Selberg trace formula - Revision history

en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-25T21:43:37Z

References: replace mr template with mr parameter in CS1 templates; using AWB

← Older revision		Revision as of 23:43, 25 September 2014
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			== Sacs Longchamps Xxl ==

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en>Richard L. Peterson: further elaboration--Γ acts on H by linear fractional transformations

2014-03-03T21:27:24Z

further elaboration--Γ acts on H by linear fractional transformations

← Older revision		Revision as of 23:27, 3 March 2014
Line 1:		Line 1:
	In the [[Mathematics\|mathematical]] field of [[algebraic geometry]], a '''singular point of an [[algebraic variety]]''' ''V'' is a point ''P'' that is 'special' (so, singular), in the geometric sense that at this point the [[tangent space]] at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of non-[[local flatness]]. A point of an algebraic variety which is not singular is said to be '''regular'''. An algebraic variety which has no singular point is said to be '''non singular''' or '''smooth'''.

	~~For example, the [[plane algebraic curve]] (a [[cubic curve]]) of equation~~
	~~:''y''<sup>2</sup> - ''x''<sup>2</sup>(''x'' + 1) = 0,~~
	~~which is plotted below, crosses itself at the origin (0,0) and the origin is thus a [[double point]] of the curve. It is ''singular'' because a single [[tangent]] may not be correctly defined there.~~

	~~[[Image:Singularptfn.JPG]]~~		When playing a new video tutorials game, read the more indulgent book. Most on-line games have a book you actually can purchase separately. You may want in consider doing this but reading it before the individual play, or even even when you are playing. This way, you can possibly get the most for of your game play.<br><br>Maintain a video game great. These can be a lot out of fun for you and your own gaming friends. You can do this online, at your house . or at a pals place. Serve awesome snacks and get the number of people as you will often involved. This is a great way to enjoy your game playing with family.<br><br>Temperance is essential in on the whole things, and enjoying clash of clans cheats is no dissimilar. Playing for hours on finish is not just good-for-you, bodily or on an emotional level. There are some games out in that respect there which know this and may include measures to remind someone to take rests. Go ahead and take initiative yourself, although! Place an alarm a person don't play for via an hour right.<br><br>On Clash of Clans Cheats (a brilliant popular personal architecture and arresting strong by Supercell) participants may well acceleration up accomplishments for example building, advance or training troops with gems that is bought for absolute money. They're basically monetizing the underlying player's impatience. Each and every amusing architecture daring Which i apperceive of manages to create it happen.<br><br>This is my testing has apparent that will this appraisement algorithm plan consists of a alternation of beeline band sectors. They are not things to consider reproductions of arced graphs. I will explain so why later.<br><br>If you loved this article and you would like to collect more info concerning [http://prometeu.net clash of clans trucos android] please visit our own page. And access it into excel, copy-paste this continued recipe into corpuscle B1. If you again obtain an majority of period of time in abnormal in corpuscle A1, the bulk wearing treasures will arise all over B1.<br><br>You don''t necessarily want one of the complex troops to win wins. A mass volume of barbarians, your first-level troop, can totally destroy an assailant village, and [http://www.answers.com/topic/strangely+it%27%27s strangely it''s] quite enjoyable to the the virtual carnage.

	~~More generally~~ a ~~plane curve defined by an [[implicit equation]]~~
	~~:''F''(''x'',''y'') = 0~~,
	~~where ''F'' is a [[smooth function]] is said to be ''singular'' at point if~~ the ~~[[Taylor series]] of ''F'' has [[Power series#Order of~~ a ~~power series\|order]] at least 2 at~~ this ~~point~~.

	~~The reason for that is that, in [[differential calculus]]~~, the ~~tangent at the point (''x''<sub>0</sub>, ''y''~~<~~sub~~>0<~~/sub~~>~~) of such~~ a ~~curve is defined by the equation~~
	~~:<math>(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0),</math>~~
	~~whose left-hand side is the term of degree one of the Taylor expansion~~. ~~Thus, if this term is zero, the tangent may not~~ be ~~defined in the standard way, either because it does not exists or~~ a ~~special definition must be provided~~.

	~~In general for a [[hypersurface]]~~

	~~:''F''(''x'', ''y'', ''z''~~, .~~..) = 0~~

	~~the '''singular points''' are those~~ at ~~which all the [[partial derivative]]s simultaneously vanish~~. ~~A general [[algebraic variety]] ''V'' being defined as~~ the ~~common zeros~~ of ~~several [[polynomial]]s, the condition on~~ a ~~point ''P'' of ''V''~~ to ~~be singular point~~ is ~~that~~ the ~~[[Jacobian matrix]] of the first order partial derivatives of the polynomials has a [[rank~~ of ~~a matrix\|rank]] at ''P'' that~~ is ~~lower than the rank at other points of the variety~~.

	~~Points of ''V'' that are~~ not ~~singular are called '''non~~-~~singular'''~~ or ~~'''regular'''~~. ~~It is always true that most points~~ are ~~non-singular~~ in ~~the sense~~ that ~~the non-singular points form~~ a ~~set that is both [[open set\|open]] and non-empty~~.<~~ref~~>~~Hartshorne, ''Algebraic Geometry'', page 33~~<~~/ref~~>

	~~In case~~ of ~~a real variety~~ (~~that is the set of the points with real coordinates of~~ a ~~variety defined~~ by ~~polynomials with real coefficients~~), the ~~variety is a [[manifold]] near~~ every ~~regular point. But~~ it ~~is important to note that a real variety may be a manifold and have singular points~~. ~~For example the equation~~ <~~math~~>~~y^3 + 2 x^2 y - x^4 = 0~~<~~/math~~> ~~defines a real [[analytic manifold]] but~~ has a ~~singular point at the origin~~.<~~ref~~>~~Milnor, pp. 12–13~~<~~/ref~~> ~~This may be explained by saying that the curve has two~~ [~~[complex conjugate~~]~~] [[branch (complex analysis)\|branch]]es that cut the real branch at the origin~~.

	~~==Singular points~~ of ~~smooth mappings==~~
	~~As the notion~~ of ~~singular points is a purely local property~~, the ~~above definition can be extended to cover the wider class of [[smooth function\|smooth]] mappings, (functions from ''M'' to '''R'''~~<~~sup~~>n<~~/sup~~> ~~where all derivatives exist)~~. ~~Analysis~~ of ~~these singular points~~ can ~~be reduced to the algebraic variety case by considering the~~ [~~[Jet (mathematics)\|jets]] of the mapping~~. ~~The~~ ''~~k''-th jet is the [[Taylor series~~]~~] of~~ the ~~mapping truncated at degree ''k'' and deleting~~ the ~~[[constant term]]~~.

	~~== See also ==~~
	*[[Singular point of a curve]]
	*[[Singularity theory]]

	~~==References==~~
	~~{{reflist}}~~
	~~{{refimprove\|date=September 2008}}~~
	* {{cite book \| title=Singular Points of Complex Hypersurfaces \| author=John Milnor \| authorlink=John Milnor \| series=Annals of Mathematics Studies \| volume=61 \| publisher=[[Princeton University Press]] \| year=1969 \| isbn=0-691-08065-8 }}

	~~[[Category:Algebraic varieties]]~~
	~~[[Category:Singularity theory]]~~

en>Duoduoduo: update redirect

2013-08-09T01:13:59Z

update redirect

← Older revision		Revision as of 03:13, 9 August 2013
Line 1:		Line 1:
	~~The title~~ of the ~~author~~ is ~~Figures~~. ~~To perform baseball~~ is the ~~hobby he will~~ by ~~no means quit doing~~. ~~Managing individuals~~ is ~~what I do and~~ the ~~wage has been really satisfying~~. ~~For~~ a ~~while she~~'s ~~been~~ in ~~South Dakota~~.<br><br>~~My web page - home std test kit~~ ([~~http:~~//~~xrambo~~.~~com~~/~~blog/127033 check out~~ the ~~post right here~~])		In the [[Mathematics\|mathematical]] field of [[algebraic geometry]], a '''singular point of an [[algebraic variety]]''' ''V'' is a point ''P'' that is 'special' (so, singular), in the geometric sense that at this point the [[tangent space]] at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of non-[[local flatness]]. A point of an algebraic variety which is not singular is said to be '''regular'''. An algebraic variety which has no singular point is said to be '''non singular''' or '''smooth'''.

			For example, the [[plane algebraic curve]] (a [[cubic curve]]) of equation
			:''y''<sup>2</sup> - ''x''<sup>2</sup>(''x'' + 1) = 0,
			which is plotted below, crosses itself at the origin (0,0) and the origin is thus a [[double point]] of the curve. It is ''singular'' because a single [[tangent]] may not be correctly defined there.

			[[Image:Singularptfn.JPG]]

			More generally a plane curve defined by an [[implicit equation]]
			:''F''(''x'',''y'') = 0,
			where ''F'' is a [[smooth function]] is said to be ''singular'' at point if the [[Taylor series]] of ''F'' has [[Power series#Order of a power series\|order]] at least 2 at this point.

			The reason for that is that, in [[differential calculus]], the tangent at the point (''x''<sub>0</sub>, ''y''<sub>0</sub>) of such a curve is defined by the equation
			:<math>(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0),</math>
			whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may not be defined in the standard way, either because it does not exists or a special definition must be provided.

			In general for a [[hypersurface]]

			:''F''(''x'', ''y'', ''z'', ...) = 0

			the '''singular points''' are those at which all the [[partial derivative]]s simultaneously vanish. A general [[algebraic variety]] ''V'' being defined as the common zeros of several [[polynomial]]s, the condition on a point ''P'' of ''V'' to be singular point is that the [[Jacobian matrix]] of the first order partial derivatives of the polynomials has a [[rank of a matrix\|rank]] at ''P'' that is lower than the rank at other points of the variety.

			Points of ''V'' that are not singular are called '''non-singular''' or '''regular'''. It is always true that most points are non-singular in the sense that the non-singular points form a set that is both [[open set\|open]] and non-empty.<ref>Hartshorne, ''Algebraic Geometry'', page 33</ref>

			In case of a real variety (that is the set of the points with real coordinates of a variety defined by polynomials with real coefficients), the variety is a [[manifold]] near every regular point. But it is important to note that a real variety may be a manifold and have singular points. For example the equation <math>y^3 + 2 x^2 y - x^4 = 0</math> defines a real [[analytic manifold]] but has a singular point at the origin.<ref>Milnor, pp. 12–13</ref> This may be explained by saying that the curve has two [[complex conjugate]] [[branch (complex analysis)\|branch]]es that cut the real branch at the origin.

			==Singular points of smooth mappings==
			As the notion of singular points is a purely local property, the above definition can be extended to cover the wider class of [[smooth function\|smooth]] mappings, (functions from ''M'' to '''R'''<sup>n</sup> where all derivatives exist). Analysis of these singular points can be reduced to the algebraic variety case by considering the [[Jet (mathematics)\|jets]] of the mapping. The ''k''-th jet is the [[Taylor series]] of the mapping truncated at degree ''k'' and deleting the [[constant term]].

			== See also ==
			*[[Singular point of a curve]]
			*[[Singularity theory]]

			==References==
			{{reflist}}
			{{refimprove\|date=September 2008}}
			* {{cite book \| title=Singular Points of Complex Hypersurfaces \| author=John Milnor \| authorlink=John Milnor \| series=Annals of Mathematics Studies \| volume=61 \| publisher=[[Princeton University Press]] \| year=1969 \| isbn=0-691-08065-8 }}

			[[Category:Algebraic varieties]]
			[[Category:Singularity theory]]

en>Suslindisambiguator: /* References */

2012-07-12T11:54:14Z

References

New page

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