Department of Mathematics

 Analysis and Applications (AaA) Seminar at UCCS

This seminar is intended to have a very informal format, and welcomes faculty and grad students from the Pikes Peak region who are interested in contemporary research in analysis (of all sorts) and applications. Areas covered include but are not limited to real and complex analysis, functional analysis, harmonic analysis, ODEs and PDEs, mathematical physics and applications to nonlinear phenomena, numerical analysis, scientific computation and other fields (too many to enumerate). 

Fall 2017  
Fridays 2-3pm, (refreshments at 1:45pm)
ENG 101, UCCS campus

Sep 29 - Inaugural Seminar: Dr. Fritz Gesztesy, Baylor U

Oct 6 - Organizational meeting 
        Oct 13 - grad student seminar

        Oct 20 - grad student seminar
Oct 27 - Seminar Speaker: Dr. Radu Cascaval,  UCCS
        Nov 3 - grad student seminar
        Nov 10 - grad student seminar: encourage attendance to the Nonlinear Days Nov 11-12;
Nov 17 - Seminar Speaker: Dr. Oksana Bihun, UCCS 
        Nov 24 - Thanksgiving (no seminar)

        Dec 1 - grad student seminar
Dec 8 - Seminar Speaker: Dr. Barbara Prinari, UCCS

Please contact Dr. Radu Cascaval (radu@uccs.edu) if you are interested to join this seminar or need more info. Limited number of parking passes will be made available to non-UCCS individuals attending this seminar.


TITLES & ABSTRACTS:

Dec 8  Seminar

Speaker: Dr. Barbara Prinari, UCCS
Title: Solitons and rogue waves for a square matrix nonlinear Schrodinger equation with nonzero boundary conditions

Abstract: In this talk we discuss the Inverse Scattering Transform (IST) under nonzero boundary conditions for a square matrix nonlinear Schrodinger equation which has been proposed as a model to describe hyperfine spin F = 1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions, or attractive interatomic interactions and ferromagnetic spin-exchange interactions. Emphasis will be given to a discussion of the soliton and rogue wave solutions one can obtain as a byproduct of the IST.

Nov 17  Seminar

Speaker: Dr. Oksana Bihun, UCCS
Title: New properties of the zeros of Krall polynomials

Abstract: We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family p_n(x), we relate the zeros of the polynomial p_N with the zeros of p_m for each m <=N (the case m = N corresponding to the relations that involve the zeros of pN only). These identities are obtained by exacting the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalises known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new remarkable identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.

Oct 27  Seminar

Speaker: Dr. Radu C. Cascaval, UCCS
Title: What do Analysis and Scientific Computation have in common ...

Abstract: Analysis, the world of the infinitesimally small, is thought to be one of the last standing outposts where humans can fight the computational invasion. In spite of this fact, computational sciences continue to benefit greatly from advances in analysis. This talk will illustrate this relationship, in particular functional analysis connections to numerical spectral methods, meshless methods, and their applications to numerical solutions to PDEs.
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Sept 29 Seminar

Speaker: Dr. Fritz Gesztesy, Baylor University
Title: The eigenvalue counting function for Krein-von Neumann extensions of elliptic operators  (Slides)

Abstract: We start by providing a historical introduction into the subject of Weyl-asymptotics for Laplacians on bounded domains in n-dimensional Euclidean space, and a brief introduction into the basic principles of self-adjoint extensions.  Subsequently, we turn to bounds on eigenvalue counting functions and derive such a bound for Krein-von Neumann extensions corresponding to a class of uniformly elliptic second order PDE operators (and their positive integer powers) on arbitrary open, bounded, n-dimensional subsets \Omega in R^n. (No assumptions on the boundary of \Omega are made; the coefficients are supposed to satisfy certain regularity conditions.)  Our technique relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of the corresponding differential operator suitably extended to all of R^n. We also consider the analogous bound for the eigenvalue counting function for the corresponding Friedrichs extension.  This is based on joint work with M. Ashbaugh, A. Laptev, M. Mitrea, and S. Sukhtaiev.
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