123680663
Representation of fundamental solutions for generalized Cauchy-Riemann equations by quasiconformal mappings
118096935
Stefan Bergman (1895--1977) in Memoriam
124090571
Bounded Functions of Two Complex Variables
115850274
Fredholm Eigenvalues and Conformal Mapping
This paper is mainly of expository nature. Its aim is to give a connected account of various results regarding the Fredholm eigenfunctions and the Fredholm eigenvalues of plane domains. The Fredholm eigenvalues of a curve system are a set of functionals of these curves whose study leads to may useful applications. They are closely related to the boundary value problem for harmonic functions. They are important in the theory of conformal mapping, in the theory of kernel functions and orthonormal series. They play a role in the theory of the Hilbert transform and in the theory of univalent functions. Their dependence on the curve system is displayed by a very elegant and convenient variational formula. Some applications of these results are given to show the significance of the identities and formulas obtained. For a more detailed development of many points the reader is referred to [10], [11] and [12].
45607022
Faber polynomials in the theory of univalent functions
Introduction. The Faber polynomials play an important role in the theory of univalent functions. Grunsky [ l] succeeded in establishing a set of conditions for a given function which are necessary and in their totality sufficient for the uni valency of this function, and in these conditions the coefficients of the Faber polynomials play an important role. Schiffer [2] gave a differential equation for univalent functions solving certain extremum problems with respect to the coefficients of such functions ; in this differential equation appears again a polynomial which is just the derivative of a Faber polynomial (cf. Schiffer [3]; see also Schaeffer-Spencer [4]). It seems, therefore, of interest to study these Faber polynomials more closely, in particular their dependence on the given function with respect to which they are defined, their variation with the latter and certain characteristic inequalities for them and their coefficients. This investigation is carried out in the present paper. In §1 we establish a generating function for all Faber polynomials with respect to a given function. In §2 we establish variation formulas for the Faber polynomials and their coefficients. In §3 we solve certain extremum problems with respect to the coefficients of these polynomials and find again all the inequalities which have been established by Grunsky. In §4 we use our method in order to generalize our results and to find inequalities for the Faber polynomials themselves.
124494402
Variation of the Green Function and Theory of the p-Valued Functions
120475501
Recovering information borne by quanta that crossed the black hole event horizon
In this essay we discuss how the statistics of quantum particles, namely, stimulated emission for bosons and the exclusion principle for fermions, provide mechanisms for correlating the radiations impinging and emanating from a black hole. In either case, the information originally borne by the incoming radiation that is partially transferred to the outging radiation is estimated within the framework of communication theory.
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.