Fluorescence correlation spectroscopy - Revision history

en>Tom.Reding: Gen fixes (page/s, endash, , et al., unicodify, concising wikilinks, etc.), ref cleanup using AWB

2014-12-26T03:58:47Z

Gen fixes (page/s, endash, &nbsp, et al., unicodify, concising wikilinks, etc.), ref cleanup using AWB

← Older revision		Revision as of 05:58, 26 December 2014
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	~~The writer~~ is ~~called Irwin~~. ~~Playing baseball~~ is the ~~pastime he will by no means stop performing~~. ~~For a whilst she's been in South Dakota. He used~~ to ~~be unemployed but now he is a meter reader~~.<br><br>~~My web-site ::~~ [http://~~netwk~~.~~hannam~~.ac.kr/xe/~~data_2~~/~~38191 netwk.hannam.ac.kr~~]		Alyson is what my husband enjoys to call me but I don't like when people use my full title. He is an information officer. To perform lacross is the factor I love most of all. Ohio is where my home is but my husband desires us to transfer.<br><br>Also visit my homepage; [http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ good psychic]

129.206.99.220: /* FCS imaging using Light sheet fluorescence microscopy */

2014-02-19T12:35:49Z

FCS imaging using Light sheet fluorescence microscopy

← Older revision		Revision as of 14:35, 19 February 2014
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	~~In [[mathematical analysis]], a '''Banach limit'''~~ is a [[continuous function\|continuous]] [[linear functional]] <math>\phi: \ell_\infty \to \mathbb{R}</math> defined on the [[Banach space]] <math>\ell_\infty</math> of all bounded [[complex number\|complex]]-valued [[sequence]]s such that for any sequences <math>x=(x_n)</math> and <math>y=(y_n)</math>, the following conditions are satisfied:		The writer is called Irwin. Playing baseball is the pastime he will by no means stop performing. For a whilst she's been in South Dakota. He used to be unemployed but now he is a meter reader.<br><br>My web-site :: [http://netwk.hannam.ac.kr/xe/data_2/38191 netwk.hannam.ac.kr]
	~~# <math>\phi(\alpha x+\beta y)=\alpha\phi(x)+\beta \phi(y)</math> (linearity);~~
	~~# if <math>x_n\geq 0</math> for all <math>n\ge1</math>, then <math>\phi(x)\geq 0</math>;~~
	~~# <math>\phi(x)=\phi(Sx)</math>, where <math>S</math>~~ is the ~~[[shift operator]] defined~~ by ~~<math>(Sx)_n=x_{n+1}</math>~~.
	~~# If <math>x</math>~~ is a ~~[[convergent sequence]], then <math>\phi(x)=\lim x</math>~~.
	~~Hence, <math>\phi~~<~~/math~~> ~~is an extension of the continuous functional~~
	<~~math~~>~~\lim x:c\mapsto \mathbb C.</math>~~

	~~In other words, a Banach limit extends the usual limits, is shift~~-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely determined in this case. However, as a consequence of the above properties, a Banach limit also satisfies:

	: ~~<math>\liminf_ {n\to\infty} x_n\le\phi(x) \le \limsup_{n\to\infty}x_n</math>~~

	~~The existence of Banach limits is usually proved using the~~ [[Hahn–Banach theorem]] (analyst's approach) or using [[ultrafilter]]s (this approach is more frequent in set-theoretical expositions). These proofs use the [[Axiom of choice]] (so called non-effective proof).

	~~==Almost convergence==~~
	~~There are non-convergent sequences which have uniquely determined Banach limits.~~
	~~For example, if <math>x=(1,0,1,0,\ldots)</math>,~~
	~~then <math>x+S(x)=(1,1,1,\ldots)<~~/~~math> is a constant sequence, and <math>2\phi(x)=\phi(x)+\phi(Sx)=1<~~/~~math> holds~~. ~~Thus for any Banach limit this sequence has limit <math>\frac 12</math>~~.

	~~A sequence <math>x</math> with the property, that for every Banach limit <math>\phi<~~/~~math> the value <math>\phi(x)<~~/~~math> is the same, is called [[almost convergent sequence\|almost convergent]].~~

	~~==Ba spaces==~~
	~~Given a sequence in ''c'', the ordinary limit of the sequence does not arise from an element of <math>\ell^1<~~/~~math>~~. ~~Thus the Banach limit on <math>\ell^\infty</math> is an example of an element of the [[continuous dual space]] of <math>\ell^\infty</math> which is not in <math>\ell^1</math>~~. ~~The dual of <math>\ell^\infty</math> is known as the [[ba space]], and consists of all finitely additive measures on the sigma-algebra of all subsets of the natural numbers~~.

	~~==External links==~~
	*{{planetmath reference\|id=7213\|title=Banach limit}}

	~~[[Category:Functional analysis]~~]

202.79.203.59: /* Common fluorescent probes */

2014-01-23T10:36:37Z

Common fluorescent probes

← Older revision		Revision as of 12:36, 23 January 2014
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	~~The author~~ is ~~recognized by~~ the ~~name~~ of ~~Figures Lint. The favorite pastime~~ for ~~my children~~ and me is ~~to play baseball but I haven't produced~~ a ~~dime with it~~. ~~Years ago we moved to North Dakota~~. In ~~her expert lifestyle she~~ is a ~~payroll clerk but she~~'s ~~always needed her personal company~~.<br><br>~~Here~~ is ~~my web page - std home test - [http:~~//~~www~~.~~ddaybeauty~~.~~com~~/~~node~~/~~14766 check out your url~~],		In [[mathematical analysis]], a '''Banach limit''' is a [[continuous function\|continuous]] [[linear functional]] <math>\phi: \ell_\infty \to \mathbb{R}</math> defined on the [[Banach space]] <math>\ell_\infty</math> of all bounded [[complex number\|complex]]-valued [[sequence]]s such that for any sequences <math>x=(x_n)</math> and <math>y=(y_n)</math>, the following conditions are satisfied:
			# <math>\phi(\alpha x+\beta y)=\alpha\phi(x)+\beta \phi(y)</math> (linearity);
			# if <math>x_n\geq 0</math> for all <math>n\ge1</math>, then <math>\phi(x)\geq 0</math>;
			# <math>\phi(x)=\phi(Sx)</math>, where <math>S</math> is the [[shift operator]] defined by <math>(Sx)_n=x_{n+1}</math>.
			# If <math>x</math> is a [[convergent sequence]], then <math>\phi(x)=\lim x</math>.
			Hence, <math>\phi</math> is an extension of the continuous functional
			<math>\lim x:c\mapsto \mathbb C.</math>

			In other words, a Banach limit extends the usual limits, is shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely determined in this case. However, as a consequence of the above properties, a Banach limit also satisfies:

			: <math>\liminf_ {n\to\infty} x_n\le\phi(x) \le \limsup_{n\to\infty}x_n</math>

			The existence of Banach limits is usually proved using the [[Hahn–Banach theorem]] (analyst's approach) or using [[ultrafilter]]s (this approach is more frequent in set-theoretical expositions). These proofs use the [[Axiom of choice]] (so called non-effective proof).

			==Almost convergence==
			There are non-convergent sequences which have uniquely determined Banach limits.
			For example, if <math>x=(1,0,1,0,\ldots)</math>,
			then <math>x+S(x)=(1,1,1,\ldots)</math> is a constant sequence, and <math>2\phi(x)=\phi(x)+\phi(Sx)=1</math> holds. Thus for any Banach limit this sequence has limit <math>\frac 12</math>.

			A sequence <math>x</math> with the property, that for every Banach limit <math>\phi</math> the value <math>\phi(x)</math> is the same, is called [[almost convergent sequence\|almost convergent]].

			==Ba spaces==
			Given a sequence in ''c'', the ordinary limit of the sequence does not arise from an element of <math>\ell^1</math>. Thus the Banach limit on <math>\ell^\infty</math> is an example of an element of the [[continuous dual space]] of <math>\ell^\infty</math> which is not in <math>\ell^1</math>. The dual of <math>\ell^\infty</math> is known as the [[ba space]], and consists of all finitely additive measures on the sigma-algebra of all subsets of the natural numbers.

			==External links==
			*{{planetmath reference\|id=7213\|title=Banach limit}}

			[[Category:Functional analysis]]

en>Nsda: /* Anomalous diffusion */ links

2012-08-14T08:02:54Z

Anomalous diffusion: links

New page

The author is recognized by the name of Figures Lint. The favorite pastime for my children and me is to play baseball but I haven't produced a dime with it. Years ago we moved to North Dakota. In her expert lifestyle she is a payroll clerk but she's always needed her personal company.<br><br>Here is my web page - std home test - [http://www.ddaybeauty.com/node/14766 check out your url],