6,558 research outputs found
On Two Kinds of Differential Operators on General Smooth Surfaces
Two kinds of differential operators that can be generally defined on an
arbitrary smooth surface in a finite dimensional Euclid space are studied, one
is termed as surface gradient and the other one as Levi-Civita gradient. The
surface gradient operator is originated from the differentiability of a tensor
field defined on the surface. Some integral and differential identities have
been theoretically studied that play the important role in the studies on
continuous mediums whose geometrical configurations can be taken as surfaces
and on interactions between fluids and deformable boundaries. The definition of
Levi-Civita gradient is based on Levi-Civita connections generally defined on
Riemann manifolds. It can be used to set up some differential identities in the
intrinsic/coordiantes-independent form that play the essential role in the
theory of vorticity dynamics for two dimensional flows on general fixed smooth
surfaces
Pseudo-factorials, elliptic functions, and continued fractions
This study presents miscellaneous properties of pseudo-factorials, which are
numbers whose recurrence relation is a twisted form of that of usual
factorials. These numbers are associated with special elliptic functions, most
notably, a Dixonian and a Weierstrass function, which parametrize the Fermat
cubic curve and are relative to a hexagonal lattice. A continued fraction
expansion of the ordinary generating function of pseudo-factorials, first
discovered empirically, is established here. This article also provides a
characterization of the associated orthogonal polynomials, which appear to form
a new family of "elliptic polynomials", as well as various other properties of
pseudo-factorials, including a hexagonal lattice sum expression and elementary
congruences.Comment: 24 pages; with correction of typos and minor revision. To appear in
The Ramanujan Journa
A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes
Recent progress in unitarity techniques for one-loop scattering amplitudes
makes a numerical implementation of this method possible. We present a
4-dimensional unitarity method for calculating the cut-constructible part of
amplitudes and implement the method in a numerical procedure. Our technique can
be applied to any one-loop scattering amplitude and offers the possibility that
one-loop calculations can be performed in an automatic fashion, as tree-level
amplitudes are currently done. Instead of individual Feynman diagrams, the
ingredients for our one-loop evaluation are tree-level amplitudes, which are
often already known. To study the practicality of this method we evaluate the
cut-constructible part of the 4, 5 and 6 gluon one-loop amplitudes numerically,
using the analytically known 4, 5 and 6 gluon tree-level amplitudes.
Comparisons with analytic answers are performed to ascertain the numerical
accuracy of the method.Comment: 29 pages with 8 figures; references updated in rsponse to readers'
suggestion
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