Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 270-272. / by Stephen Kinyon Young. / Ph.D.

Let R be an integral domain. An atom is a nonzero nonunit x of R where x = yz implies that either y or z is a unit. We say that R is an atomic domain if each nonzero nonunit is a finite product of atoms. An atomic domain with only finitely many nonassociate atoms is called a Cohen-Kaplansky (CK) domain. We will investigate atoms in integral domains R with a unique maximal ideal M. Of particular interest will be atoms that are not in M^2. After studying the atoms in integral domains, we will narrow our focus to CK domains with a unique maximal ideal M. In this pursuit, we investigate atoms in M^2 for these CK domains. We will show that the minimal number of atoms needed to have an atom in M^2 is exactly eight. This disproves a conjecture given by Cohen and Kaplansky in 1946 that the minimal number would be ten. We then classify complete local CK domains with exactly three atoms.

In the bacterium Escherichia coli, cell division involves the concerted inward growth of all three layers of the cell envelope: the cytoplasmic membrane, the peptidoglycan (PG) cell wall, and the outer membrane. This is a complex, highly regulated process that involves over 20 proteins. Four of these proteins contain a domain of ~70 amino acids known as a SPOR domain (Pfam no. 05036). One of these SPOR domains (from a protein named FtsN) has been shown previously to bind PG. In this thesis we show that six additional SPOR domains, three from E. coli and three from other bacterial species, also bind PG. Thus, PG binding is a general activity of SPOR domains. We then examine the SPOR domain from DamX of E. coli in detail. In collaboration with Dr. Andrew Fowler of the NMR Core Facility, we determined the solution structure of the domain. The domain adopts an "RNP fold," characterized by a four-stranded anti-parallel β-sheet that is buttressed on one side by α-helixes. Several mutant forms of the DamX SPOR domain were constructed and studied both in vivo and in vitro. These studies support the following inferences: 1) The β-sheet is the PG-binding site; 2) The β-sheet contains critical information for targeting the SPOR domain to the midcell; 3) The SPOR domain probably localizes to the midcell by binding preferentially to septal PG; and 4) It follows, then, that septal PG must differ from PG elsewhere around the cell. We suggest that further studies of the SPOR:PG interaction will yield novel insights into PG biogenesis during septation. This thesis also presents an in vivo characterization of several mutant forms of a cytoplasmic membrane protein named FtsW, homologs of which are found in all bacteria that contain a PG cell wall. FtsW recruits a PG synthase named FtsI to the division site and might also transport PG precursors across the cytoplasmic membrane. We systematically mutagenized each of FtsW's ten transmembrane (TM) helixes and investigated the ability of the mutant proteins to support division, localize to the division site, and recruit FtsI. This characterization leads us to propose that TM1 is involved in targeting FtsW to the division site, TM4 is involved in the putative transport activity, and TM10 is involved in recruitment of FtsI.

In this thesis, we will discuss two separate topics. First, we find a critical knot for an knot energy function. A knot is a closed curve or polygon in three space. It is possible to for a computer to simulate the flow of a knot to its minimum energy conformation. There is no guarantee, however, that a true minimizer exists near the computer's alleged minimizer. We take advantage of both the symmetry of the minimizer and the symmetry invariance of the energy function to prove that there is a critical point of the energy function near the computer's minimizer. Second, we will discuss how to determine the number of complementary domains of arrangements of algebraic curves in 2-space and ellipsoids in 3-space. In each of these situations, we supply equations that provide an upper bound for the number of complementary domains. These upper bounds are applicable even when the exact intersections between the curves or surfaces are unknown.

This dissertation consists of three parts. In the

We study the Neumann problem for the de Rham complex in a bounded domain of Rn with singularities on the boundary. The singularities may be general enough, varying from Lipschitz domains to domains with cuspidal edges on the boundary. Following Lopatinskii we reduce the Neumann problem to a singular integral equation of the boundary. The Fredholm solvability of this equation is then equivalent to the Fredholm property of the Neumann problem in suitable function spaces. The boundary integral equation is explicitly written and may be treated in diverse methods. This way we obtain, in particular, asymptotic expansions of harmonic forms near singularities of the boundary.

We show that the spectral kernel function of the dbar-Neumann problem on a non-compact strongly pseudoconvex manifold is smooth up to the boundary.

This dissertation consists of three parts. In the

Network-on-Chip (NoC) is emerging as a promising communication structure, which is scalable with respect to chip complexity. Meanwhile, latest chip designs are increasingly leveraging multiple voltage-frequency domains for energy-efficiency improvement. In this work, we propose a simultaneous task and voltage scheduling algorithm for energy minimization in NoC based designs. The energy-latency tradeoff is handled by Lagrangian relaxation. The core algorithm is a clustering based approach which not only assigns voltage levels and starting time to each task (or Processing Element) but also naturally finds voltage-frequency clusters. Compared to a recent previous work, which performs task scheduling and voltage assignment sequentially, our method leads to an average of 20 percent energy reduction.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 83-89). Also available on the Internet.