Geometric Strategy for the Optimal Quantum Search

We explore quantum search from the geometric viewpoint of a complex projective space $CP$, a space of rays. First, we show that the optimal quantum search can be geometrically identified with the shortest path along the geodesic joining a target state, an element of the computational basis, and such an initial state as overlaps equally, up to phases, with all the elements of the computational basis. Second, we calculate the entanglement through the algorithm for any number of qubits $n$ as the minimum Fubini-Study distance to the submanifold formed by separable states in Segre embedding, and find that entanglement is used almost maximally for large $n$. The computational time seems to be optimized by the dynamics as the geodesic, running across entangled states away from the submanifold of separable states, rather than the amount of entanglement itself.Comment: revtex, 10 pages, 7 eps figures, uses psfrag packag

Risk Aversion Asymptotics for Power Utility Maximization

We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to infinity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models. The limits are related to exponential and logarithmic utility. To derive these results, we combine approaches from optimal control, convex analysis and backward stochastic differential equations (BSDEs).Comment: 45 page

Electronic purchasing: determining the optimal roll-out strategy

Electronic purchasing (EP), also known as electronic ordering through catalogs is the most established form of e-procurement nowadays, yet still in its infancy. Theoretically, changing from the "traditional" way of purchasing to EP can lead to huge cost savings. However the implementation (roll-out) of EP including many commodity groups and many departments is a large and costly task. In addition, not much experience on good roll-out strategies is available yet. This paper contributes to the solution of this problem, by presenting a mathematical model for determining the optimal EP roll-out strategy into an organisation based on maximisation of the cost savings. Results from this model suggest that the optimal order of commodity groups and departments for which EP is implemented can contribute considerably to the possible savings that can be realised and is therefore an important factor for a successful implementation strategy of EP