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|Title: ||Using symmmetries to solve asymmetric problems|
|Authors: ||Gravel, Simon|
|Keywords: ||Constraint satisfaction problems|
effective pair potentials
nonlinear perturbation theory
|Issue Date: ||25-Aug-2008|
|Abstract: ||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.|
|Appears in Collections:||Theses and Dissertations (OPEN)|
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