Dissertation Defense Seminar

This dissertation has two parts. In the first part, we overview and extend the auxiliary function method for long-time averages, which computes sharp bounds on time averaged quantities in underlying dynamical variables via convex optimization and semidefinite programming. We use the extended method to study the validity of asymptotic methods for computing long-time statistics in nonlinear or nonautonomous dynamical systems. We show for the Duffing equation and the nonlinear pendulum that the harmonic balance and true solution’s mean squared amplitude agree quite well, but truncated Fourier expansions fail to accurately predict the regions of stability for a parametrically driven, coupled oscillator system. In particular, regions of stability are sensitive to the coupling effects across a broad range of modulation frequencies.



In the second part, we overview dynamic choice within the paradigms of Von Neumann-Morgenstern and discounted expected utility. We then discuss the ethical theory of utilitarianism and its connections to social choice theory while overviewing the seminal work of Kenneth Arrow and John Harsanyi. We prove a novel extension of Harsanyi's theorem to an infinite time horizon setting. Under mild assumptions, a Pareto condition is equivalent to utilitarian aggregation with unique utilitarian weights. We study the asymptotic properties of the utilitarian weights as the social discount factor or social risk attitude changes. Among other findings, we show that a higher social discount rate is associated with a more unequal assignment of utilitarian weights across generations.

Andrew's advisors are Silas Alben and Shaowei Ke. Speaker(s): Andrew McMillan (UM)