Norm Optimal Iterative Learning Control with Application to Problems in Accelerator based Free Electron Lasers and Rehabilitation Robotics

This paper gives an overview of the theoretical basis of the norm optimal approach to iterative learning control followed by results that describe more recent work which has experimentally benchmarking the performance that can be achieved. The remainder of then paper then describes its actual application to a physical process and a very novel application in stroke rehabilitation

The distribution of Yule's "nonsense correlation"

In 2017, the authors of~\citet{ernst2017yule} explicitly computed the second moment of Yule's "nonsense correlation," offering the first mathematical explanation of Yule's 1926 empirical finding of nonsense correlation.~\citep{yule1926}. The present work closes the final longstanding open question on the distribution of Yule's nonsense correlation \beqn \rho:= \frac{\int_0^1W_1(t)W_2(t) dt - \int_0^1W_1(t) dt \int_0^1 W_2(t) dt}{\sqrt{\int_0^1 W^2_1(t) dt - \parens{\int_0^1W_1(t) dt}^2} \sqrt{\int_0^1 W^2_2(t) dt - \parens{\int_0^1W_2(t) dt}^2}} \eeqn by explicitly calculating all moments of $\rho$ (up to order 16) for two {\em independent} Wiener processes, $W_1, W_2$. These lead to an approximation to the density of Yule's nonsense correlation, apparently for the first time. We proceed to explicitly compute higher moments of Yule's nonsense correlation when the two independent Wiener processes are replaced by two \textit{correlated} Wiener processes, two independent Ornstein-Uhlenbeck processes, and two independent Brownian bridges. We conclude by extending the definition of $\rho$ to the time interval $[0, T]$ for any $T > 0$ and prove a Central Limit Theorem for the case of two independent Ornstein-Uhlenbeck processes processes.Comment: 23 pages, 1 figur