Hilbert modular form - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T12:18:29Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 14:18, 5 May 2014
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	~~{{technical\|date=June 2012}}~~

	~~In [[applied mathematics]], '''highly optimized tolerance (HOT)'~~'' is ~~a method of generating [[power law]] behavior in systems by including a [[global optimization]] principle~~. ~~For some systems that display a characteristic scale,~~ a ~~global optimization term could potentially be added that would then yield power law behavior~~. ~~It has been used to generate and describe internet-~~like ~~graphs, [[forest fire]] models and may also apply to biological systems.~~

	~~==Example==~~
	~~The following~~ is ~~taken from Sornette~~'~~s book~~.

	~~Consider a [[random variable]],~~ <~~math~~>X<~~/math~~>~~, that takes on values <math>x_i</math> with probability <math>p_i</math>. Furthmore, lets assume for another parameter <math>r_i</math>~~
	~~:<math>x_i = r_i^{~~ - ~~\beta }</math>~~
	~~for some fixed <math>\beta</math>. We then want to minimize~~
	~~:<math> L = \sum_{i=0}^{N~~-~~1} p_i x_i </math>~~
	~~subject to the constraint~~
	~~:<math> \sum_{i=0}^{N~~-~~1} r_i = \kappa </math>~~
	~~Using~~ [~~[Lagrange multipliers]], this gives~~
	:~~<math> p_i \propto x_i^{ - ( 1 + 1~~/ ~~\beta) } <~~/~~math>~~
	~~giving us a power law~~. ~~The global optimization of minimizing the energy along with the power law dependence between <math>x_i<~~/~~math> and <math>r_i<~~/~~math> gives us a power law distribution in probability.~~

	~~==See also==~~
	* [[self-organized criticality]]

	~~==References==~~
	* Carlson, J. M. & Doyle, J. (1999) ''Phys. Rev. E'' '''60,''' 1412–1427.
	* Carlson, J. M. & Doyle, J. (2000) ''Phys. Rev. Lett.'' '''84,''' 2529–2532.
	* Doyle, J. & Carlson, J. M. (2000) ''Phys. Rev. Lett.'' '''84,''' 5656–5659.
	* Greene, K. (2005) ''Science News'' '''168,''' 230.
	* Li, L., Alderson, D., Tanaka, R., Doyle, J.C., Willinger, W., [http://~~arxiv~~.~~org~~/~~abs~~/~~cond-mat~~/~~0501169 Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)~~]~~. Internet Mathematics, 2005.~~
	* Robert, C., Carlson, J. M. & Doyle, ~~J. (2001) ''Phys. Rev. E'' '''63''', 56122, 1–13.~~
	* {{cite book\|last=Sornette\|first=Didier\|authorlink=Didier Sornette\|title=Critical Phenomena in Natural Sciences\|year=2000\|publisher=Springer}}
	* Zhou, T. & Carlson, J. M. (2000), ''Phys. Rev. E'' '''62,''' 3197–3204.
	* Zhou, T., Carlson, J. M. & Doyle, J. (2002) ''Proc. Natl. Acad. Sci. USA'' '''99,''' 2049–2054.



	~~[[Category:Mathematical optimization]]~~


	~~{{sci-stub}}~~

50.152.234.148: O_F definitely doesn't act on the upper half plane by multiplication. I switched it to GL_2^+ (O_F), but I feel SL_2(O_F) might be a better call?

2013-04-26T05:53:23Z

O_F definitely doesn't act on the upper half plane by multiplication. I switched it to GL_2^+ (O_F), but I feel SL_2(O_F) might be a better call?

← Older revision		Revision as of 07:53, 26 April 2013
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			{{technical\|date=June 2012}}

			In [[applied mathematics]], '''highly optimized tolerance (HOT)''' is a method of generating [[power law]] behavior in systems by including a [[global optimization]] principle. For some systems that display a characteristic scale, a global optimization term could potentially be added that would then yield power law behavior. It has been used to generate and describe internet-like graphs, [[forest fire]] models and may also apply to biological systems.

			==Example==
			The following is taken from Sornette's book.

			Consider a [[random variable]], <math>X</math>, that takes on values <math>x_i</math> with probability <math>p_i</math>. Furthmore, lets assume for another parameter <math>r_i</math>
			:<math>x_i = r_i^{ - \beta }</math>
			for some fixed <math>\beta</math>. We then want to minimize
			:<math> L = \sum_{i=0}^{N-1} p_i x_i </math>
			subject to the constraint
			:<math> \sum_{i=0}^{N-1} r_i = \kappa </math>
			Using [[Lagrange multipliers]], this gives
			:<math> p_i \propto x_i^{ - ( 1 + 1/ \beta) } </math>
			giving us a power law. The global optimization of minimizing the energy along with the power law dependence between <math>x_i</math> and <math>r_i</math> gives us a power law distribution in probability.

			==See also==
			* [[self-organized criticality]]

			==References==
			* Carlson, J. M. & Doyle, J. (1999) ''Phys. Rev. E'' '''60,''' 1412–1427.
			* Carlson, J. M. & Doyle, J. (2000) ''Phys. Rev. Lett.'' '''84,''' 2529–2532.
			* Doyle, J. & Carlson, J. M. (2000) ''Phys. Rev. Lett.'' '''84,''' 5656–5659.
			* Greene, K. (2005) ''Science News'' '''168,''' 230.
			* Li, L., Alderson, D., Tanaka, R., Doyle, J.C., Willinger, W., [http://arxiv.org/abs/cond-mat/0501169 Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)]. Internet Mathematics, 2005.
			* Robert, C., Carlson, J. M. & Doyle, J. (2001) ''Phys. Rev. E'' '''63''', 56122, 1–13.
			* {{cite book\|last=Sornette\|first=Didier\|authorlink=Didier Sornette\|title=Critical Phenomena in Natural Sciences\|year=2000\|publisher=Springer}}
			* Zhou, T. & Carlson, J. M. (2000), ''Phys. Rev. E'' '''62,''' 3197–3204.
			* Zhou, T., Carlson, J. M. & Doyle, J. (2002) ''Proc. Natl. Acad. Sci. USA'' '''99,''' 2049–2054.



			[[Category:Mathematical optimization]]


			{{sci-stub}}

en>Roentgenium111 at 17:05, 24 May 2011

2011-05-24T17:05:58Z

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