This paper analyzes the complexity of the contraction fixed point problem: compute an approximation to the fixed point V* = I*(V*) of a contraction mapping I* that maps a Banach space of continuous functions of variables into itself. We focus on quasi linear contractions where I* is a nonlinear...

We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j^{-k} with k1. For the Wiener measure occurring in many applications we have k=2. We want to...

Over the last sixty years there has been a stream of impossibility results from mathematics and theoretical computer science announcing undecidability, non-computability, and intractability. Should these impossibility results be of concern to physics? <p> Two of these impossibility results are...</p>

High-dimensional integrals are usually solved with Monte Carlo algorithms although theory suggests that low discrepancy algorithms are sometimes superior. We report on numerical testing which compares low discrepancy and Monte Carlo algorithms on the evaluation of financial derivatives. The...

I raise questions regarding the unknown and the unknowable in science. I also briefly address two topics which have been much in the news in 1996, and which might erroneously be confused with my topic. These are, first, the book by John Horgan, called The End of Science, and second, the great...

I discuss the relations among four worlds: Reality, mathematical models, computer simulation, and models of computation. Pros and cons of two models of computation, the Turing machine and the real number model, are presented. The the intrinsic difficulty of solving a mathematical model, as...

At its first meeting in 1986 the Computer Science and Technology Board, National Research Council, identified six critical national issues. <p> I discuss these six issues as they appeared in 1986 and how they appear for my 1996 vintage. <p> There was a seventh critical issue which I did not want to...</p></p>

High-dimensional integrals are usually solved with Monte Carlo algorithms although theory suggests that low-discrepancy algorithms are sometimes superior. We report on numerical testing which compares low-discrepancy and Monte Carlo algorithms on the evaluation of financial derivatives. The...