Uniform large deviation principles for Banach space valued stochastic differential equations

We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDE) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform large deviation principle over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak-$\star$ compactness of closed bounded sets in the double dual space. We prove that a modified version of our stochastic differential equation satisfies a uniform Laplace principle over weak-$\star$ compact sets and consequently a uniform over bounded sets large deviation principle. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and $2$-dimensional stochastic Navier-Stokes equations with multiplicative noise

Uniform large deviation principles for Banach space valued stochastic differential equations

We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDE) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform large deviation principle over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak-$\star$ compactness of closed bounded sets in the double dual space. We prove that a modified version of our stochastic differential equation satisfies a uniform Laplace principle over weak-$\star$ compact sets and consequently a uniform over bounded sets large deviation principle. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and $2$-dimensional stochastic Navier-Stokes equations with multiplicative noise.First author draf

Combined Loading of Composite Box-Beams with Linearized Differential Equations of Motion

The purpose of the following analysis is to develop a method for determining vertical, horizontal, and torsional displacements of arbitrarily laminated, cantilevered composite box-beam under static, transverse loading. Small deflections are assumed and the differential equations of motion have been linearized for ease of calculation. The Smith and Chopra stiffness matrix is used in the equations of motion, and software was written to solve for the stiffness matrix elements. Comparisons are made with bending slope solutions from previously published results, and the presented analysis is compared to empirical results of a composite box-beam under tip loading. Finally, an example solution for vertical, horizontal, and torsional displacements is presented for aerodynamic-type loading

Adaptive Augmentation of Non-Minimum Phase Flexible Aerospace Systems

This work demonstrates the efficacy of direct adaptive augmentation on a robotic flexible system as an analogue of a large flexible aerospace structure such as a launch vehicle or aircraft. To that end, a robot was constructed as a control system testbed. This robot, named “Penny,” contains the command and data acquisition capabilities necessary to influence and record system state data, including the flex states of its flexible structures. This robot was tested in two configurations, one with a vertically cantilevered flexible beam, and one with a flexible inverted pendulum (a flexible cart-pole system). The physical system was then characterized so that linear analysis and control design could be performed. These characterizations resulted in linear and nonlinear models developed for each testing configuration. The linear models were used to design linear controllers to regulate the nominal plant’s dynamical states. These controllers were then augmented with direct adaptive output regulation and disturbance accommodation. To accomplish this, sensor blending was used to shape the output such that the nonminimum phase open loop plant appears to be minimum phase to the controller. It was subsequently shown that augmenting linear controllers with direct adaptive output regulation and disturbance accommodation was effective in enhancing system performance and mitigating oscillation in the flexible structures through the system’s own actuation effort

The AL-Gaussian Distribution as the Descriptive Model for the Internal Proactive Inhibition in the Standard Stop Signal Task

Measurements of response inhibition components of reactive inhibition and proactive inhibition within the stop signal paradigm have been of special interest for researchers since the 1980s. While frequentist nonparametric and Bayesian parametric methods have been proposed to precisely estimate the entire distribution of reactive inhibition, quantified by stop signal reaction times(SSRT), there is no method yet in the stop-signal task literature to precisely estimate the entire distribution of proactive inhibition. We introduce an Asymmetric Laplace Gaussian (ALG) model to describe the distribution of proactive inhibition. The proposed method is based on two assumptions of independent trial type(go/stop) reaction times, and Ex-Gaussian (ExG) models for them. Results indicated that the four parametric, ALG model uniquely describes the proactive inhibition distribution and its key shape features; and, its hazard function is monotonically increasing as are its three parametric ExG components. In conclusion, both response inhibition components can be uniquely modeled via variations of the four parametric ALG model described with their associated similar distributional features.Comment: KEYWORDS Proactive Inhibition, Reaction Times, Ex-Gaussian, Asymmetric Laplace Gaussian, Bayesian Parametric Approach, Hazard functio

Effectiveness of Schema-Based Instruction for Improving Seventh-Grade Students’ Proportional Reasoning: A Randomized Experiment

This study examined the effect of schema-based instruction (SBI) on seventh-grade students’ mathematical problem solving performance. SBI is an instructional intervention that emphasizes the role of mathematical structure in word problems and also provides students with a heuristic to self-monitor and aid problem solving. Using a pretest-intervention-posttest-retention test design, the study compared the learning outcomes for 1,163 students in 42 classrooms who were randomly assigned to treatment (SBI) or control condition. After 6 weeks of instruction, results of multilevel modeling indicated significant differences favoring the SBI condition in proportion problem solving involving ratios/rates and percents on an immediate posttest (g = 1.24) and on a six-week retention test (g = 1.27). No significant difference between conditions was found for a test of transfer. These results demonstrate that SBI was more effective than students’ regular mathematics instruction

Mechanisms and Kinetics of Organic Aging and Characterization of Intermediates in High Level Waste

This presentation was given at the DOE Office of Science-Environmental Management Science Program (EMSP) High-Level Waste Workshop held on January 19-20, 2005 at the Savannah River Site