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Scalar Levin-Type Sequence Transformations
Sequence transformations are important tools for the convergence acceleration
of slowly convergent scalar sequences or series and for the summation of
divergent series. Transformations that depend not only on the sequence elements
or partial sums but also on an auxiliary sequence of so-called remainder
estimates are of Levin-type if they are linear in the , and
nonlinear in the . Known Levin-type sequence transformations are
reviewed and put into a common theoretical framework. It is discussed how such
transformations may be constructed by either a model sequence approach or by
iteration of simple transformations. As illustration, two new sequence
transformations are derived. Common properties and results on convergence
acceleration and stability are given. For important special cases, extensions
of the general results are presented. Also, guidelines for the application of
Levin-type sequence transformations are discussed, and a few numerical examples
are given.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math.,
abstract shortene
The Lagrange spectrum of a Veech surface has a Hall ray
We study Lagrange spectra of Veech translation surfaces, which are a
generalization of the classical Lagrange spectrum. We show that any such
Lagrange spectrum contains a Hall ray. As a main tool, we use the boundary
expansion developed by Bowen and Series to code geodesics in the corresponding
Teichm\"uller disk and prove a formula which allows to express large values in
the Lagrange spectrum as sums of Cantor sets.Comment: 30 pages, 5 figures. Minor revisio
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