# Research

## Classical Analysis

### Friday, April 22, 2011

Title: Analytic functions in Smirnov classes $$E^{p}$$, $$0 < p < \infty$$, with real boundary values
Speaker: Lisa De Castro
Time: 3:00pm–4:00pm
Place: PHY 120

#### Abstract

Let $$G$$ be a bounded, simply connected domain in the complex plane. We will classify the geometric characteristics of the boundary of $$G$$ that are sufficient for having non-constant analytic functions of Smirnov class $$E^{p}$$, $$p \geq 1$$, with real boundary values.

### Friday, April 15, 2011

Title: On zero distribution of some Hermite-Padé polynomials, Part III
Speaker: Evguenii Rakhmanov
Time: 4:00pm–5:00pm
Place: PHY 120

### Friday, April 8, 2011

Title: On zero distribution of some Hermite-Padé polynomials, Part II
Speaker: Evguenii Rakhmanov
Time: 4:00pm–5:00pm
Place: PHY 120

### Friday, April 1, 2011

Title: On zero distribution of some Hermite-Padé polynomials
Speaker: Evguenii Rakhmanov
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

We will derive the formula for limit zero distribution of polynomials of degree $$= 2n$$ simultaneously orthogonal to polynomials of degree $$< n$$ on two non overlapping intervals. The method is based on rather interesting combination of potential theory and theory of algebraic functions.

### Monday, March 21, 2011

Title: Quantum integrable models with elliptic $$R$$-matrices and elliptic hypergeometric series
Speaker: Anton Zabrodin, ITEP
Moscow, Russia
Time: 3:00pm–4:00pm
Place: NES 104

#### Abstract

We consider algebraic and representation theory basis for constructing quantum integrable spin chains with $$4\times 4$$ elliptic $$R$$-matrix. The algebra of observables in such models is known to be the Sklyanin algebra which is a special two-parameter deformation of $$\mathrm{gl}(2)$$. Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins $$\ell$$ and $$-\ell-1$$ are constructed using the technique of intertwining vectors for elliptic $$L$$-operator. They are expressed in terms of elliptic hypergeometric series with operator argument.

The intertwining operators obtained ($$W$$-operators) serve as building blocks for the $$R$$-matrix which intertwines tensor product of two $$L$$-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin. The Yang-Baxter equation for this $$R$$-matrix follows from simpler equations of the star-triangle type for the $$W$$-operators. A natural graphic representation of the objects and equations involved in the construction is used.

### Friday, March 11, 2011

Title: Minimal projections in $$l_\infty^n$$ and $$l_1^n$$ spaces, Part III
Speaker: Lesław Skrzypek
Time: 4:00pm–5:00pm
Place: PHY 120

### Friday, February 25, 2011

Title: Minimal projections in $$l_\infty^n$$ and $$l_1^n$$ spaces, Part II
Speaker: Lesław Skrzypek
Time: 4:00pm–5:00pm
Place: PHY 120

### Friday, February 18, 2011

Title: Minimal projections in $$l_\infty^n$$ and $$l_1^n$$ spaces
Speaker: Lesław Skrzypek
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

Finding minimal projections is usually very complex. We present a relatively simple method of determining minimal projections onto hyperplanes in $$l_\infty^n$$ and $$l_1^n$$ spaces. We construct the Chalmers-Metcalf operator for minimal projections onto hyperplanes in $$\ell_\infty^n$$ and $$\ell_1^n$$ and prove it is uniquely determined. We show how we can use Chalmers-Metcalf operator to obtain uniqueness of minimal projections. The main advantage of our approach is that it is purely algebraical and does not require consideration of the min-max problems.

### Friday, January 28, 2011

Title: Asymptotic Values and the Interaction Between Coefficient Conditions and Solution Conditions of Differential Equations in the Unit Disk
Speaker: Kari Fowler, University of Tampa
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

The interaction between the coefficients and solutions for linear differential equations and for Riccati differential equations is investigated in the unit disk in terms of their asymptotic values as described by the MacLane class. This interaction is further explored within the context of a meromorphic version of the MacLane class. Joint work with Linda Sons.

### Friday, January 21, 2011

Title: The sets of boundedness of entire functions
Speaker: Arthur Danielyan
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

In the theory of complex approximation on the general sets of the complex plane a significant difference between the natures of approximations by polynomials and approximations by entire functions occurs because of the lack of a certain boundedness principle for entire functions. However such a partial boundedness principle was suggested by Lee Rubel and the author. As its new application we give the complete description of the subsets of the complex plane on each of which an entire function can be bounded.

### Friday, January 14, 2011

Title: Universality of transitions at the point of gradient catastrophe for some integrable systems and some orthogonal polynomials
Speaker: Alex Tovbis, University of Central Florida
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

By a point of gradient catastrophe we mean a point where the leading order asymptotic behavior loses smoothness (for example, derivatives are not square integrable).

In the case of the focusing NLS (Nonlinear Schröedinger Equation), this is a point where a slowly modulated high frequency plane wave wave suddenly burst into rapid amplitude oscillation (spikes). Adjusting the nonlinear steepest descent (Deift-Zhou) method for Riemann-Hilbert problems, we give complete description of the leading order term near the point of gradient catastrophe in terms of the tritronquée solution to the Painlevé I and rational breathers for the NLS. In fact, each spike corresponds to a pole of the tritronquée and has the universal shape of a scaled rational breather.

A similar phenomenon was recently described for the asymptotic of orthogonal polynomials with a complex varying quartic weight. In this case the spikes in the asymptotics for the recurrence coefficients turn out to be unbounded.