52,204 research outputs found
The remaining cases of the Kramer-Tunnell conjecture
For an elliptic curve over a local field and a separable quadratic
extension of , motivated by connections to the Birch and Swinnerton-Dyer
conjecture, Kramer and Tunnell have conjectured a formula for computing the
local root number of the base change of to the quadratic extension in terms
of a certain norm index. The formula is known in all cases except some when
is of characteristic , and we complete its proof by reducing the positive
characteristic case to characteristic . For this reduction, we exploit the
principle that local fields of characteristic can be approximated by finite
extensions of --we find an elliptic curve defined over a
-adic field such that all the terms in the Kramer-Tunnell formula for
are equal to those for .Comment: 13 pages; final version, to appear in Compositio Mathematic
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GraphTool : a tool for interactive design and manipulation of graphs and graph algorithms
GraphTool is an interactive tool for editing graphs and visualizing the execution and results of graph algorithms. It runs under both the SunView and X Windows environments and has a full window/mouse interface which is as similar as possible for the two windowing systems. In addition, there is a standalone program called the Wrapper which simulates the Graph-Tool interface without graphics for batch processing of graph algorithms. While the primary purpose of GraphTool is to provide a means for experimentally investigating the performance of graph algorithms, it has other useful features as well. It provides features for printing graphs in a visually appealing format, which makes it easier to prepare papers for publication. It also provides a facility for "animating" algorithms, which means that it can be used in computer assisted instruction (CAI) and for preparing video presentations of algorithms
Number fields unramified away from 2
We consider finite extensions of the rationals which are unramified except
for at 2 and infinity. We show there are no such extensions of degrees 9
through 15
Extending Immersions into the Sphere
We study the problem to extend an immersed circle f in the 2-dimensional
sphere to an immersion of the disc. We analyze existence and uniqueness for
this problems in terms of the combinatorial structure of a word assigned to f.
Our techniques are based on ideas of Blank who studied the extension problem in
case of a planar target
A parallel edge orientation algorithm for quadrilateral meshes
One approach to achieving correct finite element assembly is to ensure that
the local orientation of facets relative to each cell in the mesh is consistent
with the global orientation of that facet. Rognes et al. have shown how to
achieve this for any mesh composed of simplex elements, and deal.II contains a
serial algorithm to construct a consistent orientation of any quadrilateral
mesh of an orientable manifold.
The core contribution of this paper is the extension of this algorithm for
distributed memory parallel computers, which facilitates its seamless
application as part of a parallel simulation system.
Furthermore, our analysis establishes a link between the well-known
Union-Find algorithm and the construction of a consistent orientation of a
quadrilateral mesh. As a result, existing work on the parallelisation of the
Union-Find algorithm can be easily adapted to construct further parallel
algorithms for mesh orientations.Comment: Second revision: minor change
On a general implementation of - and -adaptive curl-conforming finite elements
Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used
by the computational electromagnetics community. However, its implementation,
specially for high order methods, is not trivial, since it involves many
technicalities that are not properly described in the literature. To fill this
gap, we provide a comprehensive description of a general implementation of edge
elements of first kind within the scientific software project FEMPAR. We cover
into detail how to implement arbitrary order (i.e., -adaptive) elements on
hexahedral and tetrahedral meshes. First, we set the three classical
ingredients of the finite element definition by Ciarlet, both in the reference
and the physical space: cell topologies, polynomial spaces and moments. With
these ingredients, shape functions are automatically implemented by defining a
judiciously chosen polynomial pre-basis that spans the local finite element
space combined with a change of basis to automatically obtain a canonical basis
with respect to the moments at hand. Next, we discuss global finite element
spaces putting emphasis on the construction of global shape functions through
oriented meshes, appropriate geometrical mappings, and equivalence classes of
moments, in order to preserve the inter-element continuity of tangential
components of the magnetic field. Finally, we extend the proposed methodology
to generate global curl-conforming spaces on non-conforming hierarchically
refined (i.e., -adaptive) meshes with arbitrary order finite elements.
Numerical results include experimental convergence rates to test the proposed
implementation
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