Theta functions and symplectic groups

Reiter, Hans

doi:10.1007/bf01299149

Cited by 5 publications

(3 citation statements)

References 3 publications

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“…albopictus, each seaport and airport location was overlaid on a map of the mosquito's Old World range ( Fig. 1) (20,24,48) and classified as either inside or outside the historical distribution. Those seaports͞airports within the distribution were located on the relevant dendrogram.…”

Section: Methodsmentioning

confidence: 99%

Global traffic and disease vector dispersal

Tatem

Hay

Rogers

2006

Proc. Natl. Acad. Sci. U.S.A.

484

377

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The expansion of global air travel and seaborne trade overcomes geographic barriers to insect disease vectors, enabling them to move great distances in short periods of time. Here we apply a coupled human-environment framework to describe the historical spread of Aedes albopictus, a competent mosquito vector of 22 arboviruses in the laboratory. We contrast this dispersal with the relatively unchanged distribution of Anopheles gambiae and examine possible future movements of this malaria vector. We use a comprehensive database of international ship and aircraft traffic movements, combined with climatic information, to remap the global transportation network in terms of disease vector suitability and accessibility. The expansion of the range of Ae. albopictus proved to be surprisingly predictable using this combination of climate and traffic data. Traffic volumes were more than twice as high on shipping routes running from the historical distribution of Ae. albopictus to ports where it has established in comparison with routes to climatically similar ports where it has yet to invade. In contrast, An. gambiae has rarely spread from Africa, which we suggest is partly due to the low volume of sea traffic from the continent and, until very recently, a European destination for most flights.Aedes albopictus ͉ air travel ͉ Anopheles gambiae ͉ biological invasion ͉ shipping

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Section: Methodsmentioning

confidence: 99%

Global traffic and disease vector dispersal

Tatem

Hay

Rogers

2006

Proc. Natl. Acad. Sci. U.S.A.

484

377

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“…In this famous paper Weil introduced the metaplectic representation, which had independently been found by Shale. Weil's new methods and objects have influenced many mathematicians in their work on theta functions, most notably Cartier, Igusa and Reiter in [3,30,[50][51][52]. In his work on abelian varieties Mumford had demonstrated the relevance of the Heisenberg group in the algebraization of theta functions, cf.…”

Section: Introductionmentioning

confidence: 99%

Quantum Theta Functions and Gabor Frames for Modulation Spaces

Luef

Manin

2009

Lett Math Phys

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Representations of the celebrated Heisenberg commutation relations in quantum mechanics (and their exponentiated versions) form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we will try to bridge the two communities, represented by the two co-authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show, e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.

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“…In this way one achieves a description of / G ^o(IK'') as a sum of band-pass signals. This is what Hans Reiter really used, in [100,101]. (20) Since the choice of the window in the definition of modulation spaces gives these definition some smell of arbitrariness, some people prefer the characterization of 5o(R'') using the (quadratic) Wigner distribution as a suitable alternatively, despite the fact that from the description below it is a-priori not clear why 5o(R'') should be a linear manifold.…”

Section: It Is Continuously Embedded Into Any Other Space With This Pmentioning

confidence: 99%

Banach Gelfand Triples for Applications in Physics and Engineering

Feichtinger¹

2009

AIP Conference Proceedings

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The principle of extension is widespread within mathematics. Starting from simple objects one constructs more sophisticated ones, with a kind of natural embedding from the set of old objects to the new, enlarged set. Usually a set of operations on the old set can still be carried out, but maybe also some new ones. Done properly one obtains more completed objects of a similar kind, with additional useful properties. Let us give a simple example: While multiplication and addition can be done exactly and perfectly in the setting of Q, the rational numbers, the field R of real numbers has the advantage of being complete (Cauchy sequences have a limit ...) and hence allowing for numbers Uke : 7r or \/2. Finally the even "more complicated" field C of complex numbers allows to find solutions to equations like z^ = -1. The chain of inclusions of fields, Q C R C C is a good motivating example in the domain of "numbers".The main subject of the present survey-type article is a new theory of Banach Gelfand triples (BGTs), providing a similar setting in the context of (generalized) functions. Test functions are the simple objects, elements of the Hilbert space L^(R'*) are well suited in order to describe concepts of orthogonaUty, and they can be approximated to any given precision (in the 11 • 112-norm) by test functions. Finally one needs an even larger (Banach) space of generalized functions resp. distributions, containing among others pure frequencies and Dirac measures in order to describe various mappings between such Banach Gelfand triples in terms of the most important "elementary building blocks", in a clear analogy to the finite/discrete setting (where Dirac measures correspond to unit vectors).Our concrete Banach Gelfand triple is based on the Segal algebra 5o(R'*), which coincides with the modulation space M^(R'*) = MQ' (R**), and plays a very important and natural role for timefrequency analysis. We will point out that it provides the appropriate setting for a description of many problems in engineering or physics, including the classical Fourier transform or the Kohn-Nirenberg or Weyl calculus for pseudo-differential operators. Particular emphasis will be given to the concept of w*-convergence and w*-continuity of operators which allows to prove conceptual uniqueness results, and to give a correct interpretation to certain formal expressions coming up in various versions of the Dirac formalism.

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Theta functions and symplectic groups

Cited by 5 publications

References 3 publications

Global traffic and disease vector dispersal

Global traffic and disease vector dispersal

Quantum Theta Functions and Gabor Frames for Modulation Spaces

Banach Gelfand Triples for Applications in Physics and Engineering

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