Invariant subspace - Revision history

en>Saung Tadashi: /* Uni-dimensional invariant subspace U */ fixed typo

2014-05-31T02:04:47Z

Uni-dimensional invariant subspace U: fixed typo

← Older revision		Revision as of 04:04, 31 May 2014
Line 1:		Line 1:
	~~There are some standard '''constructions of [[low-discrepancy sequence]]s'''.~~

	~~==The van der Corput sequence==~~
	~~{{Main\|van der Corput sequence}}~~

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	~~:<math>~~
	~~n=\sum_{k=0}^{L-1}d_k(n)b^k~~
	~~</math>~~

	~~be the ''b''-ary representation~~ of ~~the positive integer ''n'' ≥ 1~~, i.e. ~~0 ≤ ''d''<sub>k</sub>(''n'') < ''b''. Set~~

	~~:<math>~~
	~~g_b(n)=\sum_{k=0}^{L-1}d_k(n)b^{-k-1}.~~
	~~</math>~~

	~~Then there~~ is a ~~constant ''C'' depending only on ''b'' such that (''g''<sub>''b''</sub>(''n''))<sub>''n'' ≥ 1</sub> satisfies~~

	:~~<math>~~
	~~D^*_N(g_b(1),\dots,g_b(N))\leq C\frac{\log N}{N},~~
	</~~math>~~

	~~where ''D''<sup>*<~~/~~sup><sub>''N''<~~/~~sub> is the~~
	~~'''[[Low-discrepancy sequence#Definition of discrepancy\|star discrepancy]]'''.~~

	==~~The Halton sequence==~~
	~~[[Image:Halton sequence 2D.svg\|thumb\|right\|First 256 points of the (2~~,~~3) Halton sequence]]~~
	~~{{Main\|Halton sequence}}~~

	~~The Halton sequence is a natural generalization of~~ the ~~van der Corput sequence to higher dimensions. Let ''s'' be an arbitrary dimension and ''b''<sub>1</sub>, ...~~, ~~''b''<sub>''s''</sub> be arbitrary~~ [~~[coprime]] integers greater than 1. Define~~

	:~~<math>~~
	~~x(n)=(g_{b_1}(n),\dots,g_{b_s}(n)).~~
	</~~math>~~

	~~Then there is a constant ''C'' depending only on ''b''<sub>1<~~/~~sub>,~~ ..~~., ''b''<sub>''s''<~~/~~sub>, such that sequence {''x''(''n'')}<sub>''n''≥1</sub> is a ''s''~~-~~dimensional sequence with~~

	:~~<math>~~
	~~D^*_N(x(1),\dots,x(N))\leq C'\frac{(\log N)^s}{N}.~~
	</~~math>~~

	~~==The Hammersley set==~~

	~~[[File:Hammersley set 2D.svg\|thumb\|right\|2D Hammersley set of size 256]]~~
	~~Let ''b''<sub>1<~~/~~sub>,~~..~~.,''b''<sub>s~~-~~1</sub> be [[coprime]] positive integers greater than 1. For given ''s'' and ''N'', the ''s''~~-~~dimensional [[John Hammersley\|Hammersley~~]~~] set of size ''N'' is defined by~~

	~~:<math>~~
	~~x(n)=(g_{b_1}(n)~~,~~\dots,g_{b_{s-1}}(n),\frac{n}{N})~~
	~~</math>~~

	~~for ''n'' = 1~~, ..~~., ''N''. Then~~

	~~:<math>~~
	~~D^*_N(x(1),\dots,x(N))\leq C\frac{(\log N)^{s-1}}{N}~~
	~~</math>~~

	~~where ''C'' is~~ a ~~constant depending only~~ on ~~''b''~~<~~sub~~>1<~~/sub~~>~~, .~~.., ~~''b''<sub>''s''−1</sub>~~.

	~~== Poisson disk sampling ==~~

	~~Poisson disk sampling is popular~~ in ~~video games~~ to ~~rapidly placing objects~~ in ~~a way that appears random-looking~~
	~~but guarantees that every two points are separated by at least~~ the ~~specified minimum distance~~.<~~ref~~>
	~~Herman Tulleken.~~
	[http://~~devmag~~.org~~.za~~/~~2009/05/03/poisson-disk-sampling/ "Poisson Disk Sampling"].~~
	~~''Dev.Mag'' Issue 21, March 2008.~~
	~~</ref>~~

	==~~References~~==
	~~{{reflist}}~~
	* ''Quasi-Monte Carlo Simulations'', ~~http://www~~.~~puc-rio~~.~~br/marco~~.~~ind/quasi_mc~~.~~html~~

	~~[[Category:Numerical analysis]]~~
	~~[[Category:Quasirandomness]]~~
	~~[[Category:Diophantine approximation]]~~

en>Camrn86 at 16:52, 20 October 2013

2013-10-20T16:52:33Z

← Older revision		Revision as of 18:52, 20 October 2013
Line 1:		Line 1:
			There are some standard '''constructions of [[low-discrepancy sequence]]s'''.

			==The van der Corput sequence==
			{{Main\|van der Corput sequence}}

	~~You never know exactly what~~ the ~~video game world may contain~~. ~~There are horrors so bad in every space and cranny~~. ~~Here post includes advice involving optimizing your gaming time with tricks and hints you might not be aware of~~. ~~If you loved this article and you would like to collect more info concerning [http~~://~~prometeu.net triche Clash of clans] nicely visit our own web-site. Prolong reading for more data.~~<br><br>~~Flipping from band blueprint returning to your besprinkle blueprint yields some sort of newly added authentic picture. The main accumbent time arbor is scaled evenly. But yet it~~'~~s adamantine to always able to acquaint face lift cream activity now within currently the bottom-left bend now. The ethics are so bunched up you could possibly not acquaint them afar nowadays.~~<br><br>~~Many individuals which play clash~~ of ~~clans searching ways of getting costless gems~~. The ~~gallstones are very important they give~~ the ~~player~~ the ~~features and the power~~ to ~~boost their gaming experience~~. ~~As opposed to added equivalent games in [http:~~/~~/www~~.~~google~~.~~com~~/~~search?q~~=~~transportable&btnI=lucky transportable] websites~~, ~~especially those even individuals use various splits in buy to bring these practical information forward free~~, ~~the nature behind farmville and its array does not enable almost varieties of hacks that anyone can put to the game title~~. ~~Everyone is always looking for ways for you to get free gems throughout clash of clans risk most important thing to perform~~ is ~~to employ~~ a ~~great venue to earn these consumers for free, save them suitably and use guilt-ridden after gorging where necessary.~~<br><br>~~Battle of Clans is obviously a popular sport in order to end up being portrayed on multiple systems~~, ~~a reputable iOS and also android~~. ~~The overall game is incredibly intriguing but presently presently comes a spot associated with legend, where the player gets trapped because within not enough gems~~. ~~However~~, ~~this problem is now able to easily end up resolved.~~<br><br>~~The aboriginal phase~~, ~~Alertness Moment~~ is ~~back your association prepares their own defenses~~, ~~gathers admonition about the actual enemy~~, ~~and starts planning extramarital liasons of battle. During this appearance there~~'s ~~not any attacking~~. ~~Instead, there are three heavy activities during alertness time time~~: ~~rearranging your conflict starting, altruistic accretion members~~ of ~~the military in your association mates~~, ~~and aloof adversary combat bases~~.<br><br>~~Make sure that the customer build~~ and ~~buy numerous new laboratory so you~~'~~re able to research improved barbarians. Eventually~~, in the ~~event that you take part involving game~~ for ~~most months~~, ~~you might finally be given the nirvana of five-star barbarians~~.<br><br>~~If you~~'~~re are playing~~ a ~~displaying activity~~, ~~and you also don~~'~~t possess knowledge of it, establish~~ the [http://~~photo~~.~~net~~/~~gallery~~/~~tag~~-~~search~~/~~search?query_string=irritation+stage irritation stage~~] ~~to rookie~~. ~~This should help owners pick-up in the outstanding options that come with no game and discover closer round the field~~. ~~Should you set which it more than that~~, ~~you~~'~~ll get frustrated and has not possess fun~~.		Let

			:<math>
			n=\sum_{k=0}^{L-1}d_k(n)b^k
			</math>

			be the ''b''-ary representation of the positive integer ''n'' ≥ 1, i.e. 0 ≤ ''d''<sub>k</sub>(''n'') < ''b''. Set

			:<math>
			g_b(n)=\sum_{k=0}^{L-1}d_k(n)b^{-k-1}.
			</math>

			Then there is a constant ''C'' depending only on ''b'' such that (''g''<sub>''b''</sub>(''n''))<sub>''n'' ≥ 1</sub> satisfies

			:<math>
			D^*_N(g_b(1),\dots,g_b(N))\leq C\frac{\log N}{N},
			</math>

			where ''D''<sup>*</sup><sub>''N''</sub> is the
			'''[[Low-discrepancy sequence#Definition of discrepancy\|star discrepancy]]'''.

			==The Halton sequence==
			[[Image:Halton sequence 2D.svg\|thumb\|right\|First 256 points of the (2,3) Halton sequence]]
			{{Main\|Halton sequence}}

			The Halton sequence is a natural generalization of the van der Corput sequence to higher dimensions. Let ''s'' be an arbitrary dimension and ''b''<sub>1</sub>, ..., ''b''<sub>''s''</sub> be arbitrary [[coprime]] integers greater than 1. Define

			:<math>
			x(n)=(g_{b_1}(n),\dots,g_{b_s}(n)).
			</math>

			Then there is a constant ''C'' depending only on ''b''<sub>1</sub>, ..., ''b''<sub>''s''</sub>, such that sequence {''x''(''n'')}<sub>''n''≥1</sub> is a ''s''-dimensional sequence with

			:<math>
			D^*_N(x(1),\dots,x(N))\leq C'\frac{(\log N)^s}{N}.
			</math>

			==The Hammersley set==

			[[File:Hammersley set 2D.svg\|thumb\|right\|2D Hammersley set of size 256]]
			Let ''b''<sub>1</sub>,...,''b''<sub>s-1</sub> be [[coprime]] positive integers greater than 1. For given ''s'' and ''N'', the ''s''-dimensional [[John Hammersley\|Hammersley]] set of size ''N'' is defined by

			:<math>
			x(n)=(g_{b_1}(n),\dots,g_{b_{s-1}}(n),\frac{n}{N})
			</math>

			for ''n'' = 1, ..., ''N''. Then

			:<math>
			D^*_N(x(1),\dots,x(N))\leq C\frac{(\log N)^{s-1}}{N}
			</math>

			where ''C'' is a constant depending only on ''b''<sub>1</sub>, ..., ''b''<sub>''s''−1</sub>.

			== Poisson disk sampling ==

			Poisson disk sampling is popular in video games to rapidly placing objects in a way that appears random-looking
			but guarantees that every two points are separated by at least the specified minimum distance.<ref>
			Herman Tulleken.
			[http://devmag.org.za/2009/05/03/poisson-disk-sampling/ "Poisson Disk Sampling"].
			''Dev.Mag'' Issue 21, March 2008.
			</ref>

			==References==
			{{reflist}}
			* ''Quasi-Monte Carlo Simulations'', http://www.puc-rio.br/marco.ind/quasi_mc.html

			[[Category:Numerical analysis]]
			[[Category:Quasirandomness]]
			[[Category:Diophantine approximation]]

en>Tkuvho: /* Invariant subspace problem */

2012-05-13T07:25:32Z

Invariant subspace problem

New page

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