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Mathematics & Statistics

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**Speaker**

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Two applications of the theory of finite fields

Giacomo Micheli

2:00pm--3:00pm

CMC 130

**Abstract**

In the first application I describe a new method to produce pseudorandom number generators using the algebraic theory of projective automorphisms (this is a joint work with F. Amadio Guidi and S. Lindqvist). Our construction beats asymptotically the classical inversive congruential generator in terms of computational cost, for the same discrepancy bounds.

In the second application, I give a brief introduction to powerline communication and permutation codes, and explain how to use classical coding theory to obtain new improved lower bounds for the size of an \((n,d)\) permutation code (this is a joint work with A. Neri).

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**Sponsor**

Tutte's integer flow conjectures

Cun-Quan (CQ) Zhang

West Virginia University

3:00pm-4:00pm

CMC 130

Lesɫaw Skrzypek

**Abstract**

Let \(G\) be a graph with an orientation \(D\). A mapping \(f:E(G)\to\{\pm1,\pm2,\dotsc,\pm(k-1)\}\) is called *a nowhere-zero \(k\)-flow* if, for every vertex \(v\in V(G)\),
\[
\sum_{e\in E^+(v)}f(e)=\sum_{e\in E^-(v)}f(e).
\]

The integer flow problem is a dual of the vertex coloring problem: it is pointed out by Tutte that a planar graph \(G\) admits a nowhere-zero \(k\)-flow if and only if \(G\) is \(k\)-face-colorable. Tutte proposed several important conjectures about integer flows, such as, 3-flow, 4-flow and 5-flow conjectures. Those conjectures, though there are some breakthrough in last 40 years, remain widely open. This talk will introduce not only some history but also some basics of Tutte's flow theory.

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Occupation Kernel Methods for System Identification and Motion Tomography

Joel Rosenfeld

2:00pm-3:00pm

CMC 130

**Abstract**

In this talk I will discuss two approximation problems that appear in dynamical systems theory. First, we will examine the gray box system identification problem, where the goal is to obtain a collection of parameters for a parameterization of a dynamical system using observed trajectories from the system. The second problem concerns flow field estimation using trajectories obtained from very simple “gliders.” Using the difference between the endpoints of the anticipated trajectory and the actual trajectory, the flow field estimation problem can be treated with the tools of motion tomography.

Each of these examples utilize occupation kernels in different ways. The system identification uses occupation kernels indirectly to obtain constraints on the parameters, whereas the motion tomography problem uses occupation kernels directly as basis functions for the estimation of the flow field. Because of the intimate connection between occupation kernels and the problems discussed, we will also spend some time talking about reproducing kernel Hilbert spaces and the estimation of the occupation kernels themselves.