1,718 research outputs found
Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation
Steepest descent methods combining complex contour deformation with numerical
quadrature provide an efficient and accurate approach for the evaluation of
highly oscillatory integrals. However, unless the phase function governing the
oscillation is particularly simple, their application requires a significant
amount of a priori analysis and expert user input, to determine the appropriate
contour deformation, and to deal with the non-uniformity in the accuracy of
standard quadrature techniques associated with the coalescence of stationary
points (saddle points) with each other, or with the endpoints of the original
integration contour. In this paper we present a novel algorithm for the
numerical evaluation of oscillatory integrals with general polynomial phase
functions, which automates the contour deformation process and avoids the
difficulties typically encountered with coalescing stationary points and
endpoints. The inputs to the algorithm are simply the phase and amplitude
functions, the endpoints and orientation of the original integration contour,
and a small number of numerical parameters. By a series of numerical
experiments we demonstrate that the algorithm is accurate and efficient over a
large range of frequencies, even for examples with a large number of coalescing
stationary points and with endpoints at infinity. As a particular application,
we use our algorithm to evaluate cuspoid canonical integrals from scattering
theory. A Matlab implementation of the algorithm is made available and is
called PathFinder
On the maximal Sobolev regularity\ud of distributions supported by subsets of Euclidean space
Given a subset of with empty interior and an integrability parameter , what is the maximal regularity for which there exists a non-zero distribution in the Bessel potential Sobolev space H^{s,p (\R^n) that is supported in ? For sets of zero Lebesgue measure we show, using results on certain set capacities from classical potential theory, that the maximal regularity is non-positive, and is characterised by the Hausdorff dimension of , improving known results. We classify all possible maximal regularities, as functions of , together with the sets of values of for which the maximal regularity is attained, and construct concrete examples for each case.\ud
\ud
For sets with positive measure the maximal regularity is non-negative, but appears more difficult to characterise in terms of geometrical properties of . We present some partial results relating to the latter case, namely lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterising the regularity that can be achieved on certain special classes of sets, such as -sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations
Homogenized boundary conditions and resonance effects\ud in Faraday cages
We present a mathematical study of two-dimensional electrostatic and electromagnetic shielding by a cage of conducting wires (the so-called `Faraday cage effect'). Taking the limit as the number of wires in the cage tends to infinity we use the asymptotic method of multiple scales to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. We show how the resulting models depend on key cage parameters such as the size and shape of the wires, and, in the electromagnetic case, on the frequency and polarisation of the incident field. In the electromagnetic case there are resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. By appropriately modifying the continuum model we calculate the modified resonant frequencies, and their associated peak amplitudes. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells
Acoustic scattering by impedance screens/cracks with fractal boundary: well-posedness analysis and boundary element approximation
We study time-harmonic scattering in () by a planar
screen (a "crack" in the context of linear elasticity), assumed to be a
non-empty bounded relatively open subset of the hyperplane
, on which impedance (Robin) boundary conditions
are imposed. In contrast to previous studies, can have arbitrarily
rough (possibly fractal) boundary. To obtain well-posedness for such
we show how the standard impedance boundary value problem and its associated
system of boundary integral equations must be supplemented with additional
solution regularity conditions, which hold automatically when
is smooth. We show that the associated system of boundary integral operators is
compactly perturbed coercive in an appropriate function space setting,
strengthening previous results. This permits the use of Mosco convergence to
prove convergence of boundary element approximations on smoother "prefractal"
screens to the limiting solution on a fractal screen. We present accompanying
numerical results, validating our theoretical convergence results, for
three-dimensional scattering by a Koch snowflake and a square snowflake
Interpolation of Hilbert and Sobolev Spaces:\ud Quantitative Estimates and Counterexamples
This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces and , for and an open . We exhibit examples in one and two dimensions of sets for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large
Indirect Collider Signals for Extra Dimensions
A recent suggestion that quantum gravity may become strong near the weak
scale has several testable consequences. In addition to probing for the new
large (submillimeter) extra dimensions associated with these theories via
gravitational experiments, one could search for the Kaluza Klein towers of
massive gravitons which are predicted in these models and which can interact
with the fields of the Standard Model. Here we examine the indirect effects of
these massive gravitons being exchanged in fermion pair production in \epem
annihilation and Drell-Yan production at hadron colliders. In the latter case,
we examine a novel feature of this theory, which is the contribution of gluon
gluon initiated processes to lepton pair production. We find that these
processes provide strong bounds, up to several TeV, on the string scale which
are essentially independent of the number of extra dimensions. In addition, we
analyze the angular distributions for fermion pair production with spin-2
graviton exchanges and demonstrate that they provide a smoking gun signal for
low-scale quantum gravity which cannot be mimicked by other new physics
scenarios.Comment: Corrected typos, added table and reference
Signals for Vector Leptoquarks in Hadronic Collisions
We analyze systematically the signatures of vector leptoquarks in hadronic
collisions. We examine their single and pair productions, as well as their
effects on the production of lepton pairs. Our results indicate that a machine
like the CERN Large Hadron Collider (LHC) will be able to unravel the existence
of vector leptoquarks with masses up to the range of -- TeV.Comment: 15 pages and 5 figures (available upon request or through anonymous
ftp), revtex3, IFUSP-P 108
Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples
This paper provides an overview of interpolation of Banach and Hilbert
spaces, with a focus on establishing when equivalence of norms is in fact
equality of norms in the key results of the theory. (In brief, our conclusion
for the Hilbert space case is that, with the right normalisations, all the key
results hold with equality of norms.) In the final section we apply the Hilbert
space results to the Sobolev spaces and
, for and an open . We exhibit examples in one and two dimensions of sets
for which these scales of Sobolev spaces are not interpolation scales. In the
cases when they are interpolation scales (in particular, if is
Lipschitz) we exhibit examples that show that, in general, the interpolation
norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio
of these two norms can be arbitrarily large
Production of Pairs of Sleptoquarks in Hadron Colliders
We calculate the cross section for the production of pairs of scalar
leptoquarks (sleptoquarks) in a supersymmetric model, at hadron
colliders. We estimate higher order corrections by including terms
induced by soft-gluon corrections. Discovery bounds on the sleptoquark mass are
estimated at collider energies of 1.8, 2, and 4 TeV (Tevatron), and 16 TeV
(LHC).Comment: 8 pages, REVTEX, (1 fig. available on request),
LAVAL-PHY-94-13/McGILL-94-26/SPhT-94-07
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