Pure Mathematics

Permanent URI for this collectionhttps://uwspace.uwaterloo.ca/handle/10012/9932

This is the collection for the University of Waterloo's Department of Pure Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 6 of 6

Algebraic Approaches to State Complexity of Regular Operations
(University of Waterloo, 2019-10-15) Davies, Sylvie; Bell, Jason; Brzozowski, John
The state complexity of operations on regular languages is an active area of research in theoretical computer science. Through connections with algebra, particularly the theory of semigroups and monoids, many problems in this area can be simplified or completely reduced to combinatorial problems. We describe various algebraic techniques for attacking state complexity problems. We present a general method for constructing witness languages for operations -- languages that attain the worst-case state complexity when used as the argument(s) of the operation. Our construction is based on full transformation monoids, which contain all functions from a finite set into itself. When a witness for an operation is known, determining the state complexity essentially becomes a counting problem. These counting problems, however, are not necessarily easy, and the witness languages produced by this method are not ideal in the sense that they have extremely large alphabets. We thus investigate some commonly used operations in detail, and look for algebraic techniques to simplify the combinatorial side of state complexity problems and to simplify the search for small-alphabet witnesses. For boolean operations (e.g., union, intersection, difference) we show that these combinatorial problems can be solved easily in special cases by studying the subgroup of permutations in the syntactic monoid of a witness candidate. If the subgroup of permutations is known to have some strong transitivity property, such as primitivity or 2-transitivity, we can draw conclusions about the worst-case state complexity when this language is used in a boolean operation. For the operations of concatenation and Kleene star (an iterated version of concatenation), we describe a “construction set” method to simplify state complexity lower-bound proofs, and determine some algebraic conditions under which this method can be applied. For the reversal operation, we show that the state complexity of the reverse of a language is closely related to the syntactic monoid of the language, and use this fact to investigate a generalized version of the reversal state complexity problem. After describing our techniques, we demonstrate them by applying them to some classical state complexity problems. We obtain complex generalizations of the classical results that would be difficult to prove without the machinery we develop.
Contributions to the Theory of Radicals for Noncommutative Rings
(University of Waterloo, 2017-06-20) Madill, Blake; Bell, Jason
We consider several radical classes of noncommutative rings. In particular, we provide new results regarding the radical theory of semigroup-graded rings, monomial algebras, and Ore extensions of derivation type. In Chapter 2 we prove that every ring graded by a torsion free nilpotent group has a homogeneous upper nilradical. Moreover, we show that a ring graded by unique-product semigroup is semiprime only when it has no nonzero nilpotent homogeneous ideals. This provides a graded-analogue of the classical result that says a ring is semiprime if and only if it has no nonzero nilpotent ideals. In Chapter 3, the class of iterative algebras are introduced. Iterative algebras serve as an interesting class of monomial algebras where ring theoretical information may be determined from combinatorial information of an associated right-infinite word. We use these monomial algebras to construct an example of a prime, semiprimitive, graded-nilpotent algebra of Gelfand-Kirillov dimension 2 which is finitely generated as a Lie algebra. In Chapter 4, the Jacobson radical of Ore extensions of derivation type is discussed. We show that if $R$ is a locally nilpotent ring which satisfies a polynomial identity, then any Ore extension of $R$ of derivation type will also be locally nilpotent. Finally, we show that the Jacobson radical of any Ore extension of derivation type of a polynomial identity ring has a nil coefficient ring.
On Hopf Ore Extensions and Zariski Cancellation Problems
(University of Waterloo, 2020-04-29) Huang, Hongdi; Bell, Jason
In this thesis, we investigate Ore extensions of Hopf algebras and the Zariski Cancellation problem for noncommutative rings. In particular, we improve upon the existing conditions for when $T=R[x; \sigma, \delta]$ is a Hopf Ore extension of a Hopf algebra $R$, and we give noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer. In Chapter 3, we study the relationship between prime ideals of $T=R[x; \sigma, \delta]$ and their contractions under $R$. In Chapter 4, we look at when $T$ is a Hopf algebra and by studying the coproduct of $x$, $\Delta(x)$, we provide a sequence of results that answers a question due to Panov; that is, given a Hopf algebra $R$, for which automorphisms $\sigma$ and $\sigma$-derivations $\delta$ does the Ore extension $T=R[x; \sigma, \delta]$ have a Hopf algebra structure extending the given Hopf algebra structure on $R$? In Chapter 5, we consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has a positive characteristic, giving a counterexample to a conjecture of Tang et al. In Chapter 6, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings.
On the Dynamical Wilf-Zeilberger Problem
(University of Waterloo, 2022-08-15) Sun, Yuxuan; Bell, Jason
In this paper, we give an algorithmic solution to a dynamical analog of the problem of certifying combinatorial identities by Wilf-Zeilberger pairs. Given two sequences generated in a dynamical setting, we calculate an upper bound N ≥ 1 such that whenever the first N terms of the two sequences agree pairwise, the two sequences agree term-by-term. Then, we give an algorithm that can be used to check whether two such sequences agree term-by-term. Our methods are mainly based on the theory of Chow rings of algebraic varieties.
Recurrence in Algebraic Dynamics
(University of Waterloo, 2020-07-28) Hossain, Ehsaan; Bell, Jason
The Dynamical Mordell--Lang Conjecture states that if a polynomial orbit has infinite intersection with a closed set in an algebraic variety, then the intersection must occur periodically. Although this problem is unsolved in general, a "weak" version of Bell--Ghioca--Tucker obtains the periodicity in the case when the orbit intersection has positive density. The thesis regards a circle of problems exhibiting a Bell--Ghioca--Tucker-type phenomenon in group theory and number theory. In Chapter 1, we explain how the BGT Theorem is essentially a noetherian version of the classic Poincare Recurrence Theorem from ergodic theory; in addition to a generalization of the BGT Theorem to amenable semigroups, we prove a combinatorial analog involving idempotent ultrafilters for any semigroup. In Chapter 2, we prove a version of the BGT Theorem for an automorphism of a polycyclic-by-finite groups, which embellishes the point that noetherian objects obey a Dynamical Mordell--Lang principle. In Chapter 3, in joint work with Bell and Chen, we study the sequence of complex numbers obtained by evaluating a rational function along an orbit; examining the intersection of this sequence with a finitely generated group of units, we again obtain a BGT-type result for such dynamical sequences, and recovering classic theorems of Methfessel, Polya, and Bezivin.
Sparse Automatic Sets
(University of Waterloo, 2020-11-26) Albayrak, Seda; Bell, Jason
The theory of automatic sets and sequences arises naturally in many different areas of mathematics, notably in the study of algebraic power series in positive characteristic, due to work of Christol, and in Derksen's classification of zero sets for sequences satisfying a linear recurrence over fields of positive characteristic. A fundamental dichotomy for automatic sets shows that they are either sparse, having counting functions that grow relatively slowly, or they are not sparse, in which case their counting functions grow reasonably fast. While this dichotomy has been known to hold for some time, there has not---to this point in time---been a systematic study of the algebraic and number theoretic properties of sparse automatic sets. This thesis rectifies this situation and gives multiple results dealing specifically with sparse automatic sets. In particular, we give a stronger version of a classical result of Cobham for automatic sets where one now specializes to sparse automatic sets; we then prove that a conjecture of Erdos and Turan holds for automatic sets, again using the theory of sparseness; finally, we give a refinement of a classical result of Christol where we consider algebraic power series whose support set is a sparse automatic set.

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Browsing Pure Mathematics by Author "Bell, Jason"