Geometric convergence to 0 of the probability that the goal has not been encountered by the nth generation is established for a class of genetic algorithms. These algorithms employ a quickly decreasing mutation rate and a crossover which restarts the algorithm in a controlled way depending on the current population and restricts execution of this crossover to occasions when progress of the algorithm is too slow. It is shown that without the crossover studied here, which amounts to a tempered restart of the algorithm, the asserted geometric convergence need not hold.

Some consequences of restarting stochastic search algorithms are studied. It is shown under reasonable conditions that restarting when certain patterns occur yields probabilities that the goal state has not been found by the nth epoch which converge to zero at least geometrically fast in n. These conditions are shown to hold for restarted simulated annealing employing a local generation matrix, a cooling schedule T n ∼ c/n and restarting after a fixed number r + 1 of duplications of energy levels of states when r is sufficiently large. For simulated annealing with logarithmic cooling these probabilities cannot decrease to zero this fast. Numerical comparisons between restarted simulated annealing and several modern variations on simulated annealing are also presented and in all cases the former performs better.

Data on the spread of invasive weeds into arid western lands are used to evaluate the environmental and economic importance of controlling invasive weed infestations early. Variable rate and constant rate infestation expansion paths are estimated. The implications of variable vs. constant infestation growth rates for projecting both biophysical and economic effects are illustrated. The projections derived from both constant and variable growth rate expansion paths support the contention that it is expedient to control new infestations early.

The Hausdorff and box dimensions for measures associated with recurrent self-similar sets generated by similitudes is explicitly given. The box dimension of the attractor associated with a class of two-dimensional affine maps is also computed.

In the following R will denote the real numbers, for a Hilbert space H, B(H) and L(H) will denote the collections of bounded linear operators on H and linear, but not necessarily bounded, operators on H respectively. Cosine Operator Functions, namely functions C:R ⟶ B(H) which satisfy D'Alembert's functional equation

(1)

and

(2)