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This thesis proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients. The stochastic problem is first transformed into a parametrized one by the use of the Karhunen--Loève expansion. This new problem is then discretized by the discontinuous Galerkin (DG) method. A priori error estimate in the energy norm for the stochastic discontinuous Galerkin solution is derived. In addition, the expected value of the numerical error is theoretically bounded in the energy norm and the L2 norm. In the second approach, the Monte Carlo method is used to generate independent identically distributed realizations of the stochastic coefficients. The resulting deterministic problems are solved by the DG method. Next, estimates are obtained for the error between the average of these approximate solutions and the expected value of the exact solution. The Monte Carlo discontinuous Galerkin method is tested numerically on several examples. Results show that the nonsymmetric DG method is stable independently of meshes and the value of penalty parameter. Symmetric and incomplete DG methods are stable only when the penalty parameter is large enough. Finally, comparisons with the Monte Carlo finite element method and the Monte Carlo discontinuous Galerkin method are presented for several cases.

Recent advances have elucidated the behavior of colloids and polymer systems in the archety pal three dimensions. However, these systems are ill-understood when confined to two dimensions. Using experimental techniques such as Langmuir-Blodgett and ellipsometry, we attempt to explain the behavior of two dimensional colloid/polymer systems via micron-scale imaging as well as measuring their surface-pressure versus area isotherms. Possible phase transition behaviors and mechanisms are uncovered and discussed. We explore the importance of proper colloidal stabilizing ligands in terms of hydrophobic forces and their impact on ellipsometry as well as Langmuir-Blodgett experiments. Lastly, we give insight for future work that still remains to be done in this area. / pages 16, 17, and 25 are missing from hardcopy

Although the importance of substrate surface topography in colloidal and bacterial adhesion is widely recognized, how it affects the adhesion process has been a controversial topic. In this study, the impact of surface topography on adhesion of biological (i.e., bacteria) and non-biological colloids was investigated using natural and engineered surfaces with well-defined surface topographic patterns. Adhesion experiments using carboxylate modified latex (CML) microspheres of 4μm in diameter and Psudomonas Aeruginosa on the taro leaf of Colocasia esculenta, a plant known for its self-cleaning property similar to that of the lotus leaf, in a 100 mM NaCl solution at pH 4 under submerged conditions showed that nanoscale surface structures on the papilla of the Colocasia esculenta leaf surface resisted adhesion by both CML and P. Aeruginosa. This resistance to adhesion was found to be independent of the wetting condition of the surface, suggesting that the surface superhydrophobicity was not the reason for the observed lack of adhesion. Interfacial force mapping by atomic force microscopy (AFM) revealed markedly lower adhesion forces over the surface area covered by these nano-structures where adhesion resistance was observed. Adhesion experiments were also performed using 6 μm CML particles on engineered micro-patterns fabricated on silicon wafers. The micro-patterned surfaces consisted cuboid pillars or pits of a wide range of sizes arranged at various spacings. Adhesion of CML particles on all micro-patterned surfaces was significantly less than on the smooth control surface. In general, adhesion decreased with decreasing pillar or pit size and spacing between the features. Adhesion was minimum on the micro-patterned surface when the dimension (pillar size) of patterns is close to/smaller than the size of the colloid when spacing between pillars was fixed to a size a bit smaller than the particle size; while the adhesion on patterns with fixed pillar size (a bit smaller than the particle size) was low for a wide range of spacings. Analysis of the spatial distribution of adhered particles on the pillar-patterned surfaces showed that more than 98% of the particles adhered on the edge of the pillars (between the pillars) when the spacing between pillars was smaller than the particle diameter; the particles adhered in the valley close to the pillars when the spacing was larger than the particle diameter. The characteristic adhesion distribution of the colloidal particles on the micro-patterned surfaces was also validated by the AFM adhesion force mapping: when spacing between pillars was smaller than the particle size, adhesion force was larger on the edge of the pillars; when spacing between pillars was larger than the particle size, adhesion force was larger on the valley. However, the AFM results could not explain the reduced adhesion on the patterned surface compared to the smooth surface.

This thesis describes self-assembly and surface modification of gold nanoparticles and nanorods. It begins with an efficient surface functionalization of gold nanoparticles with a liquid crystalline ligand that structurally resembles liquid crystal 5CB. Functionalized gold nanoparticles show increased solubility in liquid crystal compared to their alkanethiolated analogues. Surface modification technique was also applied to detoxify gold nanorods for biological applications. The process involves a ligand exchange step where cytotoxic CT AB bilayer of gold nanorods was replaced by thiolated CTAB. Covalent nature of gold-sulfur bond allows for the complete removal of CTAB via centrifugation. Functionalized gold nanorods are virtually non-toxic and they show efficient cellular uptake by cancer cells. This thesis also describes self-assembly of gold nanoparticles and nanorods into well organized colloidal crystals. Slow precipitation technique was implemented to prepare three dimensional colloidal crystals of alkanethiolated gold nanoparticles. Colloidal crystals of gold nanorods were prepared by evaporation induced self -assembly technique. Additionally, orientation of nanorods in a colloidal crystal can be altered from side-on to tip-on by controlling the hydrophobicity of the underlying substrate. Our investigation showed that the presence of excess surfactant (CTAB) strongly influences the crystallization process. The shape and size of self assembled structures were also controlled by creating patterned hydrophobic and hydrophilic areas on a substrate. A variety of self assembled structures can be prepared by controlling the dimensions of the patterns. Finally, an efficient synthesis of hybrid cadmium sulfide-gold nanowires was described.

Through polarization-dependent third hamonic generation experiments done on highly aligned carbon nanotubes on sapphire, the nonzero tensor elements of χ (3) have been extracted. The contribution of the weaker tensor elements to the overall χ (3) signal has been calculated to be approximately 1/3rd of that of the dominant [Special characters omitted.] component, which is consistent with theory and other measured values.

Image articulation manifolds (IAMs) arise in a wide variety of contexts in image processing and computer vision applications. IAMs are a natural nonlinear model for image ensembles generated by the variation of imaging parameters (scale, pose, lighting etc.). In the past, IAMs have been studied as being embedded submanifolds of higher dimensional Euclidean spaces. However, this view suffers from two major defects: lack of a meaningful metric and reliance on linear transport operators via tangent vectors. Recent work in the area indicates the existence of better nonlinear transport operators for IAMs, with optical flow based transport being a prime candidate. In this thesis, we provide a detailed theoretical analysis of optical flow based transport on IAMs. In particular, we develop new analytical tools reminiscent of differential geometry to handle the apriori data driven nature of IAMs using the notion of optical flow manifolds (OFMs). We define an appropriate metric on the IAM via a metric on the corresponding OFMs that satisfy certain local isometry conditions and we show how to use this new metric to develop a host of mathematical tools such as optical flow fields on the IAM, parallel fields and parallel transport as well as an intuitive notion of "optical curvature". We show that the space of optical flow fields along a path of constant optical curvature has a natural multiscale structure. We also consider the question of approximating non-parallel flow fields by parallel flow fields.

Many dynamic tasks have a clearly defined optimal trajectory or strategy for completion. Human operators may discover this strategy naturally through practice, but actively teaching it to them can increase their rate of performance improvement. Haptic devices, which provide force feedback to an operator, can physically guide participants through the optimal completion of a task, but this alone does not ensure that they will learn the optimal control strategy. In fact, participants may become dependent on this guidance to complete the task. This research focuses on developing and testing ways in which guidance can be modulated such that it conveys the proper task completion strategy without physically dominating the operator and thus encouraging dependency. These guidance schemes may also be applied to the real-time execution of tasks in order to convey computer-generated task completion strategies to a user without allowing the computer to physically dominate control of the task.

This thesis demonstrates the advantages of new practical implementations of compressive sensing (CS) algorithms tailored for the graphics processing unit (CPU) using a software platform called Jacket. There exist many applications which utilize CS including medical imaging, signal processing and data acquisition which have benefited from advancements in CS. However, as problems become larger not only do they become more difficult to solve but also more computationally expensive. In light of tins, existing CS algorithms are augmented for practical use on the CPU, reaping performance gains from the highly parallel architecture of the GPU. I discuss the issues associated with this transition and analyze the effects of such a movement, as well as provide results exhibiting advantages of using CPU-based methods.