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In [[mathematics]], the '''Lyusternik–Schnirelmann category''' (or, '''Lusternik–Schnirelmann category''', '''LS-category''', or simply, '''category''') of a [[topological space]] <math>X</math> is the [[homotopical invariant]] defined to be the smallest integer  number <math>k</math> such that there is an [[open covering]] <math>\{U_i\}_{1\leq i\leq k}</math> of <math>X</math> with the property that each  [[inclusion map]] <math>U_i\hookrightarrow X</math> is [[nullhomotopic]]. For example, if <math>X</math> is the [[circle]], this takes the value two.
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Sometimes a different normalization of the invariant is adopted, which is one less than the definition above.  Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).
 
In general it is not so easy to compute this invariant, which was initially introduced by [[Lazar Lyusternik]] and [[Lev Schnirelmann]] in connection with [[Calculus of variations|variational problems]]. It has a close connection with [[algebraic topology]], in particular [[cup-length]].  In the modern normalization, the cup-length is a lower bound for LS category.
 
It was, as originally defined for the case of ''X'' a [[manifold]], the lower bound for the number of [[critical point (mathematics)|critical point]]s that a real-valued function on ''X'' could possess (this should be compared with the result in [[Morse theory]] that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).
 
==See also==
* [[Ganea conjecture]]
* [[Systolic category]]
 
==References==
* [[R. H. Fox]], [http://links.jstor.org/sici?sici=0003-486X%28194104%292%3A42%3A2%3C333%3AOTLC%3E2.0.CO%3B2-V ''On the Lusternik-Schnirelmann category''], [[Annals of Mathematics]] 42 (1941), 333-370.
 
* [[Samuel Eilenberg]], [[Tudor Ganea]], [http://links.jstor.org/sici?sici=0003-486X%28195705%292%3A65%3A3%3C517%3AOTLCOA%3E2.0.CO%3B2-J ''On the Lusternik-Schnirelmann category of abstract groups''], Annals of Mathematics, 2nd Ser., 65 (1957), no. 3, 517 &ndash; 518
 
* [[F. Takens]], [http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D166832 ''The minimal number of critical points of a function on compact manifolds and the Lusternik-Schnirelmann category''], [[Inventiones Mathematicae]]  6 (1968), 197-244.
 
* Tudor Ganea, ''Some problems on numerical homotopy invariants'', Lecture Notes in Math. 249 (Springer, Berlin, 1971), pp.&nbsp;13 &ndash; 22    {{MathSciNet| id=0339147}}
 
* [[Ioan James]], [http://dx.doi.org/10.1016/0040-9383(78)90002-2 ''On category, in the sense of Lusternik-Schnirelmann''], [[Topology (journal)|Topology]] 17 (1978), 331-348.
 
* Kathryn Hess, [http://dx.doi.org/10.1016/0040-9383(91)90006-P ''A proof of Ganea's conjecture for rational spaces''], Topology 30 (1991), no. 2, 205–214.  {{MathSciNet| id=1098914}}
 
* [[Norio Iwase]], [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=19333 "Ganea's conjecture on Lusternik-Schnirelmann category"], in ''Bulletin of the [[London Mathematical Society]]'', 30 (1998), no.6, 623 &ndash; 634  {{MathSciNet| id=1642747}}
 
* Norio Iwase, [http://dx.doi.org/S0040-9383(00)00045-8 ''A<sub>∞</sub>-method in Lusternik-Schnirelmann category''], Topology 41 (2002), no. 4, 695–723. {{MathSciNet| id=1905835}}
 
* Lucile Vandembroucq, [http://dx.doi.org/10.1016/S0040-9383(02)00007-1 ''Fibrewise suspension and Lusternik-Schnirelmann category''], Topology 41 (2002), no. 6, 1239–1258. {{MathSciNet| id=1923222}}
 
* Octav Cornea, Gregory Lupton, John Oprea, Daniel Tanré, ''Lusternik-Schnirelmann category'', Mathematical Surveys and Monographs, 103. [[American Mathematical Society]], Providence, RI, 2003  ISBN 0-8218-3404-5
 
{{DEFAULTSORT:Lusternik-Schnirelmann category}}
[[Category:Algebraic topology]]
[[Category:Morse theory]]

Latest revision as of 00:25, 20 September 2014

If you compare registry cleaners there are a number of aspects to look out for. Because of the sheer amount of for registry cleaners available found on the Internet at the moment it might be very easy to be scammed. Something usually overlooked is that a few of these products can the reality is end up damaging the PC. And the registry they state they have cleaned might simply lead to more problems with a computer than the ones we began with.

Firstly, you need to employ a Antivirus or safety tool and run a scan on a computer. It is possible which a computer is infected with virus or malware that slows down your computer. If there is nothing found inside the scanning report, it may be the RAM that cause the problem.

When you compare registry cleaners you require a swiftly acting registry cleaning. It's no advantageous spending hours plus your PC waiting for the registry cleaning to complete its task. You need a cleaner to complete its task in minutes.

If which refuses to work you need to try and repair the issue with a 'registry cleaner'. What occurs on numerous computers is the fact that their registry database becomes damaged plus unable to show the computer where the DLL files which it requires are. Every Windows PC has a central 'registry' database which stores information regarding all the DLL files on your computer.

One other option whenever arresting the 1328 error is to wash out the PC's registry. The registry is important because it happens to be where settings plus files utilized by Windows for operating are stored. As it's frequently employed, breakdowns and cases of files getting corrupted are not uncommon. Also due to the means it is configured, the "registry" gets saved in the incorrect fashion consistently, which makes the system run slow, ultimately causing a PC to suffer from a series of errors. The best system 1 can use inside cleaning out registries is to use a reliable tuneup utilities 2014 program. A registry cleaner will find out and repair corrupted registry files plus settings permitting one's computer to run usually again.

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As the hub center of the computer, the important settings are stored the registry. Registry is structured as keys and each key relates to a system. The system reads the keys and uses the info to launch plus run programs. However, the big issue is the fact that there are too various unwelcome settings, useless info occuping the valuable room. It makes the program run slowly plus big amounts of settings become unreadable.

A program plus registry cleaner is downloaded within the internet. It's user friendly plus the process refuses to take long. All it does is scan plus then whenever it finds errors, it might fix plus clean those errors. An error free registry can safeguard the computer from mistakes and provide we a slow PC fix.

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