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Algebraically Determined Rings of Functions
Let R be any of the following rings: the smooth functions on R^2n with the Poisson bracket, the Hamiltonian vector fields on a symplectic manifold, the Lie algebra of smooth complex vector fields on C, or a variety of rings of functions (real or complex valued) over 2nd countable spaces. Then if H is any other Polish ring and &#966;:H &#8594;R is an algebraic isomorphism, then it is also a topological isomorphism (i.e. a homeomorphism). Moreover, many such isomorphisms between function rings induce a homeomorphism of the underlying spaces. It is also shown that there is no topology in which the ring of real analytic functions on R is a Polish ring.
Algebraically Determined Semidirect Products
Let G be a Polish group. We say that G is an algebraically determined Polish group if given any Polish group L and any algebraic isomorphism from L to G, then the algebraic isomorphism is a topological isomorphism. We will prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This will smooth the way for the proofs for some special groups. For example, let H be a separable Hilbert space and let G be a subset of the unitary group U(H) acting transitively on the unit sphere. Assume that -I in G and G is a Polish topological group in some topology such that H x G to H, (x,U) to U(x) is continuous, then H x G is a Polish topological group. Hence H x G is an algebraically determined Polish group. In addition, we apply the above the above result on the unitary group U(A) of a separable irreducible C*-algebra A with identity acting transitively on the unit sphere in a separable Hilbert space H and proved that the natural semidirect product H x U(A) is an algebraically determined Polish group. A similar theorem is true for the natural semidirect product R^{n} x G(n), where G(n) = GL(n,R), or GL^{+}(n,R), or SL(n,R), or |SL(n,R)|={A in GL(n,R) : |det(A)|=1}. On the other hand, it is known that the Heisenberg group H_{3}(R) , (R, +), (R{0}, x), and GL^{+}(n,R) are not algebraically determined Polish groups.
Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank
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.
A Comparative Study of Non Linear Conjugate Gradient Methods
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.
Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation
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 ...
Continuous Combinatorics of a Lattice Graph in the Cantor Space
We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.
Contributions to Descriptive Set Theory
Assume AD+V=L(R). In the first chapter, let W^1_1 denote the club measure on \omega_1. We analyze the embedding j_{W^1_1}\restr HOD from the point of view of inner model theory. We use our analysis to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega. In the second chapter, we provide an indiscernibles analysis for models of the form L[T_n,x]. We use our analysis to provide new proofs of the strong partition property on \delta^1_{2n+1}
Contributions to Descriptive Set Theory
In this dissertation we study closure properties of pointclasses, scales on sets of reals and the models L[T2n], which are very natural canonical inner models of ZFC. We first characterize projective-like hierarchies by their associated ordinals. This solves a conjecture of Steel and a conjecture of Kechris, Solovay, and Steel. The solution to the first conjecture allows us in particular to reprove a strong partition property result on the ordinal of a Steel pointclass and derive a new boundedness principle which could be useful in the study of the cardinal structure of L(R). We then develop new methods which produce lightface scales on certain sets of reals. The methods are inspired by Jackson’s proof of the Kechris-Martin theorem. We then generalize the Kechris-Martin Theorem to all the Π12n+1 pointclasses using Jackson’s theory of descriptions. This in turns allows us to characterize the sets of reals of a certain initial segment of the models L[T2n]. We then use this characterization and the generalization of Kechris-Martin theorem to show that the L[T2n] are unique. This generalizes previous work of Hjorth. We then characterize the L[T2n] in term of inner models theory, showing that they actually are constructible models over direct limit of mice with Woodin cardinals, a counterpart to Steel’s result that the L[T2n+1] are extender models, and finally show that the generalized contiuum hypothesis holds in these models, solving a conjecture of Woodin.
A Decomposition of the Group Algebra of a Hyperoctahedral Group
The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, Bonnafé asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.
Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups
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. ...
Determinacy-related Consequences on Limit Superiors
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.
Equivalence Classes of Subquotients of Pseudodifferential Operator Modules on the Line
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.
An Exploration of the Word2vec Algorithm: Creating a Vector Representation of a Language Vocabulary that Encodes Meaning and Usage Patterns in the Vector Space Structure
This thesis is an exloration and exposition of a highly efficient shallow neural network algorithm called word2vec, which was developed by T. Mikolov et al. in order to create vector representations of a language vocabulary such that information about the meaning and usage of the vocabulary words is encoded in the vector space structure. Chapter 1 introduces natural language processing, vector representations of language vocabularies, and the word2vec algorithm. Chapter 2 reviews the basic mathematical theory of deterministic convex optimization. Chapter 3 provides background on some concepts from computer science that are used in the word2vec algorithm: Huffman trees, neural networks, and binary cross-entropy. Chapter 4 provides a detailed discussion of the word2vec algorithm itself and includes a discussion of continuous bag of words, skip-gram, hierarchical softmax, and negative sampling. Finally, Chapter 5 explores some applications of vector representations: word categorization, analogy completion, and language translation assistance.
Fundamental Issues in Support Vector Machines
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.
Gibbs/Equilibrium Measures for Functions of Multidimensional Shifts with Countable Alphabets
Consider a multidimensional shift space with a countably infinite alphabet, which serves in mathematical physics as a classical lattice gas or lattice spin system. A new definition of a Gibbs measure is introduced for suitable real-valued functions of the configuration space, which play the physical role of specific internal energy. The variational principle is proved for a large class of functions, and then a more restrictive modulus of continuity condition is provided that guarantees a function's Gibbs measures to be a nonempty, weakly compact, convex set of measures that coincides with the set of measures obeying a form of the DLR equations (which has been adapted so as to be stated entirely in terms of specific internal energy instead of the Hamiltonians for an interaction potential). The variational equilibrium measures for a such a function are then characterized as the shift invariant Gibbs measures of finite entropy, and a condition is provided to determine if a function's Gibbs measures have infinite entropy or not. Moreover the spatially averaged limiting Gibbs measures, i.e. constructive equilibria, are shown to exist and their weakly closed convex hull is shown to coincide with the set of true variational equilibrium measures. It follows that the "pure thermodynamic phases", which correspond to the extreme points in the convex set of equilibrium measures, must be constructive equilibria. Finally, for an even smoother class of functions a method is presented to construct a compatible interaction potential and it is checked that the two different structures generate the same sets of Gibbs and equilibrium measures, respectively.
Graev Metrics and Isometry Groups of Polish Ultrametric Spaces
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.
Hermitian Jacobi Forms and Congruences
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.
Hochschild Cohomology and Complex Reflection Groups
A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and calculations using the software GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary finite groups, we show that the codimension atoms are contained in the support of every generating set for cohomology, thus yielding information about the degrees of generators for cohomology.
Irreducible Modules for Yokonuma-Type Hecke Algebras
Yokonuma-type Hecke algebras are a class of Hecke algebras built from a Type A construction. In this thesis, I construct the irreducible representations for a class of generic Yokonuma-type Hecke algebras which specialize to group algebras of the complex reflection groups and to endomorphism rings of certain permutation characters of finite general linear groups.
Kleinian Groups in Hilbert Spaces
The theory of discrete groups acting on finite dimensional Euclidean open balls by hyperbolic isometries was borne around the end of 19th century within the works of Fuchs, Klein and Poincaré. We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite dimensional separable Hilbert space. We present our investigations on the geometry of limit sets at the sphere at infinity with an attempt to highlight the differences between the finite and infinite dimensional theories. We discuss the existence of fixed points of isometries and the classification of isometries. Various notions of discreteness that were equivalent in finite dimensions, no longer turn out to be in our setting. In this regard, the robust notion of strong discreteness is introduced and we study limit sets for properly discontinuous actions. We go on to prove a generalization of the Bishop-Jones formula for strongly discrete groups, equating the Hausdorff dimension of the radial limit set with the Poincaré exponent of the group. We end with a short discussion on conformal measures and their relation with Hausdorff and packing measures on the limit set.
Maximum Likelihood Estimation of Logistic Sinusoidal Regression Models
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
Mycielski-Regular Measures
Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.
Nonparametric Estimation of Receiver Operating Characteristic Surfaces Via Bernstein Polynomials
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.
On Steinhaus Sets, Orbit Trees and Universal Properties of Various Subgroups in the Permutation Group of Natural Numbers
In the first chapter, we define Steinhaus set as a set that meets every isometric copy of another set at exactly one point. We show that there is no Steinhaus set for any four-point subset in a plane.In the second chapter, we define the orbit tree of a permutation group of natural numbers, and further introduce compressed orbit trees. We show that any rooted finite tree can be realized as a compressed orbit tree of some permutation group. In the third chapter, we investigate certain classes of closed permutation groups of natural numbers with respect to their universal and surjectively universal groups. We characterize two-sided invariant groups, and prove that there is no universal group for countable groups, nor universal group for two-sided invariant groups in permutation groups of natural numbers.
Optimal Strategies for Stopping Near the Top of a Sequence
In Chapter 1 the classical secretary problem is introduced. Chapters 2 and 3 are variations of this problem. Chapter 2, discusses the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter p and finite time horizon n. The optimal strategy (continue or stop) depends on a sequence of threshold values (critical probabilities) which has an oscillating pattern. Several properties of this sequence have been proved by Dr. Allaart. Further properties have been recently proved. In Chapter 3, a gambler will observe a finite sequence of continuous random variables. After he observes a value he must decide to stop or continue taking observations. He can play two different games A) Win at the maximum or B) Win within a proportion of the maximum. In the first section the sequence to be observed is independent. It is shown that for each n>1, theoptimal win probability in game A is bounded below by (1-1/n)^{n-1}. It is accomplished by reducing the problem to that of choosing the maximum of a special sequence of two-valued random variables and applying the sum-the-odds theorem of Bruss (2000). Secondly, it is assumed the sequence is i.i.d. The best lower bounds are provided for the winning probabilities in game B given any continuous distribution. These bounds are the optimal win probabilities of a game A which was examined by Gilbert and Mosteller (1966).
Quantum Drinfeld Hecke Algebras
Quantum Drinfeld Hecke algebras extend both Lusztig's graded Hecke algebras and the symplectic reflection algebras of Etingof and Ginzburg to the quantum setting. A quantum (or skew) polynomial ring is generated by variables which commute only up to a set of quantum parameters. Certain finite groups may act by graded automorphisms on a quantum polynomial ring and quantum Drinfeld Hecke algebras deform the natural semi-direct product. We classify these algebras for the infinite family of complex reflection groups acting in arbitrary dimension. We also classify quantum Drinfeld Hecke algebras in arbitrary dimension for the infinite family of mystic reflection groups of Kirkman, Kuzmanovich, and Zhang, who showed they satisfy a Shephard-Todd-Chevalley theorem in the quantum setting. Using a classification of automorphisms of quantum polynomial rings in low dimension, we develop tools for studying quantum Drinfeld Hecke algebras in 3 dimensions. We describe the parameter space of such algebras using special properties of the quantum determinant in low dimension; although the quantum determinant is not a homomorphism in general, it is a homomorphism on the finite linear groups acting in dimension 3.
Random Iteration of Rational Functions
It is a theorem of Denker and Urbański that if T:ℂ→ℂ is a rational map of degree at least two and if ϕ:ℂ→ℝ is Hölder continuous and satisfies the “thermodynamic expanding” condition P(T,ϕ) > sup(ϕ), then there exists exactly one equilibrium state μ for T and ϕ, and furthermore (ℂ,T,μ) is metrically exact. We extend these results to the case of a holomorphic random dynamical system on ℂ, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bogenschütz. Specifically, if (T,Ω,P,θ) is a holomorphic random dynamical system on ℂ and ϕ:Ω→ ℋα(ℂ) is a Hölder continuous random potential function satisfying one of several sets of technical but reasonable hypotheses, then there exists a unique equilibrium state of (X,P,ϕ) over (Ω,Ρ,θ).
Rankin-Cohen Brackets for Hermitian Jacobi Forms and Hermitian Modular Forms
In this thesis, we define differential operators for Hermitian Jacobi forms and Hermitian modular forms over the Gaussian number field Q(i). In particular, we construct Rankin-Cohen brackets for such spaces of Hermitian Jacobi forms and Hermitian modular forms. As an application, we extend Rankin's method to the case of Hermitian Jacobi forms. Finally we compute Fourier series coefficients of Hermitian modular forms, which allow us to give an example of the first Rankin-Cohen bracket of two Hermitian modular forms. In the appendix, we provide tables of Fourier series coefficients of Hermitian modular forms and also the computer source code that we used to compute such Fourier coefficients.
Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials
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 .
Reduced Ideals and Periodic Sequences in Pure Cubic Fields
The “infrastructure” of quadratic fields is a body of theory developed by Dan Shanks, Richard Mollin and others, in which they relate “reduced ideals” in the rings and sub-rings of integers in quadratic fields with periodicity in continued fraction expansions of quadratic numbers. In this thesis, we develop cubic analogs for several infrastructure theorems. We work in the field K=Q(), where 3=m for some square-free integer m, not congruent to ±1, modulo 9. First, we generalize the definition of a reduced ideal so that it applies to K, or to any number field. Then we show that K has only finitely many reduced ideals, and provide an algorithm for listing them. Next, we define a sequence based on the number alpha that is periodic and corresponds to the finite set of reduced principal ideals in K. Using this rudimentary infrastructure, we are able to establish results about fundamental units and reduced ideals for some classes of pure cubic fields. We also introduce an application to Diophantine approximation, in which we present a 2-dimensional analog of the Lagrange value of a badly approximable number, and calculate some examples.
The Relative Complexity of Various Classification Problems among Compact Metric Spaces
In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.
Restricting Invariants and Arrangements of Finite Complex Reflection Groups
Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. In my thesis, I extend earlier work by Douglass and Röhrle for Coxeter groups to the case where G is a complex reflection group of type G(r,p,n) in the notation of Shephard and Todd and X is in the lattice of the reflection arrangement of G. The main result characterizes when the restriction mapping is surjective in terms of the exponents of G and C and their reflection arrangements.
Semi-supervised and Self-evolving Learning Algorithms with Application to Anomaly Detection in Cloud Computing
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 ...
Strong Choquet Topologies on the Closed Linear Subspaces of Banach Spaces
In the study of Banach spaces, the development of some key properties require studying topologies on the collection of closed convex subsets of the space. The subcollection of closed linear subspaces is studied under the relative slice topology, as well as a class of topologies similar thereto. It is shown that the collection of closed linear subspaces under the slice topology is homeomorphic to the collection of their respective intersections with the closed unit ball, under the natural mapping. It is further shown that this collection under any topology in the aforementioned class of similar topologies is a strong Choquet space. Finally, a collection of category results are developed since strong Choquet spaces are also Baire spaces.
Three Topics in Descriptive Set Theory
This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpi&#324;ski, Vaintein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem of Balcerzak and G&#322;&#261;b.
Traveling Wave Solutions of the Porous Medium Equation
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.
Trees and Ordinal Indices in C(K) Spaces for K Countable Compact
In the dissertation we study the C(K) spaces focusing on the case when K is countable compact and more specifically, the structure of C() spaces for < ω1 via special type of trees that they contain. The dissertation is composed of three major sections. In the first section we give a detailed proof of the theorem of Bessaga and Pelczynski on the isomorphic classification of C() spaces. In due time, we describe the standard bases for C(ω) and prove that the bases are monotone. In the second section we consider the lattice-trees introduced by Bourgain, Rosenthal and Schechtman in C() spaces, and define rerooting and restriction of trees. The last section is devoted to the main results. We give some lower estimates of the ordinal-indices in C(ω). We prove that if the tree in C(ω) has large order with small constant then each function in the root must have infinitely many big coordinates. Along the way we deduce some upper estimates for c0 and C(ω), and give a simple proof of Cambern's result that the Banach-Mazur distance between c0 and c = C(ω) is equal to 3.
Uniformly σ-Finite Disintegrations of Measures
A disintegration of measure is a common tool used in ergodic theory, probability, and descriptive set theory. The primary interest in this paper is in disintegrating σ-finite measures on standard Borel spaces into families of σ-finite measures. In 1984, Dorothy Maharam asked whether every such disintegration is uniformly σ-finite meaning that there exists a countable collection of Borel sets which simultaneously witnesses that every measure in the disintegration is σ-finite. Assuming Gödel’s axiom of constructability I provide answer Maharam's question by constructing a specific disintegration which is not uniformly σ-finite.