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Landau quantization

2015-01-02T18:21:48Z

83.227.22.234: correct capitalization

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Modern Arabic mathematical notation

2013-07-12T09:47:56Z

83.227.51.194: /* Some features of Arabic mathematical notation */

In [[mathematics]], a '''dissipative operator''' is a [[linear operator]] ''A'' defined on a [[linear subspace]] ''D''(''A'') of [[Banach space]] ''X'', taking values in ''X'' such that for all ''λ'' > 0 and all ''x'' ∈ ''D''(''A'')
:<math>\|(\lambda I-A)x\|\geq\lambda\|x\|.</math>
A dissipative operator is called '''maximally dissipative''' if it is dissipative and for all ''λ'' > 0 the operator ''λI'' − ''A'' is surjective, meaning that the range when applied to the domain ''D'' is the whole of the space ''X''.

The main importance of dissipative operators is their appearance in the [[Lumer–Phillips theorem]] which characterizes maximally dissipative operators as the generators of [[contraction semigroup]]s.

==Properties==
A dissipative operator has the following properties<ref>Engel and Nagel Proposition II.3.14</ref>
* ''λI'' − ''A'' is [[injective]] for all ''λ'' > 0 and
:::<math>\|(\lambda I-A)^{-1}z\|\leq\frac{1}{\lambda}\|z\|</math>
::for all ''z'' in the range of ''λI'' − ''A''.
* ''λI'' − ''A'' is [[surjective]] for some ''λ'' > 0 if and only if it is surjective for all ''λ'' > 0. (This is the aforementioned maximally dissipative case.) In that case one has (0, ∞) ⊂ ''ρ''(''A'') (the [[resolvent set]] of ''A'').
* ''A'' is a [[closed operator]] if and only if the range of ''λI'' − ''A'' is closed for some (equivalently: for all) ''λ'' > 0.

==Equivalent characterization==
Define the duality set of ''x'' ∈ ''X'', a subset of the [[dual space]] ''X''' of ''X'', by
:<math>J(x):=\left\{x'\in X':\|x'\|_{X'}^2=\|x\|_{X}^2=\langle x',x\rangle \right\}.</math>
By the [[Hahn–Banach theorem]] this set is nonempty. If ''X'' is reflexive, then ''J''(''x'') consists of a single element{{Citation needed|date=November 2011}}. In the [[Hilbert space]] case (using the canonical duality between a Hilbert space and its dual) it consists of the single element ''x''.<ref>Engel and Nagel Exercise II.3.25i</ref>
Using this notation, ''A'' is dissipative if and only if<ref>Engel and Nagel Proposition II.3.23</ref> for all ''x'' ∈ ''D''(''A'') there exists a ''x''<nowiki>'</nowiki> ∈ ''J''(''x'') such that
:<math>{\rm Re}\langle Ax,x'\rangle\leq0.</math>

==Examples==

* For a simple finite-dimensional example, consider ''n''-dimensional [[Euclidean space]] '''R'''''n'' with its usual [[dot product]]. If ''A'' denotes the negative of the [[identity operator]], defined on all of '''R'''''n'', then

::<math>x \cdot A x = x \cdot (-x) = - \| x \|^{2} \leq 0,</math>

: so ''A'' is a dissipative operator.

* So long as the domain of an operator ''A'' is the whole Euclidean space, then it is dissipative if and only if it does not have any eigenvalue with positive real part, and (consequently) all such operators are maximally dissipative. An equivalent condition is that for some (and hence any) positive <math>\lambda, \lambda-A</math> has an inverse and the operator <math>(\lambda+A)(\lambda-A)^{-1}</math> is a contraction (that is, it diminishes the norm of its operand). If the time derivative of a point ''x'' in the space is given by ''Ax'', then the time evolution is governed by a [[contraction semigroup]] that constantly decreases the norm. (Note however that if the domain of ''A'' is a proper subspace, then ''A'' cannot be maximally dissipative because the range will not have a high enough dimensionality.)

* Consider ''H'' = [[Square-integrable function|''L''2]]([0, 1]; '''R''') with its usual inner product, and let ''Au'' = ''u''′ with domain ''D''(''A'') equal to those functions ''u'' in the [[Sobolev space]] ''H''1([0, 1]; '''R''') with ''u''(1) = 0. ''D''(''A'') is dense in ''L''2([0, 1]; '''R'''). Moreover, for every ''u'' in ''D''(''A''), using [[integration by parts]],

::<math>\langle u, A u \rangle = \int_{0}^{1} u(x) u'(x) \, \mathrm{d} x = - \frac1{2} u(0)^{2} \leq 0.</math>

: Hence, ''A'' is a dissipative operator. Furthermore, since there is a solution in ''D'' to <math>u-\lambda u'=f</math> for any ''f'' in ''H'', the operator ''A'' is maximally dissipative. Note that in a case of infinite dimensionality like this, the range can be the whole Banach space even though the domain is only a proper subspace thereof.

* Consider ''H'' = ''H''02(Ω; '''R''') for an [[open set|open]] and [[connected space|connected]] domain Ω &sube; '''R'''''n'' and let ''A'' = Δ, the [[Laplace operator]], defined on the dense subspace of [[compactly supported]] smooth functions on Ω. Then, using integration by parts,

::<math>\langle u, \Delta u \rangle = \int_\Omega u(x) \Delta u(x) \, \mathrm{d} x = - \int_\Omega \big| \nabla u(x) \big|^{2} \, \mathrm{d} x = - \| \nabla u \|_{L^{2} (\Omega; \mathbf{R})} \leq 0,</math>

: so the Laplacian is a dissipative operator. However in this case it is not maximally dissipative because there is no compactly supported solution to <math>u-\lambda \Delta u=f</math> if ''f'' is not compactly supported.

==Notes==

{{reflist|2}}

==References==
*{{ Cite book | last1=Engel| first1=Klaus-Jochen| last2=Nagel| first2=Rainer | title=One-parameter semigroups for linear evolution equations | year=2000| publisher=Springer}}
* {{cite book
| author = Renardy, Michael and Rogers, Robert C.
| title = An introduction to partial differential equations
| series = Texts in Applied Mathematics 13
| edition = Second
|publisher = Springer-Verlag
| location = New York
| year = 2004
| pages = 356
| isbn = 0-387-00444-0
}} (Definition 12.25)

[[Category:Operator theory]]

Fuzzy sphere

2012-10-05T00:08:20Z

83.227.31.74: fix casimir

:''This page concerns mathematician Sergei Novikov's topology conjecture. For astrophysicist Igor Novikov's conjecture regarding time travel, see [[Novikov self-consistency principle]].

The '''Novikov conjecture''' is one of the most important unsolved problems in [[topology]]. It is named for [[Sergei Novikov (mathematician)|Sergei Novikov]] who originally posed the conjecture in 1965.

The Novikov conjecture concerns the [[homotopy]] invariance of certain [[polynomial]]s in the [[Pontryagin class]]es of a [[manifold (mathematics)|manifold]], arising from the [[fundamental group]]. According to the Novikov conjecture, the ''higher signatures'', which are certain numerical invariants of smooth manifolds, are homotopy invariants.

The conjecture has been proved for finitely generated [[abelian groups]]. It is not yet known whether the Novikov conjecture holds true for all groups. There are no known counterexamples to the conjecture.

==Precise formulation of the conjecture==

Let ''G'' be a [[discrete group]] and ''BG'' its [[classifying space]], which is a [[Eilenberg–Maclane space|K(G,1)]] and therefore unique up to [[homotopy equivalence]] as a CW complex. Let

:<math>f: M\rightarrow BG</math>

be a continuous map from a closed oriented ''n''-dimensional manifold ''M'' to ''BG'', and

:<math>x \in H^{n-4i} (BG;\mathbb{Q} ).</math>

Novikov considered the numerical expression, found by evaluating the cohomology class in top dimension against the [[fundamental class]] [''M''], and known as a '''higher signature''':

:<math>\left\langle f^*(x) \cup L_i(M),[M] \right\rangle \in \mathbb{Q}</math>

where ''L''i is the ''i''th [[Hirzebruch polynomial]], or sometimes (less descriptively) as the ''i''th ''L''-polynomial. For each ''i'', this polynomial can be expressed in the Pontryagin classes of the manifold's tangent bundle. The '''Novikov conjecture''' states that the higher signature is an invariant of the oriented homotopy type of ''M'' for every such map ''f'' and every such class ''x'', in other words, if <math>h: M' \rightarrow M</math> is an orientation preserving homotopy equivalence, the higher signature associated to <math>f \circ h</math> is equal to that associated to ''f''.

==Connection with the Borel conjecture==

The Novikov conjecture is equivalent to the rational injectivity of the [[assembly map]] in [[L-theory]]. The
[[Borel conjecture]] on the rigidity of aspherical manifolds is equivalent to the assembly map being an isomorphism.

==References==

*{{Citation | last1=Davis | first1=James F. | editor1-last=Cappell | editor1-first=Sylvain | editor2-last=Ranicki | editor2-first=Andrew | editor3-last=Rosenberg | editor3-first=Jonathan | editorlink3=Jonathan Rosenberg (mathematician) | title=Surveys on surgery theory. Vol. 1 | url=http://www.indiana.edu/~jfdavis/papers/d_manc.pdf | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-691-04937-3; 978-0-691-04938-0 |mr=1747536 | year=2000 | chapter=Manifold aspects of the Novikov conjecture | pages=195–224}}

*[[J. Milnor]] and [[Jim Stasheff|J. D. Stasheff]], ''Characteristic Classes,'' Ann. Math. Stud. 76, Princeton (1974).

*S. P. Novikov, ''Algebraic construction and properties of Hermitian analogs of k-theory over rings with involution from the point of view of Hamiltonian formalism. Some applications to differential topology and to the theory of characteristic classes''. Izv.Akad.Nauk SSSR, v. 34, 1970 I N2, pp. 253-288; II: N3, pp. 475-500. English summary in Actes Congr. Intern. Math., v. 2, 1970, pp. 39-45.

==External links==

*[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Novikov_Sergi.html Biography of Sergei Novikov]
*[http://www.math.umd.edu/~jmr/NC.html Novikov Conjecture Bibliography]
*[http://www.maths.ed.ac.uk/~aar/books/novikov1.pdf Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 1]
*[http://www.maths.ed.ac.uk/~aar/books/novikov2.pdf Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 2]
*[http://www.math.uni-muenster.de/u/lueck/publ/lueck/owsemfinalextract.pdf 2004 Oberwolfach Seminar notes on the Novikov Conjecture] (pdf)
*[http://www.scholarpedia.org/article/Novikov_conjecture Scholarpedia article by S.P. Novikov] (2010)
*[http://www.map.him.uni-bonn.de/Novikov_Conjecture The Novikov Conjecture] at the Manifold Atlas
[[Category:Geometric topology]]
[[Category:Homotopy theory]]
[[Category:Conjectures]]
[[Category:Surgery theory]]