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Partial Continuous Functions and Admissible Domain Representations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2007 (English)In: Journal of logic and computation (Print), ISSN 0955-792X, E-ISSN 1465-363X, Vol. 17, no 6, 1063-1081 p.Article in journal (Refereed) Published
Abstract [en]

It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The raison detre for introducing partial continuous functions is that by passing to partial maps, we are free to consider totalities which are not dense. We show that the category of admissibly representable spaces with morphisms functions which are representable by it partial continuous function is Cartesian closed. Finally, we consider the question of effectivity.

Place, publisher, year, edition, pages
2007. Vol. 17, no 6, 1063-1081 p.
National Category
URN: urn:nbn:se:uu:diva-96206DOI: 10.1093/logcom/exm034ISI: 000252665100004OAI: oai:DiVA.org:uu-96206DiVA: diva2:170703
Available from: 2007-09-21 Created: 2007-09-21 Last updated: 2011-03-24Bibliographically approved
In thesis
1. Effective Distribution Theory
Open this publication in new window or tab >>Effective Distribution Theory
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we introduce and study a notion of effectivity (or computability) for test functions and for distributions. This is done using the theory of effective (Scott-Ershov) domains and effective domain representations.

To be able to construct effective domain representations of the spaces of test functions considered in distribution theory we need to develop the theory of admissible domain representations over countable pseudobases. This is done in the first paper of the thesis. To construct an effective domain representation of the space of distributions, we introduce and develop a notion of partial continuous function on domains. This is done in the second paper of the thesis. In the third paper we apply the results from the first two papers to develop an effective theory of distributions using effective domains. We prove that the vector space operations on each space, as well as the standard embeddings into the space of distributions effectivise. We also prove that the Fourier transform (as well as its inverse) on the space of tempered distributions is effective. Finally, we show how to use convolution to compute primitives on the space of distributions. In the last paper we investigate the effective properties of a structure theorem for the space of distributions with compact support. We show that each of the four characterisations of the class of compactly supported distributions in the structure theorem gives rise to an effective domain representation of the space. We then use effective reductions (and Turing-reductions) to study the reducibility properties of these four representations. We prove that three of the four representations are effectively equivalent, and furthermore, that all four representations are Turing-equivalent. Finally, we consider a similar structure theorem for the space of distributions supported at 0.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2007. 36 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 50
Computable mathematics, computable analysis, domain theory, domain representations, distribution theory.
National Category
Algebra and Logic
urn:nbn:se:uu:diva-8210 (URN)978-91-506-1956-0 (ISBN)
Public defence
2007-10-12, 2001, Ångströmlaboratoriet, Lägerhyddsvägen 1, Polacksbacken, Uppsala, 10:15
Available from: 2007-09-21 Created: 2007-09-21Bibliographically approved

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