Using symmmetries to solve asymmetric problems
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This dissertation describes two projects in which the treatment of a difficult and asymmetric problem is simplified by using symmetries of basic building blocks of the problem. In the first part of this dissertation we address the problem of determining the effective interaction between ions in metallic systems. Our work applies more generally to systems where effective interactions between massive particles can be calculated to take into account, in an average way, the effect of lighter particles present in the system. We find an equality relating the (asymmetric) effective interaction of two massive particles and the (symmetric) effect of a single massive particle on the density of the light particles. We show how this relation can be used to improve upon the precision of effective potentials calculated by perturbative approaches for an assortment of systems including hydrogen in metallic environment. In the second part of this dissertation we discuss constraint satisfaction problems. We provide multiple examples of constraint satisfaction problems occurring in various scientific areas. In many cases the individual constraints are highly symmetric, while the resulting constraint satisfaction problem is not; there is no symmetry common to all the constraints. We describe divide and concur, a new approach to solve constraint problems, which is based on projections to the individual constraint sets. The definition of efficient projection operators are facilitated by symmetries of the constraint sets. We show that this method is competitive with the state-of-the-art on standard benchmark problems, and in the process establish a number of records in finite disk packing problems. Many applications of the divide and concur approach are still to be explored, and we provide the reader with tools to do so, including promising applications and a list of constraint sets together with efficient projection operators.
NSERC Fellowship, FQRNT Fellowship, National Science Foundation Grant DMR-0426568, National Science Foundation Grant DMR-0601461.
Constraint satisfaction problems; effective pair potentials; iterated maps; nonlinear perturbation theory
dissertation or thesis