Brinkman number - Revision history

87.10.61.143 at 11:46, 8 January 2015

2015-01-08T11:46:54Z

← Older revision		Revision as of 13:46, 8 January 2015
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	~~{{Refimprove\|date=November 2010}}~~		Emilia Shryock is my name but you can call me something you like. For a whilst I've been in South Dakota and my parents live close by. Doing ceramics is what her family members and her appreciate. Hiring is his profession.<br><br>my page; std home test ([http://withlk.com/board_RNsI08/10421 similar web site])
	~~In [[mathematics]], '''symmetrization'''~~ is a ~~process that converts any function in ''n'' variables to a [[symmetric function]] in ''n'' variables.~~
	~~Conversely,~~ '~~''anti-symmetrization''' converts any function~~ in ~~''n'' variables into an [[antisymmetric]] function.~~

	~~==2 variables==~~
	~~Let <math>S</math> be a set~~ and <math>A</math> an [[Abelian group]]. Given a map <math>\alpha: S \times S \to A</math>, <math>\alpha</math> is termed a symmetric map if <math>\alpha(s,t) = \alpha(t,s)</math> for all <math>s,t \in S</math>.

	~~The '''symmetrization''' of a map <math>\alpha \colon S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) + \alpha(y,x)</math>.~~

	~~Conversely, the '''anti-symmetrization''' or '''skew-symmetrization''' of a map <math>\alpha \colon S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) - \alpha(y,x)</math>.~~

	~~The sum of the symmetrization and the anti-symmetrization is <math>2\alpha.</math>~~
	~~Thus, [[Localization of a ring#Terminology\|away from 2]], meaning if 2 is invertible, such as for the real numbers, one can divide~~ by ~~2 and express every function as a sum of a symmetric function and an anti-symmetric function~~.

	~~The symmetrization of a symmetric map is simply its double, while the symmetrization of an [[alternating map]]~~ is ~~zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.~~

	~~===Bilinear forms===~~
	The symmetrization and anti-symmetrization of a [[bilinear map]] are bilinear; thus away from 2, every bilinear form is a sum of a symmetric form and a skew-symmetric form, and there is no difference between a symmetric form and ~~a quadratic form~~.

	At 2, not every form can be decomposed into a symmetric form and a skew-symmetric form – for instance, over the integers, the associated symmetric form (over the rationals) may take half-integer values, while over <math>\mathbf{Z}/2,</math> a function is ~~skew-symmetric if and only if it is symmetric (as <math>1=-1</math>).~~

	~~This leads to the notion of [[e-quadratic form\|ε-quadratic form]]s and ε-symmetric forms~~.

	~~===Representation theory===~~
	~~In terms of [[representation theory]]:~~
	* exchanging variables gives a representation of the [[symmetric group]] on the space of functions in two variables,
	* the symmetric and anti-symmetric functions are the [[subrepresentation]]s corresponding to the [[trivial representation]] and the [[sign representation]], and
	* symmetrization and anti-symmetrization map a function into these subrepresentations – if one divides by 2, these yield [[projection map]]s.

	~~As the symmetric group of order two equals the [[cyclic group]] of order two (~~<~~math~~>~~S_2=C_2~~<~~/math~~>~~), this corresponds to the [[discrete Fourier transform]] of order two.~~

	~~==''n'' variables==~~
	More generally, given a function in ''n'' variables, one can symmetrize by taking the sum over all <math>n!</math> permutations of the variables,<ref>Hazewinkel (1990), {{Google books quote\|id=kwMdtnhtUMMC\|page~~=344\|text=symmetrized\|p. 344}}</ref> or anti-symmetrize by taking the sum over all <math>n!/2</math> [[even permutation]]s and subtracting the sum over all <math>n!/2</math> odd permutations.~~

	Here symmetrizing (respectively anti-symmetrizing) a symmetric (respectively anti-symmetric) function multiplies by ''n''! – thus if ''n''! is invertible, such as if one is working over the rationals or over a field of characteristic <math>p > n,</math> then these yield projections.

	In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for <math> n > 2</math> there are others – see [[representation theory of the symmetric group]] and [[symmetric polynomials]].

	~~==Bootstrapping==~~
	Given a function in ''k'' variables, one can obtain a symmetric function in ''n'' variables by taking the sum over ''k'' element subsets of the variables. In statistics, this is referred to as [[bootstrapping (~~statistics)\|bootstrapping]], and the associated statistics are called [~~[~~U-statistics]].~~

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

	~~== References ==~~
	* {{cite book \|last1=Hazewinkel \|first1=Michiel \|authorlink1=Michiel Hazewinkel \|last2= \|first2= \|authorlink2= \|title=Encyclopaedia of mathematics: an updated and annotated translation of the Soviet "Mathematical encyclopaedia" \|url=http://~~www.springer~~.com/~~mathematics/book~~/~~978-1-55608-005-0?cm_mmc=Google-_-Book%20Search-_-Springer-_-0 \|edition= \|series=Encyclopaedia of Mathematics \|volume=6 \|year=1990 \|publisher=Springer \|location= \|isbn=978-1-55608-005-0 \|id= }}~~

	~~[[Category:Symmetric functions]~~]

en>Rememberlands: Consistent math notation, Product form made more explicit

2013-06-20T22:15:41Z

Consistent math notation, Product form made more explicit

← Older revision		Revision as of 00:15, 21 June 2013
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	~~The author~~'~~s title~~ is ~~Christy Brookins~~. ~~Alaska is the only location I~~'~~ve been residing~~ in ~~but now I~~'~~m best psychic readings -~~ [~~http~~://~~cartoonkorea~~.~~com~~/~~ce002~~/~~1093612 cartoonkorea~~.~~com~~] - ~~considering other choices~~. ~~Distributing manufacturing~~ is ~~where my main earnings arrives~~ from and it's ~~something I really enjoy~~. ~~One~~ of the ~~things she enjoys most is canoeing~~ and ~~she'~~s ~~been doing it for fairly~~ a ~~whilst~~.<br><br>~~Also visit my web~~ page ... [~~http:~~//~~kard~~.dk/?p=~~24252 real psychic readings~~] ~~readings (~~[http://~~test~~.~~jeka-nn~~.ru/~~node~~/~~129 click through the up coming internet page~~])		{{Refimprove\|date=November 2010}}
			In [[mathematics]], '''symmetrization''' is a process that converts any function in ''n'' variables to a [[symmetric function]] in ''n'' variables.
			Conversely, '''anti-symmetrization''' converts any function in ''n'' variables into an [[antisymmetric]] function.

			==2 variables==
			Let <math>S</math> be a set and <math>A</math> an [[Abelian group]]. Given a map <math>\alpha: S \times S \to A</math>, <math>\alpha</math> is termed a symmetric map if <math>\alpha(s,t) = \alpha(t,s)</math> for all <math>s,t \in S</math>.

			The '''symmetrization''' of a map <math>\alpha \colon S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) + \alpha(y,x)</math>.

			Conversely, the '''anti-symmetrization''' or '''skew-symmetrization''' of a map <math>\alpha \colon S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) - \alpha(y,x)</math>.

			The sum of the symmetrization and the anti-symmetrization is <math>2\alpha.</math>
			Thus, [[Localization of a ring#Terminology\|away from 2]], meaning if 2 is invertible, such as for the real numbers, one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function.

			The symmetrization of a symmetric map is simply its double, while the symmetrization of an [[alternating map]] is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.

			===Bilinear forms===
			The symmetrization and anti-symmetrization of a [[bilinear map]] are bilinear; thus away from 2, every bilinear form is a sum of a symmetric form and a skew-symmetric form, and there is no difference between a symmetric form and a quadratic form.

			At 2, not every form can be decomposed into a symmetric form and a skew-symmetric form – for instance, over the integers, the associated symmetric form (over the rationals) may take half-integer values, while over <math>\mathbf{Z}/2,</math> a function is skew-symmetric if and only if it is symmetric (as <math>1=-1</math>).

			This leads to the notion of [[e-quadratic form\|ε-quadratic form]]s and ε-symmetric forms.

			===Representation theory===
			In terms of [[representation theory]]:
			* exchanging variables gives a representation of the [[symmetric group]] on the space of functions in two variables,
			* the symmetric and anti-symmetric functions are the [[subrepresentation]]s corresponding to the [[trivial representation]] and the [[sign representation]], and
			* symmetrization and anti-symmetrization map a function into these subrepresentations – if one divides by 2, these yield [[projection map]]s.

			As the symmetric group of order two equals the [[cyclic group]] of order two (<math>S_2=C_2</math>), this corresponds to the [[discrete Fourier transform]] of order two.

			==''n'' variables==
			More generally, given a function in ''n'' variables, one can symmetrize by taking the sum over all <math>n!</math> permutations of the variables,<ref>Hazewinkel (1990), {{Google books quote\|id=kwMdtnhtUMMC\|page=344\|text=symmetrized\|p. 344}}</ref> or anti-symmetrize by taking the sum over all <math>n!/2</math> [[even permutation]]s and subtracting the sum over all <math>n!/2</math> odd permutations.

			Here symmetrizing (respectively anti-symmetrizing) a symmetric (respectively anti-symmetric) function multiplies by ''n''! – thus if ''n''! is invertible, such as if one is working over the rationals or over a field of characteristic <math>p > n,</math> then these yield projections.

			In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for <math> n > 2</math> there are others – see [[representation theory of the symmetric group]] and [[symmetric polynomials]].

			==Bootstrapping==
			Given a function in ''k'' variables, one can obtain a symmetric function in ''n'' variables by taking the sum over ''k'' element subsets of the variables. In statistics, this is referred to as [[bootstrapping (statistics)\|bootstrapping]], and the associated statistics are called [[U-statistics]].

			== Notes ==
			<references/>

			== References ==
			* {{cite book \|last1=Hazewinkel \|first1=Michiel \|authorlink1=Michiel Hazewinkel \|last2= \|first2= \|authorlink2= \|title=Encyclopaedia of mathematics: an updated and annotated translation of the Soviet "Mathematical encyclopaedia" \|url=http://www.springer.com/mathematics/book/978-1-55608-005-0?cm_mmc=Google-_-Book%20Search-_-Springer-_-0 \|edition= \|series=Encyclopaedia of Mathematics \|volume=6 \|year=1990 \|publisher=Springer \|location= \|isbn=978-1-55608-005-0 \|id= }}

			[[Category:Symmetric functions]]

en>Vanischenu: /* References */ My sincere appologise for being irresponisble.

2012-07-19T12:41:55Z

References: My sincere appologise for being irresponisble.

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