Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles

The probabilities for gaps in the eigenvalue spectrum of the finite dimension $N \times N$ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $N \to \infty$, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlev\'e-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlev\'e-VI} transcendent.Comment: 32 pages, Latex2

Managing Water under Uncertainty and Risk: The United Nations World Water Development Report 4

This report introduces new aspects of water issues: 1) it reintroduces the 12 challenge area reports that provided the foundation for the first two World Water Development Reports (WWDR); 2) 4 new reports on water quality, groundwater, gender, and desertification, land degradation and drought; 3) in recognition that the global challenges of water can vary considerably across countries and regions, a series of 5 regional reports have been included; 4) a deeper analysis of the main external forces of freshwater resources and possibilities for their future evolution; 5) managing water under uncertainty and risk

A generalized experimental investigation of hot gas recirculation and ingestion for jet VTOL aircraft

Jet VTOL aircraft inlet temperature rise and induced recirculating flow pattern

Constructing Integrable Third Order Systems:The Gambier Approach

We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlev\'e method for ODE's and the singularity confinement method for mappings).Comment: 14 pages, TEX FIL

Partial Preference of Grazing Cattle for Contrasting Legume Swards

Yearling heifers in groups of 3 grazed 405 m2 plots made up of alternating 2.4m wide strips of white clover/birdsfoot trefoil (WC+BT) or red clover (RC) in the proportions of 80:20, 67:33, 33:67 and 20:80 for periods of 3 days over four replicates in time, balanced for effects of previous treatment. Observation of the distribution of grazing activity and biting rate were made over 3 hour periods each evening. Biting rates were consistently higher on (WC+BT) than RC (52.3 vs 46.3 ± 0.59 bites.min-1 P(0.0001). Animals initially showed partial preference for the minor sward component in each treatment but regression of the proportion of grazing activity on proportion of total area for (WC+BT) approached unity with time, indicating the development of essentially neutral behaviour as herbage on minority strips was depleted

General-relativistic Model of Magnetically Driven Jet

The general scheme for the construction of the general-relativistic model of the magnetically driven jet is suggested. The method is based on the usage of the 3+1 MHD formalism. It is shown that the critical points of the flow and the explicit radial behavior of the physical variables may be derived through the jet ``profile function."Comment: 12 pages, LaTex, no figure

Hamiltonians separable in cartesian coordinates and third-order integrals of motion

We present in this article all Hamiltonian systems in E(2) that are separable in cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-De Vries equation and the Painlev\'e transcendents.Comment: 19 pages, Will be published in J. Math. Phy