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This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is exhibited that can fill the Euclidean plane only with a tiling that contains k distinct isohedral sets of tiles, where k can be made arbitrarily large. It is shown that the construction cannot work for a simply connected 2D tile with matching rules for adjacent tiles enforced by shape alone. It is also shown that any of the following modifications allows the construction to work: (1) coloring the edges of the tiling and imposing rules on which colors can touch; (2) allowing the tile to be multiply connected; (3) requiring maximum density rather than space-filling; (4) allowing the tile to have a thickness in the third dimension.
Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with at least $binom{n}{3} - binom{n-m+1}{3} + o(n^3)$ edges contains a collection of $m$ vertex-disj
In this paper we study $k$-noncrossing matchings. A $k$-noncrossing matching is a labeled graph with vertex set ${1,...,2n}$ arranged in increasing order in a horizontal line and vertex-degree 1. The $n$ arcs are drawn in the upper halfplane subject
We produce a Bose-Einstein condensate of 39-K atoms. Condensation of this species with naturally small and negative scattering length is achieved by a combination of sympathetic cooling with 87-Rb and direct evaporation, exploiting the magnetic tunin
The order-$k$ Voronoi tessellation of a locally finite set $X subseteq mathbb{R}^n$ decomposes $mathbb{R}^n$ into convex domains whose points have the same $k$ nearest neighbors in $X$. Assuming $X$ is a stationary Poisson point process, we give expl
We perform extensive magnetic Feshbach spectroscopy of an ultracold mixture of fermionic 40K and bosonic 87Rb atoms. The magnetic-field locations of 14 interspecies resonances is used to construct a quantum collision model able to predict accurate co