Latest content added for UNT Digital Library Partner: UNT Librarieshttp://digital.library.unt.edu/explore/partners/UNT/browse/?fq=untl_decade:2010-2019&fq=str_degree_discipline:Mathematics2014-04-23T20:20:45-05:00UNT LibrariesThis is a custom feed for browsing UNT Digital Library Partner: UNT LibrariesA 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>Traveling Wave Solutions of the Porous Medium Equation2014-02-01T18:14:03-06:00http://digital.library.unt.edu/ark:/67531/metadc271876/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc271876/"><img alt="Traveling Wave Solutions of the Porous Medium Equation" title="Traveling Wave Solutions of the Porous Medium Equation" src="http://digital.library.unt.edu/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>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="http://digital.library.unt.edu/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="http://digital.library.unt.edu/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>Determinacy-related Consequences on Limit Superiors2014-02-01T18:14:03-06:00http://digital.library.unt.edu/ark:/67531/metadc271913/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc271913/"><img alt="Determinacy-related Consequences on Limit Superiors" title="Determinacy-related Consequences on Limit Superiors" src="http://digital.library.unt.edu/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>Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups2014-02-01T18:14:03-06:00http://digital.library.unt.edu/ark:/67531/metadc271792/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc271792/"><img alt="Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups" title="Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups" src="http://digital.library.unt.edu/ark:/67531/metadc271792/thumbnail/"/></a></p><p>In this thesis we study descriptive-set-theoretic and measure-theoretic properties of Polish groups, with a thematic emphasis on the contrast between groups which are locally compact and those which are not. The work is divided into three major sections. In the first, working jointly with Robert Kallman, we resolve a conjecture of Gleason regarding the Polish topologization of abstract groups of homeomorphisms. We show that Gleason's conjecture is false, and its conclusion is only true when the hypotheses are considerably strengthened. Along the way we discover a new automatic continuity result for a class of functions which behave like but are distinct from functions of Baire class 1. In the second section we consider the descriptive complexity of those subsets of the permutation group S? which arise naturally from the classical Levy-Steinitz series rearrangement theorem. We show that for any conditionally convergent series of vectors in Euclidean space, the sets of permutations which make the series diverge, and diverge properly, are ?03-complete. In the last section we study the phenomenon of Haar null sets a la Christensen, and the closely related notion of openly Haar null sets. We identify and correct a minor error in the proof of Mycielski that a countable union of Haar null sets in a Polish group is Haar null. We show the openly Haar null ideal may be distinct from the Haar null ideal, which resolves an uncertainty of Solecki. We show that compact sets are always Haar null in S? and in any countable product of locally compact non-compact groups, which extends the domain of a result of Dougherty. We show that any countable product of locally compact non-compact groups decomposes into the disjoint union of a meager set and a Haar null set, which gives a partial positive answer to a question of Darji. We display a translation property in the homeomorphism group Homeo+[0,1] which is impossible in any non-trivial locally compact group. Other related results are peppered throughout.</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="http://digital.library.unt.edu/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="http://digital.library.unt.edu/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>Nonparametric Estimation of Receiver Operating Characteristic Surfaces Via Bernstein Polynomials2013-08-13T14:47:25-05:00http://digital.library.unt.edu/ark:/67531/metadc177212/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc177212/"><img alt="Nonparametric Estimation of Receiver Operating Characteristic Surfaces Via Bernstein Polynomials" title="Nonparametric Estimation of Receiver Operating Characteristic Surfaces Via Bernstein Polynomials" src="http://digital.library.unt.edu/ark:/67531/metadc177212/thumbnail/"/></a></p><p>Receiver operating characteristic (ROC) analysis is one of the most widely used methods in evaluating the accuracy of a classification method. It is used in many areas of decision making such as radiology, cardiology, machine learning as well as many other areas of medical sciences. The dissertation proposes a novel nonparametric estimation method of the ROC surface for the three-class classification problem via Bernstein polynomials. The proposed ROC surface estimator is shown to be uniformly consistent for estimating the true ROC surface. In addition, it is shown that the map from which the proposed estimator is constructed is Hadamard differentiable. The proposed ROC surface estimator is also demonstrated to lead to the explicit expression for the estimated volume under the ROC surface . Moreover, the exact mean squared error of the volume estimator is derived and some related results for the mean integrated squared error are also obtained. To assess the performance and accuracy of the proposed ROC and volume estimators, Monte-Carlo simulations are conducted. Finally, the method is applied to the analysis of two real data sets.</p>Semi-supervised and Self-evolving Learning Algorithms with Application to Anomaly Detection in Cloud Computing2013-08-13T14:47:25-05:00http://digital.library.unt.edu/ark:/67531/metadc177238/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc177238/"><img alt="Semi-supervised and Self-evolving Learning Algorithms with Application to Anomaly Detection in Cloud Computing" title="Semi-supervised and Self-evolving Learning Algorithms with Application to Anomaly Detection in Cloud Computing" src="http://digital.library.unt.edu/ark:/67531/metadc177238/thumbnail/"/></a></p><p>Semi-supervised learning (SSL) is the most practical approach for classification among machine learning algorithms. It is similar to the humans way of learning and thus has great applications in text/image classification, bioinformatics, artificial intelligence, robotics etc. Labeled data is hard to obtain in real life experiments and may need human experts with experimental equipments to mark the labels, which can be slow and expensive. But unlabeled data is easily available in terms of web pages, data logs, images, audio, video les and DNA/RNA sequences. SSL uses large unlabeled and few labeled data to build better classifying functions which acquires higher accuracy and needs lesser human efforts. Thus it is of great empirical and theoretical interest. We contribute two SSL algorithms (i) adaptive anomaly detection (AAD) (ii) hybrid anomaly detection (HAD), which are self evolving and very efficient to detect anomalies in a large scale and complex data distributions. Our algorithms are capable of modifying an existing classier by both retiring old data and adding new data. This characteristic enables the proposed algorithms to handle massive and streaming datasets where other existing algorithms fail and run out of memory. As an application to semi-supervised anomaly detection and for experimental illustration, we have implemented a prototype of the AAD and HAD systems and conducted experiments in an on-campus cloud computing environment. Experimental results show that the detection accuracy of both algorithms improves as they evolves and can achieve 92.1% detection sensitivity and 83.8% detection specificity, which makes it well suitable for anomaly detection in large and streaming datasets. We compared our algorithms with two popular SSL methods (i) subspace regularization (ii) ensemble of Bayesian sub-models and decision tree classifiers. Our contributed algorithms are easy to implement, significantly better in terms of space, time complexity and accuracy than these two methods for semi-supervised anomaly detection mechanism.</p>Equivalence Classes of Subquotients of Pseudodifferential Operator Modules on the Line2013-03-04T14:02:27-06:00http://digital.library.unt.edu/ark:/67531/metadc149627/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc149627/"><img alt="Equivalence Classes of Subquotients of Pseudodifferential Operator Modules on the Line" title="Equivalence Classes of Subquotients of Pseudodifferential Operator Modules on the Line" src="http://digital.library.unt.edu/ark:/67531/metadc149627/thumbnail/"/></a></p><p>Certain subquotients of Vec(R)-modules of pseudodifferential operators from one tensor density module to another are categorized, giving necessary and sufficient conditions under which two such subquotients are equivalent as Vec(R)-representations. These subquotients split under the projective subalgebra, a copy of ????2, when the members of their composition series have distinct Casimir eigenvalues. Results were obtained using the explicit description of the action of Vec(R) with respect to this splitting. In the length five case, the equivalence classes of the subquotients are determined by two invariants. In an appropriate coordinate system, the level curves of one of these invariants are a pencil of conics, and those of the other are a pencil of cubics.</p>