Water model - Revision history

211.121.168.82: /* Flexible SPC water model */ English correction

2014-08-26T08:52:20Z

Flexible SPC water model: English correction

← Older revision		Revision as of 10:52, 26 August 2014
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	In [[model theory]], the '''age''' of a [[Structure (mathematical logic)\|structure]] (or model) ''A'' is the class of all finitely generated structures which are [[Embedding\|embeddable]] in ''A'' (i.e. isomorphic to substructures of ''A''). This concept is central in the construction of a [[Roland Fraisse\|Fraïssé]] limit.		They contact me Emilia. To do aerobics is a thing that I'm totally addicted to. For a while I've been in South Dakota and my parents reside close by. Hiring is my profession.<br><br>Stop by my webpage: [http://www.adosphere.com/poyocum adosphere.com]

	~~The main point of Fraïssé's construction~~ is ~~to show how one can approximate~~ a ~~structure by its finitely generated substructures. Thus for example the age of any [[List of first-order theories#Orders\|dense linear order without endpoints]] (DLO),~~
	<math>\langle\mathbb{Q},<\rangle</math> is precisely the set of all [[Wikt:finite\|finite]] linear orderings, which are distinguished up to isomorphism only by their size. Thus the age of any DLO is [[countable]]. This shows in a way that ~~a DLO is a kind of limit of finite linear orderings.~~

	~~One can easily see that any class ''K'' which is an age of some structure satisfies the following three conditions:~~

	~~;Hereditary property: If ''A'' ∈ ''K'~~' ~~and ''B'' is a finitely generated substructure of ''A'', then ''B'' is isomorphic~~ to ~~a structure in ''K''~~

	~~;[[Joint embedding property]]: If ''A'' and ''B'' are in ''K'' then there is ''C'' in ''K'' such that both ''A'' and ''B'' are embeddable in ''C''~~.

	~~Fraïssé proved that when ''K'' is any non-empty countable set of finitely generated σ-structures (with σ~~ a ~~[[signature (logic)\|signature]]) which has the above two properties, then it is an age of a countable structure.~~

	~~Furthermore, suppose that ''K'' happens to satisfy the following additional property.~~

	~~;[[Amalgamation property]]: For any structures ''A, B'' and ''C'' in ''K'' such that ''A'~~' ~~is embeddable~~ in ~~both ''B''~~ and ~~''C'', there exists ''D'' in ''K'' to which ''B'' and ''C'' are both embeddable~~ by ~~embeddings which coincide on the image of ''A'' in both structures~~.

	~~In that case, there~~ is ~~a unique countable structure, up to isomorphism, that has age ''K'' and is homogeneous~~. In this context, Homogeneous means that any isomorphism between two finitely generated substructures can be extended to an automorphism of the whole structure. Again, an example of this situation is the ordered set of rational numbers <~~math~~>~~\langle\mathbb{Q},<\rangle~~</~~math>~~. ~~It is the ''unique'' (up to isomorphism) homogenous countable structure whose age is the set of all finite linear orderings~~. Note that the ordered set of natural numbers <math>\langle\mathbb{N},<\rangle</math> has the same age as a DLO, but it is not homogenous since if we map {1, 3} to {5, 6}, it would not extend to any automorphism ''f'' since there should be an element between <math>f(1)=5</~~math> and <math>f(3)=6</math>. The same applies to integers~~.

	~~{{reflist}}~~

	~~==References==~~
	* {{ cite book \| last=Hodges \| first=Wilfrid \| authorlink=Wilfrid Hodges \| publisher=[[Cambridge University Press]] \| title=A shorter model theory \| year=1997 \| isbn=0-521-58713-1 }}

	~~[[Category:Model theory]~~]

78.14.44.58: I added another example of implicit solvation model

2014-01-03T12:45:47Z

I added another example of implicit solvation model

← Older revision		Revision as of 14:45, 3 January 2014
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	~~The title~~ of the ~~writer is Figures~~. ~~For many years he~~'~~s been residing~~ in ~~North Dakota and his family members enjoys it. I am~~ a ~~meter reader~~. ~~To collect coins~~ is ~~what his family and him appreciate~~.<br><br>~~Have~~ a ~~look at my homepage~~ ... [~~http~~:/~~/www~~.~~fanproof~~.~~com~~/~~index.php?do~~=/~~profile-115513~~/~~info/ http://www~~.~~fanproof~~.~~com~~]		In [[model theory]], the '''age''' of a [[Structure (mathematical logic)\|structure]] (or model) ''A'' is the class of all finitely generated structures which are [[Embedding\|embeddable]] in ''A'' (i.e. isomorphic to substructures of ''A''). This concept is central in the construction of a [[Roland Fraisse\|Fraïssé]] limit.

			The main point of Fraïssé's construction is to show how one can approximate a structure by its finitely generated substructures. Thus for example the age of any [[List of first-order theories#Orders\|dense linear order without endpoints]] (DLO),
			<math>\langle\mathbb{Q},<\rangle</math> is precisely the set of all [[Wikt:finite\|finite]] linear orderings, which are distinguished up to isomorphism only by their size. Thus the age of any DLO is [[countable]]. This shows in a way that a DLO is a kind of limit of finite linear orderings.

			One can easily see that any class ''K'' which is an age of some structure satisfies the following three conditions:

			;Hereditary property: If ''A'' ∈ ''K'' and ''B'' is a finitely generated substructure of ''A'', then ''B'' is isomorphic to a structure in ''K''

			;[[Joint embedding property]]: If ''A'' and ''B'' are in ''K'' then there is ''C'' in ''K'' such that both ''A'' and ''B'' are embeddable in ''C''.

			Fraïssé proved that when ''K'' is any non-empty countable set of finitely generated σ-structures (with σ a [[signature (logic)\|signature]]) which has the above two properties, then it is an age of a countable structure.

			Furthermore, suppose that ''K'' happens to satisfy the following additional property.

			;[[Amalgamation property]]: For any structures ''A, B'' and ''C'' in ''K'' such that ''A'' is embeddable in both ''B'' and ''C'', there exists ''D'' in ''K'' to which ''B'' and ''C'' are both embeddable by embeddings which coincide on the image of ''A'' in both structures.

			In that case, there is a unique countable structure, up to isomorphism, that has age ''K'' and is homogeneous. In this context, Homogeneous means that any isomorphism between two finitely generated substructures can be extended to an automorphism of the whole structure. Again, an example of this situation is the ordered set of rational numbers <math>\langle\mathbb{Q},<\rangle</math>. It is the ''unique'' (up to isomorphism) homogenous countable structure whose age is the set of all finite linear orderings. Note that the ordered set of natural numbers <math>\langle\mathbb{N},<\rangle</math> has the same age as a DLO, but it is not homogenous since if we map {1, 3} to {5, 6}, it would not extend to any automorphism ''f'' since there should be an element between <math>f(1)=5</math> and <math>f(3)=6</math>. The same applies to integers.

			{{reflist}}

			==References==
			* {{ cite book \| last=Hodges \| first=Wilfrid \| authorlink=Wilfrid Hodges \| publisher=[[Cambridge University Press]] \| title=A shorter model theory \| year=1997 \| isbn=0-521-58713-1 }}

			[[Category:Model theory]]

en>Dark Silver Crow: /* See also */ More merged content as per previous edit

2012-05-17T14:31:56Z

See also: More merged content as per previous edit

New page

The title of the writer is Figures. For many years he's been residing in North Dakota and his family members enjoys it. I am a meter reader. To collect coins is what his family and him appreciate.<br><br>Have a look at my homepage ... [http://www.fanproof.com/index.php?do=/profile-115513/info/ http://www.fanproof.com]