Variance reduction - Revision history

en>Mogism: Cleanup/Typo fixing, typo(s) fixed: 1980's → 1980s using AWB

2014-12-13T18:22:28Z

Cleanup/Typo fixing, typo(s) fixed: 1980's → 1980s using AWB

← Older revision		Revision as of 20:22, 13 December 2014
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	~~The '''Darboux derivative''' of a map between a [[manifold]] and a [[Lie group]] is a variant of the standard derivative~~. ~~In a certain sense, it is arguably a more natural generalization of~~ the ~~single-variable derivative. It allows a generalization of the single-variable [[fundamental theorem of calculus]] to higher dimensions~~, ~~in a different vein than the generalization that~~ is ~~[[Stokes' theorem]]~~.		Hi there. Allow me begin by introducing the writer, her title is Sophia. Distributing production is exactly where my primary earnings arrives from and it's some thing I truly enjoy. To play lacross is the thing I love most of all. For a while I've been in Alaska but I will have to move in a yr or two.<br><br>Stop by my blog - best psychics ([http://brazil.amor-amore.com/irboothe Going On this site])

	~~==Formal definition==~~

	~~Let <math>G</math> be a [[Lie group]], and let <math>\mathfrak{g}</math> be its [[Lie algebra]]. The [[Maurer-Cartan form]], <math>\omega_G</math>,~~ is ~~the smooth <math>\mathfrak{g}</math>-valued <math>1</math>-form on <math>G</math> (cf. [[Lie algebra valued form]]) defined by~~
	~~:<math>\omega_G(X_g) = (T_g L_g)^{-1} X_g </math>~~
	~~for all <math>g \in G</math>~~ and ~~<math>X_g \in T_g G</math>. Here <math>L_g</math> denotes left multiplication by the element <math>g \in G</math> and <math>T_g L_g</math> is its derivative at <math>g</math>.~~

	~~Let <math>f:M \to G</math> be a [[smooth function]] between a [[smooth manifold]] <math>M</math> and <math>G</math>~~. ~~Then the '''Darboux derivative''' of <math>f</math>~~ is the ~~smooth <math>\mathfrak{g}</math>-valued <math>1</math>-form~~
	~~:<math>\omega_f := f^* \omega_G,</math>~~
	~~the [[pullback (differential geometry)\|pullback]]~~ of ~~<math>\omega_G</math> by <math>f</math>~~. ~~The map <math>f</math> is called an '''integral''' or '''primitive''' of <math>\omega_f</math>.~~

	~~==More natural?==~~

	~~The reason that one might call the Darboux derivative~~ a ~~more natural generalization of the derivative of single-variable calculus is this. In single-variable calculus, the [[derivative]] <math>f~~'~~</math> of a function <math>f: \mathbb{R} \to \mathbb{R}</math> assigns to each point~~ in ~~the domain a single number. According~~ to ~~the more general manifold ideas of derivatives, the derivative assigns to each point~~ in ~~the domain~~ a ~~[[linear map]] from the tangent space at the domain point to the tangent space at the image point. This derivative encapsulates~~ two pieces of data: the image of the domain point ''and'' the linear map. In single-variable calculus, we drop some information. We retain only the linear map, in the form of a scalar multiplying agent (i.e. ~~a number).~~

	~~One way to justify this convention of retaining only the linear map aspect of the derivative is to appeal to the (very simple) Lie group structure of~~ <~~math~~>~~\mathbb{R}~~<~~/math~~> under addition. The [[tangent bundle]] of any [[Lie group]] can be trivialized via left (or right) multiplication. This means that every tangent space in <math>\mathbb{R}</math> may be identified with the tangent space at the identity, <math>0</math>, which is the [[Lie algebra]] of <math>\mathbb{R}</math>. In this case, left and right multiplication are simply translation. By post-composing the manifold-type derivative with the tangent space trivialization, for each point in the domain we obtain a linear map from the tangent space at the domain point to the Lie algebra of <math>\mathbb{R}</math>. In symbols, for each <math>x \in \mathbb{R}</math> we look at the map
	~~:<math>v \in T_x \mathbb{R} \mapsto (T_{f(x)} L_{f(x)})^{~~-~~1} \circ~~ (~~T_x f) v \in T_0 \mathbb{R}.</math>~~
	~~Since the tangent spaces involved are one-dimensional, this linear map is just multiplication by some scalar. (This scalar can change depending on what basis we use for the vector spaces, but the~~ [[Canonical units\|canonical unit]] vector field <math>\frac{\partial}{\partial t}</math> on <math>\mathbb{R}</math> gives a canonical choice of basis, and hence a canonical choice of scalar.) This scalar is what we usually denote by <math>f'(x)</math>.

	~~==Uniqueness of primitives==~~

	If the manifold <math>M</math> is connected, and <math>f,g: M \to G</math> are both primitives of <math>\omega_f</math>, i.e. <math>\omega_f = \omega_g</math>, then there exists some constant <math>C \in G</math> such that
	:~~<math>f(x) = C \cdot g(x)<~~/~~math> for all <math>x \in M<~~/~~math>.~~

	~~This constant <math>C</math> is of course the analogue of the constant that appears when taking an [[indefinite integral]]~~.

	~~==The fundamental theorem of calculus==~~

	~~Recall the '''structural equation''' for the [[Maurer~~-~~Cartan form]]:~~
	~~:<math>d \omega + \frac{1}{2} [\omega, \omega] = 0~~.</~~math>~~
	~~This means that for all vector fields <math>X</math> and <math>Y</math> on <math>G</math> and all <math>x \in G</math>, we have~~
	~~:<math>(d \omega)_x (X_x, Y_x) + [\omega_x(X_x), \omega_x(Y_x)] = 0.</math>~~
	~~For any Lie algebra-valued <math>1</math>-form on any smooth manifold, all the terms in~~ this ~~equation make sense, so for any such form we can ask whether or not it satisfies this structural equation.~~

	~~The usual [[fundamental theorem of calculus]] for single-variable calculus has the following local generalization.~~

	If a <math>\mathfrak{g}</math>-valued <math>1</math>-form <math>\omega</math> on <math>M</math> satisfies the structural equation, then every point <math>p \in M</math> has an open neighborhood <math>U</math> and a smooth map <math>f: U \to G</math> such that
	~~:<math>\omega_f = \omega\|_U,</math>~~
	~~i.e. <math>\omega</math> has a primitive defined in a neighborhood of every point of <math>M</math>.~~

	~~For a global generalization of the fundamental theorem, one needs to study certain [[monodromy]] questions in <math>M</math> and <math>G</math>.~~

	~~==References==~~

	* {{cite book\|author=R. W. Sharpe\|title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program\|year=1996\|publisher=Springer-Verlag, Berlin\|isbn=0-387-94732-9}}
	* {{cite book\|author=[[Shlomo Sternberg]]\|title=Lectures in differential geometry\|year=1964\|publisher=Prentice-Hall\|id=LCCN 64-7993\|chapter=Chapter V, Lie Groups. Section 2, Invariant forms and the Lie algebra.}}

	~~[[Category:Differential calculus]]~~
	~~[[Category:Lie groups]~~]

en>Eeeeeccccc: /* Underlying principle of the CRN technique */

2014-01-21T09:31:08Z

Underlying principle of the CRN technique

← Older revision		Revision as of 11:31, 21 January 2014
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	~~Irwin Butts~~ is ~~what my spouse enjoys~~ to ~~contact me though I don~~'~~t truly like being~~ called ~~like~~ that. ~~Hiring has been my profession~~ for some ~~time~~ but I'~~ve currently utilized~~ for ~~another~~ 1. ~~To collect cash is what her family~~ and ~~her enjoy~~. ~~My family members life~~ in ~~Minnesota and my family members enjoys~~ it.<br><br>~~Also visit my weblog~~ ... [~~http:~~//~~dateplate~~.~~com/dayzfe at home std testing~~]		The '''Darboux derivative''' of a map between a [[manifold]] and a [[Lie group]] is a variant of the standard derivative. In a certain sense, it is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable [[fundamental theorem of calculus]] to higher dimensions, in a different vein than the generalization that is [[Stokes' theorem]].

			==Formal definition==

			Let <math>G</math> be a [[Lie group]], and let <math>\mathfrak{g}</math> be its [[Lie algebra]]. The [[Maurer-Cartan form]], <math>\omega_G</math>, is the smooth <math>\mathfrak{g}</math>-valued <math>1</math>-form on <math>G</math> (cf. [[Lie algebra valued form]]) defined by
			:<math>\omega_G(X_g) = (T_g L_g)^{-1} X_g </math>
			for all <math>g \in G</math> and <math>X_g \in T_g G</math>. Here <math>L_g</math> denotes left multiplication by the element <math>g \in G</math> and <math>T_g L_g</math> is its derivative at <math>g</math>.

			Let <math>f:M \to G</math> be a [[smooth function]] between a [[smooth manifold]] <math>M</math> and <math>G</math>. Then the '''Darboux derivative''' of <math>f</math> is the smooth <math>\mathfrak{g}</math>-valued <math>1</math>-form
			:<math>\omega_f := f^* \omega_G,</math>
			the [[pullback (differential geometry)\|pullback]] of <math>\omega_G</math> by <math>f</math>. The map <math>f</math> is called an '''integral''' or '''primitive''' of <math>\omega_f</math>.

			==More natural?==

			The reason that one might call the Darboux derivative a more natural generalization of the derivative of single-variable calculus is this. In single-variable calculus, the [[derivative]] <math>f'</math> of a function <math>f: \mathbb{R} \to \mathbb{R}</math> assigns to each point in the domain a single number. According to the more general manifold ideas of derivatives, the derivative assigns to each point in the domain a [[linear map]] from the tangent space at the domain point to the tangent space at the image point. This derivative encapsulates two pieces of data: the image of the domain point ''and'' the linear map. In single-variable calculus, we drop some information. We retain only the linear map, in the form of a scalar multiplying agent (i.e. a number).

			One way to justify this convention of retaining only the linear map aspect of the derivative is to appeal to the (very simple) Lie group structure of <math>\mathbb{R}</math> under addition. The [[tangent bundle]] of any [[Lie group]] can be trivialized via left (or right) multiplication. This means that every tangent space in <math>\mathbb{R}</math> may be identified with the tangent space at the identity, <math>0</math>, which is the [[Lie algebra]] of <math>\mathbb{R}</math>. In this case, left and right multiplication are simply translation. By post-composing the manifold-type derivative with the tangent space trivialization, for each point in the domain we obtain a linear map from the tangent space at the domain point to the Lie algebra of <math>\mathbb{R}</math>. In symbols, for each <math>x \in \mathbb{R}</math> we look at the map
			:<math>v \in T_x \mathbb{R} \mapsto (T_{f(x)} L_{f(x)})^{-1} \circ (T_x f) v \in T_0 \mathbb{R}.</math>
			Since the tangent spaces involved are one-dimensional, this linear map is just multiplication by some scalar. (This scalar can change depending on what basis we use for the vector spaces, but the [[Canonical units\|canonical unit]] vector field <math>\frac{\partial}{\partial t}</math> on <math>\mathbb{R}</math> gives a canonical choice of basis, and hence a canonical choice of scalar.) This scalar is what we usually denote by <math>f'(x)</math>.

			==Uniqueness of primitives==

			If the manifold <math>M</math> is connected, and <math>f,g: M \to G</math> are both primitives of <math>\omega_f</math>, i.e. <math>\omega_f = \omega_g</math>, then there exists some constant <math>C \in G</math> such that
			:<math>f(x) = C \cdot g(x)</math> for all <math>x \in M</math>.

			This constant <math>C</math> is of course the analogue of the constant that appears when taking an [[indefinite integral]].

			==The fundamental theorem of calculus==

			Recall the '''structural equation''' for the [[Maurer-Cartan form]]:
			:<math>d \omega + \frac{1}{2} [\omega, \omega] = 0.</math>
			This means that for all vector fields <math>X</math> and <math>Y</math> on <math>G</math> and all <math>x \in G</math>, we have
			:<math>(d \omega)_x (X_x, Y_x) + [\omega_x(X_x), \omega_x(Y_x)] = 0.</math>
			For any Lie algebra-valued <math>1</math>-form on any smooth manifold, all the terms in this equation make sense, so for any such form we can ask whether or not it satisfies this structural equation.

			The usual [[fundamental theorem of calculus]] for single-variable calculus has the following local generalization.

			If a <math>\mathfrak{g}</math>-valued <math>1</math>-form <math>\omega</math> on <math>M</math> satisfies the structural equation, then every point <math>p \in M</math> has an open neighborhood <math>U</math> and a smooth map <math>f: U \to G</math> such that
			:<math>\omega_f = \omega\|_U,</math>
			i.e. <math>\omega</math> has a primitive defined in a neighborhood of every point of <math>M</math>.

			For a global generalization of the fundamental theorem, one needs to study certain [[monodromy]] questions in <math>M</math> and <math>G</math>.

			==References==

			* {{cite book\|author=R. W. Sharpe\|title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program\|year=1996\|publisher=Springer-Verlag, Berlin\|isbn=0-387-94732-9}}
			* {{cite book\|author=[[Shlomo Sternberg]]\|title=Lectures in differential geometry\|year=1964\|publisher=Prentice-Hall\|id=LCCN 64-7993\|chapter=Chapter V, Lie Groups. Section 2, Invariant forms and the Lie algebra.}}

			[[Category:Differential calculus]]
			[[Category:Lie groups]]

en>Jorgecarleitao: /* See also */ removed Vegas, added importance sampling. Vegas will be added to importance sampling's page, which it uses.

2012-07-25T08:10:44Z

See also: removed Vegas, added importance sampling. Vegas will be added to importance sampling's page, which it uses.

New page

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