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_department:Department+of+Mathematics2015-03-08T17:44:37-05:00UNT LibrariesThis is a custom feed for browsing UNT Digital Library Collection: UNT Theses and DissertationsFundamental 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>Polynomial Isomorphisms of Cayley Objects Over a Finite Field2014-08-22T18:00:56-05:00http://digital.library.unt.edu/ark:/67531/metadc331144/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc331144/"><img alt="Polynomial Isomorphisms of Cayley Objects Over a Finite Field" title="Polynomial Isomorphisms of Cayley Objects Over a Finite Field" src="http://digital.library.unt.edu/ark:/67531/metadc331144/thumbnail/"/></a></p><p>In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a
particular type.</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>Minimality of the Special Linear Groups2014-03-26T09:30:20-05:00http://digital.library.unt.edu/ark:/67531/metadc279280/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc279280/"><img alt="Minimality of the Special Linear Groups" title="Minimality of the Special Linear Groups" src="http://digital.library.unt.edu/ark:/67531/metadc279280/thumbnail/"/></a></p><p>Let F denote the field of real numbers, complex numbers, or a finite algebraic extension of the p-adic field. We prove that the special linear group SLn(F) with the usual topology induced by F is a minimal topological group. This is accomplished by first proving the minimality of the upper triangular group in SLn(F). The proof for the upper triangular group uses an induction argument on a chain of upper triangular subgroups and relies on general results for locally compact topological groups, quotient groups, and subgroups. Minimality of SLn(F) is concluded by appealing to the associated Lie group decomposition as the product of a compact group and an upper triangular group. We also prove the universal minimality of homeomorphism groups of one dimensional manifolds, and we give a new simple proof of the universal minimality of S∞.</p>Topics in Fractal Geometry2014-03-26T09:30:20-05:00http://digital.library.unt.edu/ark:/67531/metadc279332/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc279332/"><img alt="Topics in Fractal Geometry" title="Topics in Fractal Geometry" src="http://digital.library.unt.edu/ark:/67531/metadc279332/thumbnail/"/></a></p><p>In this dissertation, we study fractal sets and their properties, especially the open set condition, Hausdorff dimensions and Hausdorff measures for certain fractal constructions.</p>Multifractal Measures2014-03-26T09:30:20-05:00http://digital.library.unt.edu/ark:/67531/metadc279084/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc279084/"><img alt="Multifractal Measures" title="Multifractal Measures" src="http://digital.library.unt.edu/ark:/67531/metadc279084/thumbnail/"/></a></p><p>The purpose of this dissertation is to introduce a natural and unifying multifractal formalism which contains the above mentioned multifractal parameters, and gives interesting results for a large class of natural measures. In Part 2 we introduce the proposed multifractal formalism and study it properties. We also show that this multifractal formalism gives natural and interesting results when applied to (nonrandom) graph directed self-similar measures in Rd and "cookie-cutter" measures in R. In Part 3 we use the multifractal formalism introduced in Part 2 to give a detailed discussion of the multifractal structure of random (and hence, as a special case, non-random) graph directed self-similar measures in R^d.</p>Properties of Bicentric Circles for Three-Sided Polygons2014-03-26T09:30:20-05:00http://digital.library.unt.edu/ark:/67531/metadc278727/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc278727/"><img alt="Properties of Bicentric Circles for Three-Sided Polygons" title="Properties of Bicentric Circles for Three-Sided Polygons" src="http://digital.library.unt.edu/ark:/67531/metadc278727/thumbnail/"/></a></p><p>We define and construct bicentric circles with respect to three-sided polygons. Then using inherent properties of these circles, we explore both tangent properties, and areas generated from bicentric circles.</p>