Physical Sciences and Mathematics Commons^™

The Corona Problem For Kernel Multiplier Algebras, Eric T. Sawyer, Brett D. Wick

Mathematics Faculty Publications

We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in C, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions kx (y) of certain Hilbert function spaces H are assumed to be invertible multipliers on H and then we continue a research thread begun by Agler and McCarthy in 1999, and continued …

A Traders Guide To The Predictive Universe- A Model For Predicting Oil Price Targets And Trading On Them, Jimmie Harold Lenz

Doctor of Business Administration Dissertations

At heart every trader loves volatility; this is where return on investment comes from, this is what drives the proverbial “positive alpha.” As a trader, understanding the probabilities related to the volatility of prices is key, however if you could also predict future prices with reliability the world would be your oyster. To this end, I have achieved three goals with this dissertation, to develop a model to predict future short term prices (direction and magnitude), to effectively test this by generating consistent profits utilizing a trading model developed for this purpose, and to write a paper that anyone with …

Non-Commutative Automorphisms Of Bounded Non-Commutative Domains, John E. Mccarthy, Richard M. Timoney

Mathematics Faculty Publications

We establish rigidity (or uniqueness) theorems for non-commutative (NC) automorphisms that are natural extensions of classical results of H. Cartan and are improvements of recent results. We apply our results to NC domains consisting of unit balls of rectangular matrices.

Determinantal Representations Of Semihyperbolic Polynomials, Greg Knese

Mathematics Faculty Publications

We prove a generalization of the Hermitian version of the Helton–Vinnikov determinantal representation for hyperbolic polynomials to the class of semihyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.

A Remark On The Multipliers On Spaces Of Weak Products Of Functions, Stefan Richter, Brett D. Wick

Mathematics Faculty Publications

Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

No-Slip Billiards, Christopher Lee Cox

Arts & Sciences Electronic Theses and Dissertations

We investigate the dynamics of no-slip billiards, a model in which small rotating disks may exchange linear and angular momentum at collisions with the boundary. A general theory of rigid body collisions in is developed, which returns the known dimension two model as a special case but generalizes to higher dimensions. We give new results on periodicity and boundedness of orbits which suggest that a class of billiards (including all polygons) is not ergodic. Computer generated phase portraits demonstrate non-ergodic features, suggesting chaotic no-slip billiards cannot easily be constructed using the common techniques for generating chaos in standard billiards. However, …

Three Problems In Operator Theory And Complex Analysis, Cheng Chu

Arts & Sciences Electronic Theses and Dissertations

This thesis concerns three distinct problems in operator theory and complex analysis. In Chapter 2, we study the following problem: On the Hardy space H^2, when is the product of a Hankel operator and a Toeplitz operator compact? We give necessary and sucient conditions for when such a product H_fT_g is compact. In Chapter 3, we discuss hyponormal Toeplitz operators. We show that for those operators, there exists a lower bound for the area of the spectrum. This extends the known estimate for the spectral area of Toeplitz operators with an analytic symbol. This part is joint work with Dmitry …

On The Limiting Behavior Of Variations Of Hodge Structures, Genival Francisco Fernandes Da Silva Jr.

Arts & Sciences Electronic Theses and Dissertations

In this dissertation we will address three results concerning the limiting behavior of variations of Hodge structures. The first chapter introduces the main concepts involved and fixes some notation. In chapter two we discuss extension classes representing LMHS, compute them for a class of toric families and introduce an alternative method for the computation of VHS arising from middle convolution. The next chapter is concerned with the so called Apery constants; we provide a method of computing such constants by using higher normal functions coming from geometry. Finally, in the last chapter we analyze a family of surfaces with geometric …

Noncommutative Borsuk-Ulam Theorems, Benjamin Passer

Arts & Sciences Electronic Theses and Dissertations

The Borsuk-Ulam theorem in algebraic topology shows that there are significant restrictions on how any topological sphere interacts with the antipodal action of reflection through the origin (which maps x to -x). For example, any map f from a sphere to itself which is continuous and odd (f(-x) = -f(x)) must be homotopically nontrivial. We consider various equivalent forms of the theorem in terms of the function algebras on spheres and examine which forms generalize to certain noncommutative Banach and C*-algebras with finite group actions.

Chapter 1 contains background material on C*-algebras, K-theory, and group actions. Next, in Chapter 2, …

Polynomials With No Zeros On A Face Of The Bidisk, Jeffrey S. Geronimo, Plamen Iliev, Greg Knese

Mathematics Faculty Publications

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the “split-shift orthogonality condition” and another is an operator theoretic or matrix condition closely related to an earlier characterization due to the first two authors. This …

The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

We prove an implicit function theorem for non-commutative functions. We use this to show that if p ( X;Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of p ( X;Y ) = 0 will commute with X

Aspects Of Non-Commutative Function Theory, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.

Cyclic Polynomials In Two Variables, Catherine Bénéteau, Greg Knese, Łukasz Kosiński, Constanze Liaw, Daniel Seco, Alan Sola

Mathematics Faculty Publications

We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for polynomials, and harmonic analysis on curves.

Commutators In The Two-Weight Setting, Irina Holmes, Michael T. Lacey, Brett D. Wick

Mathematics Faculty Publications

Let R be the vector of Riesz transforms on Rn and let μ,λ∈Ap be two weights on Rn, 1p(μ)→Lp(λ)|| is shown to be equivalent to the function b being in a BMO space adapted to μ and λ. This is a common extension of a result of Coifman–Rochberg–Weiss in the case of both λ and μ being Lebesgue measure, and Bloom in the case of dimension one.

Unions Of Lebesgue Spaces And A1 Majorants, Greg Knese, John E. Mccarthy, Kabe Moen

Mathematics Faculty Publications

We study two questions. When does a function belong to the union of Lebesgue spaces, and when does a function have an A1 majorant? We provide a systematic study of these questions and show that they are fundamentally related. We show that the union ofLwp(ℝn)spaces withw∈Apis equal to the union of all Banach function spaces for which the Hardy–Littlewood maximal function is bounded on the space itself and its associate space.

The Von Neumann Inequality For 3 × 3 Matrices, Greg Knese

Mathematics Faculty Publications

This note details how recent work of Kosiński on the three point Pick interpolation problem on the polydisc can be used to prove the von Neumann inequality for d-tuples of commuting 3 x 3 contractive matrices.

Canonical Agler Decompositions And Transfer Function Realizations, Kelly Bickel, Greg Knese

Mathematics Faculty Publications

A seminal result of Agler proves that the natural de Branges-Rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift-invariant pieces. Agler's decomposition is non-constructive—a problem remedied by work of Ball-Sadosky-Vinnikov, which uses scattering systems to produce Agler decompositions through concrete Hilbert space geometry. This method, while constructive, so far has not revealed the rich structure shown to be present for special classes of functions--inner and rational inner functions. In this paper, we show that most of the important structure present in these special cases extends to general bounded analytic functions. We …