Borcherds Forms and Generalizations of Singular Moduli

We give a factorization of averages of Borcherds forms over CM points associated to a quadratic form of signature (n,2). As a consequence of this result, we are able to state a theorem like that of Gross and Zagier about which primes can occur in this factorization. One remarkable phenomenon we observe is that the regularized theta lift of a weakly holomorphic modular form is always finite

Variational Optimal-Control Problems with Delayed Arguments on Time Scales

This article deals with variational optimal-control problems on time scales in the presence of delay in the state variables. The problem is considered on a time scale unifying the discrete, the continuous and the quantum cases. Two examples in the discrete and quantum cases are analyzed to illustrate our results.Comment: To apear in Advances in Difference Equation

Massively parallel approximate Gaussian process regression

We explore how the big-three computing paradigms -- symmetric multi-processor (SMC), graphical processing units (GPUs), and cluster computing -- can together be brought to bare on large-data Gaussian processes (GP) regression problems via a careful implementation of a newly developed local approximation scheme. Our methodological contribution focuses primarily on GPU computation, as this requires the most care and also provides the largest performance boost. However, in our empirical work we study the relative merits of all three paradigms to determine how best to combine them. The paper concludes with two case studies. One is a real data fluid-dynamics computer experiment which benefits from the local nature of our approximation; the second is a synthetic data example designed to find the largest design for which (accurate) GP emulation can performed on a commensurate predictive set under an hour.Comment: 24 pages, 6 figures, 1 tabl

Robot motion planning using real-time heuristic search

Autonomous mobile robots must be able to plan quickly and stay reactive to the world around them. Currently, navigating in the presence of dynamic obstacles is a problem that modern techniques struggle to handle in a real-time manner, even when the environment is known. The solutions range from using: 1) sampling-based algorithms which cut down on the shear size of these state spaces, 2) algorithms which quickly try to plan complete paths to the goal (to avoid local minima) and 3) using real-time search techniques designed for static worlds. Each of these methods have fundamental flaws that prevent it from being used in practice. In this thesis I offer three proposed techniques to help improve planning among dynamic obstacles. First, I present a new partitioned learning technique for splitting the costs estimates used by heuristic search techniques into those caused by the static environment and those caused by the dynamic obstacles in the world. This allows for much more accurate learning. Second, I introduce a novel decaying heuristic technique for generalizing cost-to-go over states of the same pose (x. y.theta.v) in the world. Third, I show a garbage collection mechanism for removing useless states from our search to cut down on the overall memory usage. Finally, I present a new algorithm called Partitioned Learning Real-time A*. PLRTA* uses all three of these new enhancements to navigate through worlds with dynamic obstacles in a real-time manner while handling the complex situations in which other algorithms fail. I empirically compare our algorithm to other competing algorithms in a number of random instances as well as hand crafted scenarios designed to highlight desirable behavior in specific situations. I show that PLRTA* outperforms the current state-of-the-art algorithms in terms of minimizing cost over a large number of robot motion planning problems, even when planning in fairly confined environments with up to ten dynamic obstacles