Dynamic financial analysis - Revision history

en>Chris the speller: /* Cox, Ingersoll, and Ross (CIR) interest rate generator */replaced: long term → long-term, short term → short-term using AWB

2014-12-10T18:41:46Z

Cox, Ingersoll, and Ross (CIR) interest rate generator: replaced: long term → long-term, short term → short-term using AWB

← Older revision		Revision as of 20:41, 10 December 2014
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	~~In [[mathematics]],~~ '~~''hyperbolic Dehn surgery''' is an operation by which one can obtain further [[hyperbolic 3-manifold]]~~s ~~from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes [[hyperbolic geometry]] in three dimensions from other dimensions.~~		Oscar is how he's called and he totally loves this title. I am a meter reader. Years ago we moved to North Dakota and I love each working day residing right here. The favorite pastime for my children and me is to play baseball and I'm trying to make it a profession.<br><br>Look into my page [http://www.shipzula.com/blogs/post/16377 over the counter std test]

	~~Such an operation is often also~~ called ~~'''hyperbolic Dehn filling''', as [[Dehn surgery]] proper refers to a "drill~~ and ~~fill" operation on a link which consists of ''drilling'' out a neighborhood of the link and then ''filling'' back in with solid tori. Hyperbolic Dehn surgery actually only involves "filling".~~

	~~We will generally assume that a hyperbolic 3-manifold is complete.~~

	~~Suppose ''M'' is a cusped hyperbolic 3-manifold with ''n'' cusps~~. ~~''M'' can be thought of, topologically, as the interior of~~ a ~~compact manifold with toral boundary~~. ~~Suppose~~ we ~~have chosen a meridian~~ and ~~longitude for~~ each ~~boundary torus, i~~.~~e. simple closed curves that are generators~~ for the fundamental group of the torus. Let <math>M(u_1, u_2, \dots, u_n)</math> denote the manifold obtained from M by filling in the ''i''-th boundary torus with a solid torus using the slope <math>u_i = p_i/q_i</math> where each pair <math>p_i</math> and <math>q_i</math> are coprime integers. We allow a <math>u_i</math> to be <math>\infty</math> which means we do not fill in that cusp, i.e. do the "empty" Dehn filling. So ''M'' = <math>M(\infty, \dots, \infty)</math>.

	~~We equip the space ''H'' of finite volume hyperbolic 3-manifolds with the [[The geometric topology\|geometric topology]].~~

	~~'''Thurston's hyperbolic Dehn surgery theorem''' states: <math>M(u_1, u_2, \dots, u_n)</math> is hyperbolic as long as a finite set of ''exceptional slopes'' <math>E_i</math>~~ is ~~avoided for the ''i''-th cusp for each ''i''. In addition, <math>M(u_1, u_2, \dots, u_n)</math> converges~~ to '~~'M'' in ''H'' as all <math>p_i^2+q_i^2 \rightarrow \infty</math> for all <math>p_i/q_i</math> corresponding~~ to ~~non-empty Dehn fillings <math>u_i</math>.~~

	This theorem is due to [[William Thurston]] and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in ''H''. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.

	Another important result by Thurston is that volume decreases under hyperbolic Dehn filling. In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the [[Gromov norm]].

	~~Jørgensen also showed that the volume function on this space is~~ a ~~[[continuous function\|continuous]], [[proper map\|proper]] function~~. Thus by the previous results, nontrivial limits in ''H'' are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has [[ordinal number\|ordinal type]] <~~math~~>~~\omega^\omega~~<~~/math~~>~~. This result is known as the '''Thurston-Jørgensen theorem'''. Further work characterizing this set was done by [[Mikhail Gromov (mathematician)\|Gromov]].~~

	The [[Figure-eight knot (mathematics)\|figure-eight knot]] and the (-2, 3, 7) [[pretzel knot]] are the only two knots whose complements are known to have more than 6 exceptional surgeries; they have 10 and 7, respectively. [[Cameron Gordon (mathematician)\|Cameron Gordon]] conjectured that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot complement. This was proved by Marc Lackenby and Rob Meyerhoff, who show that the number of exceptional slopes is 10 for any compact orientable 3-manifold with boundary a torus and interior finite-volume hyperbolic. Their proof relies on the proof of the [[geometrization conjecture]] originated by [[Grigori Perelman]] and on [[computer-assisted proof\|computer assistance]]. However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound (except for the two knots mentioned) is 6. Agol has shown that there are only finitely many cases in which the number of exceptional slopes is 9 or 10.

	~~==References==~~
	*Ian Agol, ''Bounds on exceptional Dehn filling II'', Geom. Topol. '''14''' (2010) 1921-1940. arxiv:0803:3088
	*[[Robion Kirby]], [http://math.berkeley.edu/~kirby/problems.ps.gz ''Problems in low-dimensional topology''], (see problem 1.77, due to [[Cameron Gordon (mathematician)\|Cameron Gordon]], for exceptional slopes)
	*Marc Lackenby and Robert Meyerhoff, [http://front.math.ucdavis.edu/0808.1176 ''The maximal number of exceptional Dehn surgeries''], arXiv:0808.1176
	*William Thurston, [http://www.~~msri~~.~~org~~/~~publications~~/~~books~~/~~gt3m/ ''The geometry and topology of 3-manifolds''], Princeton lecture notes (1978–1981).~~

	~~[[Category:3-manifolds]]~~
	~~[[Category:Hyperbolic geometry]~~]

en>Jwhayles: /* Jurisdictional Risk Generator */

2014-01-07T20:51:05Z

Jurisdictional Risk Generator

← Older revision		Revision as of 22:51, 7 January 2014
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	~~Nice to satisfy you~~, ~~my title~~ is ~~Refugia~~. ~~Doing ceramics~~ is ~~what her family members~~ and ~~her appreciate~~. ~~Bookkeeping~~ is ~~what I~~ do. ~~Puerto Rico~~ is ~~where he~~'s ~~usually been living but she needs~~ to ~~transfer because~~ of ~~her family members~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my web blog;~~ [http://~~citymama~~.~~com~~.ua/~~profile_info~~.~~php?ID=38588 std testing at home~~]		In [[mathematics]], '''hyperbolic Dehn surgery''' is an operation by which one can obtain further [[hyperbolic 3-manifold]]s from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes [[hyperbolic geometry]] in three dimensions from other dimensions.

			Such an operation is often also called '''hyperbolic Dehn filling''', as [[Dehn surgery]] proper refers to a "drill and fill" operation on a link which consists of ''drilling'' out a neighborhood of the link and then ''filling'' back in with solid tori. Hyperbolic Dehn surgery actually only involves "filling".

			We will generally assume that a hyperbolic 3-manifold is complete.

			Suppose ''M'' is a cusped hyperbolic 3-manifold with ''n'' cusps. ''M'' can be thought of, topologically, as the interior of a compact manifold with toral boundary. Suppose we have chosen a meridian and longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of the torus. Let <math>M(u_1, u_2, \dots, u_n)</math> denote the manifold obtained from M by filling in the ''i''-th boundary torus with a solid torus using the slope <math>u_i = p_i/q_i</math> where each pair <math>p_i</math> and <math>q_i</math> are coprime integers. We allow a <math>u_i</math> to be <math>\infty</math> which means we do not fill in that cusp, i.e. do the "empty" Dehn filling. So ''M'' = <math>M(\infty, \dots, \infty)</math>.

			We equip the space ''H'' of finite volume hyperbolic 3-manifolds with the [[The geometric topology\|geometric topology]].

			'''Thurston's hyperbolic Dehn surgery theorem''' states: <math>M(u_1, u_2, \dots, u_n)</math> is hyperbolic as long as a finite set of ''exceptional slopes'' <math>E_i</math> is avoided for the ''i''-th cusp for each ''i''. In addition, <math>M(u_1, u_2, \dots, u_n)</math> converges to ''M'' in ''H'' as all <math>p_i^2+q_i^2 \rightarrow \infty</math> for all <math>p_i/q_i</math> corresponding to non-empty Dehn fillings <math>u_i</math>.

			This theorem is due to [[William Thurston]] and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in ''H''. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.

			Another important result by Thurston is that volume decreases under hyperbolic Dehn filling. In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the [[Gromov norm]].

			Jørgensen also showed that the volume function on this space is a [[continuous function\|continuous]], [[proper map\|proper]] function. Thus by the previous results, nontrivial limits in ''H'' are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has [[ordinal number\|ordinal type]] <math>\omega^\omega</math>. This result is known as the '''Thurston-Jørgensen theorem'''. Further work characterizing this set was done by [[Mikhail Gromov (mathematician)\|Gromov]].

			The [[Figure-eight knot (mathematics)\|figure-eight knot]] and the (-2, 3, 7) [[pretzel knot]] are the only two knots whose complements are known to have more than 6 exceptional surgeries; they have 10 and 7, respectively. [[Cameron Gordon (mathematician)\|Cameron Gordon]] conjectured that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot complement. This was proved by Marc Lackenby and Rob Meyerhoff, who show that the number of exceptional slopes is 10 for any compact orientable 3-manifold with boundary a torus and interior finite-volume hyperbolic. Their proof relies on the proof of the [[geometrization conjecture]] originated by [[Grigori Perelman]] and on [[computer-assisted proof\|computer assistance]]. However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound (except for the two knots mentioned) is 6. Agol has shown that there are only finitely many cases in which the number of exceptional slopes is 9 or 10.

			==References==
			*Ian Agol, ''Bounds on exceptional Dehn filling II'', Geom. Topol. '''14''' (2010) 1921-1940. arxiv:0803:3088
			*[[Robion Kirby]], [http://math.berkeley.edu/~kirby/problems.ps.gz ''Problems in low-dimensional topology''], (see problem 1.77, due to [[Cameron Gordon (mathematician)\|Cameron Gordon]], for exceptional slopes)
			*Marc Lackenby and Robert Meyerhoff, [http://front.math.ucdavis.edu/0808.1176 ''The maximal number of exceptional Dehn surgeries''], arXiv:0808.1176
			*William Thurston, [http://www.msri.org/publications/books/gt3m/ ''The geometry and topology of 3-manifolds''], Princeton lecture notes (1978–1981).

			[[Category:3-manifolds]]
			[[Category:Hyperbolic geometry]]

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Premium

2012-08-12T01:13:03Z

Repairing links to disambiguation pages - You can help! - Premium

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