On the virial theorem for the relativistic operator of Brown and Ravenhall, and the absence of embedded eigenvalues

A virial theorem is established for the operator proposed by Brown and Ravenhall as a model for relativistic one-electron atoms. As a consequence, it is proved that the operator has no eigenvalues greater than $\max(m c^2, 2 \alpha Z - \frac{1}{2})$, where $\alpha$ is the fine structure constant, for all values of the nuclear charge $Z$ below the critical value $Z_c$: in particular there are no eigenvalues embedded in the essential spectrum when $Z \leq 3/4 \alpha$. Implications for the operators in the partial wave decomposition are also described.Comment: To appear in Letters in Math. Physic

Selfadjoint and mm sectorial extensions of Sturm-Liouville operators

The self-adjoint and $m$-sectorial extensions of coercive Sturm-Liouville operators are characterised, under minimal smoothness conditions on the coefficients of the differential expression.Comment: accepted by IEOT, in IEOT 201

Multistage multiple-reentry turbine Patent

Multistage, multiple reentry, single rotor, axial flow turbin

Flow studies in a wet steam turbine

The design and test results of a four stage wet vapor turbine operating with slightly superheated inlet steam and expanding to 10% exit moisture are presented. High speed movies at 3000 frames per second of liquid movement on the pressure side and along the trailing edge of the last stator blade are discussed along with back lighted photographs of moisture drops as they were torn from the stator blade trailing edge. Movies at lower framing rates were also taken of the exit of the last rotating blade and the casing moisture removal slot located in line with the rotor blade shroud. Also moisture removal data are presented of casing slot removal at the exit of the third and fourth rotor blades and for slots located in the trailing edge of the last stator blade. Finally, the degradation of turbine thermodynamic performance due to condensation formation and movement is discussed

Smilansky's model of irreversible quantum graphs, II: the point spectrum

In the model suggested by Smilansky one studies an operator describing the interaction between a quantum graph and a system of K one-dimensional oscillators attached at different points of the graph. This paper is a continuation of our investigation of the case K>1. For the sake of simplicity we consider K=2, but our argument applies to the general situation. In this second paper we apply the variational approach to the study of the point spectrum.Comment: 18 page

Multistage multiple-reentry turbine Patent

Multistage multiple reentry axial flow reaction turbine with reverse flow reentry ductin

Dirac-Sobolev inequalities and estimates for the zero modes of massless Dirac operators

The paper analyses the decay of any zero modes that might exist for a massless Dirac operator H:= \ba \cdot (1/i) \bgrad + Q, where $Q$ is $4 \times 4$-matrix-valued and of order O(|\x|^{-1}) at infinity. The approach is based on inversion with respect to the unit sphere in $\R^3$ and establishing embedding theorems for Dirac-Sobolev spaces of spinors $f$ which are such that $f$ and $Hf$ lie in $(L^p(\R^3))^4, 1\le p<\infty.$Comment: 11 page

Axisymmetric Self-Similar Equilibria of Self-Gravitating Isothermal Systems

All axisymmetric self-similar equilibria of self-gravitating, rotating, isothermal systems are identified by solving the nonlinear Poisson equation analytically. There are two families of equilibria: (1) Cylindrically symmetric solutions in which the density varies with cylindrical radius as R^(-alpha), with 0 <= alpha <= 2. (2) Axially symmetric solutions in which the density varies as f(theta)/r^2, where `r' is the spherical radius and `theta' is the co-latitude. The singular isothermal sphere is a special case of the latter class with f(theta)=constant. The axially symmetric equilibrium configurations form a two-parameter family of solutions and include equilibria which are surprisingly asymmetric with respect to the equatorial plane. The asymmetric equilibria are, however, not force-free at the singular points r=0, infinity, and their relevance to real systems is unclear. For each hydrodynamic equilibrium, we determine the phase-space distribution of the collisionless analog.Comment: 13 pages, 7 figures, uses emulateapj.sty. Submitted to Ap

Making a national atlas of population by computer

This paper describes the conceptual and practical problems encountered and solved in producing a multi-colour atlas of population characteristics in Great Britain. The atlas itself is in A4 format; it consists of some thirty-four maps of Great Britain in four colours and the same number of regional maps, together with descriptive text. All maps were plotted on a laser plotter with a resolution of 127 microns. The paper describes how mapping of ratios, such as percentages, was found to be highly misleading and describes the novel probability mapping solution adopted, based on the signed chi-square statistic. In addition, the rationale for selecting the class intervals and for selecting colour schemes is described