formulasearchengine - User contributions [en]

Teichmüller space

2015-01-10T12:32:57Z

130.79.245.244: /* References */

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Mapping class group

2013-10-10T04:57:44Z

130.79.245.238: /* References */

{{Refimprove|date=December 2009}}
The '''Fitting lemma''', named after the mathematician [[Hans Fitting]], is a basic statement in [[abstract algebra]]. Suppose ''M'' is a [[module (mathematics)|module]] over some [[ring (mathematics)|ring]]. If ''M'' is [[indecomposable module|indecomposable]] and has finite [[length of a module|length]], then every [[endomorphism]] of ''M'' is either [[bijective]] or [[nilpotent]].

As an immediate consequence, we see that the [[endomorphism ring]] of every finite-length indecomposable module is [[local ring|local]].

A version of Fitting's lemma is often used in the [[group representation|representation theory of groups]]. This is in fact a special case of the version above, since every ''K''-linear representation of a group ''G'' can be viewed as a module over the [[group algebra]] ''KG''.

To prove Fitting's lemma, we take an endomorphism ''f'' of ''M'' and consider the following two sequences of submodules. The first sequence is the descending sequence im(''f''), im(''f'' 2), im(''f'' 3),..., the second sequence is the ascending sequence ker(''f''), ker(''f'' 2), ker(''f'' 3),.... Because ''M'' has finite length, the first sequence cannot be ''strictly'' decreasing forever, so there exists some ''n'' with im(''f'' ''n'') = im(''f'' ''n''+1). Likewise (as ''M'' has finite length) the second sequence cannot be ''strictly'' increasing forever, so there exists some ''m'' with ker(''f'' ''m'') = ker(''f'' ''m''+1). It is easily seen that im(''f'' ''n'') = im(''f'' ''n''+1) yields im(''f'' ''n'') = im(''f'' ''n''+1) = im(''f'' ''n''+2) = ..., and that ker(''f'' ''m'') = ker(''f'' ''m''+1) yields ker(''f'' ''m'') = ker(''f'' ''m''+1) = ker(''f'' ''m''+2) = ... . Putting ''k'' = max(''m'',''n'' ), it now follows that im(''f'' ''k'') = im(''f'' 2''k'') and ker(''f'' ''k'') = ker(''f'' 2''k''). Hence, <math>\mathrm{ker}\left(f^k\right) \cap \mathrm{im}\left(f^k\right) = 0</math> (because every <math>x\in \mathrm{ker}\left(f^k\right) \cap \mathrm{im}\left(f^k\right)</math> satisfies <math>x=f^k\left(y\right)</math> for some <math>y\in M</math> but also <math>f^k\left(x\right)=0</math>, so that <math>0=f^k\left(x\right)=f^k\left(f^k\left(y\right)\right)=f^{2k}\left(y\right)</math>, therefore <math>y\in\mathrm{ker}\left(f^{2k}\right)=\mathrm{ker}\left(f^k\right)</math> and thus <math>0=f^k\left(y\right)=x</math>) and <math>\mathrm{ker}\left(f^k\right) + \mathrm{im}\left(f^k\right) = M</math> (since for every <math>x\in M</math>, there exists some <math>y\in M</math> such that <math>f^k\left(x\right)=f^{2k}\left(y\right)</math> (since <math>f^k\left(x\right)\in\mathrm{im}\left(f^k\right)=\mathrm{im}\left(f^{2k}\right)</math>), and thus <math>f^k\left(x-f^k\left(y\right)\right)=f^k\left(x\right)-f^k\left(f^k\left(y\right)\right)=f^k\left(x\right)-f^{2k}\left(y\right)=0</math>, so that <math>x-f^k\left(y\right)\in\mathrm{ker}\left(f^k\right)</math> and thus <math>x\in \mathrm{ker}\left(f^k\right)+f^k\left(y\right)\subseteq \mathrm{ker}\left(f^k\right) + \mathrm{im}\left(f^k\right)</math>). Consequently, ''M'' is the [[direct sum of modules|direct sum]] of im(''f'' ''k'') and ker(''f'' ''k''). Because ''M'' is indecomposable, one of those two summands must be equal to ''M'', and the other must be equal to {0}. Depending on which of the two summands is zero, we find that ''f'' is bijective or nilpotent.<ref>Jacobson (2009), p. 113–114.</ref>

== Notes ==
<references/>

== References ==
* {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 2 | series= | publisher=Dover| isbn = 978-0-486-47187-7}}

{{DEFAULTSORT:Fitting Lemma}}
[[Category:Module theory]]
[[Category:Lemmas]]

Equichordal point problem

2013-04-05T13:48:48Z

130.79.245.245:

{{expert-subject|Physics|date=June 2012}}


The '''conformal cyclic cosmology''' ('''CCC''') is a [[cosmological model]] in the framework of [[general relativity]], advanced by the theoretical physicists [[Roger Penrose]] and [[Vahe Gurzadyan]].<ref>{{cite web |title=Cosmos may show echoes of events before Big Bang |first=Jason |last=Palmer |date=2010-11-27 |publisher=BBC News |url=http://www.bbc.co.uk/news/science-environment-11837869|accessdate=2010-11-27}}</ref><ref>{{cite web |url=http://accelconf.web.cern.ch/accelconf/e06/PAPERS/THESPA01.PDF |title=Before the big bang: An outrageous new perspective and its implications for particle physics |first=Roger |last=Penrose |date=June 2006 |accessdate=2010-11-27 |location=Edinburgh, Scotland |pages=2759–2767 |publisher=Proceedings of EPAC 2006}}</ref> <ref>Gurzadyan VG, Penrose R, "On CCC-predicted concentric low-variance circles in the CMB sky", Eur.Phys.J. Plus 128 (2013) 22; http://arxiv.org/abs/1302.5162</ref>
In CCC, the universe iterates through infinite cycles, with the future [[Penrose diagram|timelike infinity]] of each previous iteration being identified with the [[Big Bang]] singularity of the next.<ref>{{cite web |title=Penrose claims to have glimpsed universe before Big Bang |first=Edwin |last=Cartlidge |publisher=physicsworld.com |date=2010-11-19 |url=http://physicsworld.com/cws/article/news/44388|accessdate=2010-11-27}}</ref> Penrose popularized this theory in his 2010 book [[Cycles of Time (book)|''Cycles of Time: An Extraordinary New View of the Universe'']].

==Basic construction==
Penrose's basic construction<ref name="epac2006">{{cite journal|author=Roger Penrose |title=Before the Big Bang: An Outrageous New Perspective and its Implications for Particle Physics|year=2006|journal=Proceedings of the EPAC 2006, Edinburgh, Scotland|url=http://accelconf.web.cern.ch/accelconf/e06/PAPERS/THESPA01.PDF|pages=2759–2762}}</ref> is to connect a [[countable]] sequence of open [[FLRW]] spacetimes, each representing a [[big bang]] followed by an infinite future expansion. Penrose noticed that the past [[conformal boundary]] of one copy of FLRW spacetime can be "attached" to the future conformal boundary of another, after an appropriate [[conformal map|conformal rescaling]]. In particular, each individual FLRW metric <math>g_{ab}</math> is multiplied by the square of a conformal factor <math>\Omega</math> that approaches zero at [[Penrose diagram|timelike infinity]], effectively "squashing down" the future conformal boundary to a conformally regular [[hypersurface]] (which is [[Spacelike#Spacetime_intervals|spacelike]] if there is a positive [[cosmological constant]], as we currently believe). The result is a new solution to Einstein's equations, which Penrose takes to represent the entire Universe, and which is composed of a sequence of sectors that Penrose calls "aeons".

== Physical implications ==

The significant feature of this construction for particle physics is that, since [[boson]]s obey the laws of [[Conformal field theory|conformally invariant quantum theory]], they will behave in the same way in the rescaled aeons as in the original FLRW counterparts. (Classically, this corresponds to the fact that light cone structure is preserved under conformal rescalings.) For such particles, the boundary between aeons is not a boundary at all, but just a spacelike surface that can be passed across like any other. [[Fermions]], on the other hand, remain confined to a given aeon. This provides a convenient solution to the [[black hole information paradox]]; according to Penrose, fermions must be irreversibly converted into radiation during black hole evaporation, to preserve the smoothness of the boundary between aeons.

The curvature properties of Penrose's cosmology are also highly desirable. First, the boundary between aeons satisfies the [[Weyl curvature hypothesis]], thus providing a certain kind of low-entropy past as required by statistical mechanics and by observation. Second, Penrose has calculated that a certain amount of gravitational radiation should be preserved across the boundary between aeons. Penrose suggests this extra gravitational radiation may be enough to explain the observed [[cosmic acceleration]] without appeal to a [[dark energy]] matter field.

== Empirical tests ==
In 2010, Penrose and [[Vahe Gurzadyan]] published a [[preprint]] of a paper claiming that observations of the [[cosmic microwave background]] made by the [[Wilkinson Microwave Anisotropy Probe]] and the [[BOOMERanG experiment]] showed concentric anomalies which were consistent with the CCC hypothesis, with a low probability of the [[null hypothesis]] that the observations in question were caused by chance.<ref>{{cite arxiv |eprint=1011.3706 |author1=Gurzadyan VG |author2=Penrose R |title=Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity |class=astro-ph.CO |date=2010-11-16}}</ref> However, the statistical significance of the claimed detection has since been questioned. Three groups have independently attempted to reproduce these results, but found that the detection of the concentric anomalies was not statistically significant, in the sense that such circles would appear in a proper Gaussian simulation of the anisotropy in the CMB data.<ref>{{cite arxiv |eprint=1012.1268 |author1=Wehus IK |author2=Eriksen HK |title=A search for concentric circles in the 7-year WMAP temperature sky maps |class=astro-ph.CO |date=2010-12-07}}</ref><ref>{{cite arxiv |eprint=1012.1305 |author1=Moss A |author2=Scott D |author3=Zibin JP |title=No evidence for anomalously low variance circles on the sky |class=astro-ph.CO |date=2010-12-07}}</ref><ref>{{cite arxiv |eprint=1012.1656 |author1=Hajian A |title=Are There Echoes From The Pre-Big Bang Universe? A Search for Low Variance Circles in the CMB Sky |class=astro-ph.CO |date=2010-12-8}}</ref>

Further development of their work Gurzadyan and Penrose published in 2013.<ref>Gurzadyan VG, Penrose R, "On CCC-predicted concentric low-variance circles in the CMB sky", Eur.Phys.J. Plus 128 (2013) 22; http://arxiv.org/abs/1302.5162</ref>

== See also ==
* [[Conformal geometry]]
* [[Cyclic model]]
* [[Black hole information paradox]]
* [[White hole]]
* ''[[Cycles of Time (book)|Cycles of Time: An Extraordinary New View of the Universe]]''

== References ==
{{reflist}}

==External links==
*[http://www.ideasroadshow.com/issues/roger-penrose-2013-07-12 "The Cyclic Universe - A conversation with Roger Penrose"], ''Ideas Roadshow'', 2013

[[Category:Physical cosmology]]

Before the big bang 2 : Conformal Cyclic Cosmology Explained
www.youtube.com/watch?v=sM47acQ7pEQ

Euler sequence

2012-01-23T13:20:38Z

130.79.5.179:

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