182 research outputs found
Comments on numerical solution of boundary value problems of the Laplace equation and calculation of eigenvalues by the grid method
The mathematics involved in numerically solving for the plane boundary value of the Laplace equation by the grid method is developed. The approximate solution of a boundary value problem for the domain of the Laplace equation by the grid method consists of finding u at the grid corner which satisfies the equation at the internal corners (u=Du) and certain boundary value conditions at the boundary corners
Closed geodesics in Alexandrov spaces of curvature bounded from above
In this paper, we show a local energy convexity of maps into
spaces. This energy convexity allows us to extend Colding and
Minicozzi's width-sweepout construction to produce closed geodesics in any
closed Alexandrov space of curvature bounded from above, which also provides a
generalized version of the Birkhoff-Lyusternik theorem on the existence of
non-trivial closed geodesics in the Alexandrov setting.Comment: Final version, 22 pages, 2 figures, to appear in the Journal of
Geometric Analysi
Multiple closed geodesics on bumpy Finsler -spheres
In this paper we prove that for every bumpy Finsler metric on every
rationally homological -dimensional sphere with , there exist
always at least two distinct prime closed geodesics.Comment: 22 page
Analytic Continuation of Mellin Transforms up to two-loop Order
The analytic continuation of the Mellin transforms to complex values of N for
the basic functions of the momentum fraction x emerging in the
quantities of massless QED and QCD up to two-loop order, as the unpolarized and
polarized splitting functions, coefficient functions, and hard scattering cross
sections for space- and time-like momentum transfer are evaluated. These Mellin
transforms provide the analytic continuations of all finite harmonic sums up to
the level of the threefold sums of transcendentality four, where the basis-set
consists of products of {\sc Nielsen}-integrals up to
transcendentality four. The computer code {\tt ANCONT} is provided.Comment: 31 pages LATEX, 1 style fil
More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons
A recent precise formulation of the hoop conjecture in four spacetime
dimensions is that the Birkhoff invariant (the least maximal length of
any sweepout or foliation by circles) of an apparent horizon of energy and
area should satisfy . This conjecture together with the
Cosmic Censorship or Isoperimetric inequality implies that the length of
the shortest non-trivial closed geodesic satisfies . We have
tested these conjectures on the horizons of all four-charged rotating black
hole solutions of ungauged supergravity theories and find that they always
hold. They continue to hold in the the presence of a negative cosmological
constant, and for multi-charged rotating solutions in gauged supergravity.
Surprisingly, they also hold for the Ernst-Wild static black holes immersed in
a magnetic field, which are asymptotic to the Melvin solution. In five
spacetime dimensions we define as the least maximal area of all
sweepouts of the horizon by two-dimensional tori, and find in all cases
examined that , which we conjecture holds
quiet generally for apparent horizons. In even spacetime dimensions ,
we find that for sweepouts by the product , is
bounded from above by a certain dimension-dependent multiple of the energy .
We also find that is bounded from above by a certain
dimension-dependent multiple of the horizon area . Finally, we show that
is bounded from above by a certain dimension-dependent multiple of
the energy, for all Kerr-AdS black holes.Comment: 25 page
Mellin Representation for the Heavy Flavor Contributions to Deep Inelastic Structure Functions
We derive semi--analytic expressions for the analytic continuation of the
Mellin transforms of the heavy flavor QCD coefficient functions for neutral
current deep inelastic scattering in leading and next-to-leading order to
complex values of the Mellin variable . These representations are used in
Mellin--space QCD evolution programs to provide fast evaluations of the heavy
flavor contributions to the structure functions and
.Comment: 13 pages Letex, 1 style file, 10 eps figure
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