A Sums-of-Squares Extension of Policy Iterations

In order to address the imprecision often introduced by widening operators in static analysis, policy iteration based on min-computations amounts to considering the characterization of reachable value set of a program as an iterative computation of policies, starting from a post-fixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early research efforts relied on linear programming (LP) to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariants, relying on semidefinite programming (SDP) solvers to compute policies, and LP solvers to refine invariants. We propose here to extend the class of programs considered through the use of Sums-of-Squares (SOS) based optimization. Our approach enables the precise analysis of switched systems with polynomial updates and guards. The analysis presented has been implemented in Matlab and applied on existing programs coming from the system control literature, improving both the range of analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure

Bounded Height in Pencils of Finitely Generated Subgroups

We prove height bounds concerning intersections of finitely generated subgroups in a torus with algebraic subvarieties, all varying in a pencil. This vastly extends the previously treated constant case and involves entirely different, and more delicate, techniques

Electrodynamics of a Magnet Moving through a Conducting Pipe

The popular demonstration involving a permanent magnet falling through a conducting pipe is treated as an axially symmetric boundary value problem. Specifically, Maxwell's equations are solved for an axially symmetric magnet moving coaxially inside an infinitely long, conducting cylindrical shell of arbitrary thickness at nonrelativistic speeds. Analytic solutions for the fields are developed and used to derive the resulting drag force acting on the magnet in integral form. This treatment represents a significant improvement over existing models which idealize the problem as a point dipole moving slowly inside a pipe of negligible thickness. It also provides a rigorous study of eddy currents under a broad range of conditions, and can be used for precision magnetic braking applications. The case of a uniformly magnetized cylindrical magnet is considered in detail, and a comprehensive analytical and numerical study of the properties of the drag force is presented for this geometry. Various limiting cases of interest involving the shape and speed of the magnet and the full range of conductivity and magnetic behavior of the pipe material are investigated and corresponding asymptotic formulas are developed.Comment: 20 pages, 3 figures; computer program posted to http://www.csus.edu/indiv/p/partovimh/magpipedrag.nb Submitted to the Canadian Journal of Physic