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Topic: Pontryagin duality

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	Pontryagin duality - Wikipedia, the free encyclopedia
		The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets (i.e., the compact-open topology).
		The dual group of a locally compact abelian group is introduced as the underlying space for an abstract version of the Fourier transform.
		One use made of Pontryagin duality between compact abelian groups and discrete abelian groups is to characterize the Bohr compactification of an arbitrary abelian locally compact topological group.
	en.wikipedia.org /wiki/Pontryagin_duality (1942 words)

	Talk:Pontryagin duality - Wikipedia, the free encyclopedia
		Pontryagin duality looks good, but I wondered why you created this instead of adding to and/or rewriting dual group.
		The best solution may be a merged page under Pontryagin duality (a more accurate title) with dual group redirecting.
		If a merge was to happen, dual group suffers from not having any reasons for wanting duality (which yours covers from a couple of angles) and Pontryagin duality suffers from not actually saying what the dual group is (in terms that an undergrad maths student can handle).
	en.wikipedia.org /wiki/Talk:Pontryagin_duality (631 words)

	Pontryagin (Site not responding. Last check: 2007-11-07)
		Pontryagin's mother, Tat'yana Andreevna Pontryagina, was 29 years old when he was born and she was a remarkable woman who played a crucial role in his path to becoming a mathematician.
		Pontryagin was strongly influenced by Aleksandrov and the direction of Aleksandrov's research was to determine the area of Pontryagin's work for many years.
		Pontryagin graduated from the University of Moscow in 1929 and was appointed to the Mechanics and Mathematics Faculty.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Pontryagin.html (1514 words)

	[No title]
		Actually, at least as usually presented, the Weyl-Wigner formalism supposes a very particular kind of that duality, the Pontryagin duality, which should not be expected to be at work when the phase space is not, for each degree of freedom, the plane $\R^2$ which models vector phase spaces.
		Pontryagin duality is valid only when the group of linear symplectomorphisms (transformations preserving the phase space symplectic structure) is abelian.
		Starting from a well-established (Fourier) duality for the Heisenberg group in terms of Kac algebras, we were able to introduce two new {\em projective Kac algebras}, in terms of which a {\em projective duality} for the translation group is defined.
	mpej.unige.ch /mp_arc/papers/96-387 (9649 words)

	PONTRYAGIN DUALITY
		The theory, introduced by Lev Pontryagin and combined with Haar measure by André Weil, depends on the theory of the dual group and is its expression in the language of category theory; that is, it also includes the behaviour of frequency domains with respect to group homomorphisms.
		This duality is a symmetric relationship, since the dual group of a dual group is the original group.
		As a consequence of this, the dual of
	www.websters-online-dictionary.org /definition/PONTRYAGIN+DUALITY (1194 words)

	Injective cogenerator
		In category theory, the concept of an injective cogenerator is motivated by some major and important examples, such as Pontryagin duality[?].
		Similar statements apply to the topological (or Pontryagin) character module of continuous homomorphisms from H to R/Z (the circle group).
		The topological dual turned out to be quite useful since the dual of a discrete (untopologized) group was compact.
	www.ebroadcast.com.au /lookup/encyclopedia/in/Injective_cogenerator.html (572 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The Pontryagin dual of an abelian group $G$ is the space of characters, which is also an abelian group, though not necessarily the same.
		Duality must be understood no more as a relationship between groups, but as a relationship in a wider category.
		Since Fourier duality in its more general form is implemented in the Kac algebra structural frame \cite{ensc}, we argue that they are also able to provide a generalized Weyl prescription for quantizing a phase space on which a separable locally compact type~I group acts by symplectomorphisms.
	www.ma.utexas.edu /mp_arc/papers/96-388 (3016 words)

	Encyclopedia: Pontryagin duality (Site not responding. Last check: 2007-11-07)
		In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions.
		Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed.
		Dual graph is a concept in graph theory.
	www.nationmaster.com /encyclopedia/Pontryagin-duality (735 words)

	[No title]
		He also recalled Pontryagin's fundamental observat* *ion [Pon42, Pon47] that, for M to be such a boundary, it is necessary that all of its characteristic numbers be zero.
		In 1950, Pontryagin [Pon50] showed that the stable homotopy groups of spheres, in low dimension at least, a* *re isomorphic to the framed cobordism groups of smooth manifolds.
		Pontryagin's paper was in Russian, never translated, and it is not quoted by Thom.
	www.math.purdue.edu /research/atopology/May/history.txt (14491 words)

	Wikipedia: Duality
		In physics, this refers to the case when two different models actually turn out to be equivalent.
		In mathematics, see dual space, dual polyhedron, dual numbers, dual Boolean algebra, dual category, duality of categories, Pontryagin duality, De Morgan dual in logic, duality in projective geometry.
		In Analytical Psychology, this is an archetype, that is to say, one of the deep powerful symbols of the functioning of the psyche.
	www.factbook.org /wikipedia/en/d/du/duality.html (149 words)

	Reference.com/Encyclopedia/Lev Semenovich Pontryagin
		Lev Semenovich Pontryagin (Russian: Лев Семёнович Понтрягин) (3 September 1908 – 3 May 1988) was a Soviet Russian mathematician.
		He went on to lay foundations for the abstract theory of the Fourier transform, now called Pontryagin duality.
		This led to the introduction around 1940 of a theory of characteristic classes, now called Pontryagin classes, designed to vanish on a manifold that is a boundary.
	www.reference.com /browse/wiki/Pontryagin (228 words)

	Fourier transform
		The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality).
		See also the Pontryagin duality for the generalized underpinnings of the Fourier transform.
		In terms of signal processing, the transform takes a time series representation of a signal function and maps it into a frequency spectrum, where ω is angular frequency.
	www.brainyencyclopedia.com /encyclopedia/f/fo/fourier_transform.html (759 words)

	Wikipedia: Harmonic analysis
		The core motivating idea are the various Fourier transforms, which can be generalized to a transform of functions defined on locally compact groups.
		The theory for abelian locally compact groups is called Pontryagin duality; it is considered to be in a satisfactory state, as far as explaining the main features of harmonic analysis goes.
		Harmonic analysis studies the properties of that duality and Fourier transform; and attempts to extend those features to different settings, for instance to the case of non-abelian Lie groups.
	www.factbook.org /wikipedia/en/h/ha/harmonic_analysis.html (332 words)

	Reference.com/Encyclopedia/Pontryagin duality
		is the Haar measure on the dual group.
		iota: H rightarrow hat{G} is continuous and a homomorphism, the dual morphism
		G sim hat{hat{G}} {rightarrow} hat{H} is a morphism into a compact group which is easily shown to satisfy the requisite universal property.
	www.reference.com /browse/wiki/Pontryagin_duality (1939 words)

	Pontryagin Duality Encyclopedia Article, Description, History and Biography @ ArtisticNudity.com (Site not responding. Last check: 2007-11-07)
		Looking For pontryagin duality - Find pontryagin duality and more at Lycos Search.
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		Look for pontryagin duality - Find pontryagin duality at one of the best sites the Internet has to offer!
	www.artisticnudity.com /encyclopedia/Pontryagin_duality (2091 words)

	Untitled Document
		This is known as Pontryagin duality and extends naturally to modules over a ring, where it gives a useful duality between injective and flat modules.
		Over a commutative, Noetherian ring, Grothendieck studied duality with respect to a dualizing complex.
		For this distinguished object, it also makes sense to consider the opposite construction: morphisms from D into a module M. The derived functor RHom(D,-)$, together with its adjoint, gives a Morita-like equivalens between suitable categories.
	www.math.ku.dk /cal/events/1708.htm (166 words)

	M. Montserrat Bruguera
		A number of attempts to extend Pontryagin duality theory to categories of groups larger than that of locally compact abelian groups have been made using different approaches.
		The classical Pontryagin duality theorem states that every locally compact topological abelian group (LCA) is reflexive.
		E. Binz and H. Butzmann have succeeded to extend Pontryagin duality theory to the category of convergence abelian groups and continuous homomorphisms, CONABGRP.
	www.utm.edu /staff/jschomme/topology/c/a/a/h/10.htm (924 words)

	NBFAS Meeting at Glasgow (Site not responding. Last check: 2007-11-07)
		Abstract: For any locally compact abelian group $G$, one can consider the dual group $\hat G$ which is again a locally compact abelian group.
		Pontryagin's duality theorem states that the dual of $\hat G$ is canonically isomorphic with $G$.
		This duality is the basis of abstract harmonic analysis.
	www.mas.ncl.ac.uk /~nbfas/Nov96.html (224 words)

	Atlas: Pontryagin duality for topological Abelian groups by Salvador Hernandez (Site not responding. Last check: 2007-11-07)
		A topological abelian group G satisfies Pontryagin duality, or is Pontryagin reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism.
		The aim of this talk is to report on some recent results related to Pontryagin duality theory of topological Abelian groups.
		We present a solution to this question using the notion of "groups in duality" introduced by Varopoulos in 1964.
	atlas-conferences.com /c/a/f/q/07.htm (158 words)

	Van_Kampen (Site not responding. Last check: 2007-11-07)
		Shortly before this Pontryagin, who had been working on problems in topology and algebra, had been studying duality.
		He had proved that compact abelian groups are dual to discrete abelian groups, and von Neumann was interested in extending this result.
		Van Kampen became interested in Pontryagin's duality and wrote sixteen papers on this topic, including an excellent survey article published in 1935.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Van_Kampen.html (1081 words)

	The operator algebra approach to quantum groups -- Kustermans and Vaes 97 (2): 547 -- Proceedings of the National ... (Site not responding. Last check: 2007-11-07)
		The above discussed construction of the reduced dual of a locally compact group can be generalized to the quantum group setting.
		The construction of the dual of a Kac algebra can be found in chapter 3 of ref. 5.
		The Pontryagin duality theorem for abelian locally compact groups also has its generalization to the quantum group setting.
	intl.pnas.org /cgi/content/full/97/2/547 (3479 words)

	Untitled Document (Site not responding. Last check: 2007-11-07)
		Abstract: C^-algberas are a special class of norm-closed algerbas of operators on Hilbert space that have special connections with in topology, geometry, harmonic analysis, etc. Any COMMUTATIVE C-algebra is naturally isomorphic to the algebra of complex functions on a compact Hausdorff space naturally associated with it.
		This "Gelfand-Naimark duality" sets up an equivalence between the category of commutative C*-algebras on one hand, and compact Hausdorff spaces on the other.
		We extend this (commutative) Gelfand-Naimark duality to the general (possibly noncommutative) C^*-algebras, by identifying the noncommutative analog of compact Hausdorff spaces.
	www.math.uga.edu /seminars_conferences/Nov_30_04.htm (215 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		"One of the central results in the theory of locally compact abelian groups is the Pontryagin-van Kampen duality theorem which implies that a locally compact abelian group is completely determined by its dual and thus yields a powerful method to study the structure of such groups.
		This approach is made possible by a new and simple proof of the duality theorem, which beyond some basic facts from group theory and topology, presupposes only the Peter-Weyl theorem.
		The duality theorem is proved first for compact and discrete abelian groups and then extended to all locally compact abelian groups.
	uob-community.ballarat.edu.au /~smorris/books.htm (800 words)

	Citebase - The Dynamics of Group Codes: Dual Abelian Group Codes and Systems (Site not responding. Last check: 2007-11-07)
		Authors: Forney Jr., G. David; Trott, Mitchell D. Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed.
		Duals of sequence spaces over locally compact abelian groups may be defined via Pontryagin duality; dual group codes are orthogonal subgroups of dual sequence spaces.
		The dual of a complete code or system is finite, and the dual of a Laurent code or system is (anti-)Laurent.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:cs/0408038 (1324 words)

	Abstract (Site not responding. Last check: 2007-11-07)
		Pontryagin duality relates commutative algebra and dynamical systems.
		Dual to the Alexander module of a d-component link is a compact abelian group with d commuting automorphisms, a dynamical system with attractive properties.
		Periodic point information, for example, is related to the homology of abelian covers of the link.
	www.math.columbia.edu /~ikofman/abstracts/silverwilliams.html (225 words)

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