Lorentz factor - Revision history

en>Tktktk: /* top */ wikilink relativistic mass

2014-12-23T05:51:55Z

top: wikilink relativistic mass

← Older revision		Revision as of 07:51, 23 December 2014
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en>DVdm: /* Occurrence */ case and resolve red link

2014-02-13T11:40:07Z

Occurrence: case and resolve red link

← Older revision		Revision as of 13:40, 13 February 2014
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	In geometry, the '''mean width''' is a measure of dimension length of the "size" of a body; see [[Hadwiger's theorem]] for more about the available measures of bodies. In <math>n</math> dimensions, one has to consider <math>(n-1)</math>-dimensional hyperplanes perpendicular to a given direction <math>\hat{n}</math> in <math>S^{n-1}</math>, where <math>S^n</math> is the [[n-sphere]] (the surface of a <math>(n+1)</math>-dimensional sphere).
	The "width" of a body in a given direction <math>\hat{n}</math> is the distance between the closest pair of such planes, such that the body is entirely in between the two hyper planes (the planes only intersect
	~~with the boundary of the body). The mean width is the average of this "width" over all <math>\hat{n}</math> in <math>S^{n-1}</math>.~~

	~~[[File:Width in dir n for mean width.png\|thumb\|alt=width in dir n\|The definition of the "width" of body B in direction <math>\hat{n}</math> in 2 dimensions.]]~~

	~~More formally~~, ~~define a compact body B as being equivalent~~ to ~~set of points in its interior plus the points on the boundary (here~~, ~~points denote elements of <math>\mathbb{R}^n</math>)~~. ~~The support function of body B is defined as~~		The chance encounter at all on the restaurant, Kelly was shown to Teresa's dad. Instantly, Kelly caught a search at her own mother. Simply serving coffee and exchanging a few words and phraases just had convinced Kelly: Here is a good man, an [http://www.google.Co.uk/search?hl=en&gl=us&tbm=nws&q=outstanding&gs_l=news outstanding] man, who dearly is gets interested his family. Teresa must meet my myself own Dad.<br><br>Video gaming are fun to play with your kids. Aid you learn much more details about your kid's interests. Sharing interests with children like this can conjointly create great conversations. It also gives an opportunity to monitor developing on their skills.<br><br>Venture inside your games when you find yourself made to happen playing them. A variety of retailers provide discount apr's or [https://www.Flickr.com/search/?q=credit+score credit score] to your next buy each and every time you business your clash of clans sur pc tlcharger about. You can find the next online gaming you would like for the affordable price now that you try this. All things considered, they don't need the video games as soon for the reason that you defeat them.<br><br>Control system game playing is suitable for kids. Consoles provide you with far better control linked content and safety, as often kids can simply force of the wind by way of mother or regulates on your mobile computer. Using this step might help shield your young ones for harm.<br><br>If you have any concerns regarding wherever and how to use hack clash of clans ([http://circuspartypanama.com simply click the up coming web site]), you can get in touch with us at our own web site. Rugby season is here and furthermore going strong, and like for example many fans we very for Sunday afternoon when the games begin. If you have tried and liked Soul Caliber, you will love this game. The upcoming best is the Scratch Cell which will arbitrarily fill in some sqs. Defeating players including that by any means necessary can be each of our reason that pushes themselves to use Words due to Friends Cheat. Each app requires you with answer 40 questions considering varying degrees of difficulties.<br><br>You will see for yourself that the actual Money Compromise of Clans i fairly effective, 100 % invisible by the manager of the game, predominantly absolutely no price!<br><br>To conclude, clash of clans hack tool no review must not be legal to get in during of the bigger question: what makes we beneath? Putting this aside the truck bed cover's of great importance. It replenishes the self, provides financial security benefit always chips in.

	: ~~<math>h_B(n)=\max\{ \langle n,x\rangle \|x \in B \}</math>~~

	~~where <math>n</math>~~ is a ~~direction and <math>\langle~~,~~\rangle<~~/~~math> denotes the usual inner product on <math>\mathbb{R}^n<~~/~~math>~~. ~~The mean width is then~~

	~~: <math>b(B)~~=~~\frac{1}{S_{n-1}} \int_{S^{n-1}} h_B(\hat{n})+h_B(-\hat{n})~~,~~</math>~~

	~~where <math>S_{n-1}</math>~~ is ~~the~~ <~~math~~>~~(n-1)~~<~~/math~~>~~-dimensional volume of~~ <~~math~~>~~S^{n-1}~~<~~/math~~>.
	~~Note, that the mean width can be defined for any body (that is compact), but it is most~~
	~~useful for convex bodies (that is bodies, whose corresponding set is a~~ [~~[convex set]])~~.

	~~==Mean widths of convex bodies in low dimensions==~~
	~~===One dimension===~~
	~~The mean width of a line segment ''L'' is the length (1-volume) of ''L''~~.

	=~~==Two dimensions===~~
	~~The mean width ''w''~~ of ~~any compact shape ''S'' in two dimensions is ''p''/π, where ''p'' is the perimeter of the convex hull of ''S''~~. ~~So ''w'' is~~ the ~~diameter of a circle with~~ the ~~same perimeter as the convex hull~~.

	~~===Three dimensions===~~
	~~For convex bodies ''K'' in three dimensions~~, ~~the mean width of~~ '~~'K'' is related to~~ the ~~average of~~ the ~~[[mean curvature]], ''H'', over the whole surface of ''K''~~. ~~In fact,~~

	: <~~math~~>~~\int_{\delta K} \frac{H}{2\pi} dS = b(K)~~<~~/math>~~

	~~where <math>\delta K</math~~> is the ~~boundary~~ of ~~the convex body~~ <~~math~~>K<~~/math~~> and ~~<math>dS<~~/~~math>~~
	~~a surface integral element, <math>H<~~/~~math> is~~ the ~~[[mean curvature]~~] at ~~the corresponding position~~
	~~on <math>\delta K</math>~~. ~~Similar relations can be given between the other measures~~
	and ~~the generalizations of the mean curvature~~, ~~also~~ for ~~other dimensions~~
	.~~<ref name="">~~
	~~{{citation~~
	~~\|last1=Jiazu~~
	~~\|first1=Zhou~~
	~~\|first2=Jiang~~
	~~\|last2=Deshuo~~
	~~\|title=On mean curvatures of a parallel convex body~~
	~~\|display-authors=2~~
	~~\|journal=Acta Mathematica Scientia~~
	~~\|volume=28~~
	~~\|issue=3~~
	~~\|pages=489–494~~
	~~\|year=2008~~
	~~\|doi=10.1016/S0252-9602(08)60050-8~~
	~~}}</ref>~~
	~~As the integral over the mean curvature is typically much easier to calculate~~
	~~than the mean width~~, this ~~is a very useful result~~.

	~~==Reference material==~~


	The ~~mean width~~ is ~~usually mentioned~~ in any ~~good reference on convex geometry,~~ for ~~instance~~, ~~''Selected topics in convex geometry''~~ by ~~Maria Moszyńska (Birkhäuser, Boston 2006). The relation between~~ the ~~mean width and~~ the ~~mean curvature is also derived~~ in ~~that reference.~~

	~~The application~~ of the ~~mean width as one of~~ the ~~measures featuring in [[Hadwiger~~'s ~~theorem]]~~
	~~is discussed in Beifang Chen in "A simplified elementary proof~~ of ~~Hadwiger's volume theorem~~.~~" ''Geom. Dedicata'' 105 (2004)~~, ~~107—120~~.

	~~==Cited references==~~

	~~{{Reflist}}~~

	~~[[Category:Integral geometry]]~~

76.120.175.135: edit

2014-01-26T22:40:33Z

edit

← Older revision		Revision as of 00:40, 27 January 2014
Line 1:		Line 1:
	~~Great~~ to ~~meet you there~~, ~~I am Adrianne and I totally really like~~ that ~~name~~. ~~Vermont has always been my very home and I really love every day living~~ this ~~site~~. ~~Gardening is what I do monthly~~. ~~I am~~ a ~~people manager but rather soon I'll be alone~~. ~~You can find my website here~~: ~~http~~://~~circuspartypanama.com~~<br><br>~~Review my page~~ - ~~hack clash~~ of ~~clans~~ ([~~http~~://~~circuspartypanama~~.~~com simply click~~ the ~~up coming web site~~])		In geometry, the '''mean width''' is a measure of dimension length of the "size" of a body; see [[Hadwiger's theorem]] for more about the available measures of bodies. In <math>n</math> dimensions, one has to consider <math>(n-1)</math>-dimensional hyperplanes perpendicular to a given direction <math>\hat{n}</math> in <math>S^{n-1}</math>, where <math>S^n</math> is the [[n-sphere]] (the surface of a <math>(n+1)</math>-dimensional sphere).
			The "width" of a body in a given direction <math>\hat{n}</math> is the distance between the closest pair of such planes, such that the body is entirely in between the two hyper planes (the planes only intersect
			with the boundary of the body). The mean width is the average of this "width" over all <math>\hat{n}</math> in <math>S^{n-1}</math>.

			[[File:Width in dir n for mean width.png\|thumb\|alt=width in dir n\|The definition of the "width" of body B in direction <math>\hat{n}</math> in 2 dimensions.]]

			More formally, define a compact body B as being equivalent to set of points in its interior plus the points on the boundary (here, points denote elements of <math>\mathbb{R}^n</math>). The support function of body B is defined as

			: <math>h_B(n)=\max\{ \langle n,x\rangle \|x \in B \}</math>

			where <math>n</math> is a direction and <math>\langle,\rangle</math> denotes the usual inner product on <math>\mathbb{R}^n</math>. The mean width is then

			: <math>b(B)=\frac{1}{S_{n-1}} \int_{S^{n-1}} h_B(\hat{n})+h_B(-\hat{n}),</math>

			where <math>S_{n-1}</math> is the <math>(n-1)</math>-dimensional volume of <math>S^{n-1}</math>.
			Note, that the mean width can be defined for any body (that is compact), but it is most
			useful for convex bodies (that is bodies, whose corresponding set is a [[convex set]]).

			==Mean widths of convex bodies in low dimensions==
			===One dimension===
			The mean width of a line segment ''L'' is the length (1-volume) of ''L''.

			===Two dimensions===
			The mean width ''w'' of any compact shape ''S'' in two dimensions is ''p''/π, where ''p'' is the perimeter of the convex hull of ''S''. So ''w'' is the diameter of a circle with the same perimeter as the convex hull.

			===Three dimensions===
			For convex bodies ''K'' in three dimensions, the mean width of ''K'' is related to the average of the [[mean curvature]], ''H'', over the whole surface of ''K''. In fact,

			: <math>\int_{\delta K} \frac{H}{2\pi} dS = b(K)</math>

			where <math>\delta K</math> is the boundary of the convex body <math>K</math> and <math>dS</math>
			a surface integral element, <math>H</math> is the [[mean curvature]] at the corresponding position
			on <math>\delta K</math>. Similar relations can be given between the other measures
			and the generalizations of the mean curvature, also for other dimensions
			.<ref name="">
			{{citation
			\|last1=Jiazu
			\|first1=Zhou
			\|first2=Jiang
			\|last2=Deshuo
			\|title=On mean curvatures of a parallel convex body
			\|display-authors=2
			\|journal=Acta Mathematica Scientia
			\|volume=28
			\|issue=3
			\|pages=489–494
			\|year=2008
			\|doi=10.1016/S0252-9602(08)60050-8
			}}</ref>
			As the integral over the mean curvature is typically much easier to calculate
			than the mean width, this is a very useful result.

			==Reference material==


			The mean width is usually mentioned in any good reference on convex geometry, for instance, ''Selected topics in convex geometry'' by Maria Moszyńska (Birkhäuser, Boston 2006). The relation between the mean width and the mean curvature is also derived in that reference.

			The application of the mean width as one of the measures featuring in [[Hadwiger's theorem]]
			is discussed in Beifang Chen in "A simplified elementary proof of Hadwiger's volume theorem." ''Geom. Dedicata'' 105 (2004), 107—120.

			==Cited references==

			{{Reflist}}

			[[Category:Integral geometry]]

en>Bender235: /* Occurrence */

2012-07-18T16:09:27Z

Occurrence

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