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 $\Omega\subseteq\real^N$ 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

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

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 $\Omega\subseteq\real^N$, where the order $2m$ of the operator satisfies $N<2m$. 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 $\Omega$. 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

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

Non-parametric statistical thresholding for sparse magnetoencephalography source reconstructions.

Uncovering brain activity from magnetoencephalography (MEG) data requires solving an ill-posed inverse problem, greatly confounded by noise, interference, and correlated sources. Sparse reconstruction algorithms, such as Champagne, show great promise in that they provide focal brain activations robust to these confounds. In this paper, we address the technical considerations of statistically thresholding brain images obtained from sparse reconstruction algorithms. The source power distribution of sparse algorithms makes this class of algorithms ill-suited to "conventional" techniques. We propose two non-parametric resampling methods hypothesized to be compatible with sparse algorithms. The first adapts the maximal statistic procedure to sparse reconstruction results and the second departs from the maximal statistic, putting forth a less stringent procedure that protects against spurious peaks. Simulated MEG data and three real data sets are utilized to demonstrate the efficacy of the proposed methods. Two sparse algorithms, Champagne and generalized minimum-current estimation (G-MCE), are compared to two non-sparse algorithms, a variant of minimum-norm estimation, sLORETA, and an adaptive beamformer. The results, in general, demonstrate that the already sparse images obtained from Champagne and G-MCE are further thresholded by both proposed statistical thresholding procedures. While non-sparse algorithms are thresholded by the maximal statistic procedure, they are not made sparse. The work presented here is one of the first attempts to address the problem of statistically thresholding sparse reconstructions, and aims to improve upon this already advantageous and powerful class of algorithm