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dc.contributor.authorHaque, Sajed 15:59:30 (GMT) 15:59:30 (GMT)
dc.description.abstractThe discriminator of an integer sequence \textbf{s} = $(s(n))_{n \geq 0}$, first introduced by Arnold, Benkoski and McCabe in 1985, is the function $D_s (n)$ that maps the integer $n \geq 1$ to the smallest positive integer $m$ such that the first $n$ terms of \textbf{s} are pairwise incongruent modulo $m$. In this thesis, we provide a basic overview of discriminators, examining the background literature on the topic and presenting some general properties of discriminators. We also venture into various computational aspects relating to discriminators, such as providing algorithms to compute the discriminator, and establishing an upper bound on the discriminator growth rate. We provide a complete characterization of sequences whose discriminators are themselves, and also explore the problem of determining whether a given sequence is a discriminator of some other sequence with some partial results and algorithms. We briefly discuss some $k$-regular sequences, characterizing the discriminators for the evil and odious numbers, and show that $k$-regular sequences do not necessarily have $k$-regular discriminators. We introduce the concept of shift-invariant discriminators, i.e. discriminators that remain the same even if the original sequence is shifted, and present a class of exponential sequences with this property. Finally, we provide a complete characterization of quadratic sequences with discriminator $p^{\lceil \log_p n \rceil}$ for primes $p \neq 3$, and provide some partial results for the case of $p = 3$.en
dc.publisherUniversity of Waterlooen
dc.subjectinteger sequencesen
dc.subjectquadratic sequencesen
dc.titleDiscriminators of Integer Sequencesen
dc.typeMaster Thesisen
dc.pendingfalse R. Cheriton School of Computer Scienceen Scienceen of Waterlooen
uws-etd.degreeMaster of Mathematicsen
uws.comment.hiddenSorry, this is the second submission after the first one was already reviewed. I tried to remove the previous file as instructed (in order to replace it), but it seems I ended up removing the entire submission, so I couldn't find any other options left but to start a new submission.en
uws.contributor.advisorShallit, Jeffrey
uws.contributor.affiliation1Faculty of Mathematicsen

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