Inverse Grobner Basis Problem in Codimension Two

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, AMSLaTe

Methods for Computing Normalisations of Affine Rings

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.Comment: To Appear in "Advances in Algebra and Geometry (University of Hyderabad Conference 2001)" . Includes extensive examples. 17 page

Borel Fixed Initial Ideals of Prime Ideals in Dimension Two

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.Comment: 4 page

On the exact decomposition threshold for even cycles

A graph $G$ has a $C_k$-decomposition if its edge set can be partitioned into cycles of length $k$. We show that if $\delta(G)\geq 2|G|/3-1$, then $G$ has a $C_4$-decomposition, and if $\delta(G)\geq |G|/2$, then $G$ has a $C_{2k}$-decomposition, where $k\in \mathbb{N}$ and $k\geq 4$ (we assume $G$ is large and satisfies necessary divisibility conditions). These minimum degree bounds are best possible and provide exact versions of asymptotic results obtained by Barber, K\"uhn, Lo and Osthus. In the process, we obtain asymptotic versions of these results when $G$ is bipartite or satisfies certain expansion properties

Minimal primes of ideals arising from conditional independence statements

We consider ideals arising in the context of conditional independence models that generalize the class of ideals considered by Fink [7] in a way distinct from the generalizations of Herzog-Hibi-Hreinsdottir-Kahle-Rauh [13] and Ay-Rauh [1]. We introduce switchable sets to give a combinatorial description of the minimal prime ideals, and for some classes we describe the minimal components. We discuss many possible interpretations of the ideals we study, including as 2 \times 2 minors of generic hypermatrices. We also introduce a definition of diagonal monomial orders on generic hypermatrices and we compute some Groebner bases.Comment: We shortened and streamlined the paper from 24 to 17 pages, we improved several proofs, we updated references, and we added Groebner bases of certain ideals under t-diagonal orders on generic hypermatrices (a generalization of diagonal orders on variables in a generic matrix). The term "admissible" from previous versions is now changed to "switchable

The regularity method for graphs and digraphs

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 Thesi

A semialgebraic description of the general Markov model on phylogenetic trees

Many of the stochastic models used in inference of phylogenetic trees from biological sequence data have polynomial parameterization maps. The image of such a map --- the collection of joint distributions for a model --- forms the model space. Since the parameterization is polynomial, the Zariski closure of the model space is an algebraic variety which is typically much larger than the model space, but has been usefully studied with algebraic methods. Of ultimate interest, however, is not the full variety, but only the model space. Here we develop complete semialgebraic descriptions of the model space arising from the k-state general Markov model on a tree, with slightly restricted parameters. Our approach depends upon both recently-formulated analogs of Cayley's hyperdeterminant, and the construction of certain quadratic forms from the joint distribution whose positive (semi-)definiteness encodes information about parameter values. We additionally investigate the use of Sturm sequences for obtaining similar results.Comment: 29 pages, 0 figures; Mittag-Leffler Institute, Spring 201

Second symmetric powers of chain complexes

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.Comment: 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 Societ

Relations between semidualizing complexes

We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext^n_R(B,C) for n>>0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext^n_R(B,C)=0 for all n>>0. We introduce and investigate an equivalence relation \approx on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural action of the derived Picard group and is well-suited for the study of semidualizing complexes over nonlocal rings. We identify numerous alternate characterizations of this relation, each of which includes the condition Ext^n_R(B,C)=0 for all n>>0. Finally, we answer our original question in some special cases.Comment: final version, to appear in J. Commutative Algebra, 27 pages, uses XY-pi

Learning to lead : how does camp counseling impact leadership ability?

We live in a world where the words "me" and "I" are used most frequently and the "other" is often forgotten. Despite this, many have found that in order to lead others and eventually get what you want you must serve them and think about their needs first. This is a prevalent concept in the YMCA camping world. From my perspective, the counselors at Camp Tecumseh YMCA are held to high standards of servant leadership. I take a critical look at how camp counseling impacts a person's servant leadership abilities through research, survey, and observation.Honors CollegeThesis (B.?