Sparse Signal Representation in Digital and Biological Systems

Theories of sparse signal representation, wherein a signal is decomposed as the sum of a small number of constituent elements, play increasing roles in both mathematical signal processing and neuroscience. This happens despite the differences between signal models in the two domains. After reviewing preliminary material on sparse signal models, I use work on compressed sensing for the electron tomography of biological structures as a target for exploring the efficacy of sparse signal reconstruction in a challenging application domain. My research in this area addresses a topic of keen interest to the biological microscopy community, and has resulted in the development of tomographic reconstruction software which is competitive with the state of the art in its field. Moving from the linear signal domain into the nonlinear dynamics of neural encoding, I explain the sparse coding hypothesis in neuroscience and its relationship with olfaction in locusts. I implement a numerical ODE model of the activity of neural populations responsible for sparse odor coding in locusts as part of a project involving offset spiking in the Kenyon cells. I also explain the validation procedures we have devised to help assess the model's similarity to the biology. The thesis concludes with the development of a new, simplified model of locust olfactory network activity, which seeks with some success to explain statistical properties of the sparse coding processes carried out in the network.

Existence and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system over moving domains

This dissertation concerns the well-posedness of the Navier-Stokes-Smoluchowski system. The system models a mixture of fluid and particles in the so-called bubbling regime. The compressible Navier-Stokes equations governing the evolution of the fluid are coupled to the Smoluchowski equation for the particle density at a continuum level. First, working on fixed domains, the existence of weak solutions is established using a three-level approximation scheme and based largely on the Lions-Feireisl theory of compressible fluids. The system is then posed over a moving domain. By utilizing a Brinkman-type penalization as well as penalization of the viscosity, the existence of weak solutions of the Navier-Stokes-Smoluchowski system is proved over moving domains. As a corollary the convergence of the Brinkman penalization is proved. Finally, a suitable relative entropy is defined. This relative entropy is used to establish a weak-strong uniqueness result for the Navier-Stokes-Smoluchowski system over moving domains, ensuring that strong solutions are unique in the class of weak solutions.

Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

CHARACTERISTICS OF PRIVATE HIGH SCHOOLS FOR STUDENTS WITH EMOTIONAL DISABILITIES AND PERCEPTIONS OF THEIR PUBLIC SCHOOL CASE MANAGERS

Placement of students with disabilities in private special-education schools remains costly and controversial. This is particularly concerning, given the lack of research on the characteristics and quality of these restrictive settings. The purpose of this study was to identify the academic and vocational course offerings and behavioral supports provided in private special-education schools the serve high school students with emotional disabilities (ED). Second, the research examined the perceptions of the quality of services in these setting from the perspectives of public school case managers. Using a mixed-method design to collect data, 9 administrative heads of private special-education schools were surveyed, and 7 public school case managers were interviewed. Results indicated that (a) private special-education schools offer the basic academic core courses needed to meet graduation requirements, (b) vocational options for students enrolled in these schools are quite limited, (c) these schools provide a variety of behavioral interventions and supports, and (d) case managers are concerned with the lack of academic rigor and inconsistent programming at these schools but applauded the notion that students with ED are exiting with a high school diploma. Findings from this study may have policy implications for improving and developing programming options for high school students with ED.

Friendships of High-Achieving African American Adolescents: Relation to Ethnic Identity and Achievement Values

The academic achievement of African American adolescents is a national concern for educators and researchers especially since current reports depict the underachievement of African American students as continuing to lag behind their European American peers. Determining what factors within the school environment that contributes to the achievement gap and how it can be reduced remains an important issue in alleviating disparities seen in educational achievement and attainment. This study examined the relation between characteristics of the close friendships of high-achieving African American adolescents and students’ identity development and motivation in school. Data were collected from 217 high-achieving African American students within 10th to 12th grade from 5 public and private high schools. Each student self-reported on their ethnicity, gender, parents’ education level, grade, FARMs, GPA, perceived teacher support (emotional, academic, and instrumental support), their perception of their ethnic identity, and their perception of their achievement values. Through the use of nomination procedures, students also identified their close friends and responded to questions concerning how supportive (emotional, academic, and instrumental support) they each were. Results from multiple regression analyses showed that the provision of instrumental support from close friends related to the exploration process of the high-achieving students’ ethnic identity. In addition, there was a strong relation between the ethnic identity of close friends and that of the individual. Furthermore, although friend support was not a significant predictor of achievement values, demographic (mother’s education level, grade, and FARMS) and control (teacher support) variables predicted students’ importance and utility of school respectively. These findings add to the literature on age and socioeconomic status as they relate to student’s motivation to achieve. Overall, this study provides some evidence highlighting ways in which close friendships might relate to the self-development of high-achieving African American adolescents. This study provides a starting point for additional ways in which to explore how peer processes relate to the academic behaviors of high-achieving African American adolescents.

Multiscale Modeling and Simulation of Stepped Crystal Surfaces

A primary goal of this dissertation is to understand the links between mathematical models that describe crystal surfaces at three fundamental length scales: The scale of individual atoms, the scale of collections of atoms forming crystal defects, and macroscopic scale. Characterizing connections between different classes of models is a critical task for gaining insight into the physics they describe, a long-standing objective in applied analysis, and also highly relevant in engineering applications. The key concept I use in each problem addressed in this thesis is coarse graining, which is a strategy for connecting fine representations or models with coarser representations. Often this idea is invoked to reduce a large discrete system to an appropriate continuum description, e.g. individual particles are represented by a continuous density. While there is no general theory of coarse graining, one closely related mathematical approach is asymptotic analysis, i.e. the description of limiting behavior as some parameter becomes very large or very small. In the case of crystalline solids, it is natural to consider cases where the number of particles is large or where the lattice spacing is small. Limits such as these often make explicit the nature of links between models capturing different scales, and, once established, provide a means of improving our understanding, or the models themselves. Finding appropriate variables whose limits illustrate the important connections between models is no easy task, however. This is one area where computer simulation is extremely helpful, as it allows us to see the results of complex dynamics and gather clues regarding the roles of different physical quantities. On the other hand, connections between models enable the development of novel multiscale computational schemes, so understanding can assist computation and vice versa. Some of these ideas are demonstrated in this thesis. The important outcomes of this thesis include: (1) a systematic derivation of the step-flow model of Burton, Cabrera, and Frank, with corrections, from an atomistic solid-on-solid-type models in 1+1 dimensions; (2) the inclusion of an atomistically motivated transport mechanism in an island dynamics model allowing for a more detailed account of mound evolution; and (3) the development of a hybrid discrete-continuum scheme for simulating the relaxation of a faceted crystal mound. Central to all of these modeling and simulation efforts is the presence of steps composed of individual layers of atoms on vicinal crystal surfaces. Consequently, a recurring theme in this research is the observation that mesoscale defects play a crucial role in crystal morphological evolution.