Superbranching processes and projections of random Cantor sets

We study sequences (X ₀, X ₁, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that , for m, n0, where the X _n (m, i) are distributed as X _n and have certain properties of independence. We prove that, under appropriate conditions, X _n ^1/n almost surely and in L ¹, where =sup E(X _n )^1/n . Our principal application of this result is to study the Lebesgue measure and (Hausdorff) dimension of certain projections of sets in a class of random Cantor sets, being those obtained by repeated random subdivisions of the M-adic subcubes of [0, 1] ^d . We establish a necessary and sufficient condition for the Lebesgue measure of a projection of such a random set to be non-zero, and determine the box dimension of this projection.Work done partly whilst visiting Cornell University with the aid of a Fulbright travel grant     

Ergodicity of a measure-valued Markov chain induced by random transformations

Let Z ₁, Z ₂,... be a sequence of i.i.d. random transformations (possibly discontinuous) of a compact metric space M, and let E denote the space of normalized mass distributions on M. Given in E, let _n denote the random measure °(Z _n °...°Z ₁)^–1 (when well-defined). We construct the transition probability P of the E-valued Markov chain ( _n ), and give a necessary and sufficient condition for P to have a unique invariant measure concentrated on the degenerate mass distributions. Convergence to statistical equilibrium of the associated discrete-time stochastic flow is investigated.Research supported by NSF grants INT-8420360 and DMS-8502802. This work was done while the author was visiting the Universities of Warwick and Minnesota     

The Cauchy Problem for a Nonlinear Diffusion Equation with Absorption and Convection

In this paper we construct the large-time solution of the equation which models the effectof nonlinear absorption on thin film flow of a viscous fluidon a sloping bed. For initial data with finite support we showhow the asymptotic structure of the Cauchy problem depends onthe absorption index . An important addition motivation forthe study is to provide a basis for extending the results tothe more general equation which has applications in plasma physics.     

A Tensor product generalized ADI method for the method of planes

We consider solving separable, second order, linear elliptic prtial differential equations in three independent variables. If the partial differential opertor separates into two terms, one depending on x and y, and one depending on z, then we use the method of planes to obtain a discrete problem, which we write in tensor product from as We apply a new interative method, the tensor product generalized alternating direction implicit method, to solve the discrete problem. We study a specific implementation that uses Hermite bicubic collocation in the xy direction and symmetric finite differences in the z direction. We demostrate that this method is a fast and accurate way to solve the large linear systems arising from three-dimensional elliptic problems.     

Minimum audible movement angle as a function of signal frequency and the velocity of the source

Thresholds for the detection of the direction of travel of a moving sound source were determined in a single-interval, forced-choice paradigm. Both the rate at which the sound source is displaced (8 degrees-128 degrees/s) and the frequency of the signal to be localized (500-3700 Hz) affect dynamic spatial resolution. There is an inverse relationship between spatial resolution and the rate of travel, a finding that replicates an earlier observation on performance with sources displaced at high velocities [Perrott and Musicant, J. Acoust. Soc. Am. 62, 1463-1466 (1977)]. However, the magnitude of this effect depends on the actual velocities employed. Relatively small changes in spatial resolution are apparent for velocities below approximately 32 degrees/s. The significant frequency effect can be summarized as follows: Dynamic spatial resolution is better for signals below 1000 Hz than for signals above this value (within the range tested). Particularly poor resolution is evident for signals between 1300-2000 Hz. The present results indicate that signal frequency affects dynamic spatial resolution in a fashion similar to that which has been observed in the more common "static" localization test situation. There is no indication of an interaction between these two variables. These results provide additional support for the hypothesis that both static and dynamic spatial discrimination functions are dependent upon the same underlying mechanisms. The effects of velocity upon the spatial resolution problem, a unique aspect of the dynamic paradigm, can probably be explained without the necessity of additional hypothetical mechanisms in the auditory system (e.g., a specialized motion detector).     

Does iron inhibit cryptoendolithic microbial communities?

Photosynthetic activity of three cryptoendolithic microbial communities was studied under controlled conditions in the laboratory. In two of these communities, the dominant organisms were lichens, collected from Linnaeus Terrace and from Battleship Promontory. The third community, dominated by cyanobacteria, was collected from Battleship Promontory. Both sites are in the ice-free valleys of southern Victoria Land. Previous efforts have shown how physical conditions can influence metabolic activity in endolithic communities (Kappen and Friedmann 1983; Kappen, Friedmann, and Garty 1981; Vestal, Federle, and Friedmann 1984). Biological activity can also be strongly influenced by the chemical environment. Inorganic nutrients such as nitrate, ammonia, and phosphate are often limiting factors, so their effects on photosynthetic carbon-14 bicarbonate incorporation were investigated. Iron and manganese are two metals present in Linnaeus Terrace and Battleship Promontory sandstones, and their effects on photosynthesis were also studied. The results may add to our understanding of biogeochemical interactions within this unique microbial community.     

Progressing waves on spherical spacetimes

A family of spherically symmetrical spacetimes that are transparent to minimally coupled scalar multipole waves is investigated. The concept of transparency is reformulated in terms of the established concept of progressing waves. The set of spherically symmetric spacetimes and the family of wave equations for which the transparency condition can be shown to hold is generalized.     

Relative index theorems and supersymmetric scattering theory

We discuss supersymmetric scattering theory and employ Krein's theory of spectral shift functions to investigate supersymmetric scattering systems. This is the basis for the derivation of relative index theorems on some classes of open manifolds. As an example we discuss the de Rham complex for obstacles in ^N and asymptotically flat manifolds. It is shown that the absolute or relative Euler characteristic of an obstacle in ^N may be obtained from scattering data for the Laplace operator on forms with absolute or relative boundary conditions respectively. In the case of asymptotically flat manifolds we obtain the Chern-Gauss-Bonnet theorem for theL ²-Euler characteristic.On leave of absence from Institute of Physics, Leningrad State University, Leningrad