23,863 research outputs found
Diophantine approximation on Veech surfaces
We show that Y. Cheung's general -continued fractions can be adapted to
give approximation by saddle connection vectors for any compact translation
surface. That is, we show the finiteness of his Minkowski constant for any
compact translation surface. Furthermore, we show that for a Veech surface in
standard form, each component of any saddle connection vector dominates its
conjugates. The saddle connection continued fractions then allow one to
recognize certain transcendental directions by their developments
Commensurable continued fractions
We compare two families of continued fractions algorithms, the symmetrized
Rosen algorithm and the Veech algorithm. Each of these algorithms expands real
numbers in terms of certain algebraic integers. We give explicit models of the
natural extension of the maps associated with these algorithms; prove that
these natural extensions are in fact conjugate to the first return map of the
geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost
every real number has an infinite number of common approximants for both
algorithms.Comment: 41 pages, 10 figure
Veech surfaces with non-periodic directions in the trace field
We show that each of Veech's original examples of translation surfaces with
``optimal dynamics'' whose trace field is of degree greater than two has
non-periodic directions of vanishing SAF-invariant. Furthermore, we give
explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction
has zero SAF-invariant.Comment: 22 pages, 1 figur
Preparation of iron(0) model catalysts with iron clusters of sub-nm dimensions by decomposition of Y-zeolite adsorbed iron pentacarbonyl
Tong's spectrum for Rosen continued fractions
The Rosen fractions are an infinite set of continued fraction algorithms,
each giving expansions of real numbers in terms of certain algebraic integers.
For each, we give a best possible upper bound for the minimum in appropriate
consecutive blocks of approximation coefficients (in the sense of Diophantine
approximation by continued fraction convergents). We also obtain metrical
results for large blocks of ``bad'' approximations.Comment: 22 pages, 5 figure
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