Sequent - Revision history

en>Alan U. Kennington: Added another convenient reference for the disjunctive semantics of the right side of a sequent.

2014-07-07T13:56:48Z

Added another convenient reference for the disjunctive semantics of the right side of a sequent.

← Older revision		Revision as of 15:56, 7 July 2014
Line 1:		Line 1:
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Or, ~~whether you capital~~ to ~~build up the 1 business day bulk at 260 gems~~, ~~the band would require to acceleration added considerably~~ and ~~also 1 holiday would turn into added expensive~~.<br><br>~~Have~~ a ~~look at your child's xbox play enjoying~~. ~~Video~~ and ~~also are now rated typically like films and this also can~~ help. ~~That enables~~ you to ~~keep an eye on the information~~ your ~~kids is normally exposed to~~. ~~Dependant upon your child's age~~, ~~continue to keep my man clear of video online video media that happen to prove meant for people~~ the ~~people that are more fully grew than him~~.<br><br>~~Should~~ you ~~perform online multi~~-~~player game titles, don't forget about~~ the ~~strength~~ of ~~tone or shade~~ of ~~voice chat! A mic or headphone is a very effortless expenditure~~, and ~~having most~~ of the ~~capability~~ to ~~speak to your fellow athletes has~~ a ~~lot~~ of ~~features. You are skilled~~ to ~~create~~ more ~~influential connections with the gaming community~~ and ~~stay one specific far more successful company person when you were able connect out boisterous~~.<br><br>If you ~~want~~ to conclude, clash of clans hack tool no surveys must not be able to get in approach of the bigger question: what makes we perfect? Putting this aside the of great importance. It replenishes the self, provides financial security always chips in Here's more in regards to [http://~~prometeu~~.~~net Clash Of clans Hack ipad~~] ~~stop by~~ our ~~own webpage.~~ .		Today there are a lot of women who go for professional treatment to get this corrected. The primary purpose of the melanocyte cell is to make melanin. Stir it well to mix the beeswax thoroughly with the oil, then allow it to cool to about 90 degrees. 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en>Alejandrocaro35: /* History */

2014-02-24T23:16:33Z

History

← Older revision		Revision as of 01:16, 25 February 2014
Line 1:		Line 1:
	~~{{Other uses2\|Disc}}~~		Hospitals and clinics the Clash of Clans hack tool; there could be also hack tools with [http://Photobucket.com/images/respect respect] to other games. Men or women can check out those hacks and obtain the people which they need. It is sure the player will have lost with regards to fun once they feature the hack tool saved.<br><br>In the case when as a parent you're the one concerned with movie online game content, control what downloadable mods are put globe sport. These downloadable mods are usually written by players, perhaps not the gaming businesses, therefore there is no ranking system. You actually thought was a reasonably un-risky game can turn a lot worse offering any of these mods.<br><br>Could be little ones who have fun with video games, then you know the way challenging it really is to pull them out of the t. v.. Their eye can prove stuck towards the maintain a record of for hours as chances are they'll play their preferred quests. If you want aid regulating your your child's clash of clans Hack time, your pursuing article has some recommendations for you.<br><br>Truly possible, but the the greater part of absence one day would abatement by sixty one. 5% hailing from 260 treasures to thousand gems. Or, whether you capital to build up the 1 business day bulk at 260 gems, the band would require to acceleration added considerably and also 1 holiday would turn into added expensive.<br><br>Have a look at your child's xbox play enjoying. Video and also are now rated typically like films and this also can help. That enables you to keep an eye on the information your kids is normally exposed to. Dependant upon your child's age, continue to keep my man clear of video online video media that happen to prove meant for people the people that are more fully grew than him.<br><br>Should you perform online multi-player game titles, don't forget about the strength of tone or shade of voice chat! A mic or headphone is a very effortless expenditure, and having most of the capability to speak to your fellow athletes has a lot of features. You are skilled to create more influential connections with the gaming community and stay one specific far more successful company person when you were able connect out boisterous.<br><br>If you want to conclude, clash of clans hack tool no surveys must not be able to get in approach of the bigger question: what makes we perfect? Putting this aside the of great importance. It replenishes the self, provides financial security always chips in Here's more in regards to [http://prometeu.net Clash Of clans Hack ipad] stop by our own webpage. .
	[~~[Image~~:~~Unit disk open~~.~~svg\|thumb\|right\|An open Euclidean unit disk~~]]
	~~In [[mathematics]],~~ the ~~'''open unit disk''' (or '''disc''') around ''P'' (where ''P''~~ is ~~a given point in~~ the ~~[[plane (mathematics)\|plane]]), is~~ the ~~set of points whose distance from ''P'' is less than 1:~~

	:<~~math~~>~~D_1(P) = \{ Q : \vert P-Q\vert<1\}.\,~~<~~/math~~>

	~~The '''closed unit disk~~'~~'' around ''P'' is~~ the ~~set of points whose distance from ''P'' is less than or equal to~~ one:

	~~:<math>\bar D_1(P)=\{Q:\|P-Q\| \leq 1\}.\~~,~~</math>~~

	~~Unit disks~~ are ~~special cases of [[disk (mathematics)\|disks]] and [[unit ball]]s~~.

	~~Without further specifications~~, the ~~term ''unit disk''~~ is ~~used for the open unit disk about the [[origin (mathematics)\|origin]],~~ <~~math~~>~~D_1(0)~~<~~/math~~>, ~~with respect to~~ the ~~[[Euclidean distance\|standard Euclidean metric]]. It~~ is ~~the interior of a [[circle]]~~ of ~~radius 1, centered at~~ the ~~origin~~. ~~This set~~ can ~~be identified with~~ the ~~set~~ of ~~all [[complex number]]s of [[absolute value]] less than one. When viewed~~ as ~~a subset of the complex plane (~~''~~'C''')~~, ~~the unit disk is often denoted~~ <~~math~~>~~\mathbb{D}~~<~~/math~~>.

	~~== The open unit disk~~, the ~~plane, and~~ the ~~upper half-plane ==~~
	~~The function~~

	~~:<math>f(z)=\frac{z}{1-\|z\|^2}</math>~~

	~~is an example~~ of ~~a real [[analytic function\|analytic]] and [[bijective]] function~~ from ~~the open unit disk~~ to ~~the plane; its inverse function is also analytic~~. ~~Considered as a real 2-dimensional [[analytic manifold]]~~, ~~the open unit disk is therefore isomorphic~~ to the ~~whole plane. In particular~~, the ~~open unit disk is [[homeomorphic]]~~ to ~~the whole plane.~~

	~~There is however no [[conformal map\|conformal]] bijective map between the open unit disk~~ and ~~the plane~~. ~~Considered as~~ a ~~[[Riemann surface]], the open unit disk is therefore different from the [[complex plane]]~~.

	~~There~~ are ~~conformal bijective maps between the open unit disk~~ and ~~the open [[upper half-plane]]~~. ~~So considered as a Riemann surface,~~ the ~~open unit disk~~ is ~~isomorphic ("biholomorphic", or "conformally equivalent")~~ to ~~the upper half-plane, and the two are often used interchangeably~~.

	~~Much more generally~~, ~~the [[Riemann mapping theorem]] states that every [[simply connected]] [[open set\|open subset]]~~ of ~~the complex plane~~ that ~~is different from the complex plane itself admits a conformal and bijective map~~ to the ~~open unit disk~~.

	~~One bijective conformal map from the open unit disk to the open upper half-plane is the [[Möbius transformation]]~~

	:<~~math~~>~~g(z)=i\frac{1+z}{1-z}~~<~~/math~~> ~~  which is the inverse of the [[Cayley transform#Conformal map\|Cayley transform]].~~

	~~Geometrically~~, ~~one can imagine the real axis being bent and shrunk so that the upper half-plane becomes the disk~~'s interior and the real axis forms the disk's circumference, save for one point at the top, the "point at infinity". A bijective conformal map from the open unit disk to the open upper half-plane can also be constructed as the ~~composition~~ of ~~two [[stereographic projection]]s: first the unit disk is stereographically projected upward onto the unit upper half-sphere, taking the "south-pole"~~ of ~~the unit sphere as the projection center, and then this half-sphere~~ is ~~projected sideways onto~~ a ~~vertical half-plane touching the sphere~~, ~~taking the point on~~ the ~~half-sphere opposite~~ to ~~the touching point as projection center.~~

	~~The unit disk and the upper half-plane are not interchangeable as domains for [[Hardy spaces]]. Contributing~~ to ~~this difference is the fact that the unit circle~~ has ~~finite (one-dimensional) [[Lebesgue measure]] while the real line does not.~~

	~~==Hyperbolic space==~~
	~~The open unit disk is commonly used as a model for the [[Hyperbolic space\|hyperbolic plane]], by introducing~~ a new metric on it, the [[Poincaré metric]]. Using the above mentioned conformal map between the open unit disk and the upper half-plane, this model can be turned into the [[Poincaré half-plane model]] of ~~the hyperbolic plane~~. ~~Both the Poincaré disk and the Poincaré half-plane~~ are ~~''conformal'' models of hyperbolic space, i.e. angles measured in the model coincide with angles in hyperbolic space, and consequently the shapes (but not the sizes) of small figures are preserved.~~

	~~Another model of hyperbolic space is also built on the open unit disk: the [[Klein model]]. It is not conformal, but has the property that straight lines in the model correspond~~ to ~~straight lines in hyperbolic space.~~

	~~== Unit disks~~ with ~~respect to other metrics ==~~
	~~[[Image:Unit disc.svg\|thumb\|right\|From top to bottom: open unit disk in~~ the ~~[[Euclidean metric]], [[taxicab metric]],~~ and ~~[[Chebyshev metric]]~~.]]
	~~One also considers unit disks with respect~~ to ~~other [[Metric (mathematics)\|metric]]s. For instance~~, with the [[taxicab geometry\|taxicab metric]] and the [[Chebyshev distance\|Chebyshev metric]] disks look like squares (even though the underlying [[topological space\|topologies]] are the same as the Euclidean one).

	~~The area~~ of ~~the Euclidean unit disk is [[π]] and its [[perimeter]] is 2π. In contrast, the perimeter (relative~~ to ~~the taxicab metric)~~ of the ~~unit disk in~~ the ~~taxicab geometry is 8~~. ~~In 1932~~, ~~[[Stanisław Gołąb]] proved that~~ in ~~metrics arising from a [[norm (mathematics)\|norm]], the perimeter of the unit disk can take any value~~ in ~~between 6 and 8, and that these extremal values are obtained if and only if the unit disk is a regular [[hexagon]] or a [[parallelogram]], respectively.~~

	~~==See also==~~
	* [[Unit disk graph]]
	* [[Bieberbach conjecture]]

	~~==References==~~
	* S. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 (1932), 179.

	~~==External links==~~
	* {{mathworld \| urlname = UnitDisk \| title = Unit disk}}
	* [http://~~www~~.~~math.poly.edu/~alvarez/pdfs/disc.pdf On the Perimeter and Area of the Unit Disc~~], by J.C. ~~Álvarez Pavia and A.C. Thompson~~

	~~[[Category:Circles]]~~
	~~[[Category:One]]~~

en>LoMaPh: /* Variations */

2014-01-05T03:48:02Z

Variations

← Older revision		Revision as of 05:48, 5 January 2014
Line 1:		Line 1:
	I'~~m Lurlene and I live~~ in ~~Papenburg~~. <br>I'~~m interested in Educational Studies~~, ~~Board sports~~ and ~~Japanese art~~. ~~I like travelling and reading fantasy~~.<br><br>~~Feel free~~ to ~~visit my web site~~; ~~Oltramare~~ - [~~http~~:/~~/www~~.~~lancy~~.~~ch/agenda~~-~~21/offres~~-~~pour~~-~~les~~-~~lanceennes/promenades~~-et-~~visites/decouverte~~-~~des~~-~~richesses~~-~~naturelles~~-~~du pop over~~ to these ~~guys~~],		{{Other uses2\|Disc}}
			[[Image:Unit disk open.svg\|thumb\|right\|An open Euclidean unit disk]]
			In [[mathematics]], the '''open unit disk''' (or '''disc''') around ''P'' (where ''P'' is a given point in the [[plane (mathematics)\|plane]]), is the set of points whose distance from ''P'' is less than 1:

			:<math>D_1(P) = \{ Q : \vert P-Q\vert<1\}.\,</math>

			The '''closed unit disk''' around ''P'' is the set of points whose distance from ''P'' is less than or equal to one:

			:<math>\bar D_1(P)=\{Q:\|P-Q\| \leq 1\}.\,</math>

			Unit disks are special cases of [[disk (mathematics)\|disks]] and [[unit ball]]s.

			Without further specifications, the term ''unit disk'' is used for the open unit disk about the [[origin (mathematics)\|origin]], <math>D_1(0)</math>, with respect to the [[Euclidean distance\|standard Euclidean metric]]. It is the interior of a [[circle]] of radius 1, centered at the origin. This set can be identified with the set of all [[complex number]]s of [[absolute value]] less than one. When viewed as a subset of the complex plane ('''C'''), the unit disk is often denoted <math>\mathbb{D}</math>.

			== The open unit disk, the plane, and the upper half-plane ==
			The function

			:<math>f(z)=\frac{z}{1-\|z\|^2}</math>

			is an example of a real [[analytic function\|analytic]] and [[bijective]] function from the open unit disk to the plane; its inverse function is also analytic. Considered as a real 2-dimensional [[analytic manifold]], the open unit disk is therefore isomorphic to the whole plane. In particular, the open unit disk is [[homeomorphic]] to the whole plane.

			There is however no [[conformal map\|conformal]] bijective map between the open unit disk and the plane. Considered as a [[Riemann surface]], the open unit disk is therefore different from the [[complex plane]].

			There are conformal bijective maps between the open unit disk and the open [[upper half-plane]]. So considered as a Riemann surface, the open unit disk is isomorphic ("biholomorphic", or "conformally equivalent") to the upper half-plane, and the two are often used interchangeably.

			Much more generally, the [[Riemann mapping theorem]] states that every [[simply connected]] [[open set\|open subset]] of the complex plane that is different from the complex plane itself admits a conformal and bijective map to the open unit disk.

			One bijective conformal map from the open unit disk to the open upper half-plane is the [[Möbius transformation]]

			:<math>g(z)=i\frac{1+z}{1-z}</math>   which is the inverse of the [[Cayley transform#Conformal map\|Cayley transform]].

			Geometrically, one can imagine the real axis being bent and shrunk so that the upper half-plane becomes the disk's interior and the real axis forms the disk's circumference, save for one point at the top, the "point at infinity". A bijective conformal map from the open unit disk to the open upper half-plane can also be constructed as the composition of two [[stereographic projection]]s: first the unit disk is stereographically projected upward onto the unit upper half-sphere, taking the "south-pole" of the unit sphere as the projection center, and then this half-sphere is projected sideways onto a vertical half-plane touching the sphere, taking the point on the half-sphere opposite to the touching point as projection center.

			The unit disk and the upper half-plane are not interchangeable as domains for [[Hardy spaces]]. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) [[Lebesgue measure]] while the real line does not.

			==Hyperbolic space==
			The open unit disk is commonly used as a model for the [[Hyperbolic space\|hyperbolic plane]], by introducing a new metric on it, the [[Poincaré metric]]. Using the above mentioned conformal map between the open unit disk and the upper half-plane, this model can be turned into the [[Poincaré half-plane model]] of the hyperbolic plane. Both the Poincaré disk and the Poincaré half-plane are ''conformal'' models of hyperbolic space, i.e. angles measured in the model coincide with angles in hyperbolic space, and consequently the shapes (but not the sizes) of small figures are preserved.

			Another model of hyperbolic space is also built on the open unit disk: the [[Klein model]]. It is not conformal, but has the property that straight lines in the model correspond to straight lines in hyperbolic space.

			== Unit disks with respect to other metrics ==
			[[Image:Unit disc.svg\|thumb\|right\|From top to bottom: open unit disk in the [[Euclidean metric]], [[taxicab metric]], and [[Chebyshev metric]].]]
			One also considers unit disks with respect to other [[Metric (mathematics)\|metric]]s. For instance, with the [[taxicab geometry\|taxicab metric]] and the [[Chebyshev distance\|Chebyshev metric]] disks look like squares (even though the underlying [[topological space\|topologies]] are the same as the Euclidean one).

			The area of the Euclidean unit disk is [[π]] and its [[perimeter]] is 2π. In contrast, the perimeter (relative to the taxicab metric) of the unit disk in the taxicab geometry is 8. In 1932, [[Stanisław Gołąb]] proved that in metrics arising from a [[norm (mathematics)\|norm]], the perimeter of the unit disk can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disk is a regular [[hexagon]] or a [[parallelogram]], respectively.

			==See also==
			* [[Unit disk graph]]
			* [[Bieberbach conjecture]]

			==References==
			* S. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 (1932), 179.

			==External links==
			* {{mathworld \| urlname = UnitDisk \| title = Unit disk}}
			* [http://www.math.poly.edu/~alvarez/pdfs/disc.pdf On the Perimeter and Area of the Unit Disc], by J.C. Álvarez Pavia and A.C. Thompson

			[[Category:Circles]]
			[[Category:One]]

en>777sms at 02:26, 14 April 2012

2012-04-14T02:26:16Z

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