7,948 research outputs found
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example
Geometric Error of Finite Volume Schemes for Conservation Laws on Evolving Surfaces
This paper studies finite volume schemes for scalar hyperbolic conservation
laws on evolving hypersurfaces of . We compare theoretical
schemes assuming knowledge of all geometric quantities to (practical) schemes
defined on moving polyhedra approximating the surface. For the former schemes
error estimates have already been proven, but the implementation of such
schemes is not feasible for complex geometries. The latter schemes, in
contrast, only require (easily) computable geometric quantities and are thus
more useful for actual computations. We prove that the difference between
approximate solutions defined by the respective families of schemes is of the
order of the mesh width. In particular, the practical scheme converges to the
entropy solution with the same rate as the theoretical one. Numerical
experiments show that the proven order of convergence is optimal.Comment: 23 pages, 5 figures, to appear in Numerische Mathemati
Capturing of a Magnetic Skyrmion with a Hole
Magnetic whirls in chiral magnets, so-called skyrmions, can be manipulated by
ultrasmall current densities. Here we study both analytically and numerically
the interactions of a single skyrmion in two dimensions with a small hole in
the magnetic layer. Results from micromagnetic simulations are in good
agreement with effective equations of motion obtained from a generalization of
the Thiele approach. Skyrmion-defect interactions are described by an effective
potential with both repulsive and attractive components. For small current
densities a previously pinned skyrmion stays pinned whereas an unpinned
skyrmion moves around the impurities and never gets captured. For higher
current densities, j_c1 < j < j_c2, however, single holes are able to capture
moving skyrmions. The maximal cross section is proportional to the skyrmion
radius and to Sqrt(alpha), where alpha is the Gilbert damping. For j > j_c2 all
skyrmions are depinned. Small changes of the magnetic field strongly change the
pinning properties, one can even reach a regime without pinning, j_c2=0. We
also show that a small density of holes can effectively accelerate the motion
of the skyrmion and introduce a Hall effect for the skyrmion.Comment: 11 page
Edge instabilities and skyrmion creation in magnetic layers
We study both analytically and numerically the edge of two-dimensional
ferromagnets with Dzyaloshinskii-Moriya (DM) interactions, considering both
chiral magnets and magnets with interface-induced DM interactions. We show that
in the field-polarized ferromagnetic phase magnon states exist which are bound
to the edge, and we calculate their spectra within a continuum field theory.
Upon lowering an external magnetic field, these bound magnons condense at a
finite momentum and the edge becomes locally unstable. Micromagnetic
simulations demonstrate that this edge instability triggers the creation of a
helical phase which penetrates the field-polarized state within the bulk. A
subsequent increase of the magnetic field allows to create skyrmions close to
the edge in a controlled manner.Comment: 10 pages, 8 figures; (v2) minor corrections, published versio
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