Direct stiffness method - Revision history

en>Tensorit: /* Assembly */ Showed a more easily remembered and easily generalized form of the stiffness using direction cosines

2014-09-16T05:55:06Z

Assembly: Showed a more easily remembered and easily generalized form of the stiffness using direction cosines

← Older revision		Revision as of 07:55, 16 September 2014
Line 1:		Line 1:
	~~'''Alexander's trick''', also known~~ as ~~the '''Alexander trick''', is~~ a ~~basic result~~ in ~~[[geometric topology]], named after [[James_Waddell_Alexander_II\|J. W. Alexander]].~~		Andrew Simcox is the title his mothers and fathers gave him and he totally enjoys this title. He works as a bookkeeper. My wife and I live in Kentucky. To climb is something I truly appreciate performing.<br><br>My blog - [https://www.machlitim.org.il/subdomain/megila/end/node/12300 psychic readings online]

	~~==Statement==~~
	Two [[homeomorphism]]s of the ''n''-[[dimension]]al [[ball (mathematics)\|ball]] <math>D^n</math> which agree on the [[Boundary (topology)\|boundary]] [[sphere]] <math>S^{n-1}</math> are [[homotopy#Isotopy\|isotopic]].

	~~More generally, two homeomorphisms of ''D''<sup>''n''</sup> that are isotopic on the boundary are isotopic~~.

	~~==Proof==~~
	~~'''Base case''': every homeomorphism which fixes the boundary~~ is ~~isotopic to the identity relative to the boundary~~.

	If <~~math~~>~~f\colon D^n \to D^n~~<~~/math~~> ~~satisfies <math>f(x) = x \mbox{ for all } x \in S^{n~~-~~1}</math>, then an isotopy connecting ''f'' to the identity is given by~~
	~~:<math> J(x,t) = \begin{cases} tf(x/t), & \mbox{if } 0 \leq \\|x\\| < t, \\ x, & \mbox{if } t \leq \\|x\\| \leq 1. \end{cases} </math>~~

	~~Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' <math>f</math> down to the origin.~~ [~~[William Thurston]] calls this "combing all the tangles to one point".~~

	~~The subtlety is that at <math>t=0</math>, <math>f</math> "disappears"~~: ~~the [[Germ (mathematics)\|germ]] at the origin "jumps" from an infinitely stretched version of <math>f<~~/~~math> to the identity. Each of the steps in the homotopy could be smoothed (smooth the transition), but the homotopy (the overall map) has a singularity at <math>(x,t)=(0,0)<~~/~~math>~~. ~~This underlines that the Alexander trick is a [[Piecewise linear manifold\|PL]] construction, but not smooth~~.

	~~'''General case''': isotopic on boundary implies isotopic~~

	If <math>f,g\colon D^n \to D^n</math> are two homeomorphisms that agree on <math>S^{n-1}</math>, then <math>g^{-1}f</math> is the identity on <math>S^{n-1}</math>, so we have an isotopy <math>J</math> from the identity to <math>g^{-1}f</math>. ~~The map <math>gJ<~~/~~math> is then an isotopy from <math>g<~~/~~math> to <math>f<~~/~~math>.~~

	~~==Radial extension==~~
	~~Some authors use the term ''Alexander trick'' for the statement that every [[homeomorphism]] of <math>S^{n-1}<~~/~~math> can be extended to a homeomorphism of the entire ball <math>D^n</math>.~~

	However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true [[piecewise linear homeomorphism\|piecewise-linearly]], but not smoothly.

	~~Concretely, let <math>f\colon S^{n-1} \to S^{n-1}</math> be a homeomorphism, then~~
	~~:<math> F\colon D^n \to D^n \mbox{ with } F(rx) = rf(x) \mbox{ for all } r \in [0,1] \mbox{ and } x \in S^{n-1}<~~/~~math>~~
	~~defines a homeomorphism of the ball.~~

	~~===[[Exotic sphere]]s===~~
	~~The failure of smooth radial extension and the success of PL radial extension~~
	~~yield [[exotic sphere]]s via [[exotic sphere#Twisted spheres\|twisted spheres]].~~

	~~==References==~~
	~~{{reflist}}~~
	*{{cite book \|last=Hilden \|first=V.L. \|title=Braids and Coverings \|year=1989 \|publisher=Cambridge University Press \|isbn=0-521-38757-4}}

	~~[[Category:Geometric topology]]~~
	~~[[Category:Homeomorphisms]~~]

	~~{{unref\|date=December 2007}}~~

en>Dthomsen8: clean up, typo(s) fixed: merged together → merged using AWB

2014-01-23T02:01:57Z

clean up, typo(s) fixed: merged together → merged using AWB

← Older revision		Revision as of 04:01, 23 January 2014
Line 1:		Line 1:
	~~UGG boots are trendy~~. ~~They are the must have add-ons~~. ~~They are a single~~ of the ~~most well~~-~~liked footwear merchandise~~ on the ~~market place~~. ~~Much more women start out to put~~ on ~~these sheepskin boots~~. ~~This new statement with trend has caused a incredibly hot trend throughout~~ the ~~earth around~~ the ~~previous several years. Aussie males could in no way know their life essentials would turn out~~ to ~~be final sneakers~~. ~~People in chilly and warm alike countries like~~ to don Australian sheepskin boots. Celebs like Gwyneth Paltrow, Kate Winslet and Joe were being spotted donning these shoes on superior-finish events. Popular talk clearly show hostprah Winfrey ever regarded as the extremely type as her favourite merchandise. <br><br>~~Why are uggs well~~-~~known? How can they sweep~~ the ~~style earth by storm? Even however a very hot trend~~ is ~~replaced~~ by ~~yet another time and time yet again~~, ~~they nonetheless continue to stand the floor as one particular of the most popular pieces. Like them or loathe them~~, ~~they do realize good success other brands didn'~~t. ~~All of these can be described by their flexibility. It~~ is ~~specifically this aspect that will make sheepskin boots be a essential portion of any lady~~'~~s closet. As extensive as a girls holding up-to-date with trend~~, ~~she will by no means let these stylish shoes slip.~~<br><br>Designers hardly ever head no matter if you contact individuals parts UGG or sheepskin boots. They are relaxed as extended as they are crafted into legitimate Australian Merino sheepskin. You will locate they develop extraordinary warmth during those people chilly days. Also they are suitable shoes to ~~keeping ft dry and interesting in hotter climate~~. ~~Ft do come~~ to ~~feel hard to carry you all over whole day~~. ~~[http:~~//~~tinyurl.com/p7rre7v http~~://~~tinyurl~~.~~com/p7rre7v~~] ~~They are worthy taking very good treatment of~~. ~~Supple~~, ~~furry lining and light-weight rubber sole do promise convenience~~ on ~~ft.~~<br><br>~~Also ugg boots are very affordable. The most effective method to get reliable~~, ~~but low cost sheepskin boots~~ is ~~obtaining them~~ on the ~~web. On the web suppliers do not have~~ to ~~spend for much expenditure~~. ~~Therefore they can go components of financial savings~~ to ~~consumers~~. ~~Or you can acquire them at~~ the ~~turn~~ of ~~period. New arrivals need~~ to ~~have to be put onto~~ the ~~shelves~~. ~~Retailer homeowners have~~ to ~~clear some spot. Advertising objects on low cost rates~~ is ~~the most effective~~ and ~~speediest method~~. ~~The previous way can~~ be ~~adopted is buy them in significant portions. Stores normally deliver you~~ with ~~10%-twenty% low cost on large buy. Even while you overlook~~ all ~~probabilities to get inexpensive uggs~~, ~~you can manage them~~. ~~Unique prices on incredibly hot products are reasonably priced for most individuals~~. ~~It is significantly less than $two hundred~~.00.		'''Alexander's trick''', also known as the '''Alexander trick''', is a basic result in [[geometric topology]], named after [[James_Waddell_Alexander_II\|J. W. Alexander]].

			==Statement==
			Two [[homeomorphism]]s of the ''n''-[[dimension]]al [[ball (mathematics)\|ball]] <math>D^n</math> which agree on the [[Boundary (topology)\|boundary]] [[sphere]] <math>S^{n-1}</math> are [[homotopy#Isotopy\|isotopic]].

			More generally, two homeomorphisms of ''D''<sup>''n''</sup> that are isotopic on the boundary are isotopic.

			==Proof==
			'''Base case''': every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.

			If <math>f\colon D^n \to D^n</math> satisfies <math>f(x) = x \mbox{ for all } x \in S^{n-1}</math>, then an isotopy connecting ''f'' to the identity is given by
			:<math> J(x,t) = \begin{cases} tf(x/t), & \mbox{if } 0 \leq \\|x\\| < t, \\ x, & \mbox{if } t \leq \\|x\\| \leq 1. \end{cases} </math>

			Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' <math>f</math> down to the origin. [[William Thurston]] calls this "combing all the tangles to one point".

			The subtlety is that at <math>t=0</math>, <math>f</math> "disappears": the [[Germ (mathematics)\|germ]] at the origin "jumps" from an infinitely stretched version of <math>f</math> to the identity. Each of the steps in the homotopy could be smoothed (smooth the transition), but the homotopy (the overall map) has a singularity at <math>(x,t)=(0,0)</math>. This underlines that the Alexander trick is a [[Piecewise linear manifold\|PL]] construction, but not smooth.

			'''General case''': isotopic on boundary implies isotopic

			If <math>f,g\colon D^n \to D^n</math> are two homeomorphisms that agree on <math>S^{n-1}</math>, then <math>g^{-1}f</math> is the identity on <math>S^{n-1}</math>, so we have an isotopy <math>J</math> from the identity to <math>g^{-1}f</math>. The map <math>gJ</math> is then an isotopy from <math>g</math> to <math>f</math>.

			==Radial extension==
			Some authors use the term ''Alexander trick'' for the statement that every [[homeomorphism]] of <math>S^{n-1}</math> can be extended to a homeomorphism of the entire ball <math>D^n</math>.

			However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true [[piecewise linear homeomorphism\|piecewise-linearly]], but not smoothly.

			Concretely, let <math>f\colon S^{n-1} \to S^{n-1}</math> be a homeomorphism, then
			:<math> F\colon D^n \to D^n \mbox{ with } F(rx) = rf(x) \mbox{ for all } r \in [0,1] \mbox{ and } x \in S^{n-1}</math>
			defines a homeomorphism of the ball.

			===[[Exotic sphere]]s===
			The failure of smooth radial extension and the success of PL radial extension
			yield [[exotic sphere]]s via [[exotic sphere#Twisted spheres\|twisted spheres]].

			==References==
			{{reflist}}
			*{{cite book \|last=Hilden \|first=V.L. \|title=Braids and Coverings \|year=1989 \|publisher=Cambridge University Press \|isbn=0-521-38757-4}}

			[[Category:Geometric topology]]
			[[Category:Homeomorphisms]]

			{{unref\|date=December 2007}}

27.107.209.168: /* Assembly */

2012-01-16T08:31:32Z

Assembly

New page

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