Ph.D. Dissertations- Mathematics Department

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    Linear and non-linear operators, and the distribution of zeros of entire functions
    ( 2013) Yoshida, Rintaro ; Csordas, George
    An important chapter in the theory of distribution of zeros of entire functions pertains to the study of linear operators acting on entire functions. This dissertation presents new results involving not only linear, but also some non-linear operators.
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    On-linear coefficient-wise stability and hyperbolicity preserving transformations
    (University of Hawaii at Manoa, 2012) Grabarek, Lukasz ; Csordas, George
    We study the operation of replacing the coefficients of a real function with a non-linear combination of its coefficients. We are particularly interested in the coefficient-wise transformations that preserve the location of zeros in a prescribed region.
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    Congruence Lattices of Finite Algebras
    (University of Hawaii at Manoa, 2012) DeMeo, William J. ; Freese, Ralph
    An important and long-standing open problem in universal algebra asks whether every finite lattice is isomorphic to the congruence lattice of a finite algebra. Until this problem is resolved, our understanding of finite algebras is incomplete, since, given an arbitrary finite algebra, we cannot say whether there are any restrictions on the shape of its congruence lattice. If we find a finite lattice that does not occur as the congruence lattice of a finite algebra (as many suspect we will), then we can finally declare that such restrictions do exist. By a well known result of Palfy and Pudlak, the problem would be solved if we could prove the existence of a finite lattice that is not the congruence lattice of a transitive group action or, equivalently, is not an interval in the lattice of subgroups of a finite group. Thus the problem of characterizing congruence lattices of finite algebras is closely related to the problem of characterizing intervals in subgroup lattices. In this work, we review a number of methods for finding a finite algebra with a given congruence lattice, including searching for intervals in subgroup lattices. We also consider methods for proving that algebras with a given congruence lattice exist without actually constructing them. By combining these well known methods with a new method we have developed, and with much help from computer software like the UACalc and GAP, we prove that with one possible exception every lattice with at most seven elements is isomorphic to the congruence lattice of a finite algebra. As such, we have identified the unique smallest lattice for which there is no known representation. We examine this exceptional lattice in detail, and prove results that characterize the class of algebras that could possibly represent this lattice. We conclude with what we feel are the most interesting open questions surrounding this problem and discuss possibilities for future work.
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    Sparse ordinary graphs
    (University of Hawaii at Manoa, 2005) Kalk, Jonathan W.
    Ordinary graphs are directed graphs that can be viewed as generalizations of symmetric block designs. They were introduced by Fossorier, Jezek, Nation and Pogel in [2] in an attempt to construct new finite projective planes. In this thesis we investigate some special cases of ordinary graphs, most prominently the case where nonadjacent vertices have no common neighbors. We determine all connected graphs of this type that exist.
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    Small lattices
    (University of Hawaii at Manoa, 2000) Heeney, Xiang Xia Huang ; Freese, Ralph
    This dissertation introduces triple gluing lattices and proves that a triple gluing lattice is small if the key subcomponents are small. Then attention is turned to triple gluing irreducible small lattices. The triple gluing irreducible [Special characters omitted.] lattices are introduced. The conditions which insure [Special characters omitted.] small are discovered. This dissertation also give some triple gluing irreducible small lattices by gluing [Special characters omitted.] 's. Finally, K-structured lattices are introduced. We prove that a K-structured lattice L is triple gluing irreducible if and only if [Special characters omitted.] . We prove that no 4-element antichain lies in u 1 /v1 of a K-structured small lattice. We also prove that some special lattices with 3-element antichains can not lie in u1 /v1 of a K-structured small lattice.