Geometric group theory - Revision history

en>Frietjes: /* Examples */

2014-08-15T17:51:57Z

Examples

← Older revision		Revision as of 19:51, 15 August 2014
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	~~In [[plane geometry]]~~, ~~two lines are '''hyperbolic orthogonal''' when they are [[reflection (mathematics)\|reflections]]~~ of ~~each other over~~ the ~~asymptote of a given [[hyperbola]].~~		Have you ever heard which someone said "My computer is getting slower, Help me?" Are you seeking methods on how do I accelerate my computer? Are we tired of wasting too much time because the loading task is truly slow? If you want to recognize how to improve and speed up computer performance, then this particular article may assist show we several concepts plus tricks "What is the cause?" plus How to prevent the computer getting slower?<br><br>You should recognize certain simple and cheap methods which could solve the problem of your computer and speed it up. The earlier you fix it, the less damage the computer gets. I may tell regarding some practical techniques which usually help we to speed up we computer.<br><br>The PC could have a fragmented difficult drive or the windows registry would have been corrupted. It could equally be as a result of the dust plus dirt which should be cleaned. Whatever the issue, you are able to usually find a answer. Here are certain strategies on how to make your PC run quicker.<br><br>First, usually clean the PC plus keep it free of dust and dirt. Dirt clogs up all of the fans plus may cause the PC to overheat. You furthermore have to clean up disk area inside purchase to create your computer run faster. Delete temporary plus unnecessary files plus unused programs. Empty the recycle bin plus remove programs you're not utilizing.<br><br>The final step is to make certain which we clean the registry of your computer. The "registry" is a large database that shops significant files, settings & choices, and info. Windows reads the files it requirements in order for it to run programs through this database. If the registry gets damaged, infected, or clogged up, then Windows may not be able to correctly access the files it requires for it to load up programs. As this arises, issues plus errors like the d3d9.dll error happen. To fix this plus avoid future setbacks, we have to download and run a registry cleaning tool. The very suggested software is the "Frontline [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014]".<br><br>The software finds these problems and takes care of them inside the shortest time possible. Your hard drive could additionally cause issues sometimes, specifically in the event you have 1 which is almost maxed. When your commence the machine, the are a lot of booting processes involved plus having an almost full storage does not enable a bit. You will usually have a slow PC because there are thus many items in the hard disk being processed simultaneously. The best method to resolve this issue is to upgrade. This usually allow the PC certain time to breath and working faster swiftly.<br><br>The 'registry' is merely the central database that shops all your settings plus choices. It's a actually important part of the XP system, meaning that Windows is frequently adding and updating the files inside it. The issues happen when Windows actually corrupts & loses a few of these files. This makes your computer run slow, because it tries difficult to obtain them again.<br><br>Most folks make the mistake of struggling to fix Windows registry by hand. I strongly suggest you don't do it. Unless we are a computer expert, I bet we will spend hours plus hours understanding the registry itself, let alone fixing it. And why if you waste a precious time in understanding and fixing anything you recognize nothing about? Why not let a smart and pro registry cleaner do it for you? These software programs would be able to work in a far better method! Registry cleaners are very affordable as well; we pay a one time fee plus employ it forever. Also, many expert registry products are pretty reliable plus simple to use. If you require more info on how to fix Windows registry, merely visit my url by clicking the link below!
	~~Two~~ particular ~~hyperbolas are frequently used in~~ the ~~plane:~~
	~~:(A) ''x y'' = 1 with ''y'' = 0 as asymptote.~~
	~~:When reflected in the x-axis, a line ''y = mx'' becomes ''y'' = −''mx''.~~
	~~:In this case~~ the ~~lines are hyperbolic orthogonal if their [[slope]]s are [[additive inverse]]s.~~

	~~:(B) ''x''~~<~~sup~~>2<~~/sup~~> ~~− ''y''<sup>2</sup> = 1 with ''y'' = ''x'' as asymptote.~~
	~~:For lines ''y = mx'' with −1 < ''m'' < 1, when ''x'' = 1/''m'', then ''y'' = 1~~.
	:The ~~point (1/''m''~~ , ~~1) on~~ the ~~line is reflected across ''y'' = ''x''~~ to ~~(1, 1/''m'')~~.
	~~:Therefore~~ the ~~reflected line has slope 1/m and the slopes of hyperbolic orthogonal lines are [[multiplicative inverse\|reciprocal]]s of each other~~.

	~~The relation of hyperbolic orthogonality actually applies to classes~~ of ~~parallel lines in~~ the ~~plane, where any particular line can represent~~ the ~~class. Thus~~, ~~for~~ a ~~given hyperbola and asymptote ''A'', a pair of lines (''a,b'')~~ are ~~hyperbolic orthogonal if there is a pair (''c,d'') such that~~ <~~math~~>~~a \rVert c , b \rVert d~~ <~~/math~~>, ~~and ''c'' is~~ the ~~reflection~~ of ~~''d'' across ''A''~~.

	~~The property~~ of the ~~radius being orthogonal to the [[tangent]] at the curve, is extended from~~ the ~~circle~~ to ~~the hyperbola by the hyperbolic orthogonal concept~~. ~~See Lewis & Wilson or Felsagen for the radius-tangent orthogonality.~~

	~~Since [[Hermann Minkowski]]'s foundation for [[spacetime]] study in 1908, the concept of points in a spacetime plane being hyperbolic-orthogonal to a timeline (tangent~~ to ~~a [[World line]]) has been used to define '''simultaneity''' of events relative~~ to the ~~timeline. In Minkowski~~'~~s development the hyperbola of type (B) above is in use (see the [[wikisource]] link below)~~. ~~Two vectors~~ <~~math~~>~~x, y, z, t \quad \text {and}\quad x_1, y_1, z_ 1, t_1~~<~~/math~~> ~~are ''normal'' (meaning hyperbolic orthogonal) when~~
	~~:<math>c^{2} t \ t_1 - x \ x_1 - y \ y_1 - z \ z_1 = 0~~.~~</math>~~
	~~When ''c'' = 1 and the y's and z's are zero, x≠0~~, ~~t<sub>1</sub>~~&~~ne;0~~, ~~then <math>\frac t x = \frac{x_1} {t_1}</math>~~.

	~~Given a hyperbola with asymptote ''A'', its reflection~~ in ~~''A'' produces the ''conjugate hyperbola''~~. ~~Any diameter of~~ the ~~original hyperbola is reflected~~ to ~~a [[conjugate diameters\|conjugate diameter]]. The directions indicated by conjugate diameters are taken~~ for ~~space and time axes in relativity~~.
	As ~~[[E~~. T. ~~Whittaker]] wrote in 1910, "[the] hyperbola is unaltered when any pair of conjugate diameters are taken as new axes~~, and a ~~new unit of length~~ is ~~taken proportional to~~ the ~~length of either of these diameters.~~" ~~On this~~ [~~[principle of relativity]], he then wrote the Lorentz transformation in the modern form using [[rapidity]~~].

	~~[[Edwin Bidwell Wilson]]~~ and ~~[[Gilbert N. Lewis]] developed~~ the ~~concept within [[synthetic geometry]] in 1912~~. ~~They note (p.415) "~~in ~~our plane no pair of perpendicular [hyperbolic-orthogonal] lines~~ is ~~better suited to serve as coordinate axes than any other pair"~~.

	In the ~~terminology of [[projective geometry]]~~, the ~~operation~~ of ~~taking the hyperbolic orthogonal line is~~ a ~~[[involution (mathematics)#Projective geometry\|hyperbolic involution]]~~. ~~Suppose the slope of a vertical line is denoted ∞ so that all lines~~ have a ~~slope~~ in the ~~[[real projective line]]~~. ~~Then whichever hyperbola (A) or (B)~~ is ~~used,~~ the ~~operation is an [[real projective line#Hyperbolic involutions\|example of a hyperbolic involution]]~~.

	~~==References==~~
	* [[G. D. Birkhoff]] (1923) ''~~Relativity and Modern Physics'~~', ~~pages 62,3, [[Harvard University Press]]~~.
	* Francesco Catoni, Dino Boccaletti, & ~~Roberto Cannata (2008) ''Mathematics~~ of ~~Minkowski Space'', [[Birkhäuser Verlag]], Basel~~. ~~See page 38~~, ~~Pseudo-orthogonality~~.
	*{{Citation\|author=Minkowski, Hermann\|year=1909\|title=[[s:de:Raum und Zeit (Minkowski)\|Raum und Zeit]]\|journal=Physikalische Zeitschrift\|volume=10\|pages=75–88}}
	~~:*Various English translations on Wikisource: [[s:Space and Time\|Space and Time]]~~
	* {{Citation \| author=Whittaker, E.T. \| year=1910 \| title= 1. Edition: [http://www.archive.org/details/historyoftheorie00whitrich A History of the ~~theories~~ of ~~aether and electricity] \| place =Dublin \|publisher=Longman, Green and Co~~.~~}}(see page 441)~~
	*Bjørn Felsager (2004), [http://www.~~dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf Through the Looking Glass - A glimpse of Euclid’s twin geometry~~, the ~~Minkowski geometry]~~, ~~ICME-10 Copenhagen; pages 6 & 7~~.
	*Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time ~~Manifold of Relativity. The Non-Euclidean Geometry of Mechanics~~ and ~~Electromagnetics" Proceedings of the [[American Academy of Arts~~ and ~~Sciences]] 48:387-507~~.


	~~[[Category:Minkowski spacetime]]~~
	~~[[Category:Angle]]~~

en>Brirush: /* Modern themes and developments */

2013-11-28T17:48:01Z

Modern themes and developments

← Older revision		Revision as of 19:48, 28 November 2013
Line 1:		Line 1:
	~~Your individual Tribe is~~ the ~~quite~~ a ~~number of strong of all and discover have~~ the ~~planet~~ (~~virtual~~) ~~at your toes~~, ~~and therefore all that with solely~~ a ~~brief on-~~line ~~training video that may direct you step by step when how to get our personal cheat code for Intervene of Tribes~~.<br><br>~~Go on the internet for help~~. ~~Practically any game has its~~ ~~legion of devoted devotees~~, ~~lots of which expend countless hours crafting wide-range maps and guides~~. ~~Additionally there are newsgroups where you are speak one on one with other [http~~://~~www~~.~~alexa.com~~/~~search?q=players&r=topsites_index&p=bigtop players~~]. ~~Benefit from this lotto jackpot~~ and it is ~~easy to eventually get past~~ that ~~level of cla you have been into on forever.~~<br><br>~~Throne Rush has an exact same for just about everything in Clash~~. ~~If you have any inquiries relating~~ to the ~~place and how~~ to ~~use hack clash~~ of ~~clans,~~ [~~http://circuspartypanama~~.~~com Click On this page~~],, ~~you can get in touch with us at our own page~~. ~~Instead within your Town Hall~~, ~~it features Castle~~. ~~Instead most typically associated~~ with ~~Clans~~, ~~it has Brotherhoods~~. ~~Instead~~ of ~~Trophies, it has Morale. Perhaps~~ the ~~one part it takes~~ to the ~~next stage~~ is ~~its Immortal People. clash~~ of ~~clans has~~ a ~~Barbarian King and their Archer Queen which normally special units that could be reused in battle " they just require numerous hours~~ of ~~time~~ to ~~alleviate back to full wellness~~. ~~Throne Rush has similar heroes that could be hired~~, ~~but they are more extreme~~ and ~~more everywhere~~. ~~They play almost~~ the ~~same way, however think players will indulge~~ in ~~using four or twelve Immortal Heroes instead related just two, as~~ in ~~size~~ as ~~they dont ruin~~ the ~~balance~~ of the ~~game too severely~~.~~<br><br>In~~ the ~~first instance clash~~ of ~~clans crack tool no survey generates believe~~ in ~~among ones people~~. ~~Society~~ is ~~almost certainly definitely powered by peer pressure~~, ~~one of~~ the ~~type~~ of ~~most powerful forces in the planet~~. ~~Whenever long as peer stress~~ and ~~anxiety utilizes its power for good~~, ~~clash~~ of ~~clans hack tool no feedback survey will have its put in community~~.~~<br><br>Nfl season is here and going strong~~, and ~~similar many fans we incredibly long for Sunday afternoon when the games begin. If you have held~~ and ~~liked Soul Caliber~~, ~~you will love this skill game~~. ~~The succeeding best is the Processor Cell which will at random , fill in some sqrs~~. ~~Defeating players as if that by any involves necessary can be that this reason that pushes them to use Words that has Friends Cheat~~. ~~One particular app requires you to answer 40 questions using varying degrees~~ of ~~difficulty~~.~~<br><br>You can~~ see ~~for yourself that personal Money Compromise of Clans i fairly effective~~, ~~fully invisible by~~ the ~~executive~~ of the ~~game~~, ~~substantially absolutely no price!<br><br>Experiment with to restrain your highly important gaming to only a particular kind~~ of ~~machine~~. ~~Buying all~~ the ~~real consoles plus a gaming~~-worthy personal computer can cost up to thousands, no more than in hardware. Yet, most big titles would probably be available on near all of them. Choose one platform to assist you stick with for bargains.		In [[plane geometry]], two lines are '''hyperbolic orthogonal''' when they are [[reflection (mathematics)\|reflections]] of each other over the asymptote of a given [[hyperbola]].
			Two particular hyperbolas are frequently used in the plane:
			:(A) ''x y'' = 1 with ''y'' = 0 as asymptote.
			:When reflected in the x-axis, a line ''y = mx'' becomes ''y'' = −''mx''.
			:In this case the lines are hyperbolic orthogonal if their [[slope]]s are [[additive inverse]]s.

			:(B) ''x''<sup>2</sup> − ''y''<sup>2</sup> = 1 with ''y'' = ''x'' as asymptote.
			:For lines ''y = mx'' with −1 < ''m'' < 1, when ''x'' = 1/''m'', then ''y'' = 1.
			:The point (1/''m'' , 1) on the line is reflected across ''y'' = ''x'' to (1, 1/''m'').
			:Therefore the reflected line has slope 1/m and the slopes of hyperbolic orthogonal lines are [[multiplicative inverse\|reciprocal]]s of each other.

			The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote ''A'', a pair of lines (''a,b'') are hyperbolic orthogonal if there is a pair (''c,d'') such that <math>a \rVert c , b \rVert d </math>, and ''c'' is the reflection of ''d'' across ''A''.

			The property of the radius being orthogonal to the [[tangent]] at the curve, is extended from the circle to the hyperbola by the hyperbolic orthogonal concept. See Lewis & Wilson or Felsagen for the radius-tangent orthogonality.

			Since [[Hermann Minkowski]]'s foundation for [[spacetime]] study in 1908, the concept of points in a spacetime plane being hyperbolic-orthogonal to a timeline (tangent to a [[World line]]) has been used to define '''simultaneity''' of events relative to the timeline. In Minkowski's development the hyperbola of type (B) above is in use (see the [[wikisource]] link below). Two vectors <math>x, y, z, t \quad \text {and}\quad x_1, y_1, z_ 1, t_1</math> are ''normal'' (meaning hyperbolic orthogonal) when
			:<math>c^{2} t \ t_1 - x \ x_1 - y \ y_1 - z \ z_1 = 0.</math>
			When ''c'' = 1 and the y's and z's are zero, x≠0, t<sub>1</sub>≠0, then <math>\frac t x = \frac{x_1} {t_1}</math>.

			Given a hyperbola with asymptote ''A'', its reflection in ''A'' produces the ''conjugate hyperbola''. Any diameter of the original hyperbola is reflected to a [[conjugate diameters\|conjugate diameter]]. The directions indicated by conjugate diameters are taken for space and time axes in relativity.
			As [[E. T. Whittaker]] wrote in 1910, "[the] hyperbola is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters." On this [[principle of relativity]], he then wrote the Lorentz transformation in the modern form using [[rapidity]].

			[[Edwin Bidwell Wilson]] and [[Gilbert N. Lewis]] developed the concept within [[synthetic geometry]] in 1912. They note (p.415) "in our plane no pair of perpendicular [hyperbolic-orthogonal] lines is better suited to serve as coordinate axes than any other pair".

			In the terminology of [[projective geometry]], the operation of taking the hyperbolic orthogonal line is a [[involution (mathematics)#Projective geometry\|hyperbolic involution]]. Suppose the slope of a vertical line is denoted ∞ so that all lines have a slope in the [[real projective line]]. Then whichever hyperbola (A) or (B) is used, the operation is an [[real projective line#Hyperbolic involutions\|example of a hyperbolic involution]].

			==References==
			* [[G. D. Birkhoff]] (1923) ''Relativity and Modern Physics'', pages 62,3, [[Harvard University Press]].
			* Francesco Catoni, Dino Boccaletti, & Roberto Cannata (2008) ''Mathematics of Minkowski Space'', [[Birkhäuser Verlag]], Basel. See page 38, Pseudo-orthogonality.
			*{{Citation\|author=Minkowski, Hermann\|year=1909\|title=[[s:de:Raum und Zeit (Minkowski)\|Raum und Zeit]]\|journal=Physikalische Zeitschrift\|volume=10\|pages=75–88}}
			:*Various English translations on Wikisource: [[s:Space and Time\|Space and Time]]
			* {{Citation \| author=Whittaker, E.T. \| year=1910 \| title= 1. Edition: [http://www.archive.org/details/historyoftheorie00whitrich A History of the theories of aether and electricity] \| place =Dublin \|publisher=Longman, Green and Co.}}(see page 441)
			*Bjørn Felsager (2004), [http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf Through the Looking Glass - A glimpse of Euclid’s twin geometry, the Minkowski geometry], ICME-10 Copenhagen; pages 6 & 7.
			*Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the [[American Academy of Arts and Sciences]] 48:387-507.


			[[Category:Minkowski spacetime]]
			[[Category:Angle]]

72.196.213.226 at 02:29, 27 August 2012

2012-08-27T02:29:30Z

New page

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