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Aspects of Computable Analysis

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posted on 2021-11-15, 17:55 authored by Porter, Michelle

Computable analysis has been well studied ever since Turing famously formalised the computable reals and computable real-valued function in 1936. However, analysis is a broad subject, and there still exist areas that have yet to be explored. For instance, Sierpinski proved that every real-valued function ƒ : ℝ → ℝ is the limit of a sequence of Darboux functions. This is an intriguing result, and the complexity of these sequences has been largely unstudied. Similarly, the Blaschke Selection Theorem, closely related to the Bolzano-Weierstrass Theorem, has great practical importance, but has not been considered from a computability theoretic perspective. The two main contributions of this thesis are: to provide some new, simple proofs of fundamental classical results (highlighting the role of ∏0/1 classes), and to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpinski, and the Blaschke Selection Theorem. This thesis focuses on classical computable analysis. It does not make use of effective measure theory.

History

Copyright Date

2016-01-01

Date of Award

2016-01-01

Publisher

Te Herenga Waka—Victoria University of Wellington

Rights License

Author Retains Copyright

Degree Discipline

Mathematics

Degree Grantor

Te Herenga Waka—Victoria University of Wellington

Degree Level

Masters

Degree Name

Master of Science

ANZSRC Type Of Activity code

1 PURE BASIC RESEARCH

Victoria University of Wellington Item Type

Awarded Research Masters Thesis

Language

en_NZ

Victoria University of Wellington School

School of Mathematics, Statistics and Operations Research

Advisors

Downey, Rod; Day, Adam