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Counting prime polynomials and measuring complexity and similarity of information

This dissertation explores an analogue of the prime number theorem for polynomials over finite fields as well as its connection to the necklace factorization algorithm T-transform and the string complexity measure T-complexity. Specifically, a precise asymptotic expansion for the prime polynomial counting function is derived. The approximation given is more accurate than previous results in the literature while requiring very little computational effort. In this context asymptotic series expansions for Lerch transcendent, Eulerian polynomials, truncated polylogarithm, and polylogarithms of negative integer order are also provided. The expansion formulas developed are general and have applications in numerous areas other than the enumeration of prime polynomials.

A bijection between the equivalence classes of aperiodic necklaces and monic prime polynomials is utilized to derive an asymptotic bound on the maximal T-complexity value of a string. Furthermore, the statistical behaviour of uniform random sequences that are factored via the T-transform are investigated, and an accurate probabilistic model for short necklace factors is presented.

Finally, a T-complexity based conditional string complexity measure is proposed and used to define the normalized T-complexity distance that measures similarity between strings. The T-complexity distance is proven to not be a metric. However, the measure can be computed in linear time and space making it a suitable choice for large data sets. / Graduate / 0544 0984 0405 / nrebenich@gmail.com

Identiferoai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/7251
Date02 May 2016
CreatorsRebenich, Niko
ContributorsGulliver, T. Aaron, Neville, Stephen William
Source SetsUniversity of Victoria
LanguageEnglish, English
Detected LanguageEnglish
TypeThesis
RightsAvailable to the World Wide Web, http://creativecommons.org/licenses/by/2.5/ca/

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