In this thesis, we study the relationship between the geometric shape of a region in the plane, and certain probabilistic information about the behavior of Brownian particles inside the region. The probabilistic information is contained in the function h(r), called the harmonic measure distribution function. Consider a domain Ω in the plane, and fix a basepoint z0. Imagine lining the boundary of this domain with fly paper and releasing a million fireflies at the basepoint z0. The fireflies wander around inside this domain randomly until they hit a wall and get stuck in the fly paper. What fraction of these fireflies are stuck within a distance r of their starting point z0? The answer is given by evaluating our h-function at this distance; that is, it is given by h(r). In more technical terms, the h-function gives the probability of a Brownian first particle hitting the boundary of the domain Ω within a radius r of the basepoint z0. This function is dependent on the shape of the domain Ω, the location of the basepoint z0, and the radius r. The big question to consider is: How much information does the h-function contain about the shape of the domain’s boundary? It is known that an h-function cannot uniquely determine a domain, but is it possible to construct a domain that generates a given hfunction? This is the question we try to answer. We begin by giving some examples of domains with their h-functions, and then some examples of sequences of converging domains whose corresponding h-functions also converge to the h-function. In a specific case, we prove that artichoke domains converge to the wedge domain, and their h-functions also converge. Using another class of approximating domains, circle domains, we outline a method for constructing bounded domains from possible hfunctions f(r). We prove some results about these domains, and we finish with a possible for a proof of the convergence of the sequence of domains constructed.

Thesis (Ph. D.)--University of Rhode Island, 2005. / Typescript. Includes bibliographical references (leaf 36).

In wireless communications systems, it is well known that the instantaneous received signal is a random variable that follows a given distribution. The randomness mainly stems from e ects such as multipath fading, shadowing, and interference. The received signal is a relevant metric, such that several distributions have been used in the literature to characterize it. However, as new radio technologies emerge, the known distributions are deemed insu cient to t simulated and measure data. Subsequently, as the wireless industry moves onto the fth generation (5G), newer distributions are proposed to well represent the received signal for new wireless technologies, including those operating in the millimeter-wave (mmWave) band. These are mainly application speci c and may not be adequate to model complex 5G devices performance. Therefore, there is a need to unify and generalize the received signal distributions used for performance analysis of wireless systems. Secondly, an explosion of new radio technologies and devices operating in the same limited radio spectrum to collect and share data at alarming rates is expected. Such an explosion coupled with the 5G promise of ubiquitous connectivity and network densi cation, will thrust interference modeling in dense networks to the fore-front. Thus, interference characterization is essential when analyzing such wireless networks. Thirdly, the classical distributions used to model the received signal do not account for the inherent mobility feature for emerging radio technologies, such as avionics systems (e.g. drones), which may make the distributions inadequate as mobility e ects can no longer be ignored. Consequently, in this dissertation, we propose the use of a unifying distribution, the Fox's H-function distribution, with subsume ability to represent several traditional and future distributions, as a statistical tool to evaluate the performance of wireless communications systems. Additionally, two interference models, one with a xed number and the other with a random number of interferers, are considered to derive interference statistics, and further utilize the results to analyze system performance under the e ect of interference. Finally, we extend the classical distributions to include the mobility regime for several wireless network topologies, and perform network analysis. The analytical results are validated using computer Monte Carlo simulations. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2017. / FAU Electronic Theses and Dissertations Collection