124400876
THE COEFFICIENT PROBLEM FOR MULTIPLY-CONNECTED DOMAINS
There is sometimes considerable advantage to be gained from generalizing a problem which has been intensively studied in particular cases. At the very least, relations may be brought to light which were disregarded or covered up in the particular cases studied. These relations may not only turn out to be important in themselves, but they may also throw light on the original problems. With these thoughts in mind, we have generalized the coefficient problem for schlicht functions, a problem which has heretofore been considered mainly in the case when the function is defined over a simply-connected domain. The word "schlicht" has little meaning unless the surface is specified on which the values are carried. For example, a function may map the interior of the unit circle onto a subdomain of a Riemann surface and the subdomain may be schlicht relative to the Riemann surface but not relative to the plane. In the case of functions which are schlicht in a simply-connected domain, the meaning is that the regions of values form schlicht subdomains of the sphere. A certain symmetry is inherent in this definition because the region of definition of the function and the surface on which the values lie (namely the sphere) are about equal from the function-theory point of view. As soon as we go over to more complicated domains of definition of the function, the importance of the surface which carries the values of the function becomes evident. In this paper we have attempted to restore the symmetry by supposing that the values of the function on the multiply-connected domain lie schlicht on another multiply-connected domain. But we could go much further and choose either the domain of definition or the carrying surface or both to be domains of higher structure. By choosing the carrying surface to be of higher structure, we are brought close to the theory of subordination in the sense of Lindelof. Problems relating to domains of higher connectivity cannot be approached without first discussing the invariants of the domains including differentials. In the case of the sphere the associated invariants are of rational character, but in the case of a domain of higher structure they are transcendental. As was pointed out by Schottky [11], and later by Picard [7] and Klein [3],1 the invariants of a multiply-connected domain are best understood in terms of the so-called "double" of that domain. The double is a closed Riemann surface of genus m 1 if the domain itself is of genus zero and has m boundary components. It is the symmetry of the double which lies behind many of the relations connecting the invariants of a multiply-connected domain.
124090571
Bounded Functions of Two Complex Variables
122521355
The dielectric Green's function and quasiconformal mapping
123680663
Representation of fundamental solutions for generalized Cauchy-Riemann equations by quasiconformal mappings
124494402
Variation of the Green Function and Theory of the p-Valued Functions
124638170
Properties of Solutions of a System of Partial Differential Equations
121602047
Naked singularities cannot be observed (An information-theoretic approach to the cosmic censorship conjecture)
In this article we review some recent results obtained in the interplay between field theory in curved space-time and information theory, and explore some consequences. We shall explain how the Weyl tensor becomes responsible for the creation of quantum noise which, in turn, degrades a considerable fraction of the information which is optically transferred in a curved space-time. We conclude that causal connection between spacetime regions, although being a necessary condition for transmitting information between them, is by no means sufficient. This is because this quantum noise might preclude the observation of very distant objects in the Universe and even closer ones if it becomes very intense. We argue that naked singularities are very powerful sources of noise and, therefore, cannot be observed optically. This leads to the intriguing possibility that the ‘no-hair’ theorem might be more general than was thought before.