32,883 research outputs found
Asymptotic First Eigenvalue Estimates for the Biharmonic Operator on a Rectangle
We find an asymptotic expression for the first eigenvalue of the biharmonic
operator on a long thin rectangle. This is done by finding lower and upper
bounds which become increasingly accurate with increasing length. The lower
bound is found by algebraic manipulation of the operator, and the upper bound
is found by minimising the quadratic form for the operator over a test space
consisting of separable functions. These bounds can be used to show that the
negative part of the groundstate is small.Comment: 27 pages, 4 diagrams, 2 table
The Hardy-Rellich Inequality for Polyharmonic Operators
The Hardy-Rellich inequality given here generalizes a Hardy inequality of
Davies (1984), from the case of the Dirichlet Laplacian of a region
to that of the higher order polyharmonic operators
with Dirichlet boundary conditions. The inequality yields some immediate
spectral information for the polyharmonic operators and also bounds on the
trace of the associated semigroups and resolvents.Comment: 19 pages, 2 diagram
A Riemannian Off-Diagonal Heat Kernel Bound for Uniformly Elliptic Operators
We find a Gaussian off-diagonal heat kernel estimate for uniformly elliptic
operators with measurable coefficients acting on regions
, where the order of the operator satisfies
. The estimate is expressed using certain Riemannian-type metrics, and a
geometrical result is established allowing conversion of the estimate into
terms of the usual Riemannian metric on . Work of Barbatis is applied
to find the best constant in this expression.Comment: 29 pages, 6 diagram
Measuring Parity in Sports Leagues with Draws: Further Comments
This paper re-examines the calculation of the relative standard deviation (RSD) measure of competitive balance in leagues in which draws are possible outcomes. Some key conclusions emerging from the exchange between Cain and Haddock (2006) and Fort (2007) are reversed. There is no difference, for any given points assignment scheme, between the RSD for absolute points compared to percentages of points. However, variations in the points assignment that change the ratio of points for a win compared to a draw do result in different RSD values, although the numerical differences are minor.sports economics, competitive balance, relative standard deviation,idealized standard deviation, draws/ties
Critical Nodes In Directed Networks
Critical nodes or "middlemen" have an essential place in both social and
economic networks when considering the flow of information and trade. This
paper extends the concept of critical nodes to directed networks. We identify
strong and weak middlemen. Node contestability is introduced as a form of
competition in networks; a duality between uncontested intermediaries and
middlemen is established. The brokerage power of middlemen is formally
expressed and a general algorithm is constructed to measure the brokerage power
of each node from the networks adjacency matrix. Augmentations of the brokerage
power measure are discussed to encapsulate relevant centrality measures. We use
these concepts to identify and measure middlemen in two empirical
socio-economic networks, the elite marriage network of Renaissance Florence and
Krackhardt's advice network.Comment: 28 pages, 6 figures, 2 table
The dust and gas content of the Crab Nebula
We have constructed MOCASSIN photoionization plus dust radiative transfer
models for the Crab Nebula core-collapse supernova (CCSN) remnant, using either
smooth or clumped mass distributions, in order to determine the chemical
composition and masses of the nebular gas and dust. We computed models for
several different geometries suggested for the nebular matter distribution but
found that the observed gas and dust spectra are relatively insensitive to
these geometries, being determined mainly by the spectrum of the pulsar wind
nebula which ionizes and heats the nebula. Smooth distribution models are ruled
out since they require 16-49 Msun of gas to fit the integrated optical nebular
line fluxes, whereas our clumped models re quire 7.0 Msun of gas. A global
gas-phase C/O ratio of 1.65 by number is derived, along with a He/H number
ratio of 1.85, neither of which can be matched by current CCSN yield
predictions. A carbonaceous dust composition is favoured by the observed
gas-phase C/O ratio: amorphous carbon clumped model fits to the Crab's Herschel
and Spitzer infrared spectral energy distribution imply the presence of
0.18-0.27 Msun of dust, corresponding to a gas to dust mass ratio of 26-39.
Mixed dust chemistry models can also be accommodated, comprising 0.11-0.13 Msun
of amorphous carbon and 0.39-0.47 Msun of silicates. Power-law grain size
distributions with mass distributions that are weighted towards the largest
grain radii are derived, favouring their longer-term survival when they
eventually interact with the interstellar medium. The total mass of gas plus
dust in the Crab Nebula is 7.2 +/- 0.5 Msun, consistent with a progenitor star
mass of 9 Msun.Comment: Accepted in Ap
Periodic orbit theory for Rydberg atoms in external fields
Although hydrogen in external fields is a paradigm for the application of periodic orbits and the Gutzwiller trace formula to a real system, the trace formula has never been applied successfully to other Rydberg atoms. We show that spectral fluctuations of general Rydberg atoms are given with remarkable precision by the addition of diffractive terms. Previously unknown features in atomic spectra are exposed: there are new modulations that are neither periodic orbits nor combinations of periodic orbits; “core shadowing” generally decreases primitive periodic orbit amplitudes but can also lead to increases
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Migrant workers in the East Midlands labour market 2007
This report provides a profile of international migrants in the East Midlands and their role in the regional labour marke
The long-term evolution of photoevaporating transition discs with giant planets
Photo-evaporation and planet formation have both been proposed as mechanisms
responsible for the creation of a transition disc. We have studied their
combined effect through a suite of 2d simulations of protoplanetary discs
undergoing X-ray photoevaporation with an embedded giant planet. In a previous
work we explored how the formation of a giant planet triggers the dispersal of
the inner disc by photo-evaporation at earlier times than what would have
happened otherwise. This is particularly relevant for the observed transition
discs with large holes and high mass accretion rates that cannot be explained
by photo-evaporation alone. In this work we significantly expand the parameter
space investigated by previous simulations. In addition, the updated model
includes thermal sweeping, needed for studying the complete dispersal of the
disc. After the removal of the inner disc the disc is a non accreting
transition disc, an object that is rarely seen in observations. We assess the
relative length of this phase, to understand if it is long lived enough to be
found observationally. Depending on the parameters, especially on the X-ray
luminosity of the star, we find that the fraction of time spent as a
non-accretor greatly varies. We build a population synthesis model to compare
with observations and find that in general thermal sweeping is not effective
enough to destroy the outer disc, leaving many transition discs in a relatively
long lived phase with a gas free hole, at odds with observations. We discuss
the implications for transition disc evolution. In particular, we highlight the
current lack of explanation for the missing non-accreting transition discs with
large holes, which is a serious issue in the planet hypothesis.Comment: 11 pages, 5 figures; accepted by MNRA
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