Determination of equilibrium constants - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T09:52:06Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 11:52, 5 May 2014
Line 1:		Line 1:
	~~Friends call him Royal~~. ~~Some time in the past~~ he ~~chose to live in Idaho. The favorite pastime for my kids and me is taking part~~ in ~~crochet~~ and ~~now I~~'~~m attempting to make cash with~~ it. ~~She works as a monetary~~ officer ~~and she will not alter it whenever soon~~.<br><br>~~Check out~~ my ~~blog post ...~~ [http://~~Newdayz~~.~~de/index~~.~~php?mod=users&action=view&id=16038 http:~~//~~Newdayz.de~~/~~index.php?mod=users&action=view&id=16038~~]		Hi there, I am Sophia. To perform lacross is some thing I really enjoy performing. For years he's been residing in Mississippi and he doesn't plan on altering it. My day occupation is an information officer but I've currently applied for an additional 1.<br><br>Take a look at my web-site tarot readings [[http://netwk.hannam.ac.kr/xe/data_2/85669 more helpful hints]]

en>Hike395: Disambiguated: correlation coefficient → Pearson product-moment correlation coefficient using Dab solver

2014-02-11T06:49:59Z

Disambiguated: correlation coefficient → Pearson product-moment correlation coefficient using Dab solver

← Older revision		Revision as of 08:49, 11 February 2014
Line 1:		Line 1:
	~~{{Orphan\|date=October 2013}}~~		Friends call him Royal. Some time in the past he chose to live in Idaho. The favorite pastime for my kids and me is taking part in crochet and now I'm attempting to make cash with it. She works as a monetary officer and she will not alter it whenever soon.<br><br>Check out my blog post ... [http://Newdayz.de/index.php?mod=users&action=view&id=16038 http://Newdayz.de/index.php?mod=users&action=view&id=16038]

	~~In [[measure theory]], '''tangent measures''' are used to study the local behavior of [[Radon measure]]s,~~ in ~~much~~ the ~~same way as [[tangent space]]s are used~~ to ~~study the local behavior of [[differentiable manifold]]s~~. ~~Tangent measures (introduced by David Preiss <ref>{{cite journal~~
	~~\| last = Preiss \| first = David~~
	~~\| title = Geometry of measures in <math>R^n</math>: distribution, rectifiability,~~ and ~~densities~~
	~~\| journal = Ann. Math. \| volume = 125~~
	~~\| issue = 3 \| pages = 537–643 \| year = 1987~~
	~~\| jstor = 1971410~~
	~~\| doi = 10.2307/1971410~~
	~~}}</ref>~~ in ~~his study of [[rectifiable set]]s) are a useful tool in geometric measure theory. For example, they are used in proving [[Hausdorff density\|Marstrand’s theorem]]~~ and ~~[[Hausdorff density\|Preiss~~' ~~theorem]]~~.

	~~==Definition==~~

	~~Consider~~ a ~~Radon measure ''μ'' defined on an [[open subset]] Ω of ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>~~ and ~~let ''a'' be an arbitrary point in Ω. We can “zoom in” on a small [[open ball]] of radius ''r'' around ''a'', ''B''<sub>''r''</sub>(''a''), via the transformation~~
	~~: <math>T_{a,r}(x)=\frac{x-a}{r},</math>~~
	~~which enlarges the ball of radius ''r'' about ''a'' to a ball of radius 1 centered at 0~~. ~~With this, we may now zoom in on how ''μ'' behaves on ''B''<sub>''r''</sub>(''a'') by looking at the [[push-forward measure]] defined by~~
	~~:<math> T_{a,r \#}\mu(A)=\mu(a+rA)</math>~~
	~~where~~
	:<~~math~~>~~a+rA=\{a+rx:x\in A\}.~~<~~/math~~>
	~~As ''r'' gets smaller, this transformation on the measure ''μ'' spreads~~ out ~~and enlarges the portion of ''μ'' supported around the point ''a''~~. ~~We can get information about our measure around ''a'' by looking at what these measures tend to look like in the limit as ''r'' approaches zero~~.

	~~:'''Definition~~.''' A ''tangent measure'' of a Radon measure ''μ'' at the point ''a'' is a second Radon measure ''ν'' such that there exist sequences of positive numbers ''c''<sub>''i''</sub> > 0 and decreasing radii ''r''<sub>''i''</sub> → 0 such that

	~~:: <math>\lim_{i\rightarrow\infty} c_{i}T_{a,r_{i}\#}\mu =\nu </math>~~

	~~: where the limit is taken in the [[weak-star topology\|weak-&lowast; topology]], i.e., for any [[continuous function]] ''φ'' with~~ [~~[compact support]] in Ω,~~

	:: ~~<math> \lim_{i\rightarrow\infty}\int_{\Omega} \varphi \, \mathrm{d} (c_{i}T_{a,r_{i}\#}\mu)=\int_{\Omega} \varphi \, \mathrm{d} \nu.<~~/~~math>~~

	~~:We denote the set of tangent measures of ''μ'' at ''a'' by Tan(''μ'', ''a'').~~

	~~==Existence==~~
	The set Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the [[support (measure theory)\|support]] of ''μ'' is nonempty on mild conditions on ''μ''. By the weak compactness of Radon measures, Tan(''μ'', ''a'') is nonempty if one of the following conditions hold:

	* ''μ'' is [[Doubling measures and metric spaces\|asymptotically doubling]] at ''a'', i.e. <math>\limsup_{r\downarrow 0} \frac{\mu(B(a,2r))}{\mu(B(a,r))}<\infty</~~math>~~

	* ''μ'' has positive and finite [[Hausdorff density\|upper density]], i.e. ~~<math>0<\limsup_{r\downarrow 0}\frac{\mu(B(a,r))}{r^s}<\infty<~~/~~math> for some <math>0<s<\infty</math>~~.

	~~==Properties=~~=

	~~The collection of tangent measures at a point is closed under two types of scaling. Cones of measures were also defined by Preiss.~~

	* The set Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the support of ''μ'' is a ''cone'' of measures, i.e. if <math>\nu\in \mathrm{Tan}(\mu,a)</math> and <math>c>0</math>, then <math>c\nu\in \mathrm{Tan}(\mu,a)</math>.

	* The cone Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the support of ''μ'' is a ''d-cone'' or ''dilation invariant'', i.e. if <math>\nu\in \mathrm{Tan}(\mu,a)</math> and <math>r>0</math>, then <math>T_{0,r\#}\nu \in \mathrm{Tan}(\mu,a)</math>.

	~~At typical points in the support of a measure, the cone of tangent measures is also closed under translations.~~

	* At ''μ'' almost every ''a'' in the support of ''μ'', the cone Tan(''μ'', ''a'') of tangent measures of ''μ'' at ''a'' is ''translation invariant'', i.e. if <math>\nu\in\mathrm{Tan}(\mu,a)</math> and ''x'' is in the support of ''ν'', then <math>T_{x,1\#}\nu\in\mathrm{Tan}(\mu,a)</math>.

	~~==Examples=~~=
	*Suppose we have a circle in '''R'''<sup>2</sup> with uniform measure on that circle. Then, for any point ''a'' in the circle, the set of tangent measures will just be positive constants times 1-dimensional [[Hausdorff measure]] supported on the line tangent to the circle at that point.
	* In 1995, Toby O'Neil produced an example of a Radon measure ''μ'' on '''R'''<sup>''d''</sup> such that, for μ-almost every point ''a'' ∈ '''R'''<sup>''d''</sup>, Tan(''μ'', ''a'') consists of all nonzero Radon measures.<ref>{{cite journal
	~~\| last~~ = ~~O'Neil \| first = Toby~~
	~~\| title = A measure with a large set of tangent measures~~
	~~\| journal = Proc. of the AMS \| volume = 123~~
	~~\| issue = 7 \| pages = 2217–2220 \| year = 1995~~
	~~\| doi = 10.2307~~/~~2160960~~
	~~\| jstor = 2160960~~
	~~}}<~~/~~ref>~~

	~~==Related concepts==~~

	~~There is an associated notion of the tangent space of a measure~~. A ''k''-dimensional subspace ''P'' of '''R'''<sup>''n''</sup> is called the ''k''-dimensional tangent space of ''μ'' at ''a'' ∈ Ω if — after appropriate rescaling — ''μ'' “looks like” ''k''-dimensional [[Hausdorff measure]] ''H''<sup>''k''</~~sup> on ''P''. More precisely:~~

	~~:'''Definition~~.~~''' ''P'' is the ''k''-''dimensional tangent space'' of ''μ'' at ''a'' if there is a ''θ'' > 0 such that~~

	~~::<math>\mu_{a, r} \xrightarrow[r \to 0]{*} \theta H^{k} \lfloor_{P},</math>~~

	~~:where ''μ''<sub>''a'',''r''</sub> is the translated and rescaled measure given by~~

	~~::<math>\mu_{a, r} (A)~~ = ~~\frac1{r^{n - 1}} \mu(a + r A).</math>~~

	~~:The number ''~~&~~theta;'' is called the ''multiplicity'' of ''μ'' at ''a'', and the tangent space of ''μ'' at ''a'' is denoted T<sub>''a''</sub>(''μ'').~~

	Further study of tangent measures and tangent spaces leads to the notion of a [[varifold]].<ref>{{cite journal \| last = Röger \| first = Matthias \| title = Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation \| journal = Interfaces Free Bound. \| volume = 6 \| year = 2004 \| issue = 1 \| pages = ~~105~~&~~ndash;133 \| issn = 1463-9963 }} {{MathSciNet\|~~id=~~2047075}}</ref>~~

	~~==References==~~
	~~<references />~~

	~~[[Category:Measures (measure theory)]~~]

en>Moritz37: /* Implementations */ trans_title → title: book appeared in English

2013-07-31T17:52:18Z

Implementations: trans_title → title: book appeared in English

← Older revision		Revision as of 19:52, 31 July 2013
Line 1:		Line 1:
	~~Hello buddy~~. ~~Allow me introduce myself~~. ~~I am Luther Aubrey~~. ~~The job I~~'~~ve been occupying for years is~~ a ~~bookkeeper but I~~'~~ve already~~ [~~http~~://~~www~~.~~angieslist~~.~~com~~/~~articles~~/~~what~~-~~does~~-~~and-doesnt-your-auto-warranty-cover~~.~~htm utilized~~] ~~for another~~ [~~http~~:~~//www~~.~~Carchex.com~~/ ~~auto warranty] one. He~~ ~~auto warranty presently life in Idaho and his parents reside close~~ by. The ~~preferred hobby for him and his~~ [~~http~~:/~~/Www~~.~~Bankrate~~.~~com~~/~~finance~~/~~auto/automakers-best-car-warranties-1~~.~~aspx children]~~ is ~~to perform badminton but he~~ is ~~struggling to find time car warranty for it~~.<br><br>~~Also visit my web site: [http:~~//~~pupo~~.ch/~~mypupo~~/~~panda~~/~~panda2~~/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~161 extended auto warranty~~] ~~[http:~~//~~prolab~~.~~97048~~.~~info~~/~~UserProfile~~/~~tabid~~/~~968~~/~~userId~~/~~12295~~/~~Default~~.~~aspx extended car warranty] warranty~~ ([~~http://boatshomefurnishings~~.~~com/Default~~.~~aspx?tabid~~=97&~~userId~~=~~18651 Check This Out~~])		{{Orphan\|date=October 2013}}

			In [[measure theory]], '''tangent measures''' are used to study the local behavior of [[Radon measure]]s, in much the same way as [[tangent space]]s are used to study the local behavior of [[differentiable manifold]]s. Tangent measures (introduced by David Preiss <ref>{{cite journal
			\| last = Preiss \| first = David
			\| title = Geometry of measures in <math>R^n</math>: distribution, rectifiability, and densities
			\| journal = Ann. Math. \| volume = 125
			\| issue = 3 \| pages = 537–643 \| year = 1987
			\| jstor = 1971410
			\| doi = 10.2307/1971410
			}}</ref> in his study of [[rectifiable set]]s) are a useful tool in geometric measure theory. For example, they are used in proving [[Hausdorff density\|Marstrand’s theorem]] and [[Hausdorff density\|Preiss' theorem]].

			==Definition==

			Consider a Radon measure ''μ'' defined on an [[open subset]] Ω of ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> and let ''a'' be an arbitrary point in Ω. We can “zoom in” on a small [[open ball]] of radius ''r'' around ''a'', ''B''<sub>''r''</sub>(''a''), via the transformation
			: <math>T_{a,r}(x)=\frac{x-a}{r},</math>
			which enlarges the ball of radius ''r'' about ''a'' to a ball of radius 1 centered at 0. With this, we may now zoom in on how ''μ'' behaves on ''B''<sub>''r''</sub>(''a'') by looking at the [[push-forward measure]] defined by
			:<math> T_{a,r \#}\mu(A)=\mu(a+rA)</math>
			where
			:<math>a+rA=\{a+rx:x\in A\}.</math>
			As ''r'' gets smaller, this transformation on the measure ''μ'' spreads out and enlarges the portion of ''μ'' supported around the point ''a''. We can get information about our measure around ''a'' by looking at what these measures tend to look like in the limit as ''r'' approaches zero.

			:'''Definition.''' A ''tangent measure'' of a Radon measure ''μ'' at the point ''a'' is a second Radon measure ''ν'' such that there exist sequences of positive numbers ''c''<sub>''i''</sub> > 0 and decreasing radii ''r''<sub>''i''</sub> → 0 such that

			:: <math>\lim_{i\rightarrow\infty} c_{i}T_{a,r_{i}\#}\mu =\nu </math>

			: where the limit is taken in the [[weak-star topology\|weak-&lowast; topology]], i.e., for any [[continuous function]] ''φ'' with [[compact support]] in Ω,

			:: <math> \lim_{i\rightarrow\infty}\int_{\Omega} \varphi \, \mathrm{d} (c_{i}T_{a,r_{i}\#}\mu)=\int_{\Omega} \varphi \, \mathrm{d} \nu.</math>

			:We denote the set of tangent measures of ''μ'' at ''a'' by Tan(''μ'', ''a'').

			==Existence==
			The set Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the [[support (measure theory)\|support]] of ''μ'' is nonempty on mild conditions on ''μ''. By the weak compactness of Radon measures, Tan(''μ'', ''a'') is nonempty if one of the following conditions hold:

			* ''μ'' is [[Doubling measures and metric spaces\|asymptotically doubling]] at ''a'', i.e. <math>\limsup_{r\downarrow 0} \frac{\mu(B(a,2r))}{\mu(B(a,r))}<\infty</math>

			* ''μ'' has positive and finite [[Hausdorff density\|upper density]], i.e. <math>0<\limsup_{r\downarrow 0}\frac{\mu(B(a,r))}{r^s}<\infty</math> for some <math>0<s<\infty</math>.

			==Properties==

			The collection of tangent measures at a point is closed under two types of scaling. Cones of measures were also defined by Preiss.

			* The set Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the support of ''μ'' is a ''cone'' of measures, i.e. if <math>\nu\in \mathrm{Tan}(\mu,a)</math> and <math>c>0</math>, then <math>c\nu\in \mathrm{Tan}(\mu,a)</math>.

			* The cone Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the support of ''μ'' is a ''d-cone'' or ''dilation invariant'', i.e. if <math>\nu\in \mathrm{Tan}(\mu,a)</math> and <math>r>0</math>, then <math>T_{0,r\#}\nu \in \mathrm{Tan}(\mu,a)</math>.

			At typical points in the support of a measure, the cone of tangent measures is also closed under translations.

			* At ''μ'' almost every ''a'' in the support of ''μ'', the cone Tan(''μ'', ''a'') of tangent measures of ''μ'' at ''a'' is ''translation invariant'', i.e. if <math>\nu\in\mathrm{Tan}(\mu,a)</math> and ''x'' is in the support of ''ν'', then <math>T_{x,1\#}\nu\in\mathrm{Tan}(\mu,a)</math>.

			==Examples==
			*Suppose we have a circle in '''R'''<sup>2</sup> with uniform measure on that circle. Then, for any point ''a'' in the circle, the set of tangent measures will just be positive constants times 1-dimensional [[Hausdorff measure]] supported on the line tangent to the circle at that point.
			* In 1995, Toby O'Neil produced an example of a Radon measure ''μ'' on '''R'''<sup>''d''</sup> such that, for μ-almost every point ''a'' ∈ '''R'''<sup>''d''</sup>, Tan(''μ'', ''a'') consists of all nonzero Radon measures.<ref>{{cite journal
			\| last = O'Neil \| first = Toby
			\| title = A measure with a large set of tangent measures
			\| journal = Proc. of the AMS \| volume = 123
			\| issue = 7 \| pages = 2217–2220 \| year = 1995
			\| doi = 10.2307/2160960
			\| jstor = 2160960
			}}</ref>

			==Related concepts==

			There is an associated notion of the tangent space of a measure. A ''k''-dimensional subspace ''P'' of '''R'''<sup>''n''</sup> is called the ''k''-dimensional tangent space of ''μ'' at ''a'' ∈ Ω if — after appropriate rescaling — ''μ'' “looks like” ''k''-dimensional [[Hausdorff measure]] ''H''<sup>''k''</sup> on ''P''. More precisely:

			:'''Definition.''' ''P'' is the ''k''-''dimensional tangent space'' of ''μ'' at ''a'' if there is a ''θ'' > 0 such that

			::<math>\mu_{a, r} \xrightarrow[r \to 0]{*} \theta H^{k} \lfloor_{P},</math>

			:where ''μ''<sub>''a'',''r''</sub> is the translated and rescaled measure given by

			::<math>\mu_{a, r} (A) = \frac1{r^{n - 1}} \mu(a + r A).</math>

			:The number ''θ'' is called the ''multiplicity'' of ''μ'' at ''a'', and the tangent space of ''μ'' at ''a'' is denoted T<sub>''a''</sub>(''μ'').

			Further study of tangent measures and tangent spaces leads to the notion of a [[varifold]].<ref>{{cite journal \| last = Röger \| first = Matthias \| title = Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation \| journal = Interfaces Free Bound. \| volume = 6 \| year = 2004 \| issue = 1 \| pages = 105–133 \| issn = 1463-9963 }} {{MathSciNet\|id=2047075}}</ref>

			==References==
			<references />

			[[Category:Measures (measure theory)]]

en>Rjwilmsi: /* Implementations */fix ISBN format (endash or 13:) using AWB (7867)

2011-12-19T14:02:12Z

Implementations: fix ISBN format (endash or 13:) using AWB (7867)

New page

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