A Condensed Condensation Proof of a Determinant Evaluation Conjectured by Greg Kuperberg and Jim Propp

用硬币估计多个常数

Estimating Many Constants With a Coin

该研究在Maple中实现Jim Propp的硬币估计常数方法，扩展其功能以计算π及更多常数，突破Bruss等人的一般设定。

A new cyclic sieving phenomenon for Catalan objects

Based on computational experiments, Jim Propp and Vic Reiner suspected that there might exist a sequence of combinatorial objects $X_n$, each carrying a natural action of the cyclic group $C_{n-1}$ of order $n-1$ such that the triple $\left(X_n,C_{n-1},\frac{1}{[n+1]_q}{2n \brack n}_q\right)$ exhibits the cyclic sieving phenomenon. We prove their suspicion right.

Simulating a Random Walk with Constant Error

We analyze Jim Propp's P-machine, a simple deterministic process that simulates a random walk on $Z^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful summing. We mention three intriguing conjectures concerning sign-changes and unimodality of functions in the linear span of $\{p(\cdot,x) : x \in Z^d\}$, where $p(n,x)$ is the probability that a walk beginning from the origin arrives at $x$ at time $n$.