Leapfrog integration - Revision history

80.222.156.236: Added clarification for the note that acceleration depends only on the positions of the accelerating masses (as opposed to positions and velocities).

2014-12-09T12:11:34Z

Added clarification for the note that acceleration depends only on the positions of the accelerating masses (as opposed to positions *and velocities*).

← Older revision		Revision as of 14:11, 9 December 2014
Line 1:		Line 1:
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en>Illusional: Unbiasing: "Simple" is an opinionated or comparative term

2014-02-27T07:45:45Z

Unbiasing: "Simple" is an opinionated or comparative term

← Older revision		Revision as of 09:45, 27 February 2014
Line 1:		Line 1:
	In [[mathematics]], the '''cone of curves''' (sometimes the '''Kleiman-Mori''' cone) of an [[algebraic variety]] <math>X</math> is a [[combinatorial invariant]] of much importance to the [[birational geometry]] of <math>X</math>.

	~~==Definition==~~
	Let <math>X</math> be a [[Proper morphism\|proper]] variety. By definition, a (real) ''1-cycle'' on <math>X</math> is a formal [[linear combination]] <math>C=\sum a_iC_i</math> of irreducible, reduced and proper curves <math>C_i</math>, with coefficients <math>a_i \in \mathbb{R}</math>. ''Numerical equivalence'' of 1-cycles is defined by intersections: two 1-cycles <math>C</math> and <math>C'</math> are numerically equivalent if <math>C \cdot D = C' \cdot D</math> for every Cartier [[divisor]] <math>D</math> on <math>X</math>. Denote the [[real vector space]] of 1-cycles modulo numerical equivalence by <math>N_1(X)</math>.

	~~We define~~ the ~~''cone of curves'' of <math>X</math> to~~ be		Surely the second option would be more beneficial for any website. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. s ultimately easy to implement and virtually maintenance free. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. <br><br>Creating a website from scratch can be such a pain. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. Which is perfect for building a mobile site for business use. You can up your site's rank with the search engines by simply taking a bit of time with your site. That's a total of 180$ for each Wordpress theme if you sell 3 links and keep the designer link for your own website, or 240$ if you sell all links. <br><br>The entrepreneurs can easily captivate their readers by using these versatile themes. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. You've got invested a great cope of time developing and producing up the topic substance. These frequent updates have created menace in the task of optimization. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. The company gains commission from the customers' payment. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Internet is not only the source for information, it is also one of the source for passive income. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. If you loved this informative article in addition to you want to acquire more information about [http://flashgamer.info/profile/aibedard wordpress backup plugin] i implore you to go to our own internet site. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities.

	~~: <math>NE(X) = \left\{\sum a_i[C_i], \ 0 \leq a_i \in \mathbb{R} \right\} </math>~~

	~~where the <math>C_i</math> are irreducible, reduced, proper curves on <math>X</math>, and <math>[C_i]</math> their classes in <math>N_1(X)</math>~~. It is ~~not difficult~~ to ~~see that <math>NE(X)</math> is indeed a [[Cone_(linear_algebra)#Convex_cone\|convex cone]] in the sense of convex geometry~~.

	~~==Applications==~~
	~~One useful application of the notion of the cone of curves is the '''[[Steven Kleiman\|Kleiman]] condition'''~~, which says that a (Cartier) divisor <math>D</math> on a complete variety <math>X</math> is [[ample line bundle\|ample]] if and only if <math>D \cdot x > 0</math> for any nonzero element <math>x</math> in <math>\overline{NE(X)}</math>, ~~the closure of the cone of curves in the usual real topology. (In general, <math>NE(X)</math> need not~~ be ~~closed, so taking the closure here is important~~.)

	~~A more involved example is the role played by the cone of curves in the theory of [[minimal model]]~~s ~~of algebraic varieties~~. ~~Briefly~~, the ~~goal of~~ that ~~theory~~ is ~~as follows: given~~ a ~~(mildly singular) projective variety~~ <~~math~~>X<~~/math~~>~~, find~~ a ~~(mildly singular) variety <math>X'</math> which~~ is ~~[[birational]]~~ to ~~<math>X</math>, and whose [[canonical divisor]] <math>K_{X'}</math> is [[numerically effective\|nef]]. The great breakthrough of the early 1980s (due to [[Shigefumi Mori\|Mori]]~~ and ~~others) was~~ to ~~construct (at least morally)~~ the ~~necessary birational map from <math>X</math> to <math>X'</math> as a sequence of steps~~, ~~each of which~~ can ~~be thought of~~ as ~~contraction of~~ a ~~<math>K_x</math>-negative extremal ray of <math>NE(X)</math>~~. ~~This process encounters difficulties, however, whose resolution necessitates~~ the ~~introduction~~ of ~~the [[flip (algebraic geometry)\|flip]]~~.

	~~==A structure theorem==~~
	~~The above process of contractions could not proceed without the fundamental result on the structure of the cone of curves known as the~~ '~~''Cone Theorem'''. The first version~~ of ~~this theorem,~~ for ~~[[smooth varieties]], is due to [[Shigefumi Mori\|Mori]]; it was later generalised to a larger class of varieties by [[János_Kollár\|Kollár]], [[Miles Reid\|Reid]], [[Vyacheslav_Shokurov\|Shokurov]],~~ and ~~others. Mori's version of~~ the ~~theorem is as follows:~~

	~~'''Cone Theorem~~.~~''' Let~~ <~~math~~>X<~~/math~~> ~~be a smooth [[projective variety]]~~. ~~Then~~

	~~1. There are [[countably many]] [[rational curve]]s <math>C_i</math> on <math>X</math>, satisfying <math>0< -K_X \cdot C_i \leq \operatorname{dim} X +1 </math>, and~~

	: ~~<math>\overline{NE(X)} = \overline{NE(X)}_{K_X\geq 0} + \sum_i \mathbf{R}_{\geq0} [C_i]~~.~~</math>~~

	~~2. For any positive real number <math>\epsilon</math>~~ and ~~any [[ample divisor]] <math>H</math>,~~

	~~: <math>\overline{NE(X)} = \overline{NE(X)}_{K_X+\epsilon H\geq0} + \sum \mathbf{R}_{\geq0} [C_i],</math>~~

	~~where the sum in~~ the ~~last term is finite~~.

	~~The first assertion says that,~~ in the ~~[[closed half-space]]~~ of <~~math~~>~~N_1(X)~~<~~/math~~> ~~where intersection~~ with ~~<math>K_X</math> is nonnegative~~, we know ~~nothing, but in the complementary half~~-~~space,~~ the ~~cone is spanned by some countable collection of curves which~~ are ~~quite special:~~ they are ~~[[Rational variety\|rational]]~~, and ~~their 'degree' is bounded very tightly by the dimension of <math>X</math>~~. The ~~second assertion then tells us more: it says that, away~~ from the ~~hyperplane <math>\{C : K_X \cdot C = 0\}</math>, extremal rays of the cone cannot accumulate~~.


	If in ~~addition the variety~~ <~~math~~>X<~~/math~~> is ~~defined over a field of characteristic 0, we have~~ the ~~following assertion~~, ~~sometimes referred to as~~ the ~~'''Contraction Theorem''':~~

	3. ~~Let <math>F \subset \overline{NE(X)}</math> be~~ an ~~extremal face of the cone of curves on which <math>K_X</math> is negative~~. ~~Then there~~ is a ~~unique~~ [~~[morphism]] <math>\operatorname{cont}_F~~ : ~~X \rightarrow Z</math> to a projective variety ''Z'', such that <math>(\operatorname{cont}_F)_* \mathcal{O}_X = \mathcal{O}_Z<~~/~~math> and an irreducible curve <math>C<~~/~~math> in <math>X<~~/~~math> is mapped to a point by <math>\operatorname{cont}_F<~~/~~math> if and only if <math>[C~~] ~~\in F</math>~~.

	~~==References==~~

	* Lazarsfeld, ~~R., ''Positivity in Algebraic Geometry I'', Springer~~-~~Verlag, 2004. ISBN 3-540-22533-1~~
	* Kollár, J. and ~~Mori, S., ''Birational Geometry of Algebraic Varieties'', Cambridge University Press, 1998~~. ~~ISBN 0-521-63277-3~~

	~~[[Category:Algebraic geometry]]~~
	~~[[Category:Birational geometry]]~~

en>LutzL: Undid revision 583776602 by 193.67.17.36 (talk). "The" Runge-Kutta method is the classical order 4 method.

2013-11-29T14:43:43Z

Undid revision 583776602 by 193.67.17.36 (talk). "The" Runge-Kutta method is the classical order 4 method.

← Older revision		Revision as of 16:43, 29 November 2013
Line 1:		Line 1:
	~~I like my hobby College football~~. ~~Sounds boring? Not at all!~~<br>~~I try~~ to ~~learn Bengali~~ in ~~my spare time~~.<br><br>~~Look~~ at ~~my blog~~ ... [~~http~~://~~flashgamer~~.~~info~~/~~profile~~/~~aibedard wordpress backup plugin~~]		In [[mathematics]], the '''cone of curves''' (sometimes the '''Kleiman-Mori''' cone) of an [[algebraic variety]] <math>X</math> is a [[combinatorial invariant]] of much importance to the [[birational geometry]] of <math>X</math>.

			==Definition==
			Let <math>X</math> be a [[Proper morphism\|proper]] variety. By definition, a (real) ''1-cycle'' on <math>X</math> is a formal [[linear combination]] <math>C=\sum a_iC_i</math> of irreducible, reduced and proper curves <math>C_i</math>, with coefficients <math>a_i \in \mathbb{R}</math>. ''Numerical equivalence'' of 1-cycles is defined by intersections: two 1-cycles <math>C</math> and <math>C'</math> are numerically equivalent if <math>C \cdot D = C' \cdot D</math> for every Cartier [[divisor]] <math>D</math> on <math>X</math>. Denote the [[real vector space]] of 1-cycles modulo numerical equivalence by <math>N_1(X)</math>.

			We define the ''cone of curves'' of <math>X</math> to be

			: <math>NE(X) = \left\{\sum a_i[C_i], \ 0 \leq a_i \in \mathbb{R} \right\} </math>

			where the <math>C_i</math> are irreducible, reduced, proper curves on <math>X</math>, and <math>[C_i]</math> their classes in <math>N_1(X)</math>. It is not difficult to see that <math>NE(X)</math> is indeed a [[Cone_(linear_algebra)#Convex_cone\|convex cone]] in the sense of convex geometry.

			==Applications==
			One useful application of the notion of the cone of curves is the '''[[Steven Kleiman\|Kleiman]] condition''', which says that a (Cartier) divisor <math>D</math> on a complete variety <math>X</math> is [[ample line bundle\|ample]] if and only if <math>D \cdot x > 0</math> for any nonzero element <math>x</math> in <math>\overline{NE(X)}</math>, the closure of the cone of curves in the usual real topology. (In general, <math>NE(X)</math> need not be closed, so taking the closure here is important.)

			A more involved example is the role played by the cone of curves in the theory of [[minimal model]]s of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety <math>X</math>, find a (mildly singular) variety <math>X'</math> which is [[birational]] to <math>X</math>, and whose [[canonical divisor]] <math>K_{X'}</math> is [[numerically effective\|nef]]. The great breakthrough of the early 1980s (due to [[Shigefumi Mori\|Mori]] and others) was to construct (at least morally) the necessary birational map from <math>X</math> to <math>X'</math> as a sequence of steps, each of which can be thought of as contraction of a <math>K_x</math>-negative extremal ray of <math>NE(X)</math>. This process encounters difficulties, however, whose resolution necessitates the introduction of the [[flip (algebraic geometry)\|flip]].

			==A structure theorem==
			The above process of contractions could not proceed without the fundamental result on the structure of the cone of curves known as the '''Cone Theorem'''. The first version of this theorem, for [[smooth varieties]], is due to [[Shigefumi Mori\|Mori]]; it was later generalised to a larger class of varieties by [[János_Kollár\|Kollár]], [[Miles Reid\|Reid]], [[Vyacheslav_Shokurov\|Shokurov]], and others. Mori's version of the theorem is as follows:

			'''Cone Theorem.''' Let <math>X</math> be a smooth [[projective variety]]. Then

			1. There are [[countably many]] [[rational curve]]s <math>C_i</math> on <math>X</math>, satisfying <math>0< -K_X \cdot C_i \leq \operatorname{dim} X +1 </math>, and

			: <math>\overline{NE(X)} = \overline{NE(X)}_{K_X\geq 0} + \sum_i \mathbf{R}_{\geq0} [C_i].</math>

			2. For any positive real number <math>\epsilon</math> and any [[ample divisor]] <math>H</math>,

			: <math>\overline{NE(X)} = \overline{NE(X)}_{K_X+\epsilon H\geq0} + \sum \mathbf{R}_{\geq0} [C_i],</math>

			where the sum in the last term is finite.

			The first assertion says that, in the [[closed half-space]] of <math>N_1(X)</math> where intersection with <math>K_X</math> is nonnegative, we know nothing, but in the complementary half-space, the cone is spanned by some countable collection of curves which are quite special: they are [[Rational variety\|rational]], and their 'degree' is bounded very tightly by the dimension of <math>X</math>. The second assertion then tells us more: it says that, away from the hyperplane <math>\{C : K_X \cdot C = 0\}</math>, extremal rays of the cone cannot accumulate.


			If in addition the variety <math>X</math> is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the '''Contraction Theorem''':

			3. Let <math>F \subset \overline{NE(X)}</math> be an extremal face of the cone of curves on which <math>K_X</math> is negative. Then there is a unique [[morphism]] <math>\operatorname{cont}_F : X \rightarrow Z</math> to a projective variety ''Z'', such that <math>(\operatorname{cont}_F)_* \mathcal{O}_X = \mathcal{O}_Z</math> and an irreducible curve <math>C</math> in <math>X</math> is mapped to a point by <math>\operatorname{cont}_F</math> if and only if <math>[C] \in F</math>.

			==References==

			* Lazarsfeld, R., ''Positivity in Algebraic Geometry I'', Springer-Verlag, 2004. ISBN 3-540-22533-1
			* Kollár, J. and Mori, S., ''Birational Geometry of Algebraic Varieties'', Cambridge University Press, 1998. ISBN 0-521-63277-3

			[[Category:Algebraic geometry]]
			[[Category:Birational geometry]]

en>StevGil: /* See also */ Added link to symplectic integration, since leapfrog can be categorized as such

2012-08-12T15:08:58Z

See also: Added link to symplectic integration, since leapfrog can be categorized as such

New page

I like my hobby College football. Sounds boring? Not at all!<br>I try to learn Bengali in my spare time.<br><br>Look at my blog ... [http://flashgamer.info/profile/aibedard wordpress backup plugin]

Leapfrog integration - Revision history

80.222.156.236: Added clarification for the note that acceleration depends only on the positions of the accelerating masses (as opposed to positions *and velocities*).

en>Illusional: Unbiasing: "Simple" is an opinionated or comparative term

en>LutzL: Undid revision 583776602 by 193.67.17.36 (talk). "The" Runge-Kutta method is the classical order 4 method.

en>StevGil: /* See also */ Added link to symplectic integration, since leapfrog can be categorized as such

80.222.156.236: Added clarification for the note that acceleration depends only on the positions of the accelerating masses (as opposed to positions and velocities).