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    Duality methods in networks, computer science models, and disordered condensed matter systems

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    Mitchell_umd_0117E_15594.pdf (886.0Kb)
    No. of downloads: 282

    Date
    2014
    Author
    Mitchell, Joe Dan
    Advisor
    Galitski, Victor M
    DRUM DOI
    https://doi.org/10.13016/M2KP4B
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    Abstract
    In this thesis, I explore lattice independent duality and systems to which it can be applied. I first demonstrate classical duality on models in an external field, including the Ising, Potts, and x-y models, showing in particular how this modifies duality to be lattice independent and applicable to networks. I then present a novel application of duality on the boolean satsifiability problem, one of the most important problems in computational complexity, through mapping to a low temperature Ising model. This establishes the equivalence between boolean satisfiability and a problem of enumerating the positive solutions to a Diophantine system of equations. I continue by combining duality with a prominent tool for models on networks, belief propagation, deriving a new message passing procedure, dual belief propagation. In the final part of my thesis, I shift to propose and examine a semiclassical model, the two-component Coulomb glass model, which can explain the giant magnetoresistance peak present in disordered films near a superconductor-insulator transition as the effect of competition between single particle and localized pair transport. I numerically analyze the density of states and transport properties of this model.
    URI
    http://hdl.handle.net/1903/15829
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