Counting prime polynomials and measuring complexity and similarity of information

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dc.contributor.authorRebenich, Niko
dc.contributor.supervisorGulliver, T. Aaron
dc.contributor.supervisorNeville, Stephen William
dc.date.accessioned2016-05-02T18:03:35Z
dc.date.available2016-05-02T18:03:35Z
dc.date.copyright2016en_US
dc.date.issued2016-05-02
dc.degree.departmentDepartment of Electrical and Computer Engineeringen_US
dc.degree.levelDoctor of Philosophy Ph.D.en_US
dc.description.abstractThis 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.en_US
dc.description.proquestcode0544 0984 0405en_US
dc.description.proquestemailnrebenich@gmail.comen_US
dc.description.scholarlevelGraduateen_US
dc.identifier.bibliographicCitationN. Rebenich, U. Speidel, S. Neville and T. A. Gulliver, “FLOTT – A fast, low memory T-transform algorithm for measuring string complexity,” IEEE Trans. Comput., vol. 63, no. 4, pp. 917–926, Apr. 2014.en_US
dc.identifier.urihttp://hdl.handle.net/1828/7251
dc.languageEnglisheng
dc.language.isoenen_US
dc.rightsAvailable to the World Wide Weben_US
dc.rights.urihttp://creativecommons.org/licenses/by/2.5/ca/*
dc.subjectprime numbersen_US
dc.subjectprime polynomialsen_US
dc.subjectfinite fieldsen_US
dc.subjectirreducible polynomialsen_US
dc.subjectpolylogarithmen_US
dc.subjectLerch transcendenten_US
dc.subjectEulerian polynomialsen_US
dc.subjectT-codesen_US
dc.subjectNCDen_US
dc.subjectnecklacesen_US
dc.subjectstring complexityen_US
dc.subjectinformation distanceen_US
dc.subjectdivergent seriesen_US
dc.subjectasymptotic approximationen_US
dc.subjectrandomess testen_US
dc.subjectconditional complexityen_US
dc.subjectnecklace factorizationen_US
dc.subjectprime number theoremen_US
dc.subjectcombinatoricsen_US
dc.subjectT-complexityen_US
dc.subjectdata miningen_US
dc.subjectoptimal truncationen_US
dc.subjectfunction fieldsen_US
dc.subjectlyndon wordsen_US
dc.subjectlyndon factorizationen_US
dc.subjectsimilarity measureen_US
dc.subjectasymptotic enumerationen_US
dc.titleCounting prime polynomials and measuring complexity and similarity of informationen_US
dc.typeThesisen_US

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