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Mathematics & Statistics

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The Symbolic Dynamics of Tiling the Integers

Ethan Coven

TBA

TBA

TBA

**Abstract**

A finite collection of finite sets tiles the integers if and only if the integers can be expressed as a disjoint union of translates of members of the collection. We associate with such a tiling a doubly infinite sequence with symbols the sets in the collection. The set of all such sequences is a sofic system, called a tiling system.

For example, if \(P\) consists of the sets \(\{0\}\) and \(\{0,1\}\), then the tiling system is the collection of all doubly infinite sequences with symbols \(R\) (red, the “color” of \(\{0\}\)) and \(B\) (blue, the “color” of \(\{0,1\}\)) such that between any two consecutive appearances of \(R\), there are an even number of \(B\), i.e., the “even system”. This sofic system is closely related to the “Golden Mean” shift of finite type. Many transitive shifts of finite type, e.g., the full \(2\)-shift, cannot be realized (up to topological conjugacy) as tiling systems. However, we show that, up to powers of the shift, every shift of finite type can be realized as a tiling system.

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An Unusual Way to Generate Conic Sections and two Related Euclidean Constructions

Sam Sakmar

Department of Physics, USF

3:00pm-4:00pm

PHY 130

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Braids of surfaces in \(4\)-space

Seiichi Kamada

U. of South Alabama (and Osaka City U.)

3:00pm-4:00pm

PHY 130

**Abstract**

An \(m\)-braid is a collection of \(m\) strings in a cylinder \(D^2\times I^1\) satisfying a certain condition. The set of \(m\)-braids forms a group, called the \(m\)-braid group. This group plays an important role in knot theory. Knot theory treats of embedded closed curves in Euclidean \(3\)-space, and \(2\)-dimensional knot theory treats of embedded closed surfaces in \(4\)-space. In this talk, a generalization of \(m\)-braids is introduced, which is called a \(2\)-dimensional \(m\)-braid or a surface braid. That is a surface in a bi-disk \(D^2\times D^2\) satisfying a certain condition. The set of \(2\)-dimensional \(m\)-braids forms a monoid (a semi-group with identity). \(2\)-dimensional braids are related with \(2\)-dimensional knot theory by the following two theorems.

**Generalized Alexander's theorem**. Any closed surface in \(4\)-space is described by a closed \(2\)-dimensional braid.

**Generalized Markov's theorem**. Such a braid description is unique up to braid isotopy, conjugation and stabilization.

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The Dangers of Near-Earth Asteroids

Rudy Dvorak

Astronomy Department

University of Vienna

Vienna, Austria

2:00pm-3:00pm

PHY 108

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Some Asymptotic Results and Exponential Approximations in Semi-Markov Models

George Roussas

Professor and Associate Dean

Department of Mathematics & Statistics

University of California, Davis

2:00pm-3:00pm

PHY 118

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A Rayleigh-Ritz Method Applied to a Two-Dimensional Inverse Spectral Problem

C. Maeve McCarthy

Murray State University

3:00pm-4:00pm

PHY 120

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Groups of \(2\times 2\) Matrices

Ross Geoghegan

SUNY at Binghamton

2:00pm-3:00pm

PHY 108

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Estimation With Meyer Types and Wavelets

Marianna Pensky

3:00pm-4:00pm

PHY 118