyperbolic groups.'' Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.

			* [[Mikhail Gromov (mathematician)\|Mikhail Gromov]]. "Random walk in random groups." ''Geom. Funct. Anal.'', vol. 13 (2003), 73–146.

			[[Category:Geometric group theory]]
			[[Category:Properties of groups]]
			[[Category:Combinatorics on words]]

en>MagnInd: /* See also */

2011-10-30T20:11:20Z