Template:Modern physics

2013-11-09T23:16:30Z

41.34.254.236:

[[File:Oscar Lanford.jpg|thumb|220px|Oscar Lanford]]
'''Oscar Eramus Lanford III''' (January 6, 1940 – November 16, 2013) was an American [[mathematician]] working on [[mathematical physics]] and [[dynamical systems]] theory.<ref>{{cite web|url=http://www.math.harvard.edu/~knill/history/lanford/ |title=Oscar Lanford (1940-2013) |publisher=Math.harvard.edu |date=2013-11-16 |accessdate=2013-11-27}}</ref>

==Professional career==
Born in [[New York]], Lanford was awarded his undergraduate degree from [[Wesleyan University]] and the Ph.D. from [[Princeton University]] in 1966 under the supervision of [[Arthur Wightman]].<ref>{{mathgenealogy|id=32889}}</ref> He has served as a professor of mathematics at the [[University of California, Berkeley]], and a professor of physics at the [[Institut des Hautes Études Scientifiques]] (IHES) in [[Bures-sur-Yvette]], France. {{As of|1987}}, he has been with the department of mathematics, [[Swiss Federal Institute of Technology Zürich]] (ETH Zürich).

==Proof of the rigidity conjectures==
Lanford gave the first proof that the Feigenbaum-Cvitanovic functional equation
:<math> g(x) = T(g)(x) = -\lambda g(g(-\lambda x)), g(-2)=1, g''(0)<0,\lambda=-g(1)>0 </math>
has an even analytic solution g and that this fixed point g of the Feigenbaum renormalisation operator T is hyperbolic with a one dimensional unstable manifold. This provided the first mathematical proof of the rigidity conjectures of Feigenbaum. The proof was [[Computer-assisted proof|computer assisted]]. The hyperbolicity of the fixed point is essential to explain the Feigenbaum universality observed experimentally by [[Mitchell Feigenbaum]] and Coullet-Tresser. Feigenbaum has studied the logistic family and looked at the sequence of [[Period-doubling bifurcation|Period doubling]] bifurcations. Amazingly the asymptotic behavior near the accumulation point appeared universal in the sense that the same numerical values would appear. The [[Logistic map|logistic family]] <math> f(x) = c x(1-x) </math> of maps on the interval [0,1] for example would lead to the same asymptotic law of the ratio of the differences <math>b(n) = a(n+1)-a(n)</math> between the bifurcation values a(n) than
<math>f(x) = c \sin(\pi x)</math>. The result is that <math>\lim_{n \to \infty} b(n)/b(n+1)</math> converges to the [[Feigenbaum constants]] <math>d=4.6692016091029...</math> which is a "universal number" independent of the map f. The [[bifurcation diagram]] has become an icon of [[Chaotic map|chaos theory]].

Campanino and Epstein also gave a proof of the fixed point without computer assistance but did not establish its hyperbolicity. They cite in their paper Lanfords computer assisted proof. There are also lecture notes of Lanford from 1979 in Zurich and announcements in 1980. The hyperbolicity is essential to verify the picture discovered numerically by Feigenbaum and independently by Coullet and Tresser. Lanford later gave a shorter proof using the [[Schauder fixed point theorem|Leray-Schauder fixed point theorem]] but establishing only the fixed point without the hyperbolicity. Lyubich published in 1999 the first not computer assisted proof which also establishes hyperbolicity. Work of Sullivan later showed that the fixed point is unique in the class of real valued quadratic like germs.

==Awards and honors==
Lanford was the recipient of the 1986 [[United States National Academy of Sciences]] award in Applied Mathematics and Numerical Analysis and holds an honorary doctorate from [[Wesleyan University]].

In 2012 he became a fellow of the [[American Mathematical Society]].<ref>[http://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 2013-01-27.</ref>

=== Selected publications ===
* {{Citation|last=Lanford| first=Oscar|title=A computer-assisted proof of the Feigenbaum conjectures|
journal=Bull.Am.Math.Soc.(New Series)|volume=6|pages=427–434|year=1982|doi=10.1090/S0273-0979-1982-15008-X|issue=3}}
* {{Citation|last=Lanford|first=O.E|title=A Shorter Proof of the Existence of the Feigenbaum Fixed Point|journal=Comm. Math. Phys|volume=96|pages=521–538|year=1984|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103941912|bibcode=1984CMaPh..96..521L|doi=10.1007/BF01212533|issue=4}}

==See also==
*[[Dobrushin-Lanford-Ruelle equations]]

==References==
{{reflist}}
* {{Citation|last1=Campanino|first1=M|last2=Epstein|first2=H|title=On the existence of Feigenbaum's fixed point|journal=Commun. Math. Phys.|volume=79|pages=261–302| year=1981|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103908965|doi=10.1007/BF01942063|issue=2|bibcode = 1981CMaPh..79..261C }}
* {{Citation|last=Lyubich|first=M|title=Feigenbaum-Collet-Tresser universality and Milnor's hairness conjecture|journal=Ann. Of Math.|volume=149|pages=319–420|year=1999|url=http://www.math.sunysb.edu/~mlyubich/Archive/Selected/universe.pdf|doi=10.2307/120968|issue=2}}
* {{Citation|last=Smania|first=D|title=On the hyperbolicity of the Feigenbaum fixed point point |arxiv=math/0301118|bibcode = 2003math......1118S }}
* {{Citation|last1=Coullet|first1=P|last2=Tresser|first2=C|title=Iteration d'endomorphismes et groupe de renormalisation|journal=J. Phys. Colloque|volume=539|pages=5–25|year=1978}}
* {{Citation|last=Feigenbaum|first=M|title=Quantitative universality for a class of non linear transformations|journal=J. Stat. Phys|volume=19|year=1978|pages=25–52 |bibcode = 1978JSP....19...25F |doi = 10.1007/BF01020332 }}
* {{Citation|last1=de Melo|first1=W|last2=van Strien|first2=S|title=One-dimensional Dynamics|publisher=Springer|year=1994}}
* {{Citation|last=Sternberg|first=S|title=Dynamical systems|publisher=Dover |url=http://www.math.harvard.edu/library/sternberg/text/book.pdf}}
* {{Citation|last1=Collet|first1=P|last2=Eckmann|first2=J.P|title=Iterated maps of the interval as dynamical systems|publisher=Birkhaeuser|year=1997|edition=5 Reprint}}
*[http://www.verw.ethz.ch/cgi-win/Who.exe?ID=617&lang=engl ETH Who's who] accessed on April 29, 2007 through [http://web.archive.org/web/20040204163344/www.verw.ethz.ch/cgi-win/Who.exe?ID=617&lang=engl web archive]

==External links==
*[http://www.math.ethz.ch/~lanford/ Home page of Oscar E. Lanford III]

{{Persondata 
| NAME = Lanford III, Oscar Eramus
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = American mathematician
| DATE OF BIRTH = January 6, 1940
| PLACE OF BIRTH =
| DATE OF DEATH = November 16, 2013
| PLACE OF DEATH =
}}
{{DEFAULTSORT:Lanford III, Oscar Eramus}}
[[Category:Wesleyan University alumni]]
[[Category:Princeton University alumni]]
[[Category:University of California, Berkeley faculty]]
[[Category:20th-century mathematicians]]
[[Category:2013 deaths]]
[[Category:Fellows of the American Mathematical Society]]
[[Category:1940 births]]

Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī

2013-02-26T15:51:27Z

41.34.31.158: /* Symbolic algebra */

{{General relativity|cTopic=[[Exact solutions in general relativity|Solutions]]}}
The '''Kasner metric''' is an [[Exact solutions in general relativity|exact solution]] to [[Einstein]]'s theory of [[general relativity]]. It describes an anisotropic [[universe]] without [[matter]] (i.e., it is a [[vacuum solution]]). It can be written in any [[spacetime]] [[dimension]] <math>D>3</math> and has strong connections with the study of gravitational [[Chaos theory|chaos]].

==The Metric and Kasner Conditions==
The [[Metric (mathematics)|metric]] in <math>D>3</math> spacetime dimensions is

:<math>\text{d}s^2 = -\text{d}t^2 + \sum_{j=1}^{D-1} t^{2p_j} [\text{d}x^j]^2</math>,

and contains <math>D-1</math> constants <math>p_j</math>, called the ''Kasner exponents.'' The metric describes a spacetime whose equal-time slices are spatially flat, however space is expanding or contracting at different rates in different directions, depending on the values of the <math>p_j</math>. Test particles in this metric whose [[comoving coordinate]] differs by <math>\Delta x^j</math> are separated by a physical distance <math>t^{p_j}\Delta x^j</math>.

The Kasner metric is an exact solution to Einstein's equations in vacuum when the Kasner exponents satisfy the following ''Kasner conditions,''

:<math>\sum_{j=1}^{D-1} p_j = 1,</math>

:<math>\sum_{j=1}^{D-1} p_j^2 = 1.</math>

The first condition defines a [[plane (geometry)|plane]], the ''Kasner plane,'' and the second describes a [[sphere]], the ''Kasner sphere.'' The solutions (choices of <math>p_j</math>) satisfying the two conditions therefore lie on the sphere where the two intersect (sometimes confusingly also called the Kasner sphere). In <math>D</math> spacetime dimensions, the space of solutions therefore lie on a <math>D-3</math> dimensional sphere <math>S^{D-3}</math>.

==Features of the Kasner Metric==

There are several noticeable and unusual features of the Kasner solution:

*The volume of the spatial slices always goes like <math>t</math>. This is because their volume is proportional to <math>\sqrt{-g}</math>, and

::<math>\sqrt{-g} = t^{p_1 + p_2 + \cdots + p_{D-1}} = t</math>

:where we have used the first Kasner condition. Therefore <math>t\to 0</math> can describe either a [[Big Bang]] or a [[Big Crunch]], depending on the sense of <math>t</math>

*[[Isotropic]] [[metric expansion of space|expansion or contraction of space]] is not allowed. If the spatial slices were expanding isotropically, then all of the Kasner exponents must be equal, and therefore <math>p_j = 1/(D-1)</math> to satisfy the first Kasner condition. But then the second Kasner condition cannot be satisfied, for

::<math>\sum_{j=1}^{D-1} p_j^2 = \frac{1}{D-1} \ne 1.</math>

:The [[FRW|FLRW]] metric employed in [[cosmology]], by contrast, is able to expand or contract isotropically because of the presence of matter.

*With a little more work, one can show that at least one Kasner exponent is always negative (unless we are at one of the solutions with a single <math>p_j=1</math>, and the rest vanishing). Suppose we take the time coordinate <math>t</math> to increase from zero. Then this implies that while the volume of space is increasing like <math>t</math>, at least one direction (corresponding to the negative Kasner exponent) is actually ''contracting.''

*The Kasner metric is a solution to the vacuum Einstein equations, and so the [[Ricci tensor]] always vanishes for any choice of exponents satisfying the Kasner conditions. The [[Riemann tensor]] vanishes only when a single <math>p_j=1</math> and the rest vanish. This has the interesting consequence that ''this particular'' Kasner solution must be a solution of any extension of general relativity in which the field equations are built from the Riemann tensor.

==See also==

*[[BKL singularity]]
*[[Mixmaster universe]]

==References==

*Misner, Thorne, and Wheeler, '''Gravitation.'''

{{Relativity}}

[[Category:Exact solutions in general relativity]]
[[Category:Metric tensors]]

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Template:Modern physics

Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī