Wavelet transform - Revision history

en>Renaud difrancesco: /* See also */

2014-12-08T15:33:42Z

See also

← Older revision		Revision as of 17:33, 8 December 2014
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	~~{{Lie groups}}~~		The name of the author is Figures but it's not the most masucline title out there. He utilized to be unemployed but now he is a pc operator but his marketing never comes. It's not a common thing but what she likes performing is base jumping and now she is trying to earn cash with it. North Dakota is our beginning location.<br><br>Here is my web site [http://www.ubi-cation.com/ubication/node/6056 std testing at home]

	~~In [[mathematics]], a [[Lie algebra]] ''<math>\mathfrak{g}</math>''~~ is '~~''nilpotent''' if~~ the ~~[[lower central series]]~~

	~~:<math> \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots </math>~~

	~~becomes zero eventually~~. ~~Equivalently, <math>\mathfrak{g}</math>~~ is ~~nilpotent if~~
	~~:<math>\operatorname{ad}(x_1) \operatorname{ad}(x_2) \operatorname{ad}(x_3) ... \operatorname{ad}(x_r) = 0</math>~~
	for any sequence <math>x_i</math> of elements of <math>\mathfrak{g}</math> of sufficiently large length. (Here, <math>\operatorname{ad}(x)</math> is given by <math>\operatorname{ad}(x)y = [x, y]</math>.) Consequences are that <math>\operatorname{ad}(x)</math> is nilpotent (as a ~~linear map), and that the [[Killing form]] of a nilpotent Lie algebra is identically zero~~. ~~(In comparison,~~ a ~~Lie algebra~~ is ~~[[semisimple Lie algebra\|semisimple]] if~~ and ~~only if its Killing form is nondegenerate.)~~

	~~Every nilpotent Lie algebra~~ is ~~[[solvable Lie algebra\|solvable]]; this fact gives one of the powerful ways~~ to ~~prove the solvability of a Lie algebra since, in practice,~~ it ~~is usually easier to prove the nilpotency than the solvability~~. ~~The converse~~ is ~~not true in general~~. ~~A Lie algebra~~ <~~math~~>~~\mathfrak{g}~~<~~/math> is nilpotent if and only if its quotient over an ideal containing the center of <math>\mathfrak{g}</math> is nilpotent.~~

	Most of classic classification results on nilpotency are concerned with finite-dimensional Lie algebras over a field of characteristic 0. Let <math>\mathfrak{g}</math> be a finite-dimensional Lie algebra. <math>\mathfrak{g}</math> is nilpotent if and only if <math>\operatorname{ad}(\mathfrak{g})</math> is ~~nilpotent.~~ [~~[Engel's theorem]] states that <math>\mathfrak{g}</math> is nilpotent if and only if <math>\operatorname{ad}(x)<~~/~~math> is nilpotent for every <math>x \in \mathfrak{g}<~~/~~math>~~. ~~<math>\mathfrak{g}</math> is solvable if and only if <math>[\mathfrak{g}, \mathfrak{g}]</math> is nilpotent.~~

	~~== Examples ==~~
	*Every subalgebra and quotient of a nilpotent Lie algebra is nilpotent.
	*If <math>\mathfrak{gl}_k</~~math> is the set of <math>k\times k<~~/~~math> matrices, then the subalgebra consisting of strictly upper triangular matrices, denoted by <math>\mathfrak{n}_k<~~/~~math>, is a nilpotent Lie algebra.~~
	*A [[Heisenberg algebra]] is nilpotent.
	*A [[Cartan subalgebra]] of a Lie algebra is nilpotent and [[self-normalizing]].

	~~==References==~~
	* Humphreys, James E. ''Introduction to Lie Algebras and Representation Theory''. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. ISBN 0-387-90053-5

	~~[[Category:Properties of Lie algebras]~~]

en>Inferior Olive: Updating sentence starting with "The higher the resolution in time is required, the..."(WG)

2014-01-24T14:35:24Z

Updating sentence starting with "The higher the resolution in time is required, the..."(WG)

← Older revision		Revision as of 16:35, 24 January 2014
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	~~She~~ is ~~recognized~~ by the ~~name~~ of ~~Myrtle Shryock~~. ~~Bookkeeping~~ is ~~her working day occupation now~~. ~~For~~ a ~~while she's been~~ in ~~South Dakota~~. ~~To perform baseball~~ is the ~~hobby he will never quit doing~~.<br><br>~~Feel free to surf to my weblog~~ - [~~http:~~//~~Nuvem~~.tk/~~altergalactica~~/~~NumbersdbEstepvu at home std test~~]		{{Lie groups}}

			In [[mathematics]], a [[Lie algebra]] ''<math>\mathfrak{g}</math>'' is '''nilpotent''' if the [[lower central series]]

			:<math> \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots </math>

			becomes zero eventually. Equivalently, <math>\mathfrak{g}</math> is nilpotent if
			:<math>\operatorname{ad}(x_1) \operatorname{ad}(x_2) \operatorname{ad}(x_3) ... \operatorname{ad}(x_r) = 0</math>
			for any sequence <math>x_i</math> of elements of <math>\mathfrak{g}</math> of sufficiently large length. (Here, <math>\operatorname{ad}(x)</math> is given by <math>\operatorname{ad}(x)y = [x, y]</math>.) Consequences are that <math>\operatorname{ad}(x)</math> is nilpotent (as a linear map), and that the [[Killing form]] of a nilpotent Lie algebra is identically zero. (In comparison, a Lie algebra is [[semisimple Lie algebra\|semisimple]] if and only if its Killing form is nondegenerate.)

			Every nilpotent Lie algebra is [[solvable Lie algebra\|solvable]]; this fact gives one of the powerful ways to prove the solvability of a Lie algebra since, in practice, it is usually easier to prove the nilpotency than the solvability. The converse is not true in general. A Lie algebra <math>\mathfrak{g}</math> is nilpotent if and only if its quotient over an ideal containing the center of <math>\mathfrak{g}</math> is nilpotent.

			Most of classic classification results on nilpotency are concerned with finite-dimensional Lie algebras over a field of characteristic 0. Let <math>\mathfrak{g}</math> be a finite-dimensional Lie algebra. <math>\mathfrak{g}</math> is nilpotent if and only if <math>\operatorname{ad}(\mathfrak{g})</math> is nilpotent. [[Engel's theorem]] states that <math>\mathfrak{g}</math> is nilpotent if and only if <math>\operatorname{ad}(x)</math> is nilpotent for every <math>x \in \mathfrak{g}</math>. <math>\mathfrak{g}</math> is solvable if and only if <math>[\mathfrak{g}, \mathfrak{g}]</math> is nilpotent.

			== Examples ==
			*Every subalgebra and quotient of a nilpotent Lie algebra is nilpotent.
			*If <math>\mathfrak{gl}_k</math> is the set of <math>k\times k</math> matrices, then the subalgebra consisting of strictly upper triangular matrices, denoted by <math>\mathfrak{n}_k</math>, is a nilpotent Lie algebra.
			*A [[Heisenberg algebra]] is nilpotent.
			*A [[Cartan subalgebra]] of a Lie algebra is nilpotent and [[self-normalizing]].

			==References==
			* Humphreys, James E. ''Introduction to Lie Algebras and Representation Theory''. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. ISBN 0-387-90053-5

			[[Category:Properties of Lie algebras]]

en>Glenn: +{{Commons cat|Wavelets|Wavelets}}

2012-08-26T15:27:39Z

+{{Commons cat|Wavelets|Wavelets}}

New page

She is recognized by the name of Myrtle Shryock. Bookkeeping is her working day occupation now. For a while she's been in South Dakota. To perform baseball is the hobby he will never quit doing.<br><br>Feel free to surf to my weblog - [http://Nuvem.tk/altergalactica/NumbersdbEstepvu at home std test]