Showing papers in "Duke Mathematical Journal in 1971"

The degree of convergence for entire functions

Abstract: Consider a Doint set C in the complex plane whose complement K is connected and regular (i .e. K possesses a Green ' s function with pole at infini ty). Let denote the transfini te diameter of C . Recall that d m = l /U ' C®)! where (z) irans >C onto tl e exterior of the uni t circle. Equivalent ly , d^ = Lin | |T^(z) | Ij.] where | |g(z) | | c = max] O and type O o) by a polynomial expansion of the form where p(z) is a polynomial of degree X and ^ ( z ) is of defree A-l . The level curves of |p(z)| define a lemniscate which approximates the boundary of C . A number of lemmas are established which relate the nature of the coefficient polynomials q v (z) to the order and type of f(z). OO f ( z ) I o k ( z ) P ( Z ) k-1 k= l THE DEGREE OF CONVERGENCE FOR ENTIRE FUNCTIONS

Branching processes in Markovian environments

Abstract: : The paper generalizes the theory given in a previous paper for the Branching Process with Random Environments (BPRE) to cover situations where a certain kind of Markov dependence holds between successive environmental variables. The new process is called a Branching Process with Markovian Environments (BPME).