9,782 research outputs found
Tabulation of cubic function fields via polynomial binary cubic forms
We present a method for tabulating all cubic function fields over
whose discriminant has either odd degree or even degree
and the leading coefficient of is a non-square in , up
to a given bound on the degree of . Our method is based on a
generalization of Belabas' method for tabulating cubic number fields. The main
theoretical ingredient is a generalization of a theorem of Davenport and
Heilbronn to cubic function fields, along with a reduction theory for binary
cubic forms that provides an efficient way to compute equivalence classes of
binary cubic forms. The algorithm requires field operations as . The algorithm, examples and numerical data for
are included.Comment: 30 pages, minor typos corrected, extra table entries added, revamped
complexity analysis of the algorithm. To appear in Mathematics of Computatio
On the frontiers of polynomial computations in tropical geometry
We study some basic algorithmic problems concerning the intersection of
tropical hypersurfaces in general dimension: deciding whether this intersection
is nonempty, whether it is a tropical variety, and whether it is connected, as
well as counting the number of connected components. We characterize the
borderline between tractable and hard computations by proving
-hardness and #-hardness results under various
strong restrictions of the input data, as well as providing polynomial time
algorithms for various other restrictions.Comment: 17 pages, 5 figures. To appear in Journal of Symbolic Computatio
Distributed-memory large deformation diffeomorphic 3D image registration
We present a parallel distributed-memory algorithm for large deformation
diffeomorphic registration of volumetric images that produces large isochoric
deformations (locally volume preserving). Image registration is a key
technology in medical image analysis. Our algorithm uses a partial differential
equation constrained optimal control formulation. Finding the optimal
deformation map requires the solution of a highly nonlinear problem that
involves pseudo-differential operators, biharmonic operators, and pure
advection operators both forward and back- ward in time. A key issue is the
time to solution, which poses the demand for efficient optimization methods as
well as an effective utilization of high performance computing resources. To
address this problem we use a preconditioned, inexact, Gauss-Newton- Krylov
solver. Our algorithm integrates several components: a spectral discretization
in space, a semi-Lagrangian formulation in time, analytic adjoints, different
regularization functionals (including volume-preserving ones), a spectral
preconditioner, a highly optimized distributed Fast Fourier Transform, and a
cubic interpolation scheme for the semi-Lagrangian time-stepping. We
demonstrate the scalability of our algorithm on images with resolution of up to
on the "Maverick" and "Stampede" systems at the Texas Advanced
Computing Center (TACC). The critical problem in the medical imaging
application domain is strong scaling, that is, solving registration problems of
a moderate size of ---a typical resolution for medical images. We are
able to solve the registration problem for images of this size in less than
five seconds on 64 x86 nodes of TACC's "Maverick" system.Comment: accepted for publication at SC16 in Salt Lake City, Utah, USA;
November 201
Nonvanishing of twists of -functions attached to Hilbert modular forms
We describe algorithms for computing central values of twists of
-functions associated to Hilbert modular forms, carry out such computations
for a number of examples, and compare the results of these computations to some
heuristics and predictions from random matrix theory.Comment: 19 page
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