We continue the study of the Hermitian random matrix ensemble with external source 1where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a = 1, since the eigenvalues of M accumulate on two intervals for a > 1, and on one interval for 0 < a < 1. The case a > 1 was treated in part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0 < a < 1. As in part I we apply the Deift/Zhou steepest descent analysis to a 3 × 3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems.

Theorems describing the sharp constants for the approximation of a general class of analytic functions by rational functions are proved. Magnus's conjecture on the sharp constant for the approximation of e −z on [0, ∞] is established as a consequence. For the proof of the theorems new formulae expressing the strong asymptotics of polynomials orthogonal with respect to a varying complex weight are obtained.Bibliography: 68 titles.

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Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, f ∈ A(C \ A), A < ∞. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Padé approximants for f . The Padé approximants, which are rational functions and thus singlevalued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a singlevalued branch. Thus the domain of convergence corresponds to the maximal (in the sense of minimal boundary) domain of single-valued holomorphy for the analytic function f ∈ A(C \ A). The complete proof of Nuttall's conjecture (even in a more general setting where the set A has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive strong asymptotics for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that A is a finite set of branch points of f which have the algebro-logarithmic character and which are placed in a generic position. The last restriction means that we exclude from our consideration some degenerated "constellations" of the branch points.2000 Mathematics Subject Classification. 42C05, 41A20, 41A21.

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The previous work on the dosimetry of bone is briefly reviewed. A dosimetric theory for the response of detectors irradiated by fast neutrons is applied to the problem of bone dosimetry. In the theory the detector or cavity shape is characterised by distributions of chord lengths along which the neutron-produced charged particles travel and deposit energy. Cavities of different convex geometries can be treated. A simplified version of the theory uses a single mean chord length to characterise the cavity. The absorbed dose to individual marrow cavities in trabecular bone is calculated over a large range of marrow cavity size for monoenergetic neutrons ranging from 0.5 to 7.0 MeV and for 252Cf neutrons. The influence of cavity shape is explored by considering spheres and cylinders of different elongation. The difference in absorbed dose is not great. Also the simplified model using a single mean chord length gives results in close agreement with the results obtained with chord length distributions. The mean marrow dose to different human bones has been calculated in three ways. First by using measured chord length distributions for the marrow cavities in the bones, second by using a sphere with the same mean chord length as the measured distribution and third by applying the measured single mean chord length. The difference between the three approaches is small and the agreement is geod with results obtained by other workers who used the Monte Carlo technique. The dose to the endosteal cell layer has also been calculated by approximating the layer with an infinite slab. Dosimetry of bone irradiated by fast neutrons 957

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