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

The connection between the Riemann map and the Szego kernel is classical. But the fact that there is an efficient numerical procedure, based on the Kerzman᎐Stein integral equation, for computing the Szego kernel of a smoothly bounded domain in the plane is more recent. In this paper it is shown how to extend the results of Kerzman and Stein to certain nonsmooth domains. This provides a new method for numerically computing the Szego kernel, and hence the Riemann map, of these domains. ᮊ

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Just as the principle of uniformitarianism is a fundamental tenet of science, the past experiences of the Field Studies Council provide an insight, and possibly a signpost, to the future. During the last thirty years, there have been significant changes in the numbers and categories of visitor attending the residential Centres. Expansion between 1956 and 1976 was almost entirely due to A‐level courses in biology and geography but in the last ten years there has been considerable diversification. Adult amateur naturalists and younger, pre‐A‐level, school pupils are the obvious growth categories now. In the GCSE examination syllabuses, more than ever before, there is a commitment to ecology and to fieldwork (with the latter compulsory in all Geography syllabuses). Against such a background one might expect an optimistic prediction for the FSC's future. Unfortunately, and for a variety of reasons, many educationalists promote the view that ecology can be taught in the classroom and that first‐hand observations of animals and plants in an aquarium provide sufficient practical work. There is a growing need for those who believe in the necessity for genuine field observation of the environment, in the environment, to voice their concern at the acceptance of any such alternatives/substitutes. Although environmental awareness has been heightened in recent years one might question whether there has been a concommitant raising of environmental understanding. It is increasingly important that, as the scale and scope of decisions about the environment expand, the level of understanding of those who make decisions (and of those who elect them) should also be raised. It is no longer politic nor wise to accept Montaigne's assertion, “that men are most apt to believe what they least understand”. I suggest that our target for the future be “Breadth with Excellence”. If the FSC can move toward that goal it can face the daunting task of trying to achieve “Environmental Understanding for AH”.