Modified models of gravity - Revision history

2.31.238.14 at 14:15, 2 November 2014

2014-11-02T14:15:48Z

← Older revision		Revision as of 16:15, 2 November 2014
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en>Monkbot: Fix CS1 deprecated date parameter errors (test)

2014-02-25T20:49:41Z

Fix CS1 deprecated date parameter errors (test)

← Older revision		Revision as of 22:49, 25 February 2014
Line 1:		Line 1:
	~~In the mathematical discipline of [[descriptive set theory]]~~, a '~~''scale''' is a certain kind~~ of ~~object defined on a [[set (mathematics)\|set]] of [[point (mathematics)\|point]]s in some [[Polish space]] (for example, a scale might be defined on a set of [[real number]]s)~~. ~~Scales were originally isolated as a concept in~~ the ~~theory~~ of ~~[[uniformization~~ (~~descriptive set theory~~)~~\|uniformization]]~~,~~<ref>Kechris and Moschovakis 2008:28</ref> but have found wide applicability in descriptive set theory, with applications such as establishing bounds on the possible lengths of [[wellordering]]s of~~ a ~~given complexity, and showing (under certain assumptions) that there~~ are ~~largest [[countable set]]s~~ of ~~certain complexities.~~		To aid you make an informed selection, we've enlisted the insight of noted survival specialist and knife designer Doug Ritter. Doug is also the founder, publisher and editor of Equipped to Survive ( ), a extremely respected survival data supply. What are the basic qualities of a excellent survival knife?<br><br><br><br>I have employed mine as others use their knives, what differs is that my Strider Knife gets put away ready to go once more, other people have to have to be re-sharpened, a chip requires to be taken out of the blade or that bit of rust increasing on the hand guard might be going all the way down into the handle, eating into the tang. When I take out to use my knife, I don't have any question whether or not it will do what I am asking of it. one hundred% self-assurance. 1st, there is the knife. Practically nothing spectacular here, although the blade is incredibly very good at holding an edge, and functions fantastic for almost everything from skinning catfish to scaling bluegill to skinning & quartering wild hogs. I never heard of S30V stainless just before I handled this knife, but I now know that it is exceptionally fantastic stuff.<br><br>The [http://Lovechurch.nayana.com/xe/?document_srl=158517 purpose] the knife is applied for dictates weather it is a common, a combat or a tactical knife. If you had been to ask me what knife you need to have to invest in, I would 1st have to have to know for what reason you are looking for a knife in the 1st location. Should really you treat the knife as a weapon or as a utility tool? Will this be utilized for an emergency rescue operation or to gain access or entry to a distinct area? If you would be applying the knife for any of these purposes, then most likely what you are immediately after is a tactical knife. Forms of Tactical Knives.<br><br>When I stated immediately, I need to have worded it differently I guess. I just meant with out opening my pack or un doing my waist belt. I cant obtain a comfy place on my belt because of my pack. If you loved this article and you would like to acquire extra details with regards to [http://www.thebestpocketknifereviews.com/best-automatic-knife-discover-the-pefect-model/ best automatic knife review] kindly go to our web site. I like it on my chest but it does have an aggressive undertone I guess. I like the easy acces tho, I mostly hike with my kids and they continually want some package cut open or a walking stick trimmed. Plus I like the fixed blade for cooking or cleaning fish. Absolutely nothing worse than fish guts in the hinge of a folder. I dont don't forget the model.

	~~==Formal definition==~~
	~~Given~~ a ~~pointset ''A'' contained in some product space~~
	:<~~math~~>~~A\subseteq X=X_0\times X_1\times\ldots X_{m-1}~~<~~/math~~>
	~~where each ''X~~<~~sub~~>k<~~/sub~~>'' is ~~either~~ the ~~[[Baire space (set theory)\|Baire space]]~~ or ~~a countably infinite discrete set, we say~~ that ~~a ''norm''~~ on ~~''A'' is a map from ''A''~~ into the ~~[[ordinal number]]s~~. ~~Each norm has an associated [[prewellordering]]~~, ~~where one element of~~ '~~'A'' precedes another element if the norm~~ of ~~the first~~ is ~~less than~~ the ~~norm of the second~~.

	~~A ''scale'' on ''A'' is a countably infinite collection of norms~~
	~~:<math>(\phi_n)_{n<\omega}</math>~~
	~~with~~ the ~~following properties:~~
	~~: If the sequence ''x<sub>i</sub>'' is such that~~
	~~:: ''x<sub>i</sub>''~~ is an ~~element of ''A'' for each natural number ''i''~~, and
	~~:: ''x<sub>i</sub>'' converges~~ to ~~an element ''x''in the product space ''X''~~, ~~and~~
	~~:: for each natural number ''n'' there~~ is ~~an ordinal λ~~<~~sub~~>~~''n''~~<~~/sub~~> ~~such that φ<sub>n</sub>(''x<sub>i</sub>'')=λ<sub>''n''</sub> for all sufficiently large ''i'', then~~
	~~:''x'' is an element of ''A'', and~~
	:~~for each ''n'', φ<sub>n<~~/~~sub>(x)≤λ<sub>''n''<~~/~~sub>~~.~~<ref>Kechris and Moschovakis 2008:37<~~/~~ref>~~
	~~By itself, at least granted the [[axiom of choice]~~], the ~~existence of~~ a ~~scale on~~ a ~~pointset is trivial, as ''A'' can be wellordered and each φ<sub>''n''</sub> can simply enumerate ''A''. To make the concept useful,~~ a ~~definability criterion must be imposed on the norms (individually and together)~~. ~~Here "definability" is understood in the usual sense of descriptive set theory; it~~ need ~~not be definability~~ in ~~an absolute sense~~, ~~but rather indicates membership in some [[pointclass]] of sets of reals. The norms φ<sub>''n''</sub> themselves~~ are ~~not sets of reals, but the corresponding [[prewellordering]]s are (at least in essence).~~

	~~The idea is that,~~ for a ~~given pointclass Γ, we want the prewellorderings below a given point in ''A'' to be uniformly represented both as a set in Γ and as one~~ in the ~~dual pointclass of Γ, relative to the "larger" point being an element of ''A''~~. ~~Formally, we say that~~ the ~~φ<sub>''n''</sub> form~~ a ~~'''Γ-scale on ''A''''' if they form~~ a ~~scale on ''A'' and there are ternary relations ''S'' and ''T'' such that, if ''y'' is~~ an ~~element~~ of ~~''A''~~, then
	~~:<math>\forall n\forall x(\varphi_n(x)\leq\varphi_n(y) \iff S(n,x,y) \iff T(n,x,y))</math>~~
	~~where ''S''~~ is ~~in Γ and ''T'' is in the dual pointclass of Γ (that is, the complement~~ of ~~''T'' is in Γ)~~.<~~ref~~>~~Kechris and Moschovakis 2008:37, with harmless reworking~~<~~/ref~~> ~~Note here that we think of φ<sub>''n''</sub>(''x'') as being ∞ whenever ''x''∉''A''; thus the condition φ<sub>''n''</sub>(''x'')≤φ<sub>''n''</sub>(''y'')~~, ~~for ''y''∈''A'', also implies ''x''∈''A''~~.

	~~Note also that the definition does ''not'' imply that the collection of norms is in the intersection of Γ~~ with ~~the dual pointclass of Γ~~. ~~This is~~ because ~~the three-way equivalence is conditional on ''y'' being an element~~ of ~~''A''~~. ~~For ''y'' not in ''A'', it might be the case that one or both of ''S(n,x,y)'' or ''T(n,x,y)'' fail~~ to ~~hold, even if ''x'' is in ''A'' (and therefore automatically φ<sub>''n''<~~/~~sub>(''x'')≤φ<sub>''n''<~~/~~sub>(''y'')=∞)~~.

	~~==Applications==~~
	~~:''This section is yet to be written''~~

	~~==Scale property==~~
	~~The scale property is a strengthening of the [[prewellordering property]]~~. ~~For pointclasses of a certain form, it implies that [[relation (mathematics)\|relations]] in the given pointclass have a [[uniformization (set theory)\|uniformization]] that is also in the pointclass.~~

	~~==Periodicity==~~
	~~:''This section is yet to be written''~~

	~~==Notes==~~
	~~<references~~/>

	~~==References==~~
	* {{Citation\| author=Moschovakis, Yiannis N. \| title=Descriptive Set Theory \| publisher=North Holland \| year=1980 \|isbn=0-~~444~~-~~70199~~-~~0}}~~
	* {{Citation\|last1=Kechris\|first1=Alexander S. \|last2=Moschovakis \| first2=Yiannis N. \|editor1-~~first=Alexander S.\| editor1~~-~~last=Kechris\|editor2~~-~~first=\|editor2-last=[[Benedikt Löwe~~]~~] \|editor3-first=John R~~.~~\|editor3-last=Steel\|title=Games~~, ~~Scales~~ and ~~Suslin Cardinals: The Cabal Seminar, Volume~~ I ~~\|publisher=Cambridge University Press \|year=2008 \|pages=28–74 \|chapter=Notes on~~ the ~~theory~~ of ~~scales \|isbn=978-0-521-89951-2}}~~

	~~{{DEFAULTSORT:Scale (Descriptive Set Theory)}}~~
	~~[[Category:Descriptive set theory]]~~

en>ChrisGualtieri: Typo fixing, typos fixed: e.g, → e.g., using AWB

2012-08-07T14:42:36Z

Typo fixing, typos fixed: e.g, → e.g., using AWB

New page

In the mathematical discipline of [[descriptive set theory]], a '''scale''' is a certain kind of object defined on a [[set (mathematics)|set]] of [[point (mathematics)|point]]s in some [[Polish space]] (for example, a scale might be defined on a set of [[real number]]s). Scales were originally isolated as a concept in the theory of [[uniformization (descriptive set theory)|uniformization]],<ref>Kechris and Moschovakis 2008:28</ref> but have found wide applicability in descriptive set theory, with applications such as establishing bounds on the possible lengths of [[wellordering]]s of a given complexity, and showing (under certain assumptions) that there are largest [[countable set]]s of certain complexities.

==Formal definition==
Given a pointset ''A'' contained in some product space
:<math>A\subseteq X=X_0\times X_1\times\ldots X_{m-1}</math>
where each ''Xk'' is either the [[Baire space (set theory)|Baire space]] or a countably infinite discrete set, we say that a ''norm'' on ''A'' is a map from ''A'' into the [[ordinal number]]s. Each norm has an associated [[prewellordering]], where one element of ''A'' precedes another element if the norm of the first is less than the norm of the second.

A ''scale'' on ''A'' is a countably infinite collection of norms
:<math>(\phi_n)_{n<\omega}</math>
with the following properties:
: If the sequence ''xi'' is such that
:: ''xi'' is an element of ''A'' for each natural number ''i'', and
:: ''xi'' converges to an element ''x''in the product space ''X'', and
:: for each natural number ''n'' there is an ordinal λ''n'' such that φn(''xi'')=λ''n'' for all sufficiently large ''i'', then
:''x'' is an element of ''A'', and
:for each ''n'', φn(x)≤λ''n''.<ref>Kechris and Moschovakis 2008:37</ref>
By itself, at least granted the [[axiom of choice]], the existence of a scale on a pointset is trivial, as ''A'' can be wellordered and each φ''n'' can simply enumerate ''A''. To make the concept useful, a definability criterion must be imposed on the norms (individually and together). Here "definability" is understood in the usual sense of descriptive set theory; it need not be definability in an absolute sense, but rather indicates membership in some [[pointclass]] of sets of reals. The norms φ''n'' themselves are not sets of reals, but the corresponding [[prewellordering]]s are (at least in essence).

The idea is that, for a given pointclass Γ, we want the prewellorderings below a given point in ''A'' to be uniformly represented both as a set in Γ and as one in the dual pointclass of Γ, relative to the "larger" point being an element of ''A''. Formally, we say that the φ''n'' form a '''Γ-scale on ''A''''' if they form a scale on ''A'' and there are ternary relations ''S'' and ''T'' such that, if ''y'' is an element of ''A'', then
:<math>\forall n\forall x(\varphi_n(x)\leq\varphi_n(y) \iff S(n,x,y) \iff T(n,x,y))</math>
where ''S'' is in Γ and ''T'' is in the dual pointclass of Γ (that is, the complement of ''T'' is in Γ).<ref>Kechris and Moschovakis 2008:37, with harmless reworking</ref> Note here that we think of φ''n''(''x'') as being ∞ whenever ''x''∉''A''; thus the condition φ''n''(''x'')≤φ''n''(''y''), for ''y''∈''A'', also implies ''x''∈''A''.

Note also that the definition does ''not'' imply that the collection of norms is in the intersection of Γ with the dual pointclass of Γ. This is because the three-way equivalence is conditional on ''y'' being an element of ''A''. For ''y'' not in ''A'', it might be the case that one or both of ''S(n,x,y)'' or ''T(n,x,y)'' fail to hold, even if ''x'' is in ''A'' (and therefore automatically φ''n''(''x'')≤φ''n''(''y'')=∞).

==Applications==
:''This section is yet to be written''

==Scale property==
The scale property is a strengthening of the [[prewellordering property]]. For pointclasses of a certain form, it implies that [[relation (mathematics)|relations]] in the given pointclass have a [[uniformization (set theory)|uniformization]] that is also in the pointclass.

==Periodicity==
:''This section is yet to be written''

==Notes==
<references/>

==References==
* {{Citation| author=Moschovakis, Yiannis N. | title=Descriptive Set Theory | publisher=North Holland | year=1980 |isbn=0-444-70199-0}}
* {{Citation|last1=Kechris|first1=Alexander S. |last2=Moschovakis | first2=Yiannis N. |editor1-first=Alexander S.| editor1-last=Kechris|editor2-first=|editor2-last=[[Benedikt Löwe]] |editor3-first=John R.|editor3-last=Steel|title=Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I |publisher=Cambridge University Press |year=2008 |pages=28–74 |chapter=Notes on the theory of scales |isbn=978-0-521-89951-2}}

{{DEFAULTSORT:Scale (Descriptive Set Theory)}}
[[Category:Descriptive set theory]]