24,788 research outputs found
The stochastic thin-film equation: existence of nonnegative martingale solutions
We consider the stochastic thin-film equation with colored Gaussian
Stratonovich noise in one space dimension and establish the existence of
nonnegative weak (martingale) solutions. The construction is based on a
Trotter-Kato-type decomposition into a deterministic and a stochastic
evolution, which yields an easy to implement numerical algorithm. Compared to
previous work, no interface potential has to be included, the initial data and
the solution can have de-wetted regions of positive measure, and the
Trotter-Kato scheme allows for a simpler proof of existence than in case of
It\^o noise.Comment: 38 pages, revised version, nonnegativity proof changed, details to
time regularity and interpolation of operators adde
The Spectra of Lamplighter Groups and Cayley Machines
We calculate the spectra and spectral measures associated to random walks on
restricted wreath products of finite groups with the infinite cyclic group, by
calculating the Kesten-von Neumann-Serre spectral measures for the random walks
on Schreier graphs of certain groups generated by automata. This generalises
the work of Grigorchuk and Zuk on the lamplighter group. In the process we
characterise when the usual spectral measure for a group generated by automata
coincides with the Kesten-von Neumann-Serre spectral measure.Comment: 36 pages, improved exposition, main results slightly strengthene
On the rational subset problem for groups
We use language theory to study the rational subset problem for groups and
monoids. We show that the decidability of this problem is preserved under graph
of groups constructions with finite edge groups. In particular, it passes
through free products amalgamated over finite subgroups and HNN extensions with
finite associated subgroups. We provide a simple proof of a result of
Grunschlag showing that the decidability of this problem is a virtual property.
We prove further that the problem is decidable for a direct product of a group
G with a monoid M if and only if membership is uniformly decidable for
G-automata subsets of M. It follows that a direct product of a free group with
any abelian group or commutative monoid has decidable rational subset
membership.Comment: 19 page
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