## Research Seminars in the School of Mathematical Sciences

The School of Mathematical Sciences host regular research seminars delivered by internal and external researchers on topics across mathematical sciences. Seminars take place on the Kevin Street campus and are open to all.

## Seminar 23/04/15: Robust numerical methods

Robust numerical methods

Eugene O'Riordan

Dublin City University

Thursday 23 April 2015

1pm, Room KA3-011, DIT Kevin Street

## Seminar: 12/02/15: Mathematical Modelling of optical patterning in holographic systems

Mathematical Modelling of optical patterning in holographic systems

Paul O'Reilly

Dublin Institute of Technology

Thursday 12 February 2015

1pm, Room KA3-011, DIT Kevin Street

## Seminar 20/11/14: Three Problems in Matrix Theory

Three Problems in Matrix Theory

Anthony Cronin

UCD

Thursday 20 November 2014

1pm, Room KA3-011, DIT Kevin Street

Abstract:

In this talk I will outline three problems still outstanding in Matrix Theory.

**(1) The Nonnegative Inverse Eigenvalue Problem (NIEP).**

Given a list of n complex numbers the NIEP is the quest for necessary and sufficient conditions so that this list is the list of eigenvalues (spectrum) of an nxn nonnegative (entry wise) matrix.

**(2) Integral Similarity of Matrices**

Given two matrices A and B with integer entries can we decide if these are integrally similar i.e. can we find an invertible matrix P with integer entries such that P^{-1}AP=B?

**(3) The Pascal Matrix**

I will consider the problem of finding the multiplicative order of the matrix S_{n} which is derived from the Pascal matrix and give some new results on this order.

## Seminar 11/12/14: Schroedinger operators with slowly decaying Wigner-von Neumann potentials

Schroedinger operators with slowly decaying Wigner-von Neumann potentials

Sergey Simonov

Dublin Institute of Technology

Thursday 11 December 2014

1pm, Room KA3-011, DIT Kevin Street

Abstract:

We consider the Schroedinger operator on the half-line with a periodic background potential and a perturbation which consists of two parts: the slowly decaying Wigner-von Neumann potential csin(2ωx+d)/x^{γ} with γ from (½,1) and a summable part. The continuous spectrum of such an operator has the band-gap structure inherited from the free periodic operator, and in each band of the absolutely continuous spectrum there are two points where the operator can have eigenvalues, because at these points the eigenfunction equation has square summable solutions. These points are called critical. Existence of an embedded eigenvalue is very unstable: it disappears under a change of the boundary condition as well as under a local change of the potential. The talk is devoted to the result that in the generic case of no embedded eigenvalue the spectral density of the operator has a zeroes of the exponential type at the critical points. This extends our earlier result with Sergey Naboko for the case γ=1. In that case zeroes of the spectral density have the power type. Zeroes of the spectral density divide the continuous spectrum into parts and thus can be called pseudogaps.

## Seminar 06/11/14: A point-wise convergent method for the singularly perturbed Burgers equation

A point-wise convergent method for the singularly perturbed Burgers equation

Jason Quinn

Dublin Institute of Technology

Thursday 6 November 2014

1pm, Room KA3-011, DIT Kevin Street

Abstract:

The speaker will describe numerical methods for singularly perturbed differential equations. These are differential equations in which the highest derivative is multiplied by an arbitrary small quantity. Solutions exhibit steep gradients over parts of the problem domain (called layers) which may occur at the boundary or in the interior. The speaker will also present some of his latest results on a point-wise convergent method for Burger’s equation. The solution has an interior layer at some initially unknown point. Knowledge of this point is crucial to the successful design of a numerical method. Asymptotic expansions of this point exist. By using *close* approximations to the first two terms, we can establish a point-wise convergent numerical method for Burger’s equation. However, the use of the word *close* is open to scrutiny (and computer power!) and will, in practice, restrict the problem data under study. In the spirit of keeping the presentation light, numerical analysis details will be omitted and we will keep the discussion focused on those first two terms of the asymptotic expansion. Lots of pictures and case examples will ensue.