**Time: ** Tuesday, Oct 4, 2016, 2:00pm @ MTH1311

**Speaker: ** Prof. Stephen Casey (American University)

**Title: ** The Analysis of Periodic Point Processes

**Abstract: ** Our talk addresses the problems of extracting information from periodic point processes. These problems arise in numerous situations, from radar pulse repetition interval analysis to bit synchronization in communication systems. We divide our analysis into two cases: periodic processes created by a single source, and those processes created by several sources. We wish to extract the fundamental period of the generators, and, in the second case, to deinterleave the processes.
We first present very efficient algorithm for extracting the fundamental period from a set of sparse and noisy observations of a single source periodic process. The procedure is computationally straightforward, quickly convergent, and stable with respect to noise. Its use is justified by a theorem which shows that for a set of randomly chosen positive integers, the probability that they do not all share a common prime factor approaches one quickly as the cardinality of the set increases. The proof of this theorem rests on a probabilistic interpretation of the Riemann zeta function. We then build upon this procedure to deinterleave and then analyze data from multiple periodic processes. This relies on the probabilistic interpretation of the Riemann zeta function, the equidistribution theorem of Weyl, and techniques from spectrum analysis. We present simulations of the procedures, which were developed jointly by the speaker and Kevin Duke of American University.
We close by using our a probabilistic interpretation of the Riemann zeta function to to develop numerical approximations of the sums $\sum_{n=1}^\infty \frac{1}{n^{2k+1}}$, for k in the natural numbers.
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