In the case of an annulus, it is simple to find an orthonormal basis for the Hardy space. This allows one to write both the Szegő and Garabedian kernel functions as infinite series. These series are classical. The Ahlfors map is a two-to-one branched covering map of the annulus onto the unit disk and is given by the quotient of the Szegő and Garabedian kernels. One of the two zeros of the Ahlfors map arises from the pole of the Garabedian kernel. The other zero corresponds to the zero of the Szegő kernel. In Section 5 it is shown how to use the series for the Szegő kernel to find its zero.The boundary values of the Garabedian kernel are given in terms of those of the Szegő kernel. Hence, knowing the boundary values of the Szegő kernel is tantamount to knowing those of the Ahlfors map. A discovery of Kerzman and Stein provides an efficient numerical method for computing the boundary values of the Szegő kernel for a smoothly bounded, planar domain and hence the boundary values of the Ahlfors map. Since the Ahlfors map is a holomorphic function smooth up to the boundary, the Cauchy integral formula provides the interior values of this map. Unfortunately, this integral formula has a singular nature for interior points near the boundary. In Section 7 it is shown how to alleviate this singular behavior for the annulus, and in Section 8 graphical examples are given for the Szegő and Garabedian kernels and for the Ahlfors map. Preliminaries.Suppose that Ω is a domain in C with C ∞ smooth boundary. Let L 2 (bΩ) denote the space of square integrable functions with respect to arc length measure on the boundary bΩ of Ω, and let H 2 (bΩ) denote the subspace of L 2 (bΩ) consisting of functions that extend to be holomorphic on Ω. An inner product •, • is defined

Multiple models are discussed for ranking teams in a league and introduce a new model called the Oracle method. This is a Markovian method that can be customized to incorporate multiple team traits into its ranking. Using a foresight prediction of NFL game outcomes for the 2002-2013 seasons, it is shown that the Oracle method correctly picked 64.1% of the games under consideration, which is higher than any of the methods compared, including ESPN Power Rankings, Massey, Colley, and PageRank.

We show how to express a conformal map Φ of a general two connected domain in the plane such that neither boundary component is a point to a representative domain of the form A r = {z : |z + 1/z| < 2r}, where r > 1 is a constant. The domain A r has the virtue of having an explicit algebraic Bergman kernel function, and we shall explain why it is the best analogue of the unit disc in the two connected setting. The map Φ will be given as a simple and explicit algebraic function of an Ahlfors map of the domain associated to a specially chosen point. It will follow that the conformal map Φ can be found by solving the same extremal problem that determines a Riemann map in the simply connected case.1991 Mathematics Subject Classification. 30C35.