Latest content added for UNT Digital Library Collection: UNT Theses and Dissertationshttp://digital.library.unt.edu/explore/collections/UNTETD/browse/?fq=str_degree_discipline:Mathematics2015-08-21T05:42:39-05:00UNT LibrariesThis is a custom feed for browsing UNT Digital Library Collection: UNT Theses and DissertationsCondition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation2015-08-21T05:42:39-05:00http://digital.library.unt.edu/ark:/67531/metadc699977/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc699977/"><img alt="Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation" title="Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation" src="http://digital.library.unt.edu/ark:/67531/metadc699977/thumbnail/"/></a></p><p>A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace’s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain. Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomi-type equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented.</p>Hermitian Jacobi Forms and Congruences2015-08-21T05:42:39-05:00http://digital.library.unt.edu/ark:/67531/metadc700083/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc700083/"><img alt="Hermitian Jacobi Forms and Congruences" title="Hermitian Jacobi Forms and Congruences" src="http://digital.library.unt.edu/ark:/67531/metadc700083/thumbnail/"/></a></p><p>In this thesis, we introduce a new space of Hermitian Jacobi forms, and we determine its structure. As an application, we study heat cycles of Hermitian Jacobi forms, and we establish a criterion for the existence of U(p) congruences of Hermitian Jacobi forms. We demonstrate that criterion with some explicit examples. Finally, in the appendix we give tables of Fourier series coefficients of several Hermitian Jacobi forms.</p>Fundamental Issues in Support Vector Machines2015-03-08T17:44:37-05:00http://digital.library.unt.edu/ark:/67531/metadc500155/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc500155/"><img alt="Fundamental Issues in Support Vector Machines" title="Fundamental Issues in Support Vector Machines" src="http://digital.library.unt.edu/ark:/67531/metadc500155/thumbnail/"/></a></p><p>This dissertation considers certain issues in support vector machines (SVMs), including a description of their construction, aspects of certain exponential kernels used in some SVMs, and a presentation of an algorithm that computes the necessary elements of their operation with proof of convergence. In its first section, this dissertation provides a reasonably complete description of SVMs and their theoretical basis, along with a few motivating examples and counterexamples. This section may be used as an accessible, stand-alone introduction to the subject of SVMs for the advanced undergraduate. Its second section provides a proof of the positive-definiteness of a certain useful function here called E and dened as follows: Let V be a complex inner product space. Let N be a function that maps a vector from V to its norm. Let p be a real number between 0 and 2 inclusive and for any in V , let ( be N() raised to the p-th power. Finally, let a be a positive real number. Then E() is exp(()). Although the result is not new (other proofs are known but involve deep properties of stochastic processes) this proof is accessible to advanced undergraduates with a decent grasp of linear algebra. Its final section presents an algorithm by Dr. Kallman (preprint), based on earlier Russian work by B.F. Mitchell, V.F Demyanov, and V.N. Malozemov, and proves its convergence. The section also discusses briefly architectural features of the algorithm expected to result in practical speed increases.</p>Maximum Likelihood Estimation of Logistic Sinusoidal Regression Models2014-11-08T11:56:31-06:00http://digital.library.unt.edu/ark:/67531/metadc407796/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc407796/"><img alt="Maximum Likelihood Estimation of Logistic Sinusoidal Regression Models" title="Maximum Likelihood Estimation of Logistic Sinusoidal Regression Models" src="http://digital.library.unt.edu/ark:/67531/metadc407796/thumbnail/"/></a></p><p>We consider the problem of maximum likelihood estimation of logistic sinusoidal regression models and develop some asymptotic theory including the consistency and joint rates of convergence for the maximum likelihood estimators. The key techniques build upon a synthesis of the results of Walker and Song and Li for the widely studied sinusoidal regression model and on making a connection to a result of Radchenko. Monte Carlo simulations are also presented to demonstrate the finite-sample performance of the estimators</p>Hausdorff, Packing and Capacity Dimensions2014-08-22T18:00:56-05:00http://digital.library.unt.edu/ark:/67531/metadc330990/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc330990/"><img alt="Hausdorff, Packing and Capacity Dimensions" title="Hausdorff, Packing and Capacity Dimensions" src="http://digital.library.unt.edu/ark:/67531/metadc330990/thumbnail/"/></a></p><p>In this thesis, Hausdorff, packing and capacity dimensions are studied by evaluating sets in the Euclidean space R^. Also the lower entropy dimension is calculated for some Cantor sets. By incorporating technics of Munroe and of Saint Raymond and Tricot, outer measures are created. A Vitali covering theorem for packings is proved. Methods (by Taylor and Tricot, Kahane and Salem, and Schweiger) for determining the Hausdorff and capacity dimensions of sets using probability measures are discussed and extended. The packing pre-measure and measure are shown to be scaled after an affine transformation.
A Cantor set constructed by L.D. Pitt is shown to be dimensionless using methods developed in this thesis. A Cantor set is constructed for which all four dimensions are different. Graph directed constructions (compositions of similitudes follow a path in a directed graph) used by Mauldin and Willjams are presented. Mauldin and Williams calculate the Hausdorff dimension, or, of the object of a graph directed construction and show that if the graph is strongly connected, then the a—Hausdorff measure is positive and finite. Similar results will be shown for the packing dimension and the packing measure. When the graph is strongly connected, there is a constant so that the constant times the Hausdorff measure is greater than or equal to the packing measure when a subset of the realization is evaluated. Self—affine Sierpinski carpets, which have been analyzed by McMullen with respect to their Hausdorff dimension and capacity dimension, are analyzed with respect to their packing dimension. Conditions under which the Hausdorff measure of the construction object is positive and finite are given.</p>The Maximum Size of Combinatorial Geometries Excluding Wheels and Whirls as Minors2014-08-22T18:00:56-05:00http://digital.library.unt.edu/ark:/67531/metadc330849/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc330849/"><img alt="The Maximum Size of Combinatorial Geometries Excluding Wheels and Whirls as Minors" title="The Maximum Size of Combinatorial Geometries Excluding Wheels and Whirls as Minors" src="http://digital.library.unt.edu/ark:/67531/metadc330849/thumbnail/"/></a></p><p>We show that the maximum size of a geometry of rank n excluding the (q + 2)-point line, the 3-wheel W_3, and the 3-whirl W^3 as minor is (n - 1)q + 1, and geometries of maximum size are parallel connections of (q + 1)-point lines. We show that the maximum size of a geometry of rank n excluding the 5-point line, the 4-wheel W_4, and the 4-whirl W^4 as minors is 6n - 5, for n ≥ 3. Examples of geometries having rank n and size 6n - 5 include parallel connections of the geometries V_19 and PG(2,3).</p>Applications of Graph Theory and Topology to Combinatorial Designs2014-08-22T18:00:56-05:00http://digital.library.unt.edu/ark:/67531/metadc331968/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc331968/"><img alt="Applications of Graph Theory and Topology to Combinatorial Designs" title="Applications of Graph Theory and Topology to Combinatorial Designs" src="http://digital.library.unt.edu/ark:/67531/metadc331968/thumbnail/"/></a></p><p>This dissertation is concerned with the existence and the isomorphism of designs. The first part studies the existence of designs. Chapter I shows how to obtain a design from a difference family. Chapters II to IV study the existence of an affine 3-(p^m,4,λ) design where the v-set is the Galois field GF(p^m). Associated to each prime p, this paper constructs a graph. If the graph has a 1-factor, then a difference family and hence an affine design exists. The question arises of how to determine when the graph has a 1-factor. It is not hard to see that the graph is connected and of even order. Tutte's theorem shows that if the graph is 2-connected and regular of degree three, then the graph has a 1-factor. By using the concept of quadratic reciprocity, this paper shows that if p Ξ 53 or 77 (mod 120), the graph is almost regular of degree three, i.e., every vertex has degree three, except two vertices each have degree tow. Adding an extra edge joining the two vertices with degree tow gives a regular graph of degree three. Also, Tutte proved that if A is an edge of the graph satisfying the above conditions, then it must have a 1-factor which contains A. The second part of the dissertation is concerned with determining if two designs are isomorphic. Here the v-set is any group G and translation by any element in G gives a design automorphism. Given a design B and its difference family D, two topological spaces, B and D, are constructed. We give topological conditions which imply that a design isomorphism is a group isomorphism.</p>A Comparative Study of Non Linear Conjugate Gradient Methods2014-04-23T20:20:45-05:00http://digital.library.unt.edu/ark:/67531/metadc283864/<p><a href="http://digital.library.unt.edu/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="http://digital.library.unt.edu/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="http://digital.library.unt.edu/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="http://digital.library.unt.edu/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="http://digital.library.unt.edu/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="http://digital.library.unt.edu/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>