We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. Our methods are based on analysis of the space-time generating function, following the methods of [PW02].
International audience We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities.
California's 27 offshore oil and gas platforms will reach the end of their useful lifetimes sometime in the near future and will require decommissioning. Although existing leases require complete removal of all platforms and associated infrastructure, the underlying laws and regulations have changed in recent years to allow a number of alternative uses after decommissioning. In particular, AB 2503, signed into law in September 2010, provides for a rigs-to-reefs program that allows the state to accept ownership of decommissioned platforms in federal waters. Decisions about whether to remove platforms completely or leave them in place as artificial reefs will depend in part on the relative cost of the 2 options. In this study, we describe the design and use of a mathematical decision model that provides detailed cost estimates of complete and partial removal (to 85 feet below the water line) for California's offshore platforms. The model, PLATFORM, is loaded with Bureau of Safety and Environmental Enforcement (BSEE) and Bureau of Ocean Energy Management (BOEM) costs for complete removal, along with costs for partial removal calculated for this study and estimates of the uncertainty associated with decommissioning cost estimates. PLATFORM allows users to define a wide range of decommissioning and costing scenarios (e.g., number of platforms, choice of heavy lift vessel, shell mound removal, reef enhancement). As a benchmark cost, complete removal of all 27 offshore platforms, grouped into the 7 decommissioning projects defined by the most recent federal cost estimates produced in 2010, would cost an estimated $1.09 billion, whereas partial removal of these platforms, grouped into the same set of projects, would cost $478 million, with avoided costs of $616 million (with minor rounding).
We apply results of [BP07, BBBP08] to compute limiting probability profiles for various quantum walks in one and two dimensions. Using analytical machinery we show some features of the limit distribution that are not evident in an empirical intensity plot of the time 10,000 distribution. Some conjecutres are stated and computational techniques are discussed as well.1991 Mathematics Subject Classification. Primary 05A15; Secondary 41A60, 82C10.
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