Neural Architectures for Control

The cerebellar model articulated controller (CMAC) neural architectures are shown to be viable for the purposes of real-time learning and control. Software tools for the exploration of CMAC performance are developed for three hardware platforms, the MacIntosh, the IBM PC, and the SUN workstation. All algorithm development was done using the C programming language. These software tools were then used to implement an adaptive critic neuro-control design that learns in real-time how to back up a trailer truck. The truck backer-upper experiment is a standard performance measure in the neural network literature, but previously the training of the controllers was done off-line. With the CMAC neural architectures, it was possible to train the neuro-controllers on-line in real-time on a MS-DOS PC 386. CMAC neural architectures are also used in conjunction with a hierarchical planning approach to find collision-free paths over 2-D analog valued obstacle fields. The method constructs a coarse resolution version of the original problem and then finds the corresponding coarse optimal path using multipass dynamic programming. CMAC artificial neural architectures are used to estimate the analog transition costs that dynamic programming requires. The CMAC architectures are trained in real-time for each obstacle field presented. The coarse optimal path is then used as a baseline for the construction of a fine scale optimal path through the original obstacle array. These results are a very good indication of the potential power of the neural architectures in control design. In order to reach as wide an audience as possible, we have run a seminar on neuro-control that has met once per week since 20 May 1991. This seminar has thoroughly discussed the CMAC architecture, relevant portions of classical control, back propagation through time, and adaptive critic designs

The Kaprekar Routine and Other Digit Games for Undergraduate Exploration

The Kaprekar Routine is a famous mathematical procedure involving the digits of a positive integer. This paper offers natural generalizations of the routine, states and proves related results, and presents many open problems that are suitable for mathematical research at the undergraduate level. In the process, we shed light on some interesting facts about digit games

Unlocking Undergraduate Problem Solving

It is difficult to find good problems for undergraduates. In this article, we explore an interesting problem that can be used in virtually any mathematics course. We then offer natural generalizations, state and prove some related results, and ultimately end with several open problems suitable for undergraduate research. Finally, we attempt to shed some light on what makes a problem interesting

On some lattice computations related to moduli problems

We show how to solve computationally a combinatorial problem about the possible number of roots orthogonal to a vector of given length in $E_8$. We show that the moduli space of K3 surfaces with polarisation of degree 2d is also of general type for d=52. This case was omitted from the earlier work of Gritsenko, Hulek and the second author. We also apply this method to some related problems. In Appendix A, V. Gritsenko shows how to arrive at the case d=52 and some others directly.Comment: With an appendix by V. Gritsenk