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This MSci thesis surveys results in extremal graph theory, in particular relating to Hamilton cycles. Szem\'eredi's Regularity Lemma plays a central role. We also investigate the robust outexpansion property for digraphs. Kelly showed that every sufficiently large oriented graph on $n$ vertices with minimum in- and outdegree at least $3n/8 +o(n)$ contains any orientation of a Hamilton cycle. We use Kelly's arguments to extend his result to any robustly expanding digraph of linear degree., Comment: MSci Thesis

We study the mathematical properties of probabilistic processes in which the independent actions of $n$ players (`causes') can influence the outcome of each player (`effects'). In such a setting, each pair of outcomes will generally be statistically correlated, even if the actions of all the players provide a complete causal description of the players' outcomes, and even if we condition on the outcome of any one player's action. This correlation always holds when $n=2$, but when $n=3$ there exists a highly symmetric process, recently studied, in which each cause can influence each effect, and yet each pair of effects is probabilistically independent (even upon conditioning on any one cause). We study such symmetric processes in more detail, obtaining a complete classification for all $n \geq 3$. Using a variety of mathematical techniques, we describe the geometry and topology of the underlying probability space that allows independence and influence to coexist., Comment: 24 pages, 3 figures

We investigate Buchbaum and Eisenbud's construction of the second symmetric power S^2_R(X) of a chain complex X of modules over a commutative ring R. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following version of a result of Avramov, Buchweitz, and Sega: Let R \to S be a module-finite ring homomorphism such that R is noetherian and local, and such that 2 is a unit in R. Let X be a complex of finite rank free S-modules such that X_n = 0 for each n < 0. If \cup_n Ass_R(H_n(X \otimes_S X)) \subseteq Ass(R) and if X_P \simeq S_P for each P \in Ass(R), then X \simeq S., 25 pages, uses xypic. v.2: introduction revised, Theorem 3.1 generalized with part in new Corollary 3.2, and minor changes made throughout. v3: significantly rewritten, final version to appear in Bulletin of the Iranian Mathematical Society

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger., 4 pages

Our main purpose is to give multiple examples for using the available implementations for computing the normalization of an affine ring, computing the minimial generators of the normalization as an algebra over the original ring and integral closures of ideals. Some such examples have been published for Singular, but not for Macaulay 2 and we present both in this paper. We also briefly describe the implementations., To Appear in "Advances in Algebra and Geometry (University of Hyderabad Conference 2001)" . Includes extensive examples. 17 pages

Generic linkage is used to compute a prime ideal such that the radical of the initial ideal of the prime ideal is equal to the radical of a given codimension two monomial ideal that has a Cohen-Macaulay quotient ring., Comment: 18 pages, AMSLaTeX