ρ(I) = sup{m/r | I^(m) ⊈ I^r}.

In particular, if m/r > ρ(I), then I^(m) ⊆ I^r. An interesting problem, then, is to compute ρ(I) for various classes of ideals. Much of the work that has been done on this question involves examining ideals of points in P^N. In Chapter 2 we investigate such questions for an ideal defining a certain configuration of points in P² using a certain k-vector space basis of k[P²] compatible with I^(m) and I^r. We are also able to use this approach to verify several conjectures of Harbourne-Huneke and Bocci-Cooper-Harbourne for our particular class of ideals, and compute some well-known invariants of these ideals, such as α(I^(m)), γ(I), the Castelnuovo-Mumford regularity and the saturation degree. In Chapter 3, we consider a question raised in Bocci and Chiantini's paper which is related to the computation of γ(I). Bocci and Chiantini classify configurations of points in P² based on the difference t = α(I⁽²⁾) − α(I), where I = I(Z) and Z ⊆ P² is a finite set of points. When t = 1, Z is either a set of collinear points or a star configuration of points. We extend that result to configurations of lines in P³.

Adviser: Brian Harbourne

Advisor: Srikanth Iyengar

Advisor: Mikil Foss

1) The projective line over the integers (Chapter 2),

2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3),

3) Fiber products and connected sums of local rings (Chapter 4),

In the first chapter we introduce basic terminology used in this thesis for all three topics.

In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj(Z[h,k]) over the integers Z, and we interpret this poset as Spec(Z[x]) U Spec(Z[1/x]) with an appropriate identification.

We have some new results that support Aihua Li and Sylvia Wiegand's conjecture regarding the characterization of Proj(Z[h,k]). In particular we show that a possible axiom for Proj(Z[h,k]) proposed by Arnavut, Li and Wiegand holds for some previously unknown cases.

We study the sets of prime ideals of polynomial rings, power series rings and mixed power series-polynomial rings in Chapter 3. Let R be a one-dimensional Noetherian domain and let x and y be indeterminates. We describe the prime spectra of certain two-dimensional quotients of mixed power series/polynomial rings over R, that is, Spec(R[[x]][y]/Q) and Spec(R[y][[x]]/Q'), where Q and Q' are certain height-one prime ideals of R[[x]][y] and R[y][[x]] respectively.

In the last chapter we describe some ring-theoretic and homological properties of fiber products and connected sums of local rings. For Gorenstein Artin k-algebras R and S where k is a field, the connected sum, R#_k S, is a quotient of the classical fiber product RX_k S. We give basic properties of connected sums over a field and show that certain Gorenstein local k-algebras decompose as connected sums. We generalize structure theorems given by Sally, Elias and Rossi that show two types of Gorenstein local k-algebras are connected sums.

Advisers: Sylvia Wiegand and Mark Walker

Adviser: Thomas Marley

A linear extension of a poset might be considered "good'' if incomparable elements appear near to one another. The linear discrepancy of a poset is a natural way of measuring just how good the best linear extension of that poset can be, i.e. ld(P)=min_L max _{x || y} |L(x)-L(y)|, where L ranges over all linear extensions of P mapping P to {1,2,..., |P|}. In certain situations, it makes sense to weaken the definition of a linear extension by allowing elements of the poset to be sent to the same integer, while still requiring that x<y implies L(x) < L(y). This is known as a weak labeling. Similar to linear discrepancy, the weak discrepancy measures how nicely we can weakly label the elements of the poset. In this dissertation, we calculate the weak discrepancy of grids, the permutohedron, the partition lattice, and the two-dimensional Young's lattice.

Adviser: Jamie Radcliffe

Adviser: Lynn Erbe and Allan Peterson

In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for Cirsium canescens. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to changes in kernel elements and show that these sensitivities can also vary considerably between the two models.

In Chapter 3 we study the global asymptotic stability of a general model for a plant population with an age-structured seed bank. We show how different assumptions for density-dependent seed production (contest vs. scramble competition) can change whether or not the equilibrium population is globally asymptotically stable. Finally, we consider a more difficult model that does not give rise to a positive system, complicating the global stability proof.

Finally, in Chapter 4 we develop a stochastic integral projection model for a disturbance specialist plant and its seed bank. In years without a disturbance, the population relies solely on its seed bank to persist. Disturbances and a seed's depth in the soil affect the survival and germination probability of seeds in the seed bank, which in turn also affect population dynamics. We show that increasing the frequency of disturbances increases the long-term viability of the population but the relationship between the mean depth of disturbance and the long-term viability of the population is not necessarily monotone for all parameter combinations. Specifically, an increase in the probability of disturbance increases the long-term mean of the total seed-bank population and decreases the probability of quasi-extinction. However, if the probability of disturbance is too low, a larger mean depth of disturbance can actually yield a smaller mean total seed-bank population and a larger quasi-extinction probability, a relationship that switches as the probability of disturbance increases.

Advisers: Richard Rebarber and Brigitte Tenhumberg

In the second chapter, we consider single-parameter function spaces and extend a fundamental integration formula of Paley, Wiener, and Zygmund for an important class of functionals on this space. In the third chapter, we discuss measures on very general function spaces and introduce the specific example of a generalized Wiener space of several parameters; this will be the setting for the fourth chapter, where we extend some interesting results of Cameron and Storvick. In the final chapter, we apply the work of the preceding chapters to the question of reflection principles for single-parameter and multiple-parameter Gaussian stochastic processes.

u_tt – Δu + g₁(u_t) = f₁(u, v)

v_tt – Δv + g₂(v_t) = f₂(u, v);

in a bounded domain Ω subset of the real numbers (Rⁿ) with a nonlinear Robin boundary condition on u and a zero boundary conditions on v. The nonlinearities f₁(u, v) and f₂(u, v) are with supercritical exponents representing strong sources, while g₁(u_t) and g₂(v_t) act as damping. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H¹(Ω) × L²(∂Ω) with boundary data from L²(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this dissertation are non-dissipative and are not locally Lipschitz from H¹(Ω) into L²(Ω) or L²(∂Ω). By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system.

Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good" part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. Moreover, we prove a blow up result for weak solutions with nonnegative initial energy. Finally, we establish important generalization of classical results by H. Brézis in 1972 on convex integrals on Sobolev spaces. These results allowed us to overcome a major technical difficulty that faced us in the proof of the local existence of weak solutions.

Adviser: Allan Donsig

Adviser: Thomas Marley

Adviser: Lynn Erbe and Allan Peterson

First, I will consider extremal problems for trees. In these questions we examine the trees that maximize or minimize various invariants. For instance the number of independent sets, the number of matchings, the number of subtrees, the sum of pairwise distances, the spectral radius, and the number of homomorphisms to a fixed graph. I have two general approaches to these problems. To find the extremal trees in the collection of trees on n vertices with a fixed degree bound I use the certificate method. The certificate is a branch invariant, related to, but not the same as, the original invariant. We exploit the recursive structure of the problem. The second approach is geared towards finding the trees with given degree sequence that are extremal. I have a common approach involving labelings of the vertices corresponding to each invariant; the canonical example of which is labeling the vertices by the components of the leading eigenvector. This approach yields strictly stronger results when combined with a majorization result.

Second, I will consider two problems in graphs reconstruction. For these problems we are given limited information about a graph and decide whether the graph is uniquely determined by this data. The first problem is reconstruction of trees from their k-subtree matrix; a generalization of the Wiener matrix. This includes the problem of reconstruction from the Wiener matrix which was an open problem. Two vertices are adjacent if the corresponding entry is the largest in either its row or its column. The second problem is reconstructing graphs from metric balls of their vertices. I give a solution to the conjecture that every graph with no pendant vertices and girth at least 2r + 3 can be reconstructed from its metric balls of radius r. We do so by examining the intersections of metric balls and their sizes.

Adviser: Jamie Radclie

Adviser: Mohammad A. Rammaha

Adviser: Stephen G. Hartke

Adviser: Srikanth B. Iyengar