Thesis (M.Sc. (Mathematics) -- University of Limpopo, 2010 / Summary not available

The classical Riemann-Roch theorem is extended to solutions of elliptic equations on manifolds with conical points.

In his paper, Borcherds introduced a theta lift which allowed him to lift classical modular forms with poles at cusps to automorphic forms on the orthogonal group O(2, l). The resulting automorphic forms, called Borcherds products, possess an infinite product expansion and have their singularities located along certain arithmetic divisors, the so-called Heegner divisors. Mainly based on the work of Bruinier, we study the question whether every automorphic form having its divisor along the Heegner divisors can be realized as a Borcherds product.

Factorization theory is concerned with the decomposition of mathematical objects. Such an object could be a polynomial, a number in the set of integers, or more generally an element in a ring. A classic example of a ring is the set of integers. If we take any two integers, for example 2 and 3, we know that $2 \cdot 3=3\cdot 2$, which shows that multiplication is commutative. Thus, the integers are a commutative ring. Also, if we take any two integers, call them $a$ and $b$, and their product $a\cdot b=0$, we know that $a$ or $b$ must be $0$. Any ring that possesses this property is called an integral domain. If there exist two nonzero elements, however, whose product is zero we call such elements zero divisors. This thesis focuses on factorization in commutative rings with zero divisors. In this work we extend the theory of factorization in commutative rings to polynomial rings with zero divisors. For a commutative ring $R$ with identity and its polynomial extension $R[X]$ the following questions are considered: if one of these rings has a certain factorization property, does the other? If not, what conditions must be in place for the answer to be yes? If there are no suitable conditions, are there counterexamples that demonstrate a polynomial ring can possess one factorization property and not another? Examples are given with respect to the properties of atomicity and ACCP. The central result is a comprehensive characterization of when $R[X]$ is a unique factorization ring.

This thesis will survey a group of problems related to certain number-theoretic functions. In particular, for said functions, these problems take the form of when and how often they are equal over consecutive integers, n and n+1. The first chapter will introduce the functions and the histories of the related problems. The second chapter will take on a variant of the Ruth-Aaron pairs problem, which asks how often sums of primes of two consecutive integers are equal. The third chapter will examine, in depth, a proof by D.R. Heath-Brown of the infinitude of consecutive integer pairs with the same number of divisors---i.e. such that d(n)=d(n+1). After that we examine a similar proof of the infinitude of pairs with the same number of prime factors---ω(n)=ω(n+1).

This thesis will survey a group of problems related to certain number-theoretic functions. In particular, for said functions, these problems take the form of when and how often they are equal over consecutive integers, n and n+1. The first chapter will introduce the functions and the histories of the related problems. The second chapter will take on a variant of the Ruth-Aaron pairs problem, which asks how often sums of primes of two consecutive integers are equal. The third chapter will examine, in depth, a proof by D.R. Heath-Brown of the infinitude of consecutive integer pairs with the same number of divisors---i.e. such that d(n)=d(n+1). After that we examine a similar proof of the infinitude of pairs with the same number of prime factors---ω(n)=ω(n+1).

The Atiyah Conjecture originates from a paper written 40 years ago by Sir Michael Atiyah, a famous mathematician and Fields medalist. Since publication of the paper, mathematicians have been working to solve many questions related to the conjecture, but it is still open. The conjecture is about certain topological invariants attached to a group 𝐺. There are examples showing that the conjecture does not hold in general. These examples involve something like the lamplighter group (the wreath product ℤ/2ℤ ≀ ℤ). We are interested in looking at examples where this is not the case. We are interested in the specific case where 𝐺 is a finitely generated group in which the Prüfer group can be embedded as the center. The Prüfer group is a 𝑝-group for some prime 𝑝 and its finite subgroups have unbounded order, in particular the finite subgroups of G will have unbounded order. To understand whether any form of the Atiyah conjecture is true for 𝐺, it will first help to determine whether the group ring 𝑘𝐺 of the group 𝐺 has a classical ring of quotients for some field 𝑘. To determine this we will need to know the zero divisors for the group ring 𝑘𝐺. Our investigations will be divided into two cases, namely when the characteristic of the field 𝑘 is the same as the prime p for the Prüfer group and when it is different. / Master of Science

This thesis generalizes to digraphs certain recent results about graphs. There are special digraphs C such that AxC is isomorphic to BxC for some pair of distinct digraphs A and B. Lovasz named these digraphs C zero-divisors and completely characterized their structure. Knowing that all directed cycles are zero-divisors, we focus on the following problem: Given any directed cycle D and any digraph A, enumerate all digraphs B such that AxD is isomorphic to BxD. From our result for cycles, we generalize to an arbitrary zero-divisor C, developing upper and lower bounds for the collection of digraphs B satisfying AxC isomorphic to BxC.

This thesis discusses the Kaplansky zero divisor conjecture. The conjecture states that a group ring of a torsion-free group over a field has no nonzero zero divisors. There are situations for which this conjecture is known to hold, such as linearly orderable groups, unique product groups, solvable groups, and elementary amenable groups. This paper considers the possibility that the conjecture is false and there is some counterexample in existence. The approach to searching for such a counterexample discussed here is to first find a torsion-free group that has subsets A and B such that AB has no unique product. We do this by exhaustively searching for the subsets A and B with fixed small sizes. When |A| = 1 or 2 and |B| is arbitrary we know that AB contains a unique product, but when |A| is larger, not much was previously known. After an example is found we then verify that the sets are contained in a torsion-free group and further investigate whether the group ring yields a nonzero zero divisor. Together with Dr. Pace P. Nielsen, assistant math professor of Brigham Young University, we created code that was implemented in Magma, a computational algebra system, for the purpose of considering each size of A and B and running through each case. Along the way we check for the possibility of torsion elements and for other conditions that lead to contradictions, such as a decrease in the size of A or B. Our results are the following: If A and B are sets of the sizes below contained in a torsion-free group, then they must contain a unique product. |A| = 3 and |B| ≤ 16; |A| = 4 and |B| ≤ 12; |A| = 5 and |B| ≤ 9; |A| = 6 and |B| ≤ 7. We have continued to run cases of larger size and hope to increase the size of B for each size of A. Additionally, we found a torsion-free group containing sets A and B, both of size 8, where AB has no unique product. Though this group does not yield a counterexample for the Kaplansky zero divisor conjecture, it is the smallest explicit example of a non-uniqueproduct group in terms of the size of A and B.

The study of factorization in integral domains has a long history. Unique factorization domains, like the integers, have been studied extensively for many years. More recently, mathematicians have turned their attention to generalizations of this such as Dedekind domains or other domains which have weaker factorization properties. Many authors have sought to generalize the notion of factorization in domains. One particular method which has encapsulated many of the generalizations into a single study is that of tau-factorization, studied extensively by A. Frazier and D.D. Anderson. Another generalization comes in the form of studying factorization in rings with zero-divisors. Factorization gets quite complicated when zero-divisors are present due to the existence of several types of associate relations as well as several choices about what to consider the irreducible elements. In this thesis, we investigate several methods for extending the theory of tau-factorization into rings with zero-divisors. We investigate several methods including: 1) the approach used by A.G. Agargun and D.D. Anderson, S. Chun and S. Valdes-Leon in several papers; 2) the method of U-factorization developed by C.R. Fletcher and extended by M. Axtell, J. Stickles, and N. Baeth and 3) the method of regular factorizations and 4) the method of complete factorizations. This thesis synthesizes the work done in the theory of generalized factorization and factorization in rings with zero-divisors. Along the way, we encounter several nice applications of the factorization theory. Using tau_z-factorizations, we discover a nice relationship with zero-divisor graphs studied by I. Beck as well as D.D. Anderson, D.F. Anderson, A. Frazier, A. Lauve, and P. Livingston. Using tau-U-factorization, we are able to answer many questions that arise when discussing direct products of rings. There are several benefits to the regular factorization factorization approach due to the various notions of associate and irreducible coinciding on regular elements greatly simplifying many of the finite factorization property relationships. Complete factorization is a very natural and effective approach taken to studying factorization in rings with zero-divisors. There are several nice results stemming from extending tau-factorization in this way. Lastly, an appendix is provided in which several examples of rings satisfying the various finite factorization properties studied throughout the thesis are given.