A new and flexible method for constructing designs for computer experiments

We develop a new method for constructing "good" designs for computer experiments. The method derives its power from its basic structure that builds large designs using small designs. We specialize the method for the construction of orthogonal Latin hypercubes and obtain many results along the way. In terms of run sizes, the existence problem of orthogonal Latin hypercubes is completely solved. We also present an explicit result showing how large orthogonal Latin hypercubes can be constructed using small orthogonal Latin hypercubes. Another appealing feature of our method is that it can easily be adapted to construct other designs; we examine how to make use of the method to construct nearly orthogonal and cascading Latin hypercubes.Comment: Published in at http://dx.doi.org/10.1214/09-AOS757 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics

Motivated by questions in mass-action kinetics, we introduce the notion of vertexical family of differential inclusions. Defined on open hypercubes, these families are characterized by particular good behavior under projection maps. The motivating examples are certain families of reaction networks -- including reversible, weakly reversible, endotactic, and strongly endotactic reaction networks -- that give rise to vertexical families of mass-action differential inclusions. We prove that vertexical families are amenable to structural induction. Consequently, a trajectory of a vertexical family approaches the boundary if and only if either the trajectory approaches a vertex of the hypercube, or a trajectory in a lower-dimensional member of the family approaches the boundary. With this technology, we make progress on the global attractor conjecture, a central open problem concerning mass-action kinetics systems. Additionally, we phrase mass-action kinetics as a functor on reaction networks with variable rates.Comment: v5: published version; v3 and v4: minor additional edits; v2: contains more general version of main theorem on vertexical families, including its accompanying corollaries -- some of them new; final section contains new results relating to prior and future research on persistence of mass-action systems; improved exposition throughou

Entanglement and quantum combinatorial designs

We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way than quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called $k$-uniform, i.e. multipartite pure states such that every reduction to $k$ parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite $k$-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.Comment: 14 pages, 3 figures. Comments are very welcome

Parallelization of adaptive MC Integrators

Monte Carlo (MC) methods for numerical integration seem to be embarassingly parallel on first sight. When adaptive schemes are applied in order to enhance convergence however, the seemingly most natural way of replicating the whole job on each processor can potentially ruin the adaptive behaviour. Using the popular VEGAS-Algorithm as an example an economic method of semi-micro parallelization with variable grain-size is presented and contrasted with another straightforward approach of macro-parallelization. A portable implementation of this semi-micro parallelization is used in the xloops-project and is made publicly available.Comment: 10 pages, LaTeX2e, 1 pstricks-figure included and 2 eps-figures inserted via epsfig. To appear in Comput. Phys. Commu