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Nuclei for totally point determining graphs
AbstractThe nucleus (edge nucleus) of a point determining graph is defined by Geoffroy and Sumner to be the set of all points (edges) whose removal leaves the graph point determining. It is the purpose of this paper to develop the analogous concepts for totally point determining graphs, that is, graphs in which distinct points have distinct neighborhoods and closed neighborhoods
Cubulating hyperbolic free-by-cyclic groups: the general case
Let be an automorphism of the finite-rank free group
. Suppose that is word-hyperbolic. Then acts
freely and cocompactly on a CAT(0) cube complex.Comment: 36 pages, 11 figures. Version 2 contains minor corrections. Accepted
to GAF
Impact of Interaction Range and Curvature on Crystal Growth of Particles Confined to Spherical Surfaces
When colloidal particles form a crystal phase on a spherical template, their
packing is governed by the effective interaction between them and the elastic
strain of bending the growing crystal. For example, if growth commences under
appropriate conditions, and the circular crystal that forms reaches a critical
size, growth continues by incorporation of defects to alleviate elastic strain.
Recently it was found experimentally that, if defect formation is somehow not
possible, the crystal instead continues growing in ribbons that protrude from
the original crystal. Here we report on computer simulations in which we
observe both the formation of ribbons at short interaction ranges and packings
that incorporate defects if the interaction is longer-ranged. The ribbons only
form above some critical crystal size, below which the nucleus is roughly
spherically shaped. We find that the scaling of the critical crystal size
differs slightly from the one proposed by the Manoharan group, and reason this
is because the actual process is a two-step heterogeneous nucleation of ribbons
on top of roughly circular crystals.Comment: 24 pages, 11 figure
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