Browsing by Subject "Functional Analysis"

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    Pattern formation in the wake of external mechanisms
    (2016-06) Goh, Ryan
    Pattern formation in nature has intrigued humans for centuries, if not millennia. In the past few decades researchers have become interested in harnessing these processes to engineer and manufacture self-organized and self-regulated devices at various length scales. Since many natural pattern forming processes nucleate or grow from a homogeneous unstable state, they typically create defects, caused by thermal and other inherent sources of noise, which can hamper effectiveness in applications. One successful experimental method for controlling the pattern forming process is to use an external mechanism which moves through a system, transforming it from a stable state to an unstable state from which the pattern forming dynamics can take hold. In this thesis, we rigorously study partial differential equations which model how such triggering mechanisms can select and control patterns. We first use dynamical systems techniques to study the case where a spatial trigger perturbs a pattern forming freely invading front in a scalar partial differential equation. We study such perturbations for the two generic types of scalar invasion fronts, known as pulled and pushed fronts, which roughly correspond to fronts which invade either through a linear or nonlinear mechanism. Our results give the existence of perturbed fronts and provide expansions in the speed of the triggering mechanism for the wavenumber perturbation of the pattern formed. With the hope of moving towards the more complicated geometries which can arise in two spatial dimensions, where many dynamical systems methods cannot be readily applied, we also develop a functional analytic method for the study of Hopf bifurcation in the presence of continuous spectrum. Our method, while still giving computable information about the bifurcating solution, is more direct than previously proposed methods. We develop this method in the context of a triggered Cahn-Hilliard equation, in one spatial dimension, which has been used to study many triggered pattern forming systems. Furthermore, we use these abstract results to characterize an explicit example and also use our method to give a simplified proof of the bifurcation of oscillatory shock solutions in viscous conservation laws.
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    Using functional analysis methodology to evaluate neuroleptic medication effects on positively and negatively reinforced severe problem behavior.
    (2010-08) Danov, Stacy
    Clinically prescribed atypical neuroleptic medication (aripiprazole) was evaluated using a randomized AB multiple baseline, double-blind, placebo controlled design in the treatment of severe problem behavior (SPB) in children with intellectual and developmental disabilities. A pre-treatment screening procedure identified participants whose behavior was maintained by two behavioral mechanisms, positive or negative reinforcement. Functional analysis (FA) was conducted concurrent with the medication evaluation to determine how SPB is differentially affected by the medication under common environmental situations. Weekly rating scales were completed by parent/guardian. Data were analyzed using descriptive statistics, visual inspection, and inferential statistics. Results indicated that aripiprazole had differential effects across behavioral function and behavioral topography. This study demonstrated how functional analysis may provide information on those conditions and behaviors that are most likely to be affected by a specific medication.