Electromagnetically induced transparency - Revision history

en>BG19bot: /* Slow light and stopped light */WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (10093)

2014-05-09T06:06:39Z

Slow light and stopped light: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (10093)

← Older revision		Revision as of 08:06, 9 May 2014
Line 1:		Line 1:
	~~In [[mathematics]], a '''moment problem''' arises as the result~~ of ~~trying to invert the mapping that takes a [[measure (mathematics)\|measure]] μ to the sequences of [[Moment (mathematics)\|moment]]s~~		The name of the writer is Garland. Some time in the past he chose to live in Idaho. The job he's been occupying for years is a messenger. Playing crochet is something that I've carried out for many years.<br><br>Feel free to surf to my web blog ... [http://Gamescap.com/profile/scgeake Gamescap.com]

	~~:<math>m_n = \int_{-\infty}^\infty x^n \,d\mu(x)\,.\,\!</math>~~

	~~More generally, one may consider~~
	~~:<math>m_n = \int_{-\infty}^\infty M_n(x) \,d\mu(x)\,\!</math>~~
	~~for an arbitrary sequence of functions ''M''<sub>''n''</sub>.~~

	~~== Introduction ==~~

	~~In the classical setting, μ is a measure on the [[real line]], and ''M'' is in the sequence { ''x''<sup>''n''</sup> : ''n'' = 0, 1, 2, ... } In this form~~ the ~~question appears in [[probability theory]], asking whether there~~ is ~~a [[probability measure]] having specified [[mean]], [[variance]] and so on, and whether it is unique~~.

	~~There are three named classical moment problems: the [[Hamburger moment problem]]~~ in ~~which~~ the ~~[[support (mathematics)\|support]] of μ is allowed~~ to be the whole real line; the [[Stieltjes moment problem]], for <nowiki>[0, +∞)</nowiki>; and the [[Hausdorff moment problem]] for a bounded interval, which [[without loss of generality]] may be taken as [0, 1].

	~~==Existence==~~

	~~A sequence of numbers~~ '~~'m''<sub>''n''</sub>~~ is ~~the sequence of moments of~~ a ~~measure ''μ'' if and only if a certain positivity condition is fulfilled; namely, the [[Hankel matrices]] ''H''<sub>''n''</sub>,~~

	~~:<math>(H_n)_{ij} = m_{i+j}\,,\,\!</math>~~

	~~should be [[positive-definite matrix\|positive semi-definite]]~~. ~~A condition of similar form~~ is ~~necessary and sufficient for the existence of a measure <math>\mu</math> supported on a given interval [''a'', ''b''].~~

	~~One way to prove these results is to consider the linear functional <math>\scriptstyle\varphi</math>~~ that ~~sends a polynomial~~

	~~: <math>P(x) = \sum_k a_k x^k \,\!</math>~~

	to

	~~: <math>\sum_k a_k m_k.\,\!</math>~~

	~~If ''m''<sub>''kn''</sub> are the moments of some measure ''μ'' supported on [''a'', ''b''], then evidently~~

	~~:{{NumBlk\|:\|''φ'~~'~~(''P'') ≥ 0~~ for ~~any polynomial ''P'' that is non-negative on [''a'', ''b'']~~.~~\|{{EquationRef\|1}}}}~~

	~~Vice versa, if ({{EquationNote\|1}}) holds, one can apply the [[Riesz–Markov–Kakutani representation theorem\|M. Riesz extension theorem]] and extend~~ <~~math~~>~~\phi~~<~~/math~~> to ~~a functional on the space of continuous functions with compact support ''C''<sub>0</sub>([''a'', ''b'']), so that~~

	~~:{{NumBlk\|:\|<math>\qquad \varphi(f) \ge 0\text{ for any } f \in C_0([a,b])</math>\|{{EquationRef\|2}}}}~~

	~~such that ''&fnof;'' ≥ 0 on [''a'', ''b'']~~.

	~~By the [[Riesz_representation_theorem#The_representation_theorem_for_linear_functionals_on_Cc~~.~~28X~~.~~29\| Riesz representation theorem]], ({{EquationNote\|2}}) holds iff there exists a measure ''μ'' supported on~~ [~~''a'', ''b''], such that~~

	: ~~<math> \phi(f) = \int f \, d\mu\,\!<~~/~~math>~~

	~~for every ''&fnof;'' ∈ ''C''<sub>0<~~/~~sub>([''a'', ''b''])~~.

	~~Thus the existence of the measure <math>\mu<~~/math> is equivalent to ({{EquationNote\|1}}). Using a representation theorem for positive polynomials on [''a'', ''b''], <!-- This is due to Riesz or Fejer (or maybe both); a ref. is needed (maybe Szego's book?) --> one can reformulate ({{EquationNote\|1}}) as a condition on [[Hankel matrices]].

	~~See Refs. 1–3. for more details.~~

	~~== Uniqueness (or determinacy) ==~~

	The uniqueness of μ in the Hausdorff moment problem follows from the [[Weierstrass approximation theorem]], which states that [[polynomial]]s are [[dense set\|dense]] under the [[uniform norm]] in the space of [[continuous functions]] on [0, 1]. ~~For the problem on an infinite interval, uniqueness is a more delicate question; see [[Carleman's condition]], [[Krein's condition]] and Ref. 2.~~

	~~== Variations ==~~

	An important variation is the [[truncated moment problem]], which studies the properties of measures with fixed first ''k'' moments (for a finite ''k''). Results on the truncated moment problem have numerous applications to [[extremal problems]], optimisation and limit theorems in [[probability theory]]. See also: [[Chebyshev–Markov–Stieltjes inequalities]] and Ref. 3.

	~~==References==~~
	~~:'''1'''. Shohat, James Alexander; [[Jacob Tamarkin\|Tamarkin, J. D.]]; ''The Problem of Moments'', American mathematical society, New York, 1943.~~
	:'''2'''. [[Naum Akhiezer\|Akhiezer, N. I.]], ''The classical moment problem and some related questions in analysis'', translated from the Russian by N. Kemmer, Hafner Publishing Co., New York 1965 x+253 pp.
	:'''3'''. Krein, M. G.; Nudelman, A. A.; ''The Markov moment problem and extremal problems. Ideas and problems of P. L. Chebyshev and A. A. Markov and their further development.'' Translated from the Russian by D. Louvish. Translations of Mathematical Monographs, Vol. 50. American Mathematical Society, Providence, R.I., 1977. v+417 pp.

	~~[[Category:Mathematical analysis]]~~
	~~[[Category:Hilbert space]]~~
	~~[[Category:Probability theory]]~~
	~~[[Category:Theory of probability distributions]]~~
	~~[[Category:Mathematical problems]]~~
	~~[[Category:Real algebraic geometry]]~~
	~~[[Category:Optimization in vector spaces]~~]

203.55.120.150: /* Slow light and stopped light */

2013-12-29T06:12:59Z

Slow light and stopped light

← Older revision		Revision as of 08:12, 29 December 2013
Line 1:		Line 1:
	~~<br><br>Luke is really~~ a ~~superstar within~~ the ~~producing as well as occupation expansion initial second~~ to ~~his third place recording, And , will be~~ the ~~confirmation. He burst~~ to the ~~picture in 2015 regarding his funny combination~~ of ~~lower~~-~~residence accessibility~~, ~~video legend fantastic looks and words~~, ~~is [http~~://~~lukebryantickets~~.~~neodga.com luke bryan tour dates~~] ~~defined t~~ in ~~a main way~~. ~~The latest~~ a ~~in the region graph~~ and ~~#2 in the pop maps~~, ~~earning~~ it the ~~next top debut during those times~~ of ~~2010~~ for ~~any~~ [~~http:~~/~~/www.banburycrossonline.com luke bryan presale tickets~~] ~~nation performer~~. <br><br>~~The boy~~ of the , ~~knows determination~~ and ~~dedication are important elements in terms~~ of ~~an excellent occupation-~~ . ~~His initially album~~, ~~Continue~~ to ~~be Me~~, ~~made~~ the ~~Top reaches “All My Friends “Country and Say” Guy~~,~~” although his hard work~~, ~~Doin’ Issue~~, ~~discovered~~ the ~~artist-about three straight No~~. ~~6 single men~~ [~~http~~:/~~/www~~.~~hotelsedinburgh~~.~~org luke bryan concert schedule~~] ~~and women~~: ~~More Contacting Is usually~~ a ~~Fantastic Point~~.”<br><br>~~While~~ in the ~~fall of 1999~~, ~~Concert tours: Luke Bryan &~~ that ~~have a remarkable~~ set of , ~~including Downtown~~. ~~“It’s much like you are obtaining~~ a ~~authorization to go to~~ the ~~next level~~, ~~states those musicians that~~ have ~~been a part~~ of the ~~Concert toursabove right into a larger sized degree~~ of ~~designers~~.~~” It twisted as one of the best~~ ~~tours in~~ the ~~twenty-calendar year historical past~~.~~<br><br>Take a look at my website -~~ [~~http~~:~~//lukebryantickets.pyhgy.com coldplay concert~~]		In [[mathematics]], a '''moment problem''' arises as the result of trying to invert the mapping that takes a [[measure (mathematics)\|measure]] μ to the sequences of [[Moment (mathematics)\|moment]]s

			:<math>m_n = \int_{-\infty}^\infty x^n \,d\mu(x)\,.\,\!</math>

			More generally, one may consider
			:<math>m_n = \int_{-\infty}^\infty M_n(x) \,d\mu(x)\,\!</math>
			for an arbitrary sequence of functions ''M''<sub>''n''</sub>.

			== Introduction ==

			In the classical setting, μ is a measure on the [[real line]], and ''M'' is in the sequence { ''x''<sup>''n''</sup> : ''n'' = 0, 1, 2, ... } In this form the question appears in [[probability theory]], asking whether there is a [[probability measure]] having specified [[mean]], [[variance]] and so on, and whether it is unique.

			There are three named classical moment problems: the [[Hamburger moment problem]] in which the [[support (mathematics)\|support]] of μ is allowed to be the whole real line; the [[Stieltjes moment problem]], for <nowiki>[0, +∞)</nowiki>; and the [[Hausdorff moment problem]] for a bounded interval, which [[without loss of generality]] may be taken as [0, 1].

			==Existence==

			A sequence of numbers ''m''<sub>''n''</sub> is the sequence of moments of a measure ''μ'' if and only if a certain positivity condition is fulfilled; namely, the [[Hankel matrices]] ''H''<sub>''n''</sub>,

			:<math>(H_n)_{ij} = m_{i+j}\,,\,\!</math>

			should be [[positive-definite matrix\|positive semi-definite]]. A condition of similar form is necessary and sufficient for the existence of a measure <math>\mu</math> supported on a given interval [''a'', ''b''].

			One way to prove these results is to consider the linear functional <math>\scriptstyle\varphi</math> that sends a polynomial

			: <math>P(x) = \sum_k a_k x^k \,\!</math>

			to

			: <math>\sum_k a_k m_k.\,\!</math>

			If ''m''<sub>''kn''</sub> are the moments of some measure ''μ'' supported on [''a'', ''b''], then evidently

			:{{NumBlk\|:\|''φ''(''P'') ≥ 0 for any polynomial ''P'' that is non-negative on [''a'', ''b''].\|{{EquationRef\|1}}}}

			Vice versa, if ({{EquationNote\|1}}) holds, one can apply the [[Riesz–Markov–Kakutani representation theorem\|M. Riesz extension theorem]] and extend <math>\phi</math> to a functional on the space of continuous functions with compact support ''C''<sub>0</sub>([''a'', ''b'']), so that

			:{{NumBlk\|:\|<math>\qquad \varphi(f) \ge 0\text{ for any } f \in C_0([a,b])</math>\|{{EquationRef\|2}}}}

			such that ''&fnof;'' ≥ 0 on [''a'', ''b''].

			By the [[Riesz_representation_theorem#The_representation_theorem_for_linear_functionals_on_Cc.28X.29\| Riesz representation theorem]], ({{EquationNote\|2}}) holds iff there exists a measure ''μ'' supported on [''a'', ''b''], such that

			: <math> \phi(f) = \int f \, d\mu\,\!</math>

			for every ''&fnof;'' ∈ ''C''<sub>0</sub>([''a'', ''b'']).

			Thus the existence of the measure <math>\mu</math> is equivalent to ({{EquationNote\|1}}). Using a representation theorem for positive polynomials on [''a'', ''b''], <!-- This is due to Riesz or Fejer (or maybe both); a ref. is needed (maybe Szego's book?) --> one can reformulate ({{EquationNote\|1}}) as a condition on [[Hankel matrices]].

			See Refs. 1–3. for more details.

			== Uniqueness (or determinacy) ==

			The uniqueness of μ in the Hausdorff moment problem follows from the [[Weierstrass approximation theorem]], which states that [[polynomial]]s are [[dense set\|dense]] under the [[uniform norm]] in the space of [[continuous functions]] on [0, 1]. For the problem on an infinite interval, uniqueness is a more delicate question; see [[Carleman's condition]], [[Krein's condition]] and Ref. 2.

			== Variations ==

			An important variation is the [[truncated moment problem]], which studies the properties of measures with fixed first ''k'' moments (for a finite ''k''). Results on the truncated moment problem have numerous applications to [[extremal problems]], optimisation and limit theorems in [[probability theory]]. See also: [[Chebyshev–Markov–Stieltjes inequalities]] and Ref. 3.

			==References==
			:'''1'''. Shohat, James Alexander; [[Jacob Tamarkin\|Tamarkin, J. D.]]; ''The Problem of Moments'', American mathematical society, New York, 1943.
			:'''2'''. [[Naum Akhiezer\|Akhiezer, N. I.]], ''The classical moment problem and some related questions in analysis'', translated from the Russian by N. Kemmer, Hafner Publishing Co., New York 1965 x+253 pp.
			:'''3'''. Krein, M. G.; Nudelman, A. A.; ''The Markov moment problem and extremal problems. Ideas and problems of P. L. Chebyshev and A. A. Markov and their further development.'' Translated from the Russian by D. Louvish. Translations of Mathematical Monographs, Vol. 50. American Mathematical Society, Providence, R.I., 1977. v+417 pp.

			[[Category:Mathematical analysis]]
			[[Category:Hilbert space]]
			[[Category:Probability theory]]
			[[Category:Theory of probability distributions]]
			[[Category:Mathematical problems]]
			[[Category:Real algebraic geometry]]
			[[Category:Optimization in vector spaces]]

en>Mets501: explain this better

2012-07-23T05:07:39Z

explain this better

New page

Luke is really a superstar within the producing as well as occupation expansion initial second to his third place recording, And , will be the confirmation. He burst to the picture in 2015 regarding his funny combination of lower-residence accessibility, video legend fantastic looks and words, is [http://lukebryantickets.neodga.com luke bryan tour dates] defined t in a main way. The latest a in the region graph and #2 in the pop maps, earning it the next top debut during those times of 2010 for any [http://www.banburycrossonline.com luke bryan presale tickets] nation performer. The boy of the , knows determination and dedication are important elements in terms of an excellent occupation- . His initially album, Continue to be Me, made the Top reaches “All My Friends “Country and Say” Guy,” although his hard work, Doin’ Issue, discovered the artist-about three straight No. 6 single men [http://www.hotelsedinburgh.org luke bryan concert schedule] and women: More Contacting Is usually a Fantastic Point.” While in the fall of 1999, Concert tours: Luke Bryan & that have a remarkable set of , including Downtown. “It’s much like you are obtaining a authorization to go to the next level, states those musicians that have been a part of the Concert toursabove right into a larger sized degree of designers.” It twisted as one of the best tours in the twenty-calendar year historical past. Take a look at my website - [http://lukebryantickets.pyhgy.com coldplay concert]