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 …

Robbins' problem is an optimal stopping problem where one seeks to minimize the expected rank of their observations among all observations. We examine random walk analogs to Robbins' problem in both discrete and continuous time. In discrete time, we consider full information and relative ranks versions of this problem. For three step walks, we give the optimal stopping rule and the expected rank for both versions. We also give asymptotic upper bounds for the expected rank in discrete time. Finally, we give upper and lower bounds for the expected rank in continuous time, and we show that the expected rank in the continuous time problem is at least as large as the normalized asymptotic expected rank in the full information discrete time version.

Spaces such as the closed interval [0, 1] do not have the property of being homogeneous, strongly locally homogeneous (SLH) or countable dense homogeneous (CDH), but the Hilbert cube has all three properties. We investigate subsets X of real numbers to determine when their countable product is homogeneous, SLH, or CDH. We give necessary and sufficient conditions for the product to be homogeneous. We also prove that the product is SLH if and only if X is zero-dimensional or an interval. And finally we show that for a Borel subset X of real numbers the product is CDH iff X is a G-delta zero-dimensional set or an interval.

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 …

In this dissertation we focus on the application of thermodynamic formalism to non-autonomous and random dynamical systems. Specifically we use the thermodynamic formalism to investigate the dimension of various fractal constructions via the, now standard, technique of Bowen which he developed in his 1979 paper on quasi-Fuchsian groups. Bowen showed, roughly speaking, that the dimension of a fractal is equal to the zero of the relevant topological pressure function. We generalize the results of Rempe-Gillen and Urbanski on non-autonomous iterated function systems to the setting of non-autonomous graph directed Markov systems and then show that the Hausdorff dimension of the fractal limit set is equal to the zero of the associated pressure function provided the size of the alphabets at each time step do not grow too quickly. In trying to remove these growth restrictions, we present several other systems for which Bowen's formula holds, most notably ascending systems. We then use these various constructions to investigate the Hausdorff dimension of various subsets of the Julia set for different large classes of transcendental meromorphic functions of finite order which have been perturbed non-autonomously. In particular we find lower and upper bounds for the dimension of the subset of the Julia …

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.

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.

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.

In prophet problems, two players with different levels of information make decisions to optimize their return from an underlying optimal stopping problem. The player with more information is called the "prophet" while the player with less information is known as the "gambler." In this thesis, as in the majority of the literature on such problems, we assume that the prophet is omniscient, and the gambler does not know future outcomes when making his decisions. Certainly, the prophet will get a better return than the gambler. But how much better? The goal of a prophet problem is to find the least upper bound on the difference (or ratio) between the prophet's return, M, and the gambler's return, V. In this thesis, we present new prophet problems where we seek the least upper bound on M-V when there is a fixed cost per observations. Most prophet problems in the literature compare M and V when prophet and gambler buy (or sell) one asset. The new prophet problems presented in Chapters 3 and 4 treat a scenario where prophet and gambler optimize their return from selling two assets, when there is a fixed cost per observation. Sharp bounds for the problems on small …

In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring. We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, …

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.

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 …

We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.

The study of Borel equivalence relations on Polish spaces has become a major area of focus within descriptive set theory. Primarily, work in this area has been carried out using the standard methods of descriptive set theory. In this work, however, we develop a model-theoretic framework suitable for the study of Borel equivalence relations, introducing a class of objects we call Borel structurings. We then use these structurings to examine conditions under which marker sets for Borel equivalence relations can be concluded to exist or not exist, as well as investigating to what extent the Compactness Theorem from first-order logic continues to hold for Borel structurings.

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.

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.

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}

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.

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.

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.

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 …

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.

First, we show that the Rothberger and 2-Rothberger games are equivalent. Then we adjust the former proof and introduce another game, the restricted Menger game, in order to obtain a broader result. This provides an answer in the context of Hausdorff spaces for an open question posed by Aurichi, Bella, and Dias.

In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.