For a graph $G$, let $\alpha (G)$ be the cardinality of a maximum independent set, let $\mu (G)$ be the cardinality of a maximum matching and let $\xi (G)$ be the number of vertices belonging to all maximum independent sets. Boros, Golumbic and Levit showed that in connected graphs where the independence number $\alpha (G)$ is greater than the matching number $\mu (G)$, $\xi (G) \geq 1 + \alpha(G) - \mu (G)$. For any graph $G$, we will show there is a distinguished induced subgraph $G[X]$ such that, under weaker assumptions, $\xi (G) \geq 1 + \alpha (G[X]) - \mu (G[X])$. Furthermore $1 + \alpha (G[X]) - \mu (G[X]) \geq 1 + \alpha (G) - \mu (G)$ and the difference between these bounds can be arbitrarily large. Lastly some results toward a characterization of graphs with equal independence and matching numbers is given.

An agent based model was developed to examine the growth of a tumor in a healthy cell population. Response to radiation and impact of mutation and bystander effects were studied. In the growth model, the cancer cells proliferated outward becoming invasive. The mass of cancer cells developed a necrotic core. Various treatment protocols of radiation were compared. Timing of treatments was critical to the success of therapy. The event of mutation was rare. When mutation occurred, either unsuccessful treatment or re-growth could result. Multiple rounds of radiation potentially led to increased mutation. Low levels of the bystander effect had little impact on the overall behavior of the system when considered alone and in combination with mutation. Higher levels of bystander effect and increased affected area resulted in significant cell death. Increasing the radius of the bystander effect in combination with mutation resulted in large numbers of mutation and unsuccessful treatment.

Zeta potential is one of the tools to measure the surface charge of materials, and Alvarez et al. have developed a microchannel device to measure zeta potentials in real time for label-free sensing using immobilized receptors on microchannel surfaces. However, the challenge has been the charge interference of surface modifiers on analyte detection. Therefore, it was necessary to find the best strategy to regenerate minimal surface charge after modifying the channel with polymer films that would anchor the affinity groups for the analyte. It was demonstrated that adsorption of positively and negatively charged analytes were monitored via real time zeta potential measurements by using surface-immobilized polystyrene nanospheres, and the best discrimination of analyte binding on the nanoparticles was observed when the underlying film was a non-ionic polymer. Titanium oxide nanoparticles (TiO2) were immobilized on microchannels modified with non-ionic polymers to investigate if the surface charge of the microchannel was induced by the concomitant surface charge reactions of the TiO2 nanoparticles upon UV exposure. Analysis by XPS indicates that desorption of proteins monitored by zeta potential changes are induced by UV exposure.

We will explore the topic of stochastic differential equations (SDEs) first by developing a foundation in probability theory and It\^o calculus. Formulas are then derived to simulate these equations analytically as well as numerically. These formulas are then applied to a basic population model as well as a logistic model and the various methods are compared. Finally, we will study a model for low dose anthrax exposure which currently implements a stochastic probabilistic uptake in a deterministic differential equation, and analyze how replacing the probablistic uptake with an SDE alters the dynamics.

Metallic nanowires have useful applications in scanning tunneling microscopes and atomic force microscopes due to their unique sensitivity to force and electricity. These unique properties arise because of the large surface area to volume ratio. One of these properties is that introducing twinning planes the mechanical properties of metallic nanowires can be altered. The effects of twinning planes on metallic nanowires were studied using molecular dynamics simulations. Silver, copper, and nickel nanowires with and without twinning planes were simulated with engineering strain until the first yielding stress was obtained. The radial simulations showed that as the radius of twinned nanowires increased, the strength gained by introducing the twinning planes increased. The temperature simulations showed that nanowires with twinning planes were stronger than their un-twinned counterparts as temperature increased. The purpose of this investigation was to better understand the effect twinning planes had on metallic nanowires, so that future technological advances would benefit from the results.

Over the years, a lot has been written about the three more common graph products (Cartesian product, Direct product and the Strong product), as all three of these are commutative products. This thesis investigates a non-commutative product graph, H, G, we call the Semi-Strong graph product, also referred in the literature as the Augmented Tensor and/or the Strong Tensor. We will start by discussing its basic properties and then focus on embeddings where the second factor, G, is a regular graph. We will use permutation voltage graphs and their graph coverings to compute the minimum genus for several families of graphs. The results follow work started first by A T White [12], extended by Ghidewon Abay Asmerom [1],[2], and follows the lead of Pisanski [9]. The strategy we use starts with an embedding of a graph H and then modifying H creating a pseudograph H*. H* is a voltage graph whose covering is HxG. Given the graph product HxG, where G is a regular graph and H meets certain conditions, we will use the embedding of H to study topological properties, particularly the surface on which HxG is minimally embedded.

A large portion of decision analysis lies in a decision maker’s uncertainty about an outcome and what they perceive is the chance (probability) of that outcome occurring (in other words, an individual’s “degree of belief” that an outcome will occur). However, thinking probabilistically can be difficult and we rely on “rather primitive cognitive techniques to make” such assessments (these techniques are termed heuristics) (Clemen & Reilly, 2001 p.311). Heuristics are simple and intuitive but tend to result in probabilities that are biased. This thesis will connect the literature available from both the psychology behind the biases and the mathematical problems associated with the probability elicitation itself. Additionally, this thesis will present a better understanding of the biases that distort the probability elicitation for the decision maker along with suggestions for improving such assessments.

Understanding hydrophobic effects at different length scales is relevant to many complex and poorly understood everyday phenomena in materials science and biology. In this thesis, a variety of theory/computational methods in statistical physics and statistical mechanics are used to address three separated, but interconnected problems: (1) How solvation free energy scales with a partical size that is charged? This problem has never been attempted to solve before despite its immense importance in colloidal and protein solutions (J. Wang, D. Bratko, K. Leung and A. Luzar, Hydrophobic hydration at different length-scales: manipulating the crossover by charges, to be submitted to J. Stat. Phys. (Special issue on Water and Associated Liquids)); (2) Can onset to capillary evaporation, seen in some protein complexes with large hydrophobic areas be predicted in a simple way? A simple coarse-grained model of water/protein system, which is developed here shows the conditions for wet and dry hydrophobic protein cavities, and is able to reproduce all atom simulation results. The method should serve as an intermediate step between the initial screening of protein hydrophobic cavities and expensive molecular simulations (J. Wang, S. Kudesia, and A. Luzar, Computational probe of dewetting events in protein systems, in preparation for submission to J. Phys. Chem. B); (3) How to predicts hydrophobicity of a mixed surface from the knowledge of its pure constituents? To this end, wetting free energy on synthetic and biological heterogeneous surfaces is studied. Two distinct mechanisms responsible for their non-additivity have been identified in each case (J. Wang, D. Bratko and A. Luzar, Probing surface tension additivity on heterogenous surfaces: a molecular approach, Proc. Natl. Acad. Sci, under revision)

Gallium nitride is an important III-V semiconductor which is used in many optoelectronic and high-frequency devices. The nature of the GaN surface and its electrical characteristics can impact the performance of such devices. In this study, several GaN surfaces are locally charged using an atomic force microscope, and then subsequently studied by measuring the surface potential with scanning Kelvin probe microscopy (SKPM). The charging and discharging behavior of the surface appears to be strongly influenced by surface preparation and the presence of a surface oxide layer. If a substantial oxide layer exists, then both positive and negative charging is possible on n-type and p-type samples. Surface treatments and photoemission spectroscopy (XPS) data confirm the presence and influence of the oxide layer on surface charging behavior. In the case of forward-bias charging, a small change in surface contact potential (0.1 – 0.3 eV) is observed that is primarily due to a small voltage drop across the surface oxide. Reverse-bias charging produces a substantially larger change in surface potential (~1 – 3 eV) that must be explained by a large increase in surface band bending. Temperature-dependent SKPM measurements also indicate that the decay behavior of deposited surface charge in dark involves a thermionic mechanism.

Silica based sol-gel thin films have been extensively studied because of their advantages that include optical transparency, high biodegradability and low intrinsic fluorescence. However, one concern with silica based sol-gel derived materials is their long term stability in aqueous solutions. Another concern is their limited porosity. These two concerns limit the application of these materials in catalysis and separations. The main objective of this study is to prepare porous, thin films using titanium alkoxides as precursors and evaluate their long term stability in aqueous solutions. Colloidal crystal templating will be used to introduce macrosized pores into the titania network. The materials will be characterized using SEM and AFM. To prove that the templated films provide access to the underlying surface of the electrode, a conducting metal like copper was electrodeposited inside the nano-wells. The stability of titania and silica films will be evaluated over a two month period using cyclic voltammetry with three redox probes.