Hilbert scales and Sobolev spaces defined by associated Legendre functions. (arXiv:0909.4266v1 [math.CA])

In this paper we study the Hilbert scales defined by the associated Legendre
functions for arbitrary integer values of the parameter. This problem is
equivalent to study the left-definite spectral theory associated to the
modified Legendre equation. We give several characterizations of the spaces as
weighted Sobolev spaces and prove identities among the spaces corresponding to
lower regularity index.

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Original source : http://arxiv.org/abs/0909.4266...

A Numerical Algorithm for Zero Counting. II: Distance to Ill-posedness and Smoothed Analysis. (arXiv:0909.4101v1 [cs.NA])

We show a Condition Number Theorem for the condition number of zero counting
for real polynomial systems. That is, we show that this condition number equals
the inverse of the normalized distance to the set of ill-posed systems (i.e.,
those having multiple real zeros). As a consequence, a smoothed analysis of
this condition number follows.

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Original source : http://arxiv.org/abs/0909.4101...

Numerical simulations of interfaces in relativistic hydrodynamics. (arXiv:0909.4217v1 [gr-qc])

We consider models of relativistic matter containing sharp interfaces across
which the matter model changes. These models will be relevant for neutron stars
with crusts, phase transitions, or for viscous boundaries where the length
scale is too short to be modelled smoothly. In particular we look at numerical
techniques that allow us to evolve stable interfaces, for the interfaces to
merge, and for strong waves and shocks to interact with the interfaces. We test
these techniques for ideal hydrodynamics in special and general relativity for
simple equations of state, finding that simple level set-based methods extend
well to relativistic hydrodynamics.

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Original source : http://arxiv.org/abs/0909.4217...