An asymptotic analysis of the kick-out diffusion mechanism

The kick-out model for impurity diffusion in semiconductors is studied. The kick-out mechanism is thought to play an important rôle in a number of applications, including the diffusion of zinc and chromium in gallium arsenide. Asymptotic solutions are derived for both one- and two-dimensional surface source problems. In the one-dimensional case, a mechanism for the destruction of self-interstitials is also incorporated. The calculated diffusion profiles have shapes which are typical of diffusion systems in which the kick-out mechanism is believed to be active. For the two-dimensional problem, contours of constant impurity concentration are calculated and some are found to have the bird's beak shape which is frequently observed in experiments.     

Automatic structures,rational growth,and geometrically finite hyperbolic groups

Summary We show that the set of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupG is dense in the product of the sets over all maximal parabolic subgroupsP. The set of equivalence classes of biautomatic structures onG is isomorphic to the product of the sets over the cusps (conjugacy classes of maximal parabolic subgroups) ofG. Each maximal parabolicP is a virtually abelian group, so and were computed in [NS1].We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics forG is regular. Moreover, the growth function ofG with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.Oblatum 14-VI-1993 & 4-I-1994Both authors acknowledge support from the NSF for this research.     

Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations , whereT corresponds to the closure of the graph of .     

On the growth of meromorphic functions of infinite order

Letf be a meromorphic function of infinite order,T(r, f) its Nevanlinna (or Ahlfors-Shimizu) characteristic, andM(r, f) its maximum modulus. It is proved that $$\mathop {\lim \inf }\limits_{r \to \infty } \frac{{\log M(r,f)}}{{rT'(r,f)}} \leqslant \pi and\mathop {\lim \inf }\limits_{r \to \infty } \frac{{\log M(r,f)}}{{T(r,f)\psi (log T(r,f))}} = 0$$ . if ? (x)/x is non-decreasing, ?′(x)<-√?(x) and ∝^∞ dx/?(x) < ∞.