19,450 research outputs found
Height bounds and the Siegel property
Let be a reductive group defined over and let
be a Siegel set in . The Siegel property tells us that there are
only finitely many of bounded determinant and
denominator for which the translate intersects
. We prove a bound for the height of these which is
polynomial with respect to the determinant and denominator. The bound
generalises a result of Habegger and Pila dealing with the case of , and
has applications to the Zilber-Pink conjecture on unlikely intersections in
Shimura varieties.
In addition we prove that if is a subset of , then every Siegel set
for is contained in a finite union of -translates of a
Siegel set for .Comment: 24 pages, minor revision
Constrained hyperbolic divergence cleaning in smoothed particle magnetohydrodynamics with variable cleaning speeds
We present an updated constrained hyperbolic/parabolic divergence cleaning
algorithm for smoothed particle magnetohydrodynamics (SPMHD) that remains
conservative with wave cleaning speeds which vary in space and time. This is
accomplished by evolving the quantity instead of . Doing so
allows each particle to carry an individual wave cleaning speed, , that
can evolve in time without needing an explicit prescription for how it should
evolve, preventing circumstances which we demonstrate could lead to runaway
energy growth related to variable wave cleaning speeds. This modification
requires only a minor adjustment to the cleaning equations and is trivial to
adopt in existing codes. Finally, we demonstrate that our constrained
hyperbolic/parabolic divergence cleaning algorithm, run for a large number of
iterations, can reduce the divergence of the field to an arbitrarily small
value, achieving to machine precision.Comment: 23 pages, 16 figures, accepted for publication in Journal of
Computational Physic
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