M.Sc.

M.Sc. (Mathematics) / Any sensible piece of writing has an intended readership. Conversely, any piece of writing that has no intended readership has no sense. These are axioms of authorship and necessary directions to any prospective author. The aim of this dissertation was to serve as an experimental exposition of the analysis of the resolvent operator. Its intended readership is therefore graduate-level students in operator theory and Banach algebras. The analysis included in this dissertation is of a specific kind: it includes and occasionally extends beyond the analysis of a function at certain of its singularities of finite order. The exposition is experimental in the sense that it does not even aim at a comprehensive review of analysis of the resolvent operator, but it is concerned with that part of it which seems to have interesting and useful results and which appears to be the most suggestive of further research. In order to obtain an exhaustive exposition, we still lack a study of the properties of the resolvent operator where it is differentiable (which seemingly entails little more than undergraduate-level complex analysis), and a study of essential singularities of the resolvent operator (which seems too difficult for the expository style). A brief overview of the contents of this dissertation is in order: a chapter introducing some analytic concepts used throughout this dissertation; a chapter on poles of order 1 follows (so-called simple poles), where the Gelfand theorem (2.1.1) is the most important result; a chapter on poles of higher order, where the Hille theorem is the most prominent; and lastly some topics that have arisen out of the study of poles of the resolvent, collected in chapter 4. I should make it abundantly clear to the reader that although this dissertation is my work, it does not for the most part follow that the result are my own. What is my own is the arrangement, but as it is a literature study, the results are mainly those of other authors. My own addition has been mostly notes, usually in italics. The literature study has benefited very much from Zemanek's paper (Zemanek,[54]), and I am deeply indebted to him for it. Incidentally, this has also been a chance to exhibit my style of citation; the number corresponds to the number of the citation in the bibliography. There are numerous instances where I have indicated possible extensions and recumbent studies that could be roused effectively, but which have swelled this volume unnecessarily. For instance, the last subsection is little more than such indications.

We use a resolvent method to study atomic and molecular negative ions in a constant electric field potential which is linear. When a linear potential is applied, it makes the shape of the original potential of the system slanted into one side and thus changes the time evolution of the system. In particular, a bound state can be changed into a state, so called 'quasibound' state, which is not bound anymore and decays into the continuum due to the presence of the linear potential. For an atomic system, we use an attractive delta function fixed at the origin for the interaction potential and solve the single particle Schrodinger equation. For an actual system, we choose the Hydrogen negative ion, and determine the strength of the delta function so that the bound state energy can simulate the electron affinity of the Hydrogen. We find the resolvent of the system and the poles of the resolvent in the analytically continued region. From the patterns of the location of the poles, we can view the one delta function system as a combination of three simple systems. Though they are not exactly the same, this view gives some insight on the system. From the residue at each pole, complex eigenstates are constructed and used for the calculation of the survival probability of an initial state. For the same initial state, we calculate the photodetachment rate when a time-periodic potential is applied. The plot for the photodetachment rate shows peaks at certain incident photon energies. These are compared with an experimental data and give a good agreement although our model is just one dimensional. For a molecular system, two delta function model is suggested by us as an extension of the one delta function model. We find the resolvent of the system and the pole structure from the resolvent. The complex eigenstates are constructed from the residue of the resolvent at each pole. We try to model Oxygen molecular negative ion and determine the strength of the delta function and the distance between the delta functions so that they are consistent with the electron affinity and the internuclear distance of the Oxygen molecule. We also calculate the survival probability and the photodetachment rate of an initial state and find that the plot of the photodetachment rate has similar shape to that of the one delta function model. / text

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Distributed optimization over multi-agent networks has become an increasingly popular research topic as it incorporates many applications from various areas such as consensus optimization, distributed control, network resource allocation, large scale machine learning, etc. Parallel distributed solution algorithms are highly desirable as they are more scalable, more robust against agent failure, align more naturally with either underlying agent network topology or big-data parallel computing framework. In this dissertation, we consider a multi-agent optimization formulation where the global objective function is the summation of individual local objective functions with respect to local agents' decision variables of different dimensions, and the constraints include both local private constraints and shared coupling constraints. Employing and extending tools from the monotone operator theory (including resolvent operator, operator splitting, etc.) and fixed point iteration of nonexpansive, averaged operators, a series of distributed solution approaches are proposed, which are all iterative algorithms that rely on parallel agent level local updates and inter-agent coordination. Some of the algorithms require synchronizations across all agents for information exchange during each iteration while others allow asynchrony and delays. The algorithms' convergence to an optimal solution if one exists are established by first characterizing them as fixed point iterations of certain averaged operators under certain carefully designed norms, then showing that the fixed point sets of these averaged operators are exactly the optimal solution set of the original multi-agent optimization problem. The effectiveness and performances of the proposed algorithms are demonstrated and compared through several numerical examples.<br>