Database storage structures - Revision history

68.97.1.59: /* Heap files */

2014-02-11T05:33:46Z

Heap files

← Older revision		Revision as of 07:33, 11 February 2014
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	'''Kravchuk polynomials''' or '''Krawtchouk polynomials''' (also written using several other transliterations of the Ukrainian name "Кравчу́к") are [[discrete orthogonal polynomials\|discrete]] [[orthogonal polynomials]] associated with the [[binomial distribution]], introduced by {{harvs\|txt\|authorlink=Mikhail Kravchuk\|first=Mikhail\|last=Kravchuk\|year= 1929}}.		The writer is called Araceli Gulledge. Interviewing is how I make a living and it's something I really appreciate. Delaware has always been my living place and will by no means transfer. Climbing is what adore doing.<br><br>Also visit my page: [http://nationalheritagemuseum.org/UserProfile/tabid/43/userId/31686/Default.aspx extended warranty car]
	The ~~first few polynomials are:~~
	* <math>\mathcal{K}_0(x; n) = 1</math>
	* <math>\mathcal{K}_1(x; n) = -2x + n</math>
	* <math>\mathcal{K}_2(x; n) = 2x^2 - 2nx + {n\choose 2}</math>
	* <math>\mathcal{K}_3(x; n) = -\frac{4}{3}x^3 + 2nx^2 - (n^2 - n + \frac{2}{3})x + {n \choose 3}.~~</math>~~

	~~The Kravchuk polynomials are~~ a ~~special case of the [[Meixner polynomials]] of the first kind.~~

	~~==Definition==~~
	~~For any [[prime power]] ''q''~~ and ~~positive integer~~ '~~'n'', define the Kravchuk polynomial~~

	~~:<math>\mathcal{K}_k(x; n) = \mathcal{K}_k(x) = \sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \binom {x}{j} \binom{n-x}{k-j}, \quad k=0,1, \ldots, n~~.~~</math>~~

	~~==Properties==~~
	~~The Kravchuk polynomial~~ has ~~following alternative expressions:~~

	~~:<math>\mathcal{K}_k(x; n) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}~~. <~~/math~~>
	:<~~math~~>~~\mathcal{K}_k(x; n) = \sum_{j=0}^{k}(-1)^j q^{k-j} \binom {n-k+j}{j} \binom{n-x}{k-j}. </math>~~

	~~===Orthogonality relations===~~
	~~For nonnegative integers ''r'', ''s'',~~

	:~~<math>\sum_{i=0}^n\binom{n}{j}(q-1)^i\mathcal{K}_r(i; n)\mathcal{K}_s(i; n) = q^n(q-1)^r\binom{n}{r}\delta_{r,s}. </math>~~

	~~==See also==~~
	* [[~~Hermite polynomials]]~~
	~~==References==~~
	~~<references/>~~
	*{{Citation \| last1=Kravchuk \| first1=M. \| \| authorlink = Mikhail Kravchuk \| title=Sur une généralisation des polynomes d'Hermite. \| url=http://~~gallica.bnf.fr/ark:/12148/bpt6k3142j.pleinepage.f620.langEN \| language=French \| jfm=55.0799.01 \| year=1929 \| journal=Comptes Rendus Mathematique \| volume=189 \| pages=620–622}}~~
	*{{dlmf\|id=18.19\|title=Hahn Class: Definitions\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}}
	*{{citation
	~~\| last1 = Nikiforov \| first1 = A. F.~~
	~~\| last2 = Suslov \| first2 = S. K.~~
	~~\| last3 = Uvarov \| first3 = V. B.~~
	~~\| isbn = 3-540-51123-7~~
	~~\| location = Berlin~~
	~~\| mr = 1149380~~
	~~\| publisher = Springer-Verlag~~
	~~\| series = Springer Series in Computational Physics~~
	~~\| title = Classical Orthogonal Polynomials of a Discrete Variable~~
	~~\| year = 1991}}.~~
	*{{citation
	~~\| last = Levenshtein \| first = Vladimir I. \| author-link = Vladimir Levenshtein~~
	~~\| doi = 10.1109/18.412678~~
	~~\| issue = 5~~
	~~\| journal = IEEE Transactions on Information Theory~~
	~~\| mr = 1366326~~
	~~\| pages = 1303–1321~~
	~~\| title = Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces~~
	~~\| volume = 41~~
	~~\| year = 1995}}.~~
	*{{Citation \| author=F. J. MacWilliams \| coauthors=N. ~~J. A. Sloane \| title=The Theory of Error-Correcting Codes \| publisher=North-Holland \| year=1977 \| isbn=0-444-85193-3}}~~

	~~==External links==~~
	*[http://~~orthpol.narod.ru~~/ ~~Krawtchouk Polynomials Home Page]~~
	*[http://~~mathworld.wolfram.com~~/~~KrawtchoukPolynomial~~.~~html "Krawtchouk polynomial"] at [[MathWorld]]~~

	~~[[Category:Orthogonal polynomials]~~]

en>ChrisGualtieri: /* Data orientation */General Fixes using AWB

2013-10-24T01:49:05Z

Data orientation: General Fixes using AWB

← Older revision		Revision as of 03:49, 24 October 2013
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	The ~~writer is known as Wilber Pegues~~. For ~~a whilst I~~'~~ve been in Alaska but I will have to transfer in a yr or two~~. ~~To climb is some thing I really enjoy performing~~. ~~Credit authorising is how he tends to make money~~.<br><br>~~Here is my web~~-~~site~~ ... ~~clairvoyants ([https~~://~~www~~.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 visit my home page~~])		'''Kravchuk polynomials''' or '''Krawtchouk polynomials''' (also written using several other transliterations of the Ukrainian name "Кравчу́к") are [[discrete orthogonal polynomials\|discrete]] [[orthogonal polynomials]] associated with the [[binomial distribution]], introduced by {{harvs\|txt\|authorlink=Mikhail Kravchuk\|first=Mikhail\|last=Kravchuk\|year= 1929}}.
			The first few polynomials are:
			* <math>\mathcal{K}_0(x; n) = 1</math>
			* <math>\mathcal{K}_1(x; n) = -2x + n</math>
			* <math>\mathcal{K}_2(x; n) = 2x^2 - 2nx + {n\choose 2}</math>
			* <math>\mathcal{K}_3(x; n) = -\frac{4}{3}x^3 + 2nx^2 - (n^2 - n + \frac{2}{3})x + {n \choose 3}.</math>

			The Kravchuk polynomials are a special case of the [[Meixner polynomials]] of the first kind.

			==Definition==
			For any [[prime power]] ''q'' and positive integer ''n'', define the Kravchuk polynomial

			:<math>\mathcal{K}_k(x; n) = \mathcal{K}_k(x) = \sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \binom {x}{j} \binom{n-x}{k-j}, \quad k=0,1, \ldots, n.</math>

			==Properties==
			The Kravchuk polynomial has following alternative expressions:

			:<math>\mathcal{K}_k(x; n) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}. </math>
			:<math>\mathcal{K}_k(x; n) = \sum_{j=0}^{k}(-1)^j q^{k-j} \binom {n-k+j}{j} \binom{n-x}{k-j}. </math>

			===Orthogonality relations===
			For nonnegative integers ''r'', ''s'',

			:<math>\sum_{i=0}^n\binom{n}{j}(q-1)^i\mathcal{K}_r(i; n)\mathcal{K}_s(i; n) = q^n(q-1)^r\binom{n}{r}\delta_{r,s}. </math>

			==See also==
			* [[Hermite polynomials]]
			==References==
			<references/>
			*{{Citation \| last1=Kravchuk \| first1=M. \| \| authorlink = Mikhail Kravchuk \| title=Sur une généralisation des polynomes d'Hermite. \| url=http://gallica.bnf.fr/ark:/12148/bpt6k3142j.pleinepage.f620.langEN \| language=French \| jfm=55.0799.01 \| year=1929 \| journal=Comptes Rendus Mathematique \| volume=189 \| pages=620–622}}
			*{{dlmf\|id=18.19\|title=Hahn Class: Definitions\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}}
			*{{citation
			\| last1 = Nikiforov \| first1 = A. F.
			\| last2 = Suslov \| first2 = S. K.
			\| last3 = Uvarov \| first3 = V. B.
			\| isbn = 3-540-51123-7
			\| location = Berlin
			\| mr = 1149380
			\| publisher = Springer-Verlag
			\| series = Springer Series in Computational Physics
			\| title = Classical Orthogonal Polynomials of a Discrete Variable
			\| year = 1991}}.
			*{{citation
			\| last = Levenshtein \| first = Vladimir I. \| author-link = Vladimir Levenshtein
			\| doi = 10.1109/18.412678
			\| issue = 5
			\| journal = IEEE Transactions on Information Theory
			\| mr = 1366326
			\| pages = 1303–1321
			\| title = Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces
			\| volume = 41
			\| year = 1995}}.
			*{{Citation \| author=F. J. MacWilliams \| coauthors=N. J. A. Sloane \| title=The Theory of Error-Correcting Codes \| publisher=North-Holland \| year=1977 \| isbn=0-444-85193-3}}

			==External links==
			*[http://orthpol.narod.ru/ Krawtchouk Polynomials Home Page]
			*[http://mathworld.wolfram.com/KrawtchoukPolynomial.html "Krawtchouk polynomial"] at [[MathWorld]]

			[[Category:Orthogonal polynomials]]

87.64.208.204 at 12:31, 28 May 2011

2011-05-28T12:31:29Z

New page

The writer is known as Wilber Pegues. For a whilst I've been in Alaska but I will have to transfer in a yr or two. To climb is some thing I really enjoy performing. Credit authorising is how he tends to make money.<br><br>Here is my web-site ... clairvoyants ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 visit my home page])