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Let S be a finite set of points in the plane and let T (S) be the set of intersection points between pairs of lines passing through any two points in S. We characterize all configurations of points S such that iteration of the above operation produces a dense set. We also discuss partial results on the characterization of those finite point-sets with… (More)

We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X, A) = B, where w is a finite word in the alphabet {X, A}. We think of A, B ∈ G as fixed coefficients, and X ∈ G as the unknown. Certain word equations, such as XAXAX = B, have solutions in terms of radicals: X =… (More)

- Kelli F Talaska, Sergey Fomin, Samuel D Epstein, Karen E Smith, John R Stembridge, Thomas F Y Lam +36 others
- 2010

This dissertation is dedicated to the memory of my mother, Kathleen Marie Harter. ii ACKNOWLEDGEMENTS I have come this far largely because I have had a great deal of support, encouragement , and companionship from amazing teachers, friends, and colleagues, and I would like to take this opportunity to thank some of those people. Graduate school at Michigan… (More)

- Lionel David Jerision, Scott Levine, Sheffield, Christopher J Hillar, Lionel Levine, Darren Rhea +6 others
- 2010

Equations solvable by radicals in a uniquely divisible group. arXiv:1004.5239 2011 4. Lionel Levine, Sandpile groups and spanning trees of directed line graphs. Parallel chip-firing on the complete graph: devil's staircase and Poincaré rotation number. Scaling limits for internal aggregation models with multiple sources. Strong spherical asymptotics for… (More)

We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X, A) = B, where w is a finite word in the alphabet {X, A}. We think of A, B ∈ G as fixed coefficients, and X ∈ G as the unknown. Certain word equations, such as XAXAX = B, have solutions in terms of radicals: X =… (More)

- David Jerision, Lionel Levine, Scott Sheffield, Christopher, J Hillar, Darren Rhea
- 2010

Discrete probability, statistical mechanics, algebraic combinatorics. Equations solvable by radicals in a uniquely divisible group. arXiv:1004.5239 2011 4. Lionel Levine, Sandpile groups and spanning trees of directed line graphs.

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