List decoding - Revision history

202.120.38.184: /* Theorem (list-decoding capacity) */

2014-07-16T08:47:21Z

Theorem (list-decoding capacity)

← Older revision		Revision as of 10:47, 16 July 2014
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	<li>[http://yaoxm.net/forum.php?mod=viewthread&tid=2432171 http://yaoxm.net/forum.php?mod=viewthread&tid=2432171]</li>		<li>[http://yaoxm.net/forum.php?mod=viewthread&tid=2432171 http://yaoxm.net/forum.php?mod=viewthread&tid=2432171]</li>

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129.132.57.154: /* External links */

2014-02-11T15:18:43Z

External links

Show changes

128.220.159.17: deleted extra word

2013-11-11T02:48:52Z

deleted extra word

← Older revision		Revision as of 04:48, 11 November 2013
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	~~Golda is what~~'s ~~created~~ on ~~my birth certificate although~~ it is ~~not~~ the ~~title~~ on ~~my birth certification~~. ~~Since he was 18 he~~'s ~~been operating as an information officer but he plans~~ on ~~changing it~~. ~~What me~~ and ~~my family members adore~~ is to ~~climb but I~~'~~m thinking on beginning some thing new~~. ~~Ohio is where her house~~ is.<br><br>~~Also visit my webpage~~: [~~http:~~//~~c045~~.~~danah~~.co.kr/~~home~~/~~index~~.~~php?document_srl~~=~~1356970&mid~~=~~qna live psychic reading~~]		In the [[mathematics\|mathematical]] field of [[Riemannian geometry]], [[Mikhail Gromov (mathematician)\|M. Gromov]]'s '''systolic inequality''' bounds the length of the shortest [[contractible space\|non-contractible]] loop on a [[Riemannian manifold]] in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983;<ref>see {{harvtxt\|Gromov\|1983}}</ref> it can be viewed as a generalisation, albeit non-optimal, of [[Loewner's torus inequality]] and [[Pu's inequality\|Pu's inequality for the real projective plane]].

			Technically, let ''M'' be an [[essential manifold\|essential]] Riemannian manifold of dimension ''n''; denote by sys''π''<sub>1</sub>(''M'') the homotopy 1-systole of ''M'', that is, the least length of a non-contractible loop on ''M''. Then Gromov's inequality takes the form

			:<math> \left(\operatorname{sys\pi}_1(M)\right)^n \leq C_n \operatorname{vol}(M),</math>

			where ''C''<sub>''n''</sub> is a universal constant only depending on the dimension of ''M''.

			==Essential manifolds==
			{{main\|essential manifold}}

			A closed manifold is called ''essential'' if its [[fundamental class]] defines a nonzero element in the [[homology (mathematics)\|homology]] of its [[fundamental group]], or more precisely in the homology of the corresponding [[Eilenberg–MacLane space]]. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.

			Examples of essential manifolds include [[aspherical manifold]]s, [[real projective space]]s, and [[lens space]]s.

			==Proofs of Gromov's inequality==

			Gromov's original 1983 proof is about 35 pages long. It relies on a number of techniques and inequalities of global Riemannian geometry. The starting point of the proof is the imbedding of X into the Banach space of Borel functions on X, equipped with the sup norm. The imbedding is defined by mapping a point ''p'' of ''X'', to the real function on ''X'' given by the distance from the point ''p''. The proof utilizes the [[coarea formula\|coarea inequality]], the [[isoperimetry\|isoperimetric inequality]], the cone inequality, and the deformation theorem of [[Herbert Federer]].

			==Filling invariants and recent work==

			One of the key ideas of the proof is the introduction of filling invariants, namely the [[filling radius]] and the filling volume of ''X''. Namely, Gromov proved a sharp inequality relating the systole and the filling radius,

			:<math>\mathrm{sys\pi}_1 \leq 6\; \mathrm{FillRad}(X),</math>

			valid for all essential manifolds ''X''; as well as an inequality

			:<math>\mathrm{FillRad}(X) \leq C_n \mathrm{vol}_n{}^{\tfrac{1}{n}}(X),</math>

			valid for all closed manifolds ''X''.

			It was shown by {{harvtxt\|Brunnbauer\|2008}} that the filling invariants, unlike the systolic invariants, are independent of the topology of the manifold in a suitable sense.

			{{harvtxt\|Guth\|2011}} and {{harvtxt\|Ambrosio\|2011}} developed approaches to the proof of Gromov's systolic inequality for essential manifolds.

			==Inequalities for surfaces and polyhedra==

			Stronger results are available for surfaces, where the asymptotics when the genus tends to infinity are by now well understood, see [[systoles of surfaces]]. A uniform inequality for arbitrary 2-complexes with non-free fundamental groups is available, whose proof relies on the [[Grushko theorem\|Grushko decomposition theorem]].

			==Notes==
			{{Reflist}}

			==See also==
			*[[Filling area conjecture]]
			*[[Gromov's inequality]]
			*[[Gromov's inequality for complex projective space]]
			*[[Loewner's torus inequality]]
			*[[Pu's inequality]]
			*[[Systolic geometry]]

			==References==
			*{{citation
			\| mr = 2803853
			\| last1 = Ambrosio \| first1 = Luigi
			\| author1-link = Luigi Ambrosio
			\| last2 = Katz \| first2 = Mikhail
			\| author2-link = Mikhail Katz
			\| arxiv = 1004.1374
			\| doi = 10.4171/CMH/234
			\| issue = 3
			\| journal = [[Commentarii Mathematici Helvetici]]
			\| pages = 557–592
			\| title = Flat currents modulo p in metric spaces and filling radius inequalities
			\| volume = 86
			\| year = 2011}}.
			*{{Citation\|last=Brunnbauer\|first=M.\|title=Filling inequalities do not depend on topology\|volume=624\|year=2008\|pages=217–231\|journal=J. Reine Angew. Math.}}
			*{{Citation\|last=Gromov\|first=M.\|title=Filling Riemannian manifolds\|journal=J. Diff. Geom.\|volume=18\|year=1983\|pages=1–147\|mr=0697984\|id={{Euclid\|euclid.jdg/1214509283}}\|zbl=0515.53037}}
			*{{Citation\|mr=2753599\|last=Guth\|first=Larry
			\|author1-link= Larry Guth
			\|title=Volumes of balls in large Riemannian manifolds\|journal=Ann. Of Math. (2)\|volume=173\|year=2011\|issue=1\|pages=51–76\|doi=10.4007/annals.2011.173.1.2}}
			*{{Citation \| last1=Katz \| first1=Mikhail G. \| title=Systolic geometry and topology\|pages=19 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=Mathematical Surveys and Monographs \| isbn=978-0-8218-4177-8 \| year=2007 \| volume=137}}

			{{Systolic geometry navbox}}

			[[Category:Geometric inequalities]]
			[[Category:Riemannian geometry]]
			[[Category:Systolic geometry]]

86.6.155.171: /* Sketch of proof */

2012-05-04T18:23:07Z

Sketch of proof

New page

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