Latest content added for UNT Digital Library Collection: UNT Theses and Dissertationshttp://digital.library.unt.edu/explore/collections/UNTETD/browse/?fq=untl_institution:UNT&fq=dc_language:eng&fq=str_degree_discipline:Mathematics2014-04-23T20:20:45-05:00UNT LibrariesThis is a custom feed for browsing UNT Digital Library Collection: UNT Theses and DissertationsA Comparative Study of Non Linear Conjugate Gradient Methods2014-04-23T20:20:45-05:00http://digital.library.unt.edu/ark:/67531/metadc283864/<p><a href="/ark:/67531/metadc283864/"><img alt="A Comparative Study of Non Linear Conjugate Gradient Methods" title="A Comparative Study of Non Linear Conjugate Gradient Methods" src="/ark:/67531/metadc283864/thumbnail/"/></a></p><p>We study the development of nonlinear conjugate gradient methods, Fletcher Reeves (FR) and Polak Ribiere (PR). FR extends the linear conjugate gradient method to nonlinear functions by incorporating two changes, for the step length αk a line search is performed and replacing the residual, rk (rk=b-Axk) by the gradient of the nonlinear objective function. The PR method is equivalent to FR method for exact line searches and when the underlying quadratic function is strongly convex. The PR method is basically a variant of FR and primarily differs from it in the choice of the parameter βk. On applying the nonlinear Rosenbrock function to the MATLAB code for the FR and the PR algorithms we observe that the performance of PR method (k=29) is far better than the FR method (k=42). But, we observe that when the MATLAB codes are applied to general nonlinear functions, specifically functions whose minimum is a large negative number not close to zero and the iterates too are large values far off from zero the PR algorithm does not perform well. This problem with the PR method persists even if we run the PR algorithm for more iterations or with an initial guess closer to the actual minimum. To improve the PR algorithm we suggest finding a better weighing parameter βk, using better line search method and/or using specific line search for certain functions and identifying specific restart criteria based on the function to be optimized.</p>Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank2014-04-23T20:20:45-05:00http://digital.library.unt.edu/ark:/67531/metadc283833/<p><a href="/ark:/67531/metadc283833/"><img alt="Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank" title="Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank" src="/ark:/67531/metadc283833/thumbnail/"/></a></p><p>Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extension of work of Bringmann, Conley, and Richter in the rank 1case.</p>Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems2014-03-26T09:30:20-05:00http://digital.library.unt.edu/ark:/67531/metadc278917/<p><a href="/ark:/67531/metadc278917/"><img alt="Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems" title="Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems" src="/ark:/67531/metadc278917/thumbnail/"/></a></p><p>In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular,
these results hold for a fairly nonrestrictive class of triangular configurations of
scatterers.</p>On Groups of Positive Type2014-03-24T20:07:29-05:00http://digital.library.unt.edu/ark:/67531/metadc277804/<p><a href="/ark:/67531/metadc277804/"><img alt="On Groups of Positive Type" title="On Groups of Positive Type" src="/ark:/67531/metadc277804/thumbnail/"/></a></p><p>We describe groups of positive type and prove that a group G is of positive type if and only if G admits a non-trivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero.</p>Polish Spaces and Analytic Sets2014-03-24T20:07:29-05:00http://digital.library.unt.edu/ark:/67531/metadc277605/<p><a href="/ark:/67531/metadc277605/"><img alt="Polish Spaces and Analytic Sets" title="Polish Spaces and Analytic Sets" src="/ark:/67531/metadc277605/thumbnail/"/></a></p><p>A Polish space is a separable topological space that can be metrized by means
of a complete metric. A subset A of a Polish space X is analytic if there is a Polish
space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that
each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.</p>Physical Motivation and Methods of Solution of Classical Partial Differential Equations2014-03-24T20:07:29-05:00http://digital.library.unt.edu/ark:/67531/metadc277898/<p><a href="/ark:/67531/metadc277898/"><img alt="Physical Motivation and Methods of Solution of Classical Partial Differential Equations" title="Physical Motivation and Methods of Solution of Classical Partial Differential Equations" src="/ark:/67531/metadc277898/thumbnail/"/></a></p><p>We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.</p>Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials2014-02-01T18:14:03-06:00http://digital.library.unt.edu/ark:/67531/metadc271768/<p><a href="/ark:/67531/metadc271768/"><img alt="Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials" title="Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials" src="/ark:/67531/metadc271768/thumbnail/"/></a></p><p>Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family .</p>Graev Metrics and Isometry Groups of Polish Ultrametric Spaces2014-02-01T18:14:03-06:00http://digital.library.unt.edu/ark:/67531/metadc271898/<p><a href="/ark:/67531/metadc271898/"><img alt="Graev Metrics and Isometry Groups of Polish Ultrametric Spaces" title="Graev Metrics and Isometry Groups of Polish Ultrametric Spaces" src="/ark:/67531/metadc271898/thumbnail/"/></a></p><p>This dissertation presents results about computations of Graev metrics on free groups and characterizes isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces. In Chapter 2, computations of Graev metrics are performed on free groups. One of the related results answers an open question of Van Den Dries and Gao. In Chapter 3, isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces are characterized. The notion of generalized tree is defined and a correspondence between the isomorphism group of a generalized tree and the isometry group of a Heine-Borel Polish ultrametric space is established. The concept of a weak inverse limit is introduced to capture the characterization of isomorphism groups of generalized trees. In Chapter 4, partial results of isometry groups of uncountable compact ultrametric spaces are given. It turns out that every compact ultrametric space has a unique countable orbital decomposition. An orbital space consists of disjoint orbits. An orbit subspace of an orbital space is actually a compact homogeneous ultrametric subspace.</p>Traveling Wave Solutions of the Porous Medium Equation2014-02-01T18:14:03-06:00http://digital.library.unt.edu/ark:/67531/metadc271876/<p><a href="/ark:/67531/metadc271876/"><img alt="Traveling Wave Solutions of the Porous Medium Equation" title="Traveling Wave Solutions of the Porous Medium Equation" src="/ark:/67531/metadc271876/thumbnail/"/></a></p><p>We prove the existence of a one-parameter family of solutions of the porous medium equation, a nonlinear heat equation. In our work, with space dimension 3, the interface is a half line whose end point advances at constant speed. We prove, by using maximum principle, that the solutions are stable under a suitable class of perturbations. We discuss the relevance of our solutions, when restricted to two dimensions, to gravity driven flows of thin films. Here we extend the results of J. Iaia and S. Betelu in the paper "Solutions of the porous medium equation with degenerate interfaces" to a higher dimension.</p>Determinacy-related Consequences on Limit Superiors2014-02-01T18:14:03-06:00http://digital.library.unt.edu/ark:/67531/metadc271913/<p><a href="/ark:/67531/metadc271913/"><img alt="Determinacy-related Consequences on Limit Superiors" title="Determinacy-related Consequences on Limit Superiors" src="/ark:/67531/metadc271913/thumbnail/"/></a></p><p>Laczkovich proved from ZF that, given a countable sequence of Borel sets on a perfect Polish space, if the limit superior along every subsequence was uncountable, then there was a particular subsequence whose intersection actually contained a perfect subset. Komjath later expanded the result to hold for analytic sets. In this paper, by adding AD and sometimes V=L(R) to our assumptions, we will extend the result further. This generalization will include the increasing of the length of the sequence to certain uncountable regular cardinals as well as removing any descriptive requirements on the sets.</p>