Mathematics Commons^™

Real Complex Functions, Stephan Ramon Garcia, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to operator theory.

Cameron-Liebler Line Classes And Partial Difference Sets, Uthaipon Tantipongipat

Honors Theses

The work consists of three parts. The first is a study of Cameron-Liebler line classes which receive much attention recently. We studied a new construction of infinite family of Cameron-Liebler line classes presented in the paper by Tao Feng, Koji Momihara, and Qing Xiang (rst introduced in 2014), and summarized our attempts to generalize this construction to discover any new Cameron-Liebler line classes or partial difference sets (PDSs) resulting from the Cameron-Liebler line classes. The second is our approach to finding PDS in non-elementary abelian groups. Our attempt eventually led to the same general construction of PDS presented in John …

Nonexistence Of Nonquadratic Kerdock Sets In Six Variables, John Clikeman

Honors Theses

Kerdock sets are maximally sized sets of boolean functions such that the sum of any two functions in the set is bent. This paper modifies the methodology of a paper by Phelps (2015) to the problem of finding Kerdock sets in six variables containing non-quadratic elements. Using a computer search, we demonstrate that no Kerdock sets exist containing non-quadratic six- variable bent functions, and that the largest bent set containing such functions has size 8.

Introduction To Model Spaces And Their Operators, William T. Ross, Stephan Ramon Garcia, Javad Mashreghi

Bookshelf

The study of model spaces, the closed invariant subspaces of the backward shift operator, is a vast area of research with connections to complex analysis, operator theory and functional analysis. This self-contained text is the ideal introduction for newcomers to the field. It sets out the basic ideas and quickly takes the reader through the history of the subject before ending up at the frontier of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further.

Chebyshev Polynomials And The Frohman-Gelca Formula, Heather M. Russell, Hoel Queffelec

Department of Math & Statistics Faculty Publications

Using Chebyshev polynomials, C. Frohman and R. Gelca introduced a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones–Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which emphasizes and demystifies the role played by Chebyshev polynomials.

A Survey On Reverse Carleson Measures, Emmanuel Fricain, Andreas Hartmann, William T. Ross

Department of Math & Statistics Faculty Publications

This is a survey on reverse Carleson measures for various Hilbert spaces of analytic functions. These spaces include the Hardy, Bergman, certain harmonically weighted Dirichlet, Paley-Wiener, Fock, model (backward shift invariant), and de Branges-Rovnyak spaces. The reverse Carleson measure for backward shift invariant subspaces in the non-Hilbert situation is new.

Partial Orders On Partial Isometries, William T. Ross, Stephan Ramon Garcia, Robert T. W. Martin

Department of Math & Statistics Faculty Publications

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBranges-Rovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate.

Weak Parallelogram Laws On Banach Spaces And Applications To Prediction, William T. Ross, R. Cheng

Department of Math & Statistics Faculty Publications

This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.

The Robinson-Schensted Correspondence And A2-Web Bases, Heather M. Russell, Matthew Housley, Julianna Tymoczko

Department of Math & Statistics Faculty Publications

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n; n; n]: the reduced web basis associated to Kuperberg's combinatorial description of the spider category; and the left cell basis for the left cell construction of Kazhdan and Lusztig. In the case of [n; n], the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the image of these bases under classical maps: the Robinson-Schensted algorithm between permutations and Young tableaux and Khovanov-Kuperberg's bijection between …

Weak Parallelogram Laws On Banach Spaces And Applications To Prediction, R. Cheng, William T. Ross

Department of Math & Statistics Faculty Publications

This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.

Statistical Analysis Of The Variability And Reliability Of Eye-Tracking Test In Measuring Mild Traumatic Brain Injury, Xi He

Honors Theses

Saccadic eye-tracking tests have been advocated as a useful tool to distinguish mTBI patients from healthy people. However, intra-individual variances sometimes interfere with the interpretation of eye-tracking results, especially in experiments when group size is restricted. This study analyzes eye-tracking results of 14 mTBI patients taking the test twice with no medical administration in between. Using more accurate models to fit each individual's result, variables such as asymptote (of the fit func­tions) and hypothetical values for peak velocity, peak acceleration, and duration are derived for variability analysis. We conclude that the asymptotes for peak velocity and peak acceleration are the …

A Reduced Set Of Moves On One-Vertex Ribbon Graphs Coming From Links, Heather M. Russell, Susan Abernathy, Cody Armond, Moshe Cohen, Oliver T. Dasbach, Hannah Manuel, Chris Penn, Neal W. Stoltzfus

Department of Math & Statistics Faculty Publications

Every link in R³ can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.

Promoting Reu Participation From Students In Underrepresented Groups, Heather M. Russell, Heather A. Dye

Department of Math & Statistics Faculty Publications

Research experiences for undergraduates (REUs) are an important component of undergraduate education. However, at the 2012 Trends in Undergraduate Research in the Mathematical Sciences conference, questions were raised about why many REU programs see few applications from students that are members of underrepresented groups. We examine the benefits of REUs and factors preventing or promoting participation in REUs.

Stability And Bifurcation Of Equilibria For The Axisymmetric Averaged Mean Curvature Flow, Jeremy Lecrone

Department of Math & Statistics Faculty Publications

We study the averaged mean curvature ow, also called the volume preserving mean curvature ow, in the particular setting of axisymmetric surfaces embedded in R3 satisfying periodic boundary conditions. We establish analytic well-posedness of the ow within the space of little-Holder continuous surfaces, given rough initial data. We also establish dynamic properties of equilibria, including stability, instability, and bifurcation behavior of cylinders, where the radius acts as a bifurcation parameter.

C*-Algebras Generated By Truncated Toeplitz Operators, William T. Ross, Stephan Ramon Garcia, Warren R. Wogen

Department of Math & Statistics Faculty Publications

We obtain an analogue of Coburn’s description of the Toeplitz algebra in the setting of truncated Toeplitz operators. As a byproduct, we provide several examples of complex symmetric operators which are not unitarily equivalent to truncated Toeplitz operators having continuous symbols.

Bad Boundary Behavior In Star Invariant Subspaces I, William T. Ross, Andreas Hartmann

Department of Math & Statistics Faculty Publications

We discuss the boundary behavior of functions in star invariant subspaces (BH²)¹, where B is a Blaschke product. Extending some results of Ahern and Clark, we are particularly interested in the growth rates of functions at points of the spectrum of B where B does not admit a derivative in the sense of Carathéodory.

A Twisted Dimer Model For Knots, Heather M. Russell, Moshe Cohen, Oliver Dasbach

Department of Math & Statistics Faculty Publications

We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group.

Model Spaces: A Survey, William T. Ross, Stephan Ramon Garcia

Department of Math & Statistics Faculty Publications

This is a brief and gentle introduction, aimed at graduate students, to the subject of model subspaces of the Hardy space.

Power Distribution In The European Union, Dayton Steele

Honors Theses

The Treaty of Lisbon, the latest treaty governing law-making in the European Union (EU), was ratified in 2009 and goes fully into effect in 2014. This treaty, with its change to voting procedures in the Council of Ministers, claims to make decision-making in the EU more democratic and more efficient. Since the EU serves as an economic and political entity, we will assess these claims by comparing each member state's GDP and population to its power as modeled using the concept of a power index from the game theory literature. We will utilize the normalized Banzhaf index, the Shapley-Shubik index, …

Difference Sets In Non-Abelian Groups Of Order 256, Taylor Applebaum

Honors Theses

This paper considers the problem of determining which of the 56092 groups of order 256 contain (256; 120; 56; 64) difference sets. John Dillon at the National Security Agency communicated 724 groups which were still open as of August 2012. In this paper, we present a construction method for groups containing a normal subgroup isomorphic to Z4 Z4 Z2 . This construction method was able to produce difference sets in 643 of the 649 unsolved groups with the correct normal subgroup. These constructions elimated approximately 90% of the open cases, leaving 81 remaining unsolved groups.

Direct And Reverse Carleson Measure For Hb Spaces, William T. Ross, Alain Blandigneres, Emmanuel Fricain, Frederic Gaunard, Andreas Hartmann

Department of Math & Statistics Faculty Publications

In this paper we discuss direct and reverse Carleson measures for the de Branges-Rovnyak spaces H(b), mainly when b is a non-extreme point of the unit ball of H^∞.

Oddification Of The Cohomology Of Type A Springer Varieties, Heather M. Russell, Aaron D. Lauda

Department of Math & Statistics Faculty Publications

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are ‘odd’ analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient of the skew polynomial ring by the left ideal of nonconstant odd symmetric functions. The top degree component of the odd cohomology of Springer varieties is identifiedwith the …

An Explicit Bijection Between Semistandard Tableaux And Non-Elliptic Sl3 Webs, Heather M. Russell

Department of Math & Statistics Faculty Publications

The sl₃ spider is a diagrammatic category used to study the representation theory of the quantum group U_q(sl₃). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov- Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and joins of webs. We build on Tymoczko’s bijection to give a simple and explicit algorithm for constructing all …

Model Spaces: A Survey, William T. Ross, Stephan Ramon Garcia

Department of Math & Statistics Faculty Publications

This is a brief and gentle introduction, aimed at graduate students, to the subject of model subspaces of the Hardy space.

On Well-Posedness, Stability, And Bifurcation For The Axisymmetric Surface Diffusion Flow, Jeremy Lecrone, Gieri Simonett

Department of Math & Statistics Faculty Publications

We study the axisymmetric surface diffusion (ASD) flow, a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2+α)-little-Holder regular surfaces of revolution embedded in R3 and satisfying periodic boundary conditions. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability, and bifurcation behavior, with the radius acting as a bifurcation parameter.

Truncated Toeplitz Operators And Boundary Values In Nearly Invariant Subspaces, William T. Ross, Andreas Hartmann

Department of Math & Statistics Faculty Publications

We consider truncated Toeplitz operator on nearly invariant subspaces of the Hardy space H². Of some importance in this context is the boundary behavior of the functions in these spaces which we will discuss in some detail.

On A Theorem Of Livsic, William T. Ross, Alexandru Aleman, R. T. W. Martin

Department of Math & Statistics Faculty Publications

The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators such as Schrodinger operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions such as model subspaces of Hardy spaces, deBranges-Rovnyak spaces and Herglotz spaces, ordinary differential operators (including Schrodinger operators from quantum mechanics), Toeplitz operators, and infinite Jacobi matrices.

In this paper we develop a general representation theory of simple symmetric operators with equal deficiency indices, and …

Recent Progress On Truncated Toeplitz Operators, William T. Ross, Stephan Ramon Garcia

Department of Math & Statistics Faculty Publications

This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason’s seminal paper [88] from 2007.

Reverse Carleson Embeddings For Model Spaces, William T. Ross, Alain Blandigneres, Emmanuel Fricain, Frederic Gaunard, Andreas Hartmann

Department of Math & Statistics Faculty Publications

The classical embedding theorem of Carleson deals with finite positive Borel measures μ on the closed unit disk for which there exists a positive constant c such that for all f∈H², the Hardy space of the unit disk. Lefèvre et al. examined measures μ for which there exists a positive constant c such that for all f∈H². The first type of inequality above was explored with H² replaced by one of the model spaces (Θ H²)^⊥ by Aleksandrov, Baranov, Cohn, Treil, and Vol'berg. In this paper, we discuss …

An Extremal Problem For Characteristic Functions, William T. Ross, Isabelle Chalendar, Stephan Ramon Garcia, Dan Timotin

Department of Math & Statistics Faculty Publications

Suppose E is a subset of the unit circle T and H^∞C L^∞ is the Hardy subalgebra. We examine the problem of Finding the distance from the characteristic function of E to zⁿ H^∞. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.