6,462 research outputs found
Apollonian structure in the Abelian sandpile
The Abelian sandpile process evolves configurations of chips on the integer
lattice by toppling any vertex with at least 4 chips, distributing one of its
chips to each of its 4 neighbors. When begun from a large stack of chips, the
terminal state of the sandpile has a curious fractal structure which has
remained unexplained. Using a characterization of the quadratic growths
attainable by integer-superharmonic functions, we prove that the sandpile PDE
recently shown to characterize the scaling limit of the sandpile admits certain
fractal solutions, giving a precise mathematical perspective on the fractal
nature of the sandpile.Comment: 27 Pages, 7 Figure
Equivalent trace sets for arithmetic Fuchsian groups
We show that the modular group has an infinite family of finite index
subgroups, each of which has the same trace set as the modular group itself.
Various congruence subgroups of the modular group, and the Bianchi groups, are
also shown to have this property. In the case of the modular group, we
construct examples of such finite index subgroups.Comment: 14 pages, 5 figures. To appear in Proceedings of the AM
On walls of marginal stability in N=2 string theories
We study the properties of walls of marginal stability for BPS decays in a
class of N=2 theories. These theories arise in N=2 string compactifications
obtained as freely acting orbifolds of N=4 theories, such theories include the
STU model and the FHSV model. The cross sections of these walls for a generic
decay in the axion-dilaton plane reduce to lines or circles. From the
continuity properties of walls of marginal stability we show that central
charges of BPS states do not vanish in the interior of the moduli space. Given
a charge vector of a BPS state corresponding to a large black hole in these
theories, we show that all walls of marginal stability intersect at the same
point in the lower half of the axion-dilaton plane. We isolate a class of
decays whose walls of marginal stability always lie in a region bounded by
walls formed by decays to small black holes. This enables us to isolate a
region in moduli space for which no decays occur within this class. We then
study entropy enigma decays for such models and show that for generic values of
the moduli, that is when moduli are of order one compared to the charges,
entropy enigma decays do not occur in these models.Comment: 40 pages, 2 figure
Naturality of Heegaard Floer invariants under positive rational contact surgery
For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we
consider a contact structure obtained by positive rational contact surgery. We
prove that in this situation the Heegaard Floer contact invariant of Y is
mapped by a surgery cobordism to the contact invariant of the result of contact
surgery. In addition we characterize the spin-c structure on the cobordism that
induces the relevant map. As a consequence we determine necessary and
sufficient conditions for the nonvanishing of the contact invariant after
rational surgery when Y is the standard 3-sphere, generalizing previous results
of Lisca-Stipsicz and Golla. In fact our methods allow direct calculation of
the contact invariant in terms of the rational surgery mapping cone of
Ozsv\'ath and Szab\'o. The proof involves a construction called reducible open
book surgery, which reduces in special cases to the capping-off construction
studied by Baldwin.Comment: Extended the main results from surgery coefficients that are at least
1 to all positive surgery coefficients. This version to appear in Journal of
Differential Geometr
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