8,963 research outputs found
Knots and Numbers in Theory to 7 Loops and Beyond
We evaluate all the primitive divergences contributing to the 7--loop
\/--function of theory, i.e.\ all 59 diagrams that are free of
subdivergences and hence give scheme--independent contributions. Guided by the
association of diagrams with knots, we obtain analytical results for 56
diagrams. The remaining three diagrams, associated with the knots ,
, and , are evaluated numerically, to 10 sf. Only one
satellite knot with 11 crossings is encountered and the transcendental number
associated with it is found. Thus we achieve an analytical result for the
6--loop contributions, and a numerical result at 7 loops that is accurate to
one part in . The series of `zig--zag' counterterms,
,
previously known for loops, is evaluated to 10 loops, corresponding
to 17 crossings, revealing that the \/--loop zig--zag term is , where are the Catalan numbers, familiar in knot theory. The investigations
reported here entailed intensive use of REDUCE, to generate
lines of code for multiple precision FORTRAN computations, enabled by Bailey's
MPFUN routines, running for CPUhours on DecAlpha machines.Comment: 6 pages plain LaTe
Convergence Acceleration via Combined Nonlinear-Condensation Transformations
A method of numerically evaluating slowly convergent monotone series is
described. First, we apply a condensation transformation due to Van Wijngaarden
to the original series. This transforms the original monotone series into an
alternating series. In the second step, the convergence of the transformed
series is accelerated with the help of suitable nonlinear sequence
transformations that are known to be particularly powerful for alternating
series. Some theoretical aspects of our approach are discussed. The efficiency,
numerical stability, and wide applicability of the combined
nonlinear-condensation transformation is illustrated by a number of examples.
We discuss the evaluation of special functions close to or on the boundary of
the circle of convergence, even in the vicinity of singularities. We also
consider a series of products of spherical Bessel functions, which serves as a
model for partial wave expansions occurring in quantum electrodynamic bound
state calculations.Comment: 24 pages, LaTeX, 12 tables (accepted for publication in Comput. Phys.
Comm.
Level truncation and the quartic tachyon coupling
We discuss the convergence of level truncation in bosonic open string field
theory. As a test case we consider the calculation of the quartic tachyon
coupling . We determine the exact contribution from states up to
level L=28 and discuss the extrapolation by means of the BST
algorithm. We determine in a self-consistent way both the coupling and the
exponent of the leading correction to at finite that we
assume to be . The results are and
.}Comment: 17 pages, 2 eps figure
Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations
We consider the numerical discretization of the time-domain Maxwell's
equations with an energy-conserving discontinuous Galerkin finite element
formulation. This particular formulation allows for higher order approximations
of the electric and magnetic field. Special emphasis is placed on an efficient
implementation which is achieved by taking advantage of recurrence properties
and the tensor-product structure of the chosen shape functions. These
recurrences have been derived symbolically with computer algebra methods
reminiscent of the holonomic systems approach.Comment: 16 pages, 1 figure, 1 table; Springer Wien, ISBN 978-3-7091-0793-
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