191,645 research outputs found
The Convex Hull Problem in Practice : Improving the Running Time of the Double Description Method
The double description method is a simple but widely used algorithm for computation of extreme points in polyhedral sets. One key aspect of its implementation is the question of how to efficiently test extreme points for adjacency. In this dissertation, two significant contributions related to adjacency testing are presented. First, the currently used data structures are revisited and various optimizations are proposed. Empirical evidence is provided to demonstrate their competitiveness. Second, a new adjacency test is introduced. It is a refinement of the well known algebraic test featuring a technique for avoiding redundant computations. Its correctness is formally proven. Its superiority in multiple degenerate scenarios is demonstrated through experimental results. Parallel computation is one further aspect of the double description method covered in this work. A recently introduced divide-and-conquer technique is revisited and considerable practical limitations are demonstrated
Rational Solutions of the Painlev\'e-II Equation Revisited
The rational solutions of the Painlev\'e-II equation appear in several
applications and are known to have many remarkable algebraic and analytic
properties. They also have several different representations, useful in
different ways for establishing these properties. In particular,
Riemann-Hilbert representations have proven to be useful for extracting the
asymptotic behavior of the rational solutions in the limit of large degree
(equivalently the large-parameter limit). We review the elementary properties
of the rational Painlev\'e-II functions, and then we describe three different
Riemann-Hilbert representations of them that have appeared in the literature: a
representation by means of the isomonodromy theory of the Flaschka-Newell Lax
pair, a second representation by means of the isomonodromy theory of the
Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner
related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and
Bertola-Bothner Riemann-Hilbert representations of the rational Painlev\'e-II
functions are explicitly connected to each other. Finally, we review recent
results describing the asymptotic behavior of the rational Painlev\'e-II
functions obtained from these Riemann-Hilbert representations by means of the
steepest descent method
Archimedean cohomology revisited
Archimedean cohomology provides a cohomological interpretation for the
calculation of the local L-factors at archimedean places as zeta regularized
determinant of a log of Frobenius. In this paper we investigate further the
properties of the Lefschetz and log of monodromy operators on this cohomology.
We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian
connection whose residue is the log of the monodromy. We also present a
dictionary of analogies between the geometry of a tubular neighborhood of the
``fiber at arithmetic infinity'' of an arithmetic variety and the complex of
nearby cycles in the geometry of a degeneration over a disk, and we recall
Deninger's approach to the archimedean cohomology through an interpretation as
global sections of a analytic Rees sheaf. We show that action of the Lefschetz,
the log of monodromy and the log of Frobenius on the archimedean cohomology
combine to determine a spectral triple in the sense of Connes. The archimedean
part of the Hasse-Weil L-function appears as a zeta function of this spectral
triple. We also outline some formal analogies between this cohomological theory
at arithmetic infinity and Givental's homological geometry on loop spaces.Comment: 28 pages LaTeX 3 eps figure
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