On optimizing over lift-and-project closures

The lift-and-project closure is the relaxation obtained by computing all lift-and-project cuts from the initial formulation of a mixed integer linear program or equivalently by computing all mixed integer Gomory cuts read from all tableau's corresponding to feasible and infeasible bases. In this paper, we present an algorithm for approximating the value of the lift-and-project closure. The originality of our method is that it is based on a very simple cut generation linear programming problem which is obtained from the original linear relaxation by simply modifying the bounds on the variables and constraints. This separation LP can also be seen as the dual of the cut generation LP used in disjunctive programming procedures with a particular normalization. We study some properties of this separation LP in particular relating it to the equivalence between lift-and-project cuts and Gomory cuts shown by Balas and Perregaard. Finally, we present some computational experiments and comparisons with recent related works

Arithmetically defined dense subgroups of Morava stabilizer groups

For every prime $p$ and integer $n\ge 3$ we explicitly construct an abelian variety A/\F_{p^n} of dimension $n$ such that for a suitable prime $l$ the group of quasi-isogenies of A/\F_{p^n} of $l$-power degree is canonically a dense subgroup of the $n$-th Morava stabilizer group at $p$. We also give a variant of this result taking into account a polarization. This is motivated by a perceivable generalization of topological modular forms to more general topological automorphic forms. For this, we prove some results about approximation of local units in maximal orders which is of independent interest. For example, it gives a precise solution to the problem of extending automorphisms of the $p$-divisible group of a simple abelian variety over a finite field to quasi-isogenies of the abelian variety of degree divisible by as few primes as possible.Comment: major revision, main results slightly changed; final version, to appear in Compositio Mat

Enhancing Approximations for Regular Reachability Analysis

This paper introduces two mechanisms for computing over-approximations of sets of reachable states, with the aim of ensuring termination of state-space exploration. The first mechanism consists in over-approximating the automata representing reachable sets by merging some of their states with respect to simple syntactic criteria, or a combination of such criteria. The second approximation mechanism consists in manipulating an auxiliary automaton when applying a transducer representing the transition relation to an automaton encoding the initial states. In addition, for the second mechanism we propose a new approach to refine the approximations depending on a property of interest. The proposals are evaluated on examples of mutual exclusion protocols

Optimal prediction for moment models: Crescendo diffusion and reordered equations

A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.Comment: Revised version: 17 pages, 6 figures, presented at Workshop on Moment Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor correction

Operator splittings and spatial approximations for evolution equations

The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end a variant of Chernoff's product formula is proved. The methods are applied to abstract partial delay differential equations.Comment: to appear in J. Evol. Equations. Reviewers comments are incorporate