Weighted Majority Algorithm - Revision history

en>Rjwilmsi: Journal cites, Added 1 doi to a journal cite using AWB (10331)

2014-08-08T06:34:59Z

Journal cites, Added 1 doi to a journal cite using AWB (10331)

← Older revision		Revision as of 08:34, 8 August 2014
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	In [[complex analysis]], a branch of [[mathematics]], the '''antiderivative''', or '''primitive''', of a [[complex number\|complex]]-valued [[function (mathematics)\|function]] ''g'' is a function whose [[complex derivative]] is ''g''. More precisely, given an [[open set]] <math>U</math> in the complex plane and a function <math>g:U\to ~~\mathbb C,</math> the antiderivative of <math>g</math> is a function <math>f:U\~~to ~~\mathbb C</math> that satisfies <math>\frac{df}{dz}=g</math>.~~		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. To play lacross is something he would never give up. He is an information officer. Some time ago he chose to live in North Carolina and he doesn't plan on altering it.<br><br>Feel free to visit my site - spirit messages ([http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 view it])

	~~As such~~, ~~this concept is the complex-variable version of the [[antiderivative]] of a [[real number\|real]]-valued function~~.

	~~==Uniqueness==~~

	~~The derivative of a constant function~~ is ~~zero~~. ~~Therefore, any constant~~ is an ~~antiderivative of the zero function~~. If <math>U</math> is a [[connected set]], then the constants are the only antiderivatives of the zero function. Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each [[connected set\|connected component]] of <math>U</math> (those constants need not be equal).

	~~This observation implies that if a function <math>g:U\to \mathbb C</math> has an antiderivative, then that antiderivative is unique [[up~~ to~~]] addition of a function which is constant on each connected component of <math>U</math>.~~

	~~==Existence==~~
	~~One can characterize the existence of antiderivatives via path integrals~~ in ~~the complex plane, much like the case of functions of a real variable. Perhaps not surprisingly, ''g'' has an antiderivative ''f'' if~~ and ~~only if, for every γ path from~~ '~~'a'' to ''b'', the path integral~~

	~~:<math> \int_{\gamma} g(\zeta) d \zeta = f(b) - f(a)~~.<~~/math>~~

	~~Equivalently,~~

	~~:<math~~> ~~\int_{\gamma} g(\zeta) d \zeta = 0,~~<~~/math~~>

	~~for any closed path γ.~~

	However, this formal similarity notwithstanding, possessing a complex-antiderivative is a much more restrictive condition than its real counterpart. While it is possible for a discontinuous real function to ~~have an anti-derivative, anti~~-~~derivatives can fail to exist even for [[holomorphic function\|''holomorphic'' functions]] of a complex variable. For example, consider the reciprocal function, ''g''~~(''z'') = 1/''z'' which is holomorphic on the punctured plane '''C'''\{0}. A direct calculation shows that the integral of ''g'' along any circle enclosing the origin is non-zero. So ''g'' fails the condition cited above.

	In fact, holomorphy is characterized by having an antiderivative ''locally'', that is, ''g'' is holomorphic if for every ''z'' in its domain, there is some neighborhood ''U'' of ''z'' such that ''g'' has an antiderivative on ''U''. Furthermore, holomorphy is a necessary condition for a function to have an antiderivative, since the derivative of any holomorphic function is holomorphic.

	~~Various versions of~~ [~~[Cauchy integral theorem]], an underpinning result of Cauchy function theory, which makes heavy use of path integrals, gives sufficient conditions under which, for a holomorphic ''g'',~~

	:~~<math> \int_{\gamma} g(\zeta) d \zeta<~~/~~math>~~

	~~vanishes for any closed path γ (which may be, for instance, that the domain of ''g'' be simply connected or star-convex).~~

	~~===Necessity===~~
	~~First we show that if ''f'' is an antiderivative of ''g'' on ''U'', then it has the path integral property given above. Given any piecewise [[smooth function\|''C''<sup>1<~~/~~sup>]] [[path (topology)\|path]] γ : [''a'', ''b''] → ''U'', one can express the [[line integral\|path integral]] of ''g'' over γ as~~

	~~:<math>\int_\gamma g(z)\,dz=\int_a^b g(\gamma(t))\gamma'(t)\, dt=\int_a^b f'(\gamma(t))\gamma'(t)\,dt.</math>~~

	~~By the [[chain rule]] and the [[fundamental theorem of calculus]] one then has~~

	~~:<math>\int_\gamma g(z)\,dz=\int_a^b \frac{d}{dt}f\left(\gamma(t)\right)\,dt=f\left(\gamma(b)\right)~~-~~f\left(\gamma(a)\right).</math>~~

	~~Therefore the integral of ''g'' over γ does ''not'' depend on the actual path γ, but only on its endpoints, which is what we wanted to show~~.

	~~===Sufficiency===~~
	~~Next we show that if ''g'' is holomorphic, and the integral of ''g'' over any path depends only on the endpoints, then ''g'' has an antiderivative~~. ~~We will do so by finding an anti-derivative explicitly~~.

	~~Without loss of generality, we can assume that the domain ''U'' of ''g'' is connected, as otherwise one can prove the existence of an antiderivative on each connected component~~. ~~With this assumption, fix a point ''z''<sub>0</sub> in ''U'' and for any ''z'' in ''U'' define the function~~

	~~: <math>f(z)=\int_{\gamma}\! g(\zeta)\, d\zeta<~~/~~math>~~

	~~where γ is any path joining ''z''<sub>0<~~/~~sub> to ''z''. Such a path exists since ''U'' is assumed to be an open connected set. The function ''f'' is well-defined because the integral depends only on the endpoints of γ.~~

	~~That this ''f'' is an antiderivative of ''g'' can be argued in the same way as the real case. We have, for a given ''z'' in ''U'',~~

	~~:<math>\begin{align}~~
	~~\left\| \frac{f(w) - f(z)}{ w-z } - g(z) \right\|&~~= ~~\left\| \int_z^w \frac{ g(\zeta) d\zeta}{w -z} - \int_z^w \frac{ g(z) d\zeta}{w -z} \right\|\\~~
	~~&\leq \int_z^w \frac{ \| g(\zeta) - g (z) \|}{\|w -z\|} d\zeta \\~~
	~~&\leq \max_{ \zeta \in [w, z~~]~~} \| g(\zeta) - g(z~~) \|,
	~~\end{align}</math>~~

	~~where [''z'', ''w''] denotes the line segment between ''z'' and ''w''. By continuity of ''g'', the final expression goes to zero as ''w'' approaches ''z''. In other words, ''f′'' = ''g''.~~

	~~==References==~~

	* {{cite book \| author=Ian Stewart, [[David O. Tall]] \| title=Complex Analysis \| publisher=Cambridge University Press \| year=Mar 10, 1983 \| isbn= 0-521-28763-4}}
	* {{cite book \| author=Alan D Solomon \| title=The Essentials of Complex Variables I \| publisher=Research & Education Assoc \| year=Jan 1, 1994 \| isbn= 0-87891-661-X}}

	~~==External links==~~
	* {{planetmath reference\|id=7140\|title=Antiderivative of a complex function}}
	* {{MathWorld \| urlname= FundamentalTheoremsofCalculus \| title= Fundamental Theorems of Calculus}}
	* [http://math.fullerton.edu/mathews/c2003/LiouvilleMoreraGaussMod.html Fundamental Theorems of Integration by John H. Mathews]

	~~[[Category:Complex analysis]]~~

en>Awaterl: Add a see also section.

2013-05-13T02:25:36Z

Add a see also section.

← Older revision		Revision as of 04:25, 13 May 2013
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	~~Nice to satisfy you~~, ~~my name~~ is ~~Araceli Oquendo but I don~~'~~t like when people use my full title~~. ~~Her family life~~ in ~~Idaho~~. One of the ~~things I adore most~~ is ~~greeting card gathering but I don~~'t have the ~~time lately~~. ~~She works~~ as a ~~monetary officer~~ and ~~she~~ will ~~not alter it whenever soon~~.<br><br>~~my blog~~ :: [http://~~www~~.~~gettingtherefromhere~~.~~info~~/~~User-Profile~~/~~userId~~/~~13851 gettingtherefromhere~~.~~info~~]		In [[complex analysis]], a branch of [[mathematics]], the '''antiderivative''', or '''primitive''', of a [[complex number\|complex]]-valued [[function (mathematics)\|function]] ''g'' is a function whose [[complex derivative]] is ''g''. More precisely, given an [[open set]] <math>U</math> in the complex plane and a function <math>g:U\to \mathbb C,</math> the antiderivative of <math>g</math> is a function <math>f:U\to \mathbb C</math> that satisfies <math>\frac{df}{dz}=g</math>.

			As such, this concept is the complex-variable version of the [[antiderivative]] of a [[real number\|real]]-valued function.

			==Uniqueness==

			The derivative of a constant function is zero. Therefore, any constant is an antiderivative of the zero function. If <math>U</math> is a [[connected set]], then the constants are the only antiderivatives of the zero function. Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each [[connected set\|connected component]] of <math>U</math> (those constants need not be equal).

			This observation implies that if a function <math>g:U\to \mathbb C</math> has an antiderivative, then that antiderivative is unique [[up to]] addition of a function which is constant on each connected component of <math>U</math>.

			==Existence==
			One can characterize the existence of antiderivatives via path integrals in the complex plane, much like the case of functions of a real variable. Perhaps not surprisingly, ''g'' has an antiderivative ''f'' if and only if, for every γ path from ''a'' to ''b'', the path integral

			:<math> \int_{\gamma} g(\zeta) d \zeta = f(b) - f(a).</math>

			Equivalently,

			:<math> \int_{\gamma} g(\zeta) d \zeta = 0,</math>

			for any closed path γ.

			However, this formal similarity notwithstanding, possessing a complex-antiderivative is a much more restrictive condition than its real counterpart. While it is possible for a discontinuous real function to have an anti-derivative, anti-derivatives can fail to exist even for [[holomorphic function\|''holomorphic'' functions]] of a complex variable. For example, consider the reciprocal function, ''g''(''z'') = 1/''z'' which is holomorphic on the punctured plane '''C'''\{0}. A direct calculation shows that the integral of ''g'' along any circle enclosing the origin is non-zero. So ''g'' fails the condition cited above.

			In fact, holomorphy is characterized by having an antiderivative ''locally'', that is, ''g'' is holomorphic if for every ''z'' in its domain, there is some neighborhood ''U'' of ''z'' such that ''g'' has an antiderivative on ''U''. Furthermore, holomorphy is a necessary condition for a function to have an antiderivative, since the derivative of any holomorphic function is holomorphic.

			Various versions of [[Cauchy integral theorem]], an underpinning result of Cauchy function theory, which makes heavy use of path integrals, gives sufficient conditions under which, for a holomorphic ''g'',

			:<math> \int_{\gamma} g(\zeta) d \zeta</math>

			vanishes for any closed path γ (which may be, for instance, that the domain of ''g'' be simply connected or star-convex).

			===Necessity===
			First we show that if ''f'' is an antiderivative of ''g'' on ''U'', then it has the path integral property given above. Given any piecewise [[smooth function\|''C''<sup>1</sup>]] [[path (topology)\|path]] γ : [''a'', ''b''] → ''U'', one can express the [[line integral\|path integral]] of ''g'' over γ as

			:<math>\int_\gamma g(z)\,dz=\int_a^b g(\gamma(t))\gamma'(t)\, dt=\int_a^b f'(\gamma(t))\gamma'(t)\,dt.</math>

			By the [[chain rule]] and the [[fundamental theorem of calculus]] one then has

			:<math>\int_\gamma g(z)\,dz=\int_a^b \frac{d}{dt}f\left(\gamma(t)\right)\,dt=f\left(\gamma(b)\right)-f\left(\gamma(a)\right).</math>

			Therefore the integral of ''g'' over γ does ''not'' depend on the actual path γ, but only on its endpoints, which is what we wanted to show.

			===Sufficiency===
			Next we show that if ''g'' is holomorphic, and the integral of ''g'' over any path depends only on the endpoints, then ''g'' has an antiderivative. We will do so by finding an anti-derivative explicitly.

			Without loss of generality, we can assume that the domain ''U'' of ''g'' is connected, as otherwise one can prove the existence of an antiderivative on each connected component. With this assumption, fix a point ''z''<sub>0</sub> in ''U'' and for any ''z'' in ''U'' define the function

			: <math>f(z)=\int_{\gamma}\! g(\zeta)\, d\zeta</math>

			where γ is any path joining ''z''<sub>0</sub> to ''z''. Such a path exists since ''U'' is assumed to be an open connected set. The function ''f'' is well-defined because the integral depends only on the endpoints of γ.

			That this ''f'' is an antiderivative of ''g'' can be argued in the same way as the real case. We have, for a given ''z'' in ''U'',

			:<math>\begin{align}
			\left\| \frac{f(w) - f(z)}{ w-z } - g(z) \right\|&= \left\| \int_z^w \frac{ g(\zeta) d\zeta}{w -z} - \int_z^w \frac{ g(z) d\zeta}{w -z} \right\|\\
			&\leq \int_z^w \frac{ \| g(\zeta) - g (z) \|}{\|w -z\|} d\zeta \\
			&\leq \max_{ \zeta \in [w, z]} \| g(\zeta) - g(z) \|,
			\end{align}</math>

			where [''z'', ''w''] denotes the line segment between ''z'' and ''w''. By continuity of ''g'', the final expression goes to zero as ''w'' approaches ''z''. In other words, ''f′'' = ''g''.

			==References==

			* {{cite book \| author=Ian Stewart, [[David O. Tall]] \| title=Complex Analysis \| publisher=Cambridge University Press \| year=Mar 10, 1983 \| isbn= 0-521-28763-4}}
			* {{cite book \| author=Alan D Solomon \| title=The Essentials of Complex Variables I \| publisher=Research & Education Assoc \| year=Jan 1, 1994 \| isbn= 0-87891-661-X}}

			==External links==
			* {{planetmath reference\|id=7140\|title=Antiderivative of a complex function}}
			* {{MathWorld \| urlname= FundamentalTheoremsofCalculus \| title= Fundamental Theorems of Calculus}}
			* [http://math.fullerton.edu/mathews/c2003/LiouvilleMoreraGaussMod.html Fundamental Theorems of Integration by John H. Mathews]

			[[Category:Complex analysis]]

en>Chire: −Category:Machine learning; +Category:Machine learning algorithms using HotCat

2011-10-27T15:38:48Z

−Category:Machine learning; +Category:Machine learning algorithms using HotCat

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Nice to satisfy you, my name is Araceli Oquendo but I don't like when people use my full title. Her family life in Idaho. One of the things I adore most is greeting card gathering but I don't have the time lately. She works as a monetary officer and she will not alter it whenever soon.<br><br>my blog :: [http://www.gettingtherefromhere.info/User-Profile/userId/13851 gettingtherefromhere.info]