The quantitative application of Fermi-Dirac statistics involves the evaluation of certain integrals which have not previously been tabulated. In this paper, tables are given of the values of the basic integrals most frequently required , with a view to placing Fermi-Dirrac statistics on as firm a numerical basis as is Maxwell-Boltzmann statistics. T e expression for the energy distribution of particles subject to Fermi-Dirrac statistics may be written in the form dN He) de e<*+Pe -)-1 ’ wherev(e) is the number of states per unit energy range, and dN is the number of particles in the energy range e to e--de. In the statistical treatment, the parameters ot and fi, which are usually introduced as undetermined multipliers in a variational equation, are to be determined from two equations expressing conditions imposed by the total number of particles, and the total energy of the system. By linking up the statistical and thermodynamical treatments, interpretation can be given to a and b this is expressed by P**:l IkT, a = -C lk T ,

Abstract. The purpose of this paper is to analyse a class of quadratic extremal problems defined on various Hilbert spaces of analytic functions, thereby generalizing an extremal problem on the Dirichlet space which was solved by S.D. Fisher. Each extremal problem considered here is shown to be connected with a system of orthogonal polynomials. The orthogonal polynomials then determine properties of the extremal function, and provide information about the existence of extremals.

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A JS surface is a minimal graph over a polygonal domain that becomes infinite in magnitude at the domain boundary. Jenkins and Serrin characterized the existence of these minimal graphs in terms of the signs of the boundary values and the side-lengths of the polygon. For a convex polygon, there can be essentially only one JS surface, but a non-convex domain may admit several distinct JS surfaces. We consider two families of JS surfaces corresponding to different boundary values, namely JS 0 and JS 1 , over domains in the form of regular stars. We give parameterizations for these surfaces as lifts of harmonic maps, and observe that all previously constructed JS surfaces have been of type JS 0 . We give an example of a JS 1 surface that is a new complete embedded minimal surface generalizing Scherk's doubly periodic surface, and show also that the JS 0 surface over a regular convex 2n-gon is the limit of JS 1 surfaces over non-convex stars. Finally we consider the construction of other JS surfaces over stars that belong neither to JS 0 nor to JS 1 .

A general version of the Radó-Kneser-Choquet theorem implies that a piecewise constant sensepreserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.  2004 Elsevier Inc. All rights reserved.