McDonald–Kreitman test: Difference between revisions

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|align=right
|width=33%
|quote=The question recently arose in conversation whether a dissertation of 2 lines could deserve and get a Fellowship. ... Cayley's projective definition of length is a clear case if we may interpret "2 lines" with reasonable latitude. ... With Cayley the importance of the idea is obvious at first sight.
|source= {{harvtxt|Littlewood|1986|pp=39–40}}
}}
In mathematics, a '''Cayley–Klein metric''' is a [[metric (mathematics)|metric]] on the complement of a fixed [[quadric]] in [[projective space]] defined using a [[cross-ratio]]. The construction originated with [[Arthur Cayley]]'s essay "On the theory of distance"<ref>A. Cayley (1859) "Sixth Memoir on Conics", p 82, §§209 to 229</ref> where he calls the quadric the '''absolute'''. The construction was developed in further detail by [[Felix Klein]] in papers in 1871 and 1873, and in his book ''Vorlesungen über Nicht-Euklidischen Geometrie''(1928). The Cayley-Klein metrics are a unifying idea in geometry since the method is used to provide metrics in [[hyperbolic geometry]], [[elliptic geometry]], and [[Euclidean geometry]]. The field of [[non-Euclidean geometry]] rests largely on the footing provided by Cayley-Klein metrics.
 
==Definition==
 
Suppose that ''Q'' is a fixed [[quadric]] in projective space.
If ''p'' and ''q'' are 2 points then the line through ''p'' and ''q'' intersects the quadric ''Q'' in 2 further points ''a'' and ''b''. The Cayley–Klein distance ''d''(''p'',''q'') from ''p'' to ''q'' is proportional to the logarithm of the [[cross-ratio]]:
:<math>d(p,q)=C \log \frac{|qa||bp|}{|pa||bq|}</math>
for some fixed constant ''C''.
==Foundations==
The [[Karl von Staudt#Algebra of throws|algebra of throws]] is an approach to geometry that is independent of [[metric (mathematics)|metric]]. [[Karl von Staudt]]'s idea was to use the relation of [[projective harmonic conjugate]]s as fundamental to the measure on a line. Klein made reference to Staudt's development,<ref>Klein, ''Vorlesungen über Nicht-Euklidische Geometrie'', page 163</ref> thus basing the new metric on logarithm, and the [[cross-ratio]] as a number generated by the geometric arrangement of four points. This procedure is necessary to avoid a circular definition of distance if cross-ratio is merely a double ratio of previously defined distances.<ref>[[Bertrand Russell]] (1898) ''An Essay on the Foundations of Geometry'', page 32, re-issued 1956 by [[Dover Books]]</ref>
 
Cayley-Klein plane geometry depends upon the selection of a [[conic section|conic]] that becomes the ''absolute'' of the space. For example, the [[unit circle]] is the absolute of the [[Poincare disk model]] and the [[Beltrami-Klein model]] in [[hyperbolic geometry]]. Similarly, the [[real line]] is the absolute of the [[Poincare half-plane model]].
 
For [[Euclidean space]], Klein described the absolute as the "sphere-circle" consisting of the sphere of radius zero in the [[hyperplane at infinity]]:<ref>Felix Klein, [[Robert Hermann (mathematician)|Robert Hermann]] translator (1979) ''Development of Mathematics in the 19th Century'', p 132, Math Sci Press</ref>
:<math>\lbrace (\xi , \eta, \zeta, \tau ) :\quad \tau = 0,\quad \xi^2 + \eta^2 + \zeta^2 = 0 \rbrace \subset P^3(R) .</math>
 
Klein also related the metric construction to physical science:<ref>Klein/Hermann p 138</ref>
:The case <math>x^2 + y^2 + z^2 - t^2 = 0</math> in the four-dimensional world or <math>dx^2 + dy^2 + dz^2 - dt^2 = 0</math> (to remain in three-dimensions and use [[homogeneous coordinates]]) has recently won special significance through the relativity theory of physics.
Evidently Klein is referring here to <math>(\frac {dx}{dt})^2 + (\frac {dy}{dt})^2 + (\frac {dz}{dt})^2 = 1</math> where physical velocities are bounded by the [[speed of light]] c, so that for any physical velocity ''v'', the ratio ''v''/c is confined to the interior of a unit sphere, and the surface of the sphere forms the Cayley absolute for the geometry
 
==See also==
* [[Hilbert metric]]
 
==References==
{{Reflist}}
*{{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title= A sixth memoir upon quantics | jstor=108690 | year=1859 | journal=[[Philosophical Transactions of the Royal Society of London]] | issn=0080-4614 | volume=149 | pages=61–90}}
*{{Citation | last1=Klein | first1=Felin | title=Ueber die sogenannte Nicht-Euklidische Geometrie | publisher=Springer Berlin / Heidelberg | language=German | doi=10.1007/BF02100583 | jfm=03.0231.02 | year=1871 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=4 | pages=573–625}}
*{{Citation | last1=Klein | first1=Felix | title=Ueber die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz) | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01443189 | jfm= 05.0271.01 | year=1873 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=6 | pages=112–145}}
* Klein, Felix (1928) ''Vorlesungen über Nicht-Euklidische Geometrie'', [[Julius Springer]], Berlin.
*{{Citation | last1=Littlewood | first1=J. E. | title=Littlewood's miscellany | origyear=1953 | url=http://books.google.com/books?id=MjVgeT7Laf8C | publisher=[[Cambridge University Press]] | isbn=978-0-521-33058-9 | mr=872858 | year=1986}}
* [[Duncan Sommerville]] (1910/11) "Cayley-Klein metrics in n-dimensional space", [[Proceedings of the Edinburgh Mathematical Society]] 28:25–41.
*{{Citation | last1=Struve | first1=Horst | last2=Struve | first2=Rolf | title=Projective spaces with Cayley-Klein metrics | doi=10.1007/s00022-004-1679-5 | mr=2134074 | year=2004 | journal=Journal of Geometry | issn=0047-2468 | volume=81 | issue=1 | pages=155–167}}
*{{Citation | last1=Struve | first1=Horst | last2=Struve | first2=Rolf | title=Non-euclidean geometries: the Cayley-Klein approach |doi=10.1007/s00022-010-0053-z | mr=2739193 | year=2010 | journal=Journal of Geometry | issn=0047-2468 | volume=89 | issue=1 | pages=151–170}}
 
 
{{DEFAULTSORT:Cayley-Klein metric}}
[[Category:Projective geometry]]
[[Category:Metric geometry]]

Latest revision as of 14:46, 20 October 2014

Air compressor acts as a central power supply that is driven by higher air pressure and are handy to use in unique applications. While the Bulldog definitely doesn't have the strength of these huge monsters, you can use it all evening at a time with no operator fatigue-or tool failure. Incidentally, this Bosch can also be my 2-year old son's favorite tool ever since that threshhold demolition-he calls it the "wowee mammer" and continuously pleads to make use of it. Attach a cutting bit to the rotary tool for clean cuts in drywall or ceramic tile.

Regardless of the differences I felt on the strong rod cutoff, I assume the Milwaukee M12 Cordless Rotary Tool and the Dremel 8200 Cordless Rotary Tool are each outstanding possibilities. If you are currently working with other Milaukee M12 merchandise, then the choice to go with the M12 rotary is a no-brainer. One more nice feature to note is that this tool does involve Milwaukee's safety cut off to guard the tool from overload.

To find a regional retailer, merely click the "businesses" link of the web app toolbar and it will show local firms in your area, with map, that sell the Dremel 400 XPR Rotary Tool. Once you have saved funds on a Dremel 400 XPR Rotary Tool with the free of charge web app, you will have to promise to tell others about it and how you got it for absolutely free! While functioning with a Dremel rotary tool generally stay alert. Don't apply as well a lot stress-let the tool and the bit do the cutting.

I am not certain it is a massive enough alter to upgrade from a various recent model, but if you only have an older tool or have under no circumstances tried a Dremel, you must appear into this series! This Dremel 7300-N/eight MiniMite 4.8-volt cordless two-speed rotary tool is an excellent decision, according to quite a few raters on , who give the tool an typical rating of 4 stars. It's a terrific production tool for any Porter Cable Belt Disc Sander Critiques amateur (or professional) videographer.

The mat protects the rotary cutter from harm, prevents harm of the surface exactly where you are cutting and helps hold the fabric from slipping about. For the reason that of this razor sharp blade, the rotary cutter is a tool that demands to be handled with care. The rotary cutters themselves come in different sizes but I personally uncover that the ideal cutter for cutting quilt fabric is a rotary cutter that has a 45 mm blade. The tool was light in the case extremely effortless to carry - a plus!

Dremel not too long ago released a funky seeking tool referred to as the Trio (and had been good sufficient to send us a single to critique). When you boil the Dremel Trio down to it is mechanics, it is a rotary tool, just like the million - and - a half other rotary tools that Dremel has released over the years. Dremel may have addressed this, as we noticed a disclaimer on the Dremel web site saying that the Trio is not compatible with typical rotary tool accessories.

You nonetheless want to get the Dremel MM40-01 two. If you beloved this write-up and you would like to receive a lot more data relating to www.bestoscillatingtoolreviews.com kindly go to our own web-page. five Amp Higher Functionality Oscillating Tool Kit?TheDremel MM40-01 2.5 Amp Higher Functionality Oscillating Tool Kit is generally a incredibly excellent item. Lol i like it, eight speeds and it comes with alot of bits and tools and even a flex shaft and tool hangar, if your searching to get, this is the one particular. This rotary tool would be nicely worth the cash even if it came devoid of the accessories.

The VersaPak Wizard cordless rotary tool is no longer in production, but is nevertheless in circulation. The VP940 came as a kit that integrated 63 accessories, a battery and a battery charger. Which rotary tool to choose will depend in significant component on the way it will be utilised by the consumer. Due to the fact it is a very versatile tool that can assist when large conventional tools won't do, a rotary tool may perhaps be made use of a lot more usually than other folks in the toolbox.

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