Napoleon points - Revision history

en>David Eppstein: /* Isosceles triangles */ Vecten points

2014-11-05T05:46:48Z

Isosceles triangles: Vecten points

← Older revision		Revision as of 07:46, 5 November 2014
Line 1:		Line 1:
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en>Rhododendrites: Reverted 1 edit by 173.160.157.41 identified as test/vandalism using STiki

2014-02-14T22:26:28Z

Reverted 1 edit by 173.160.157.41 identified as test/vandalism using STiki

← Older revision		Revision as of 00:26, 15 February 2014
Line 1:		Line 1:
	~~In [[model theory]], a branch of mathematical logic, '''U~~-~~rank''' is one measure of the complexity of a~~ (~~complete~~) ~~type, in the context of [[stable theory\|stable theories]]. As usual, higher U~~-~~rank indicates less restriction, and the existence of a U~~-~~rank for all types over all sets is equivalent to an important model~~-~~theoretic condition: in this case, [[Stable theory#Superstable theories\|superstability]].~~		35 year-old Diving Teacher (Open water ) Curtis Rowen from Saint-Mathieu-de-Beloeil, has hobbies and interests such as hunting, [http://natasasrnka2213.tumblr.com/post/100146948786/marvel-puzzle-quest-hack-android-ios Marvel Puzzle Quest Hack] and bottle tops collecting. Finds the world an interesting place after working 8 days at Arabian Oryx Sanctuary Delisted.

	~~== Definition ==~~
	~~U-rank is defined inductively~~, as ~~follows~~, ~~for any (complete) n-type p over any set A~~:

	* ''U''(''p'') ≥ 0
	* If ''δ'' is a limit ordinal, then ''U''(''p'') ≥ ''δ'' precisely when ''U''(''p'') ≥ ''α'' for all ''α'' less than ''δ''
	* For any ''α'' = ''β'' + 1, ''U''(''p'') ≥ ''α'' precisely when there is a forking extension ''q'' of ''p'' with ''U''(''q'') ≥ ''β''

	~~We say that ''U''(''p'') = ''α'' when the ''U''(''p'') ≥ ''α'' but not ''U''(''p'') ≥ ''α'' + 1~~.

	~~If ''U''(''p'') ≥ ''α'' for all ordinals ''α'', we say the U-rank is unbounded, or ''U''(''p'') = ∞~~.

	~~Note: U-rank is formally denoted <math>U_n(p)<~~/~~math>, where p is really p(x), and x is a tuple of variables of length n. This subscript is typically omitted when no confusion can result.~~

	~~== Ranking theories ==~~

	U-rank is '''[[Monotonic function#Monotonicity in order theory\|monotone]]''' in its domain. That is, suppose ''p'' is a complete type over ''A'' and ''B'' is a subset of ''A''. Then for ''q'' the restriction of ''p'' to ''B'', ''U''(''q'') ≥ ''U''(''p'').

	If we take ''B'' (above) to be empty, then we get the following: if there is an ''n''-type ''p'', over some set of parameters, with rank at least ''α'', then there is a type over the empty set of rank at least ''α''. Thus, we can define, for a complete (stable) theory ''T'', <math>U_n(T)=\sup \{ U_n(p) : p\in S(T) \}</~~math>.~~

	~~We then get a concise characterization of superstability; a stable theory ''T'' is superstable if and only if <math>U_n(T)<\infty<~~/~~math> for every ''n''.~~

	~~== Properties ==~~

	* As noted above, U-~~rank is monotone in its domain.~~
	* If ''p'' has U-~~rank ''α'', then for any ''β'' < ''α'', there is a forking extension ''q'' of ''p'' with U~~-~~rank ''β''.~~
	* If ''p'' is the type of ''b'' over ''A'', there is some set ''B'' extending ''A'', with ''q'' the type of ''b'' over ''B''.
	* If ''p'' is unranked (that is, ''p'' has U-~~rank ∞), then there is a forking extension ''q'' of ''p'' which is also unranked.~~
	* Even in the absence of superstability, there is an ordinal ''β'' which is the maximum rank of all ranked types, and for any ''α'' < ''β'', there is a type ''p'' of rank ''α'', and ~~if the rank of ''p'' is greater than ''β'', then it must be ∞.~~

	~~== Examples ==~~

	* ''U''(''p'') > 0 precisely when ''p'' is nonalgebraic.
	* If ''T'' is the theory of [[algebraically closed field]]s (of any fixed characteristic) then <math>U_1(T)=1</math>. ~~Further, if ''A'' is any set of parameters and ''K'' is~~ the ~~field generated by ''A'', then a 1-type ''p'' over ''A'' has rank 1 if (all realizations of) ''p'' are transcendental over ''K'', and 0 otherwise. More generally,~~ an ~~''n''-type ''p'' over ''A'' has U-rank ''k'', the transcendence degree (over ''K'') of any realization of it~~.

	~~== References ==~~

	~~{{cite book \|last1=Pillay \|first1=Anand \|title=An Introduction to Stability Theory \|year=2008 \|origyear=1983 \|publisher=Dover \|isbn=978-0-486-46896-9 \|page=57}}~~

	~~{{reflist}}~~

	~~[[Category:Model theory\| ]]~~

en>Monkbot: /* Generalization to isosceles triangles */Fix CS1 deprecated date parameter errors

2014-01-27T09:58:14Z

Generalization to isosceles triangles: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 11:58, 27 January 2014
Line 1:		Line 1:
	~~A νegetable diet with ρlenty~~ of ~~varietу~~ is one of the ~~best աaƴs to be ultimately healthy and strong~~. ~~Of ϲourse~~, ~~many people dο not get enough vegetables in their diet~~ and ~~lack vital nutrients that will help thеm get~~ the ~~necessary vitamins and minerals foг comρlete~~ all ~~around healtɦ~~. ~~Spirulina~~ is ~~an excellent addition to~~ a ~~healthy vegetable based diet~~, ~~and can be~~ the ~~substitute you need~~ for ~~getting vitamins. This algae was also consumed by Mayans and Aztecs to pгomote~~ all ~~around complete body health and energy~~.<br>~~Ѕpirulina іs~~ a ~~Ьlue~~-~~greеn algae that comes~~ in a ~~number~~ of ~~forms~~. ~~Some people eat it in~~ the ~~freshest form~~ of the ~~algae~~, ~~but it comes in~~ a ~~pill form as well as powder. Many foߋd products contain thiѕ algae alreɑdy. There are chips and evеn protein powders thаt ϲontain~~ the ~~algae and all~~ of ~~the nutrients it contains~~.<br><br>~~<br>There are spеcial աell flavoured protein powders that contain hearty amounts~~ of ~~the algae. Adding spirulina to your diet daily can give you~~ a ~~skin boost~~ and ~~аn internal health boost too~~. ~~Getting plenty of it~~ is ~~important though~~. If ~~you օnly consume it sporadically~~ then ~~it ԁoesntt have time to really benefit you. However~~, ~~drinking it in ɑ protein smoothie can be comparable to drinking~~ a ~~multi~~-~~vitamin~~. ~~Spirulina powder has a ɡood amount of vitamins that are easily аbsorbed into~~ the ~~body.<br><br><br>3 grams~~ of the ~~algae provides the nutritional bеnefits~~ of ~~two servіngs of vegetables~~. ~~That means~~ that ~~in 3 grams~~, ~~easily added to~~ a ~~smoothie, you can get аll~~ of ~~the nutrition of twο ѕervings of leafy greens. Ѕpirulina~~ is also ~~a very raгe plant sοurce οf b-12 and еasily absorbаble iron~~. ~~On top of all~~ of ~~thаt~~, ~~60% of the ρrotein іn the algae~~ is ~~absorbed into~~ the ~~body~~, ~~compared to much less~~ for ~~meаt products.~~<~~br><br><br>To toƿ things off~~, ~~thiѕ magical algae is also ρacked with omega fatty acids known as GLA. This algae~~ is a ~~miracle food~~ and ~~can provide 20 times more protein ƿer acre~~ than ~~soybeans~~, ~~and a whopping 200 times more protein than beef~~. ~~It іs amazіng how a small and easy to grow algae can pаck so much nutrition. You can also easily digest them, unlike trying to digest more complex foods~~. [~~http:~~/~~/Oilandgasstaffunion~~.~~com/people/nox70 Spirulina Side Effects]~~ is ~~ѵery low in calories. Not only do you get an amazing ɑbοut~~ of ~~vitamins, mineralѕ,~~ and ~~proteins~~, ~~but also~~ a ~~low fat low~~-~~cɑlorie power food.~~ A ~~500mg tablet~~ of ~~the algae contains оnly tաo calories maximum.<br><br><br>If you~~ are ~~looking for a qսick health boost ог long teгm hеalth~~, ~~then this ƅeautiful algae can Ьe the answer you are looҡing for it~~. ~~It mаkes a supeгior addition to a diet~~, ~~espeϲially іf you are unable to eat veցetables often. Sincе this~~ has ~~everƴthing you need~~, ~~consuming~~ it ~~oncе daily ѡill certainly provide your bodʏ with everything it needs~~ to ~~thrivе and be healthy.~~		In [[model theory]], a branch of mathematical logic, '''U-rank''' is one measure of the complexity of a (complete) type, in the context of [[stable theory\|stable theories]]. As usual, higher U-rank indicates less restriction, and the existence of a U-rank for all types over all sets is equivalent to an important model-theoretic condition: in this case, [[Stable theory#Superstable theories\|superstability]].

			== Definition ==
			U-rank is defined inductively, as follows, for any (complete) n-type p over any set A:

			* ''U''(''p'') ≥ 0
			* If ''δ'' is a limit ordinal, then ''U''(''p'') ≥ ''δ'' precisely when ''U''(''p'') ≥ ''α'' for all ''α'' less than ''δ''
			* For any ''α'' = ''β'' + 1, ''U''(''p'') ≥ ''α'' precisely when there is a forking extension ''q'' of ''p'' with ''U''(''q'') ≥ ''β''

			We say that ''U''(''p'') = ''α'' when the ''U''(''p'') ≥ ''α'' but not ''U''(''p'') ≥ ''α'' + 1.

			If ''U''(''p'') ≥ ''α'' for all ordinals ''α'', we say the U-rank is unbounded, or ''U''(''p'') = ∞.

			Note: U-rank is formally denoted <math>U_n(p)</math>, where p is really p(x), and x is a tuple of variables of length n. This subscript is typically omitted when no confusion can result.

			== Ranking theories ==

			U-rank is '''[[Monotonic function#Monotonicity in order theory\|monotone]]''' in its domain. That is, suppose ''p'' is a complete type over ''A'' and ''B'' is a subset of ''A''. Then for ''q'' the restriction of ''p'' to ''B'', ''U''(''q'') ≥ ''U''(''p'').

			If we take ''B'' (above) to be empty, then we get the following: if there is an ''n''-type ''p'', over some set of parameters, with rank at least ''α'', then there is a type over the empty set of rank at least ''α''. Thus, we can define, for a complete (stable) theory ''T'', <math>U_n(T)=\sup \{ U_n(p) : p\in S(T) \}</math>.

			We then get a concise characterization of superstability; a stable theory ''T'' is superstable if and only if <math>U_n(T)<\infty</math> for every ''n''.

			== Properties ==

			* As noted above, U-rank is monotone in its domain.
			* If ''p'' has U-rank ''α'', then for any ''β'' < ''α'', there is a forking extension ''q'' of ''p'' with U-rank ''β''.
			* If ''p'' is the type of ''b'' over ''A'', there is some set ''B'' extending ''A'', with ''q'' the type of ''b'' over ''B''.
			* If ''p'' is unranked (that is, ''p'' has U-rank ∞), then there is a forking extension ''q'' of ''p'' which is also unranked.
			* Even in the absence of superstability, there is an ordinal ''β'' which is the maximum rank of all ranked types, and for any ''α'' < ''β'', there is a type ''p'' of rank ''α'', and if the rank of ''p'' is greater than ''β'', then it must be ∞.

			== Examples ==

			* ''U''(''p'') > 0 precisely when ''p'' is nonalgebraic.
			* If ''T'' is the theory of [[algebraically closed field]]s (of any fixed characteristic) then <math>U_1(T)=1</math>. Further, if ''A'' is any set of parameters and ''K'' is the field generated by ''A'', then a 1-type ''p'' over ''A'' has rank 1 if (all realizations of) ''p'' are transcendental over ''K'', and 0 otherwise. More generally, an ''n''-type ''p'' over ''A'' has U-rank ''k'', the transcendence degree (over ''K'') of any realization of it.

			== References ==

			{{cite book \|last1=Pillay \|first1=Anand \|title=An Introduction to Stability Theory \|year=2008 \|origyear=1983 \|publisher=Dover \|isbn=978-0-486-46896-9 \|page=57}}

			{{reflist}}

			[[Category:Model theory\| ]]

en>Melchoir: remove supercategory of Category:Triangle centers

2012-08-10T09:10:30Z

remove supercategory of Category:Triangle centers

New page

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