Relativistic Comparison Theorems

Comparison theorems are established for the Dirac and Klein--Gordon equations. We suppose that V^{(1)}(r) and V^{(2)}(r) are two real attractive central potentials in d dimensions that support discrete Dirac eigenvalues E^{(1)}_{k_d\nu} and E^{(2)}_{k_d\nu}. We prove that if V^{(1)}(r) \leq V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered E^{(1)}_{k_d\nu} \leq E^{(2)}_{k_d\nu}. This result generalizes an earlier more restrictive theorem that required the wave functions to be node free. For the the Klein--Gordon equation, similar reasoning also leads to a comparison theorem provided in this case that the potentials are negative and the eigenvalues are positive.Comment: 6 page

Semiclassical energy formulas for power-law and log potentials in quantum mechanics

We study a single particle which obeys non-relativistic quantum mechanics in R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2, then q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\ell} may be represented exactly by the semiclassical expression E_{n\ell}(q) = min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) = ln(r). By writing one power as a smooth transformation of another, and using envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are monotone increasing. Recent refinements to the comparison theorem of QM in which comparison potentials can cross over, allow us to prove for n = 1 that Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q} is monotone decreasing. Thus P(q) cannot increase too slowly. This result yields some sharper estimates for power-potential eigenvlaues at the bottom of each angular-momentum subspace.Comment: 20 pages, 5 figure

Photoelectric observations of Mars and Jupiter with a scanning polarimeter

Photoelectric observations of Mars and Jupiter with scanning polarimete

The principle of equivalence and projective structure in space-times

This paper discusses the extent to which one can determine the space-time metric from a knowledge of a certain subset of the (unparametrised) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the space-time concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated metric is uniquely determined up to a choice of units of measurement, by the specification of these geodesics. It is further demonstrated that if two space-times share the same unparametrised geodesics and only one is assumed vacuum then their Levi-Civita connections are again equal (and so the other metric is also a vacuum metric) and the first result above is recovered.Comment: 23 pages, submitted to Classical and Quantum Gravit

Coulomb plus power-law potentials in quantum mechanics

We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for the Coulomb plus power-law potential V(r)=-1/r+ beta sgn(q)r^q, where beta > 0, q > -2 and q \ne 0. We show by envelope theory that the discrete eigenvalues E_{n\ell} of H may be approximated by the semiclassical expression E_{n\ell}(q) \approx min_{r>0}\{1/r^2-1/(mu r)+ sgn(q) beta(nu r)^q}. Values of mu and nu are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, psi(r)= r^{\ell+1}e^{-(xr)^{q}}. We give detailed results for V(r) = -1/r + beta r^q, q = 0.5, 1, 2 for n=1, \ell=0,1,2, along with comparison eigenvalues found by direct numerical methods.Comment: 11 pages, 3 figure

Recent Studies of Nonequilibrium Flows at the Cornell Aeronautical Laboratory

Vibrational relaxation in supersonic nozzle diatomic gas flow, nonequilibrium effects in high enthalpy airflow over thick wedge flat plates, and reentry nonequilibrium flow field

Global public health training in the UK: preparing for the future.

BACKGROUND: Many major public health issues today are not confined by national boundaries. However, the global public health workforce appears unprepared to confront the challenges posed by globalization. We therefore sought to investigate whether the current UK public health training programme adequately prepares its graduates to operate in a globalized world. METHODS: We used mixed methods involving an online cross-sectional survey of UK public health trainees on the international content of the Faculty of Public Health's written examination, a qualitative review of the Faculty's 2007 training curriculum and a questionnaire survey of all training deaneries in the UK. RESULTS: We found that global health issues are not addressed by the current training curriculum or in the written examination despite trainee interest for this. Many of the deaneries were also unreceptive to international placements. CONCLUSIONS: Despite the recognized educational legitimacy of global health placements and the favourable UK policy context, the opportunities and international content of public health training remain limited. In order to retain its position as a leader in the field of public health, the UK needs to adapt its training programme to better reflect today's challenges

Software fault characteristics: A synthesis of the literature

Faults continue to be a significant problem in software. Understanding the nature of these faults is important for practitioners and researchers. There are many published fault characteristics schemes but no one scheme dominates. Consequently it is difficult for practitioners to effectively evaluate the nature of faults in their software systems, and it is difficult for researchers to compare the types of faults found by different fault detection techniques. In this paper we synthesise previous fault characteristics schemes into one comprehensive scheme. Our scheme provides a richer view of faults than the previous schemes published and presents a comprehensive, unified approach which accommodates the many previous schemes. A characteristics-based view of faults should be considered by future researchers in the analysis of software faults and in the design and evaluation of new fault detection tools. We recommend that our fault characteristics scheme be used as a benchmark scheme

Investigation of advanced counterrotation blade configuration concepts for high speed turboprop systems, task 1: Ducted propfan analysis

The time-dependent three-dimensional Euler equations of gas dynamics were solved numerically to study the steady compressible transonic flow about ducted propfan propulsion systems. Aerodynamic calculations were based on a four-stage Runge-Kutta time-marching finite volume solution technique with added numerical dissipation. An implicit residual smoothing operator was used to aid convergence. Two calculation grids were employed in this study. The first grid utilized an H-type mesh network with a branch cut opening to represent the axisymmetric cowl. The second grid utilized a multiple-block mesh system with a C-type grid about the cowl. The individual blocks were numerically coupled in the Euler solver. Grid systems were generated by a combined algebraic/elliptic algortihm developed specifically for ducted propfans. Numerical calculations were initially performed for unducted propfans to verify the accuracy of the three-dimensional Euler formulation. The Euler analyses were then applied for the calculation of ducted propfan flows, and predicted results were compared with experimental data for two cases. The three-dimensional Euler analyses displayed exceptional accuracy, although certain parameters were observed to be very sensitive to geometric deflections. Both solution schemes were found to be very robust and demonstrated nearly equal efficiency and accuracy, although it was observed that the multi-block C-grid formulation provided somewhat better resolution of the cowl leading edge region