An extremal property of normal distributions is that they have the smallest Fisher Information for location among all distributions with the same variance. A new test of normality proposed by Terrell (1995) utilizes the above property by finding that density of maximum likelihood constrained on having the expected Fisher Information under normality based on the sample variance. The test statistic is then constructed as a ratio of the resulting likelihood against that of normality. Since the asymptotic distribution of this test statistic is not available, the critical values for n = 3 to 200 have been obtained by simulation and smoothed using polynomials. An extensive power study shows that the test has superior power against distributions that are symmetric and leptokurtic (long-tailed). Another advantage of the test over existing ones is the direct depiction of any deviation from normality in the form of a density estimate. This is evident when the test is applied to several real data sets. Testing of normality in residuals is also investigated. Various approaches in dealing with residuals being possibly heteroscedastic and correlated suffer from a loss of power. The approach with the fewest undesirable features is to use the Ordinary Least Squares (OLS) residuals in place of independent observations. From simulations, it is shown that one has to be careful about the levels of the normality tests and also in generalizing the results. / Ph. D.

This dissertation focuses on the study of decentralized coordination algorithms of multiple autonomous vehicles. Here, the term decentralized coordination is used to refer to the behavior that a group of vehicles reaches the desired group behavior via local interaction. Research is conducted towards designing and analyzing distributed coordination algorithms to achieve desired group behavior in the presence of none, one, and multiple group reference states. Decentralized coordination in the absence of any group reference state is a very active research topic in the systems and controls society. We first focus on studying decentralized coordination problems for both single-integrator kinematics and double-integrator dynamics in a sampled-data setting because real systems are more appropriate to be modeled in a sampled-data setting rather than a continuous setting. Two sampled-data consensus algorithms are proposed and the conditions to guarantee consensus are presented for both fixed and switching network topologies. Because a number of coordination algorithms can be employed to guarantee coordination, it is important to study the optimal coordination problems. We further study the optimal consensus problems in both continuous-time and discrete-time settings via an linear-quadratic regulator (LQR)-based approach. Noting that fractional-order dynamics can better represent the dynamics of certain systems, especially when the systems evolve under complicated environment, the existing integer-order coordination algorithms are extended to the fractional-order case. Decentralized coordination in the presence of one group reference state is also called coordinated tracking, including both consensus tracking and swarm tracking. Consensus tracking refers to the behavior that the followers track the group reference state. Swarm tracking refers to the behavior that the followers move cohesively with the external leader while avoiding inter-vehicle collisions. In this part, consensus tracking is studied in both discrete-time setting and continuous-time settings while swarm tracking is studied in a continuous-time setting. Decentralized coordination in the presence of multiple group reference states is also called containment control, where the followers will converge to the convex hull, i.e., the minimal geometric space, formed by the group references states via local interaction. In this part, the containment control problem is studied for both single-integrator kinematics and double-integrator dynamics. In addition, experimental results are provided to validate some theoretical results.

Two major applications of lambda calculi in computer science are functional programming languages and mechanized reasoning systems (or, proof assistants). According to the Curry--Howard correspondence, it is possible, in principle, to design a unified language based on a typed lambda calculus for both logical reasoning and programming. However, the different requirements of programming languages and reasoning systems make it difficult to design such a unified language that provides both. Programming languages usually extend lambda calculi with programming-friendly features (e.g., recursive datatypes, general recursion) for supporting the flexibility to model various computations, while sacrificing logical consistency. Logical reasoning systems usually extend lambda calculi with logic-friendly features (e.g., induction principles, dependent types) for paradox-free inference over fine-grained properties, while being more restrictive in modeling computations. In this dissertation, we design and implement a language called Nax that embraces benefits of both. Nax accepts all recursive datatypes, thus, allowing the same flexibility of defining recursive datatypes as in functional languages. Nax supports a number of Mendler-style recursion schemes that can express various kinds of recursive computations and also guarantee termination. Nax supports term-indexed types to support specifications of fine-grained properties. In addition, Nax supports a conservative extension of Hindley--Milner type inference. The theoretical contributions of this dissertation include theories for Mendler-style recursion schemes and term-indexed types, which we developed to establish strong normalization and logical consistency of Nax.

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Students are entering college having earned credit for college Calculus 1 based on their scores on the College Board's Advanced Placement (AP) Calculus AB exam. Despite being granted credit for college Calculus 1, it is unclear whether these students are adequately prepared for college Calculus 2. College calculus classes are often taught from a more theoretical perspective rather than AP Calculus courses taught in high schools, and many students who enter college mathematics with Calculus 2—who possess AP credit for Calculus 1—have found the theoretical perspective of college Calculus 2 courses to be overwhelming. Consequently, these students have not performed well in Calculus 2. This has led to a belief that students with AP Calculus credit for Calculus 1 do not perform as well in college Calculus 2 in comparison to their peers who earned credit for college Calculus 1. Simultaneously, a contradicting belief exists: Students with AP Calculus credit for college Calculus 1 are the strongest students in college Calculus 2, outperforming their peers. The goal of this quantitative study was to compare the learning outcomes of students in college Calculus 2 of students with and without AP Calculus AB credit for college Calculus 1. In analyzing the data, four distinct entry points into college mathematics on a path to college Calculus 2 were identified: Calculus 2 having earned credit for Calculus 1 by means of the AP Calculus AB, Calculus 1 despite having taken AP Calculus in high school, Calculus 1 having not taken AP Calculus in high school, and Pre-Calculus. Each of these entry points were analyzed to identify measures of success in high school and college which are associated with success in college Calculus 2. The results of this study suggest that students with AP Calculus credit for college Calculus 1 do outperform their peers in Calculus 2. Furthermore, the higher the entry point into college mathematics, the better a student is likely to do in Calculus 2. Measures of success that were found to be positively associated with success in Calculus 2 include high school cumulative grade point average and college Calculus 1 grade. A measure that was found to be negatively associated with Calculus 2 success was the number of times a student repeated Calculus 1 prior to enrolling in Calculus 2.

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This thesis consists of four chapters. In Chapter 1, we focus on a class of transportation inequalities known as the transportation-information inequalities. These inequalities bound optimal transportation costs in terms of relative Fisher information, and are known to characterize certain concentration properties of Markov processes around their invariant measures. We provide a characterization of the quadratic transportation-information inequality in terms of a dimension-free concentration property for i.i.d. copies of the underlying Markov process, identifying the precise high-dimensional concentration property encoded by this inequality. We also illustrate how this result is an instance of a general convex-analytic tensorization principle. In Chapter 2, we study the entropic gradient flow property of McKean--Vlasov diffusions via a stochastic analysis approach. We formulate a trajectorial version of the relative entropy dissipation identity for these interacting diffusions, which describes the rate of relative entropy dissipation along every path of the diffusive motion. As a first application, we obtain a new interpretation of the gradient flow structure for the granular media equation. Secondly, we show how the trajectorial approach leads to a new derivation of the HWBI inequality. In Chapter 3, we further extend the trajectorial approach to a class of degenerate diffusion equations that includes the porous medium equation. These equations are posed on a bounded domain and are subject to no-flux boundary conditions, so that their corresponding probabilistic representations are stochastic differential equations with normal reflection on the boundary. Our stochastic analysis approach again leads to a new derivation of the Wasserstein gradient flow property for these nonlinear diffusions, as well as to a simple proof of the HWI inequality in the present context. Finally, in Chapter 4, we turn our attention to mean field approximation -- a method widely used to study the behavior of large stochastic systems of interacting particles. We propose a new approach to deriving quantitative mean field approximations for any strongly log-concave probability measure. Our framework is inspired by the recent theory of nonlinear large deviations, for which we offer an efficient non-asymptotic perspective in log-concave settings based on functional inequalities. We discuss three implications, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems.

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