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In this paper we investigate the smallest eigenvalue, denoted as λ N , of a (N +1)×(N +1) Hankel or moments matrix, associated with the weight, w(x) = exp(−x β ), x > 0, β > 0, in the large N limit. Using a previous result, the asymptotics for the polynomials, P n (z), z / ∈ [0, ∞), orthonormal with respect to w, which are required in the determination of λ N are found. Adopting an argument of Szegö the asymptotic behaviour of λ N , for β > 1/2 where the related moment problem is determinate, is derived. This generalises the result given by Szegö for β = 1. It is shown that for β > 1/2 the smallest eigenvalue of the infinite Hankel matrix is zero, while for 0 < β < 1/2 it is greater then a positive constant. This shows a phase transition in the corresponding Hermitian random matrix model as the parameter β varies with β = 1/2 identified as the critical point. The smallest eigenvalue at this point is conjectured.

Aim  The aim of this study was to explore and understand the leadership experiences of modern matrons. Background  Modern matrons were re-introduced to the National Health Service in 2002, and effective leadership has been identified as being essential for the role to be successful. However, there is minimal evidence of how modern matrons experience effective leadership. Methods  The study used a descriptive generic qualitative methodology; one-to-one semi-structured interviews were conducted with nine matrons. This was subjected to an inductive thematic analysis. Results  Three themes were found to influence modern matron's leadership experiences: leadership behaviours, negative influences and leadership investment. They did not follow one leadership style but adapted this to their situation. Various factors appeared to restrict their leadership effectiveness. Conclusions  The findings suggest that exposure to a range of leadership styles should be included in preparation and CPD for the modern matron role and a more consistent job description and job purpose should be developed. Implications for nursing management  Leadership styles such as transformational leadership alone do not meet the complex demands of nursing leaders, and therefore there is a requirement for greater flexibility in leadership development for all health care professionals.

ABSTRACT. In this paper, we study the asymptotic eigenvalue density of large n x n random Hermitian matrices. The eigenvalue density can be interpreted in the context of orthogonal polynomials as the density of zeros. We adopt two approaches; the first, using a recent theorem, gives the density of zeros as an integral representation with the (appropriately scaled) recurrence coefficients as input. The second makes use of the Coulomb fluid approach pioneered by Dyson where the weight with respect to which the polynomials are orthogonal is the input.The zero density of the Stieltjes-Wigert, g -1 -Hermite, q-Laguerre polynomials and a constructed set of orthogonal polynomials are obtained. In the last two cases, the density can be expressed in terms of complete and incomplete elliptic integrals of various kinds.We also compute, in some cases, the effective potentials from the densities.