Many combinatorial problems that arises from real-world applications contain uncertainty in its parameters. The framework called the optimization with queries handles the uncertainty by modeling the parameters drawn from stochastic distributions, and conducting queries to reveal these realized values. The objective of this framework is to obtain high-quality solutions by conducting a small number of queries. In this paper, we introduce a general framework to derive efficient query strategies for packing-type combinatorial optimization problems. The main technique used in our framework is called the witness cover, that is a combination of linear programming duality and counting argument.

In this article, we review recent progress in the understanding of the transition from laminar to turbulent flow in shear flows. We describe why and how the idea of directed percolation can be applied to these transitions in different flows, and how the idea was tested by experiments and simulations.

Numerical computations give approximations to solutions of problems in general. Verified computations yield intervals containing the solutions. The author has established a framework for verified computations of solutions to matrix equations. This manuscript introduces the framework with matrix square roots being an example.