195 research outputs found
Subword complexity and Laurent series with coefficients in a finite field
Decimal expansions of classical constants such as , and
have long been a source of difficult questions. In the case of
Laurent series with coefficients in a finite field, where no carry-over
difficulties appear, the situation seems to be simplified and drastically
different. On the other hand, Carlitz introduced analogs of real numbers such
as , or . Hence, it became reasonable to enquire how
"complex" the Laurent representation of these "numbers" is. In this paper we
prove that the inverse of Carlitz's analog of , , has in general a
linear complexity, except in the case , when the complexity is quadratic.
In particular, this implies the transcendence of over \F_2(T). In the
second part, we consider the classes of Laurent series of at most polynomial
complexity and of zero entropy. We show that these satisfy some nice closure
properties
Searchable Sky Coverage of Astronomical Observations: Footprints and Exposures
Sky coverage is one of the most important pieces of information about
astronomical observations. We discuss possible representations, and present
algorithms to create and manipulate shapes consisting of generalized spherical
polygons with arbitrary complexity and size on the celestial sphere. This shape
specification integrates well with our Hierarchical Triangular Mesh indexing
toolbox, whose performance and capabilities are enhanced by the advanced
features presented here. Our portable implementation of the relevant spherical
geometry routines comes with wrapper functions for database queries, which are
currently being used within several scientific catalog archives including the
Sloan Digital Sky Survey, the Galaxy Evolution Explorer and the Hubble Legacy
Archive projects as well as the Footprint Service of the Virtual Observatory.Comment: 11 pages, 7 figures, submitted to PAS
On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof
We exhibit a family of graphs that develop turning singularities (i.e. their
Lipschitz seminorm blows up and they cease to be a graph, passing from the
stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem
where the permeability is given by a nonnegative step function. We study the
influence of different choices of the permeability and different boundary
conditions (both at infinity and considering finite/infinite depth) in the
development or prevention of singularities for short time. In the general case
(inhomogeneous, confined) we prove a bifurcation diagram concerning the
appearance or not of singularities when the depth of the medium and the
permeabilities change. The proofs are carried out using a combination of
classical analysis techniques and computer-assisted verification.Comment: 30 pages, 6 figure
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