This thesis is intended to fulfill the requirements of the Math and Physics departments at Harvey Mudd College. We begin with a brief introduction to the study of surfactant dynamics followed by some background on the experimental framework our work is related to. We then go through a derivation of the model we use, and explore in depth the nature of the Equation of State (EoS), the relationship between the surface tension on a fluid and the surfactant concentration. We consider the effect of using an empirical equation of state on the results of the simulations and compare the new results against the results produced using a multilayer (EoS) as well as experimental observations. We find that the empirical EoS leads to two new behaviors - preserving of large gradients of surfactant concentration and the occurrence of dynamics in distinct regimes. These behaviors suggest that the empirical EoS improves the agreement of the model’s prediction with experiment.

With the increased presence of image-based data in modern applications, the need for robust methods of image denoising grows greater. The work presented herein considers two of the most ubiquitous approaches towards image denoising: variational and non-local methods. The effectiveness of these methods is assessed using quantitatively using peak signal-to-noise ratio and structural similarity index measure metrics. This study employs ��−means clustering, an unsupervised machine learning algorithm, to isolate the most dominant cluster centroids within the incoming data and propose the introduction of a new adaptive parameter into the non-local means framework. Motivated by the fact that a majority of discrepancies between clean and denoised images occur at feature edges, this study examines several convolution-based edge detection methods to isolate relevant feature. The resultant gradient and edge information is used to further parameterize the ��−means non-local method. An additional hybrid method involving the combined contributions of variational and ��−means non-local denoising is proposed, with the weighting determined by edge intensities. This method outperforms the other methods outlined in the study, both conventional and newly presented.

This dissertation presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, for triangular meshes, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree. Finally, for rectangular meshes, the test space is reduced, yet the convergence is recovered regardless of parity.

A variety of two-dimensional fluid systems, known as dipole-mediated systems, exhibit a dipole-dipole interaction between their fluid constituents. The com- petition of this repulsive dipolar force with the cohesive fluid forces cause these systems to form intricate and patterned structures in their boundaries. In this thesis, we show that the microscopic details of any such system are irrelevant in the macroscopic limit and contribute only to a constant offset in the system’s energy. A numeric model is developed, and some important stable domain morphologies are characterized. Previously unresolved bifurcating branches are explored. Finally, by applying a random energy background to the numer- ics, we recover the smörgåsbord of diverse domain morphologies that are seen in experiment. We develop an empirical description of these domains and use it to demonstrate that the system's nondimensional parameter, which is the ratio of the line tension to the dipole–dipole density, can be extracted for any domain using only its shape.

This paper will demonstrate the principles and important facts of the randomized Kaczmarz algorithm as well as its extended version proposed by Zouzias and Ferris. Through the analysis made by Strohmer and Vershynin as well as Needell, it can be shown that the randomized Kaczmarz method is theoretically applicable in solving over-determined linear systems with or without noise. The extension of the randomized Kaczmarz algorithm further applies to the linear systems with non-unique solutions. In the experiment section of this paper, we compare the accuracies of the algorithms discussed in the paper in terms of making real-world macroeconomic analyses and predictions. The extended randomized Kaczmarz method outperforms both the randomized Kaczmarz method and the randomized Gauss-Seidel method on our data sets.

This paper explores the potential applications of existing spectral clustering algorithms to real life problems through experiments on existing road traffic data. The analysis begins with an overview of previous unsupervised machine learning techniques and constructs an effective spectral clustering algorithm that demonstrates the analytical power of the method. The paper focuses on the spectral embedding method’s ability to project non-linearly separable, high dimensional data into a more manageable space that allows for accurate clustering. The key step in this method involves solving a normalized eigenvector problem in order to construct an optimal representation of the original data. While this step greatly enhances our ability to analyze the relationships between data points and identify the natural clusters within the original dataset, it is difficult to comprehend the eigenvalue representation of the data in terms of the original input variables. The later sections of this paper will explore how the careful framing of questions with respect to available data can help researchers extract tangible decision driving results from real world data through spectral clustering analysis.

This paper is based on Landau's book "Foundations of Analysis" which constitutes a development of the number system founded on the Peano axioms for natural numbers. In order to show mastery of the subject matter this paper gives a somewhat different organization of material and modified or more detailed proofs of theorems. In situations where proofs become rather routine re pet it ions of previously noted techniques the proofs are omitted. The following symbols and notation are used. Natural numbers are denoted by lower case letters such as a,b,c, ... x,y,z. Sets are denoted by upper case letters such as M, N, ... X, Y, Z. If a is an element of M, this will be written atM, The denial of this is written at M. The symbol 3 /x is read "There exists an unique x". If x and y are names for the same number we write x=y. It is assumed that the relation= is an equivalence relation; i.e., (1) x=x, (2) if x=y, then y=x, (3) u x=y and y=z, then x=z. Throughout this paper there will be no special attempt to distinguish between the name of a number and the number itself. For example, the phrase" if xis a number" will be used in place of "if x is the name of a number."

Four-dimensional variational data assimilation (4D-Var) provides an estimate to the state of a dynamical system through the minimization of a cost functional that measures the distance to a prior state (background) estimate and observations over a time window. The analysis fit to each information input component is determined by the specification of the error covariance matrices in the data assimilation system (DAS). Weak-constraint 4D-Var (w4D-Var) provides a theoretical framework to account for modeling errors in the analysis scheme. In addition to the specification of the background error covariance matrix, the w4D-Var formulation requires information on the model error statistics and specification of the model error covariance. Up to now, the increased computational cost associated with w4D-Var has prevented its practical implementation. Various simplifications to reduce the computational burden have been considered, including writing the model error covariance as a scalar multiple of the background error covariance and modeling the model error. In this thesis, the main objective is the development of computationally feasible techniques for the improved representation of the model error statistics in a data assimilation system. Three new approaches are considered. A Monte Carlo method that uses an ensemble of w4D-Var systems to obtain flow-dependent estimates to the model error statistics. The evaluation of statistical diagnostic equations involving observation residuals to estimate the model error covariance matrix. An adaptive tuning procedure based on the sensitivity of a short-range forecast error measure to the model error DAS parametrization. The validity and benefits of these approaches are shown in two stages of numerical experiments. A proof-of-concept is shown using the Lorenz multi-scale model and the shallow water equations for a one-dimensional domain. The results show the potential of these methodologies to produce improved state estimates, as compared to other approaches in data assimilation. It is expected that the techniques presented will find an extended range of applications to assess and improve the performance of a w4D-Var system.

Partially Recursive Acceptance Rejection (PRAR) and bounding chains used in conjunction with coupling from the past (CFTP) are two perfect simulation protocols which can be used to sample from a variety of unnormalized target distributions. This paper first examines and then implements these two protocols to sample from the hardcore gas process. We empirically determine the subset of the hardcore process's parameters for which these two algorithms run in polynomial time. Comparing the efficiency of these two algorithms, we find that PRAR runs much faster for small values of the hardcore process's parameter whereas the bounding chain approach is vastly superior for large values of the process's parameter.

Have you ever wondered if you are better than the average bowler? If so, there are a variety of ways to compute the average score of a bowling game, including methods that account for a bowler’s skill level. In this thesis, we discuss several different ways to generate bowling scores randomly. For each distribution, we give results for the expected value and standard deviation of each frame's score, the expected value of the game’s final score, and the correlation coefficient between the score of the first and second roll of a single frame. Furthermore, we shall generalize the results in each distribution for an frame game on pins. Additionally, we shall generalize the number of possible games when bowling frames on pins. Then, we shall derive the frequency distribution of each frame’s scores and the arithmetic mean for frames on pins. Finally, to summarize the variety of distributions, we shall make tables that display the results obtained from each distribution used to model a particular bowler’s score. We evaluate the special case when bowling 10 frames on 10 pins, which represents a standard bowling game.