en>Monsterman222 at 22:47, 18 July 2013

2013-07-18T22:47:41Z

← Older revision		Revision as of 00:47, 19 July 2013
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	== ~~Piazza del Comune Hollister Melbourne Fl~~ ==		In mathematical [[invariant theory]], a '''perpetuant''' is informally an irreducible covariant of a form or infinite degree. More precisely, the dimension of the space of irreducible covariants of given degree and weight for a binary form stabilizes provided the degree of the form is larger than the weight of the covariant, and the elements of this space are called perpetuants. Perpetuants were introduced and named by {{harvs\|txt\|last=Sylvester\|year=1882\|loc=p.105}}. {{harvs\|txt\|last=MacMahon\|year1=1884\|year2=1885\|year3=1894}} and {{harvs\|txt\|last=Stroh\|year=1888}} classified the perpetuants. {{harvtxt\|Elliott\|1907}} describes the early history of perpetuants and gives an annotated bibliography. There are very few papers after about 1910 discussing perpetuants; {{harv\|Littlewood\|1944}} is one of the few exceptions.

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			MacMahon conjectured and Stroh proved that the dimension of the space of perpetuants of degree ''n''>2 and weight ''w'' is the coefficient of ''x''<sup>''w''</sup> of
	~~<li>[~~http://~~www~~.~~kueiwa~~.~~com/forum~~.~~php?mod~~=~~viewthread&tid~~=~~334336~~ http://www.~~kueiwa~~.~~com~~/~~forum~~.~~php?mod~~=~~viewthread&tid~~=~~334336]</li>~~		:<math>\frac{x^{2^{n-1}-1}}{(1-x^2)(1-x^3)\cdots(1-x^n)}</math>
			For ''n''=1 there is just one perpetuant, of weight 0, and for ''n''=2 the number is given by the coefficient of ''x''<sup>''w''</sup> of ''x''<sup>2</sup>/(1-''x''<sup>2</sup>).
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			==References==
	~~<li>[http:~~//~~enseignement~~-~~lsf~~.~~com/spip~~.~~php?article66#forum24711395~~ http://~~enseignement-lsf~~.~~com~~/~~spip~~.~~php?article66#forum24711395]<~~/~~li>~~
			*{{Citation \| last1=Cayley \| first1=Arthur \| author1-link=Arthur Cayley \| title=A Memoir on Seminvariants \| url=http://www.jstor.org/stable/2369456 \| publisher=The Johns Hopkins University Press \| year=1884 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=7 \| issue=1 \| pages=1–25}}
	~~<li>~~[~~http://www.film~~-~~video-dvd-production~~.~~com/spip~~.~~php?article6/~~ http://www.~~film-video-dvd-production~~.~~com~~/~~spip.php?article6~~/]~~</li>~~		*{{Citation \| last1=Elliott \| first1=Edwin Bailey \| title=An introduction to the algebra of quantics \| url=http://archive.org/details/introductiontoal00elliiala \| publisher=Oxford, Clarendon Press \| id=Reprinted by Chelsea Scientific Books 1964 \| year=1895}}
			*{{Citation \| last1=Elliott \| first1=Edwin Bailey \| title=On Perpetuants and Contra-Perpetuants \| url=http://books.google.com/books?id=a7xLAAAAYAAJ&pg=PA228 \| doi=10.1112/plms/s2-4.1.228 \| year=1907 \| journal=Proc. London Math. Soc. \| volume=4 \| issue=1 \| pages=228–246}}
	~~</ul>~~		*{{Citation \| last1=Grace \| first1=J. H. \| last2=Young \| first2=Alfred \| author2-link=Alfred Young \| title=The algebra of invariants \| url=http://archive.org/details/algebraofinvaria00graciala \| publisher=[[Cambridge University Press]] \| year=1903}}
			*{{Citation \| last1=Littlewood \| first1=D. E. \| title=Invariant theory, tensors and group characters \| url=http://www.jstor.org/stable/91389 \| id={{MR\|0010594}} \| year=1944 \| journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences \| issn=0080-4614 \| volume=239 \| pages=305–365}}
			*{{Citation \| last1=MacMahon \| first1=P. A. \| title=On Perpetuants \| url=http://www.jstor.org/stable/2369457 \| publisher=The Johns Hopkins University Press \| year=1884 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=7 \| issue=1 \| pages= 26–46}}
			*{{Citation \| last1=MacMahon \| first1=P. A. \| title=A Second Paper on Perpetuants \| url=http://www.jstor.org/stable/2369271 \| publisher=The Johns Hopkins University Press \| year=1885 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=7 \| issue=3 \| pages=259–263}}
			*{{Citation \| last1=MacMahon \| first1=P. A. \| title= The Perpetuant Invariants of Binary Quantics \| doi=10.1112/plms/s1-26.1.262 \| year=1894 \| journal= Proc. London Math. Soc. \| volume=26 \| issue=1 \| pages=262–284}}
			*{{Citation \| last1=Stroh \| first1=E. \| title=Ueber eine fundamentale Eigenschaft des Ueberschiebungs-processes und deren Verwerthung in der Theorie der binären Formen \| url=http://dx.doi.org/10.1007/BF01444111 \| publisher=Springer Berlin / Heidelberg \| year=1888 \| journal=[[Mathematische Annalen]] \| issn=0025-5831 \| volume=33 \| pages=61–107}}
			*{{Citation \| last1=Sylvester \| first1=James Joseph \| title=On Subvariants, i.e. Semi-Invariants to Binary Quantics of an Unlimited Order \| url=http://www.jstor.org/stable/2369536 \| publisher=The Johns Hopkins University Press \| year=1882 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=5 \| issue=1 \| pages=79–136}}

			[[Category:Invariant theory]]

98.210.156.37: Removed extra "is" (grammar fix).

2012-08-18T09:11:11Z

Removed extra "is" (grammar fix).

New page

== Piazza del Comune Hollister Melbourne Fl ==

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Bayes error rate - Revision history

en>Monsterman222 at 22:47, 18 July 2013

98.210.156.37: Removed extra "is" (grammar fix).