## 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 22/2/19: Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions

Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions

Adrian Rodriguez Sanjurjo

University College Cork

Friday 22 February 2019

1pm, Blue Room, 4th floor, Main Building, TU Dublin, Kevin Street, City Campus

Abstract:

The aim of this talk is to provide a rigorous although accessible analysis of some nonlinear surface waves in the presence of a depth-invariant zonal current constituting a generalisation of Pollard's waves. These oceanic water waves, which are nonlinear solutions of the geophysical f-plane equations accounting for rotational effects, have an exact and explicit representation in the Lagrangian framework. This is indeed a remarkable discovery; however, in order to rigorously show their mathematical validity, the three-dimensional Lagrangian flow-map prescribing these solutions needs to be a global diffeomorphism. This constitutes the main result of the talk and it will be proven by applying a mixture of analytical and degree-theoretical arguments and by imposing certain conditions on the physical and Lagrangian labelling parameters.

## Seminar 8/2/19: Integrability and chaos in figure skating

Integrability and chaos in figure skating

Prof Vakhtang Putkaradze

University of Alberta, Canada

Friday 8 February 2019

1pm, Blue Room, 4th floor, Main Building, TU Dublin Kevin Street Campus

Abstract:

We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories. We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

Joint work with Vaughn Gzenda (UofA). The work was supported by NSERC Discovery Grant program (VP), USRA (VG) and the University of Alberta. This talk has also been made possible by the awarding of a James M Flaherty Visiting Professorship from the Ireland Canada University Foundation, with the assistance of the Government of Canada/avec l’appui du gouvernement du Canada.

## Seminar 28/9/18: Handling missing network data

Handling missing network data

Robert Krause

University of Groningen

Friday 28 September 2018

2pm, Blue Room, 4th floor, Main Building, DIT Kevin Street

Abstract:

My work focuses on handling missing data in cross-sectional, longitudinal and multiplex networks. We focus on the development and evaluation of multiple imputation procedures for the current state of the art model families (Stochastic Actor-Oriented Models and Exponential Random Graph Models). In this talk I will give an overview about our work.

## Seminar 7/12/18: Infinite mixtures of infinite factor analysers (IMIFA)

Infinite Mixtures of Infinite Factor Analysers (IMIFA)

Isobel Claire Gormley

Insight Centre for Data Analytics, University College Dublin

Friday 7 December 2018

1pm, Blue Room, 4th floor, Main Building, DIT Kevin Street

Abstract:

Factor-analytic Gaussian mixture models are often employed as a model-based approach to clustering high-dimensional data. Typically, the numbers of clusters and latent factors must be specified in advance of model fitting, and the optimal pair selected using a model choice criterion. For computational reasons, models in which the number of latent factors is common across clusters are generally considered.

Here the infinite mixture of infinite factor analysers (IMIFA) model is introduced. IMIFA employs a Poisson-Dirichlet process prior to facilitate automatic inference on the number of clusters. Further, IMIFA employs shrinkage priors to allow cluster specific numbers of factors, automatically inferred via an adaptive Gibbs sampler. IMIFA is presented as the flagship of a family of factor-analytic mixture models, providing flexible approaches to clustering highdimensional data.

Applications to benchmark and real data sets illustrate the IMIFA model and its advantageous features: IMIFA obviates the need for model selection criteria, reduces model search and associated computational burden, improves clustering performance by allowing cluster-specific numbers of factors, and quantifies uncertainty in the numbers of clusters and cluster-specific factors.

## Seminar 27/4/18: On mathematical models for microelectromechanical systems

On mathematical models for microelectromechanical systems

Prof Joachim Escher

Leibniz University Hannover

Friday 27 April 2018

1pm, Blue Room, 4th floor, Main Building, DIT Kevin Street

Abstract:

A review of some recent results on mathematical models for microelectromechanical systems with general permittivity profile will be presented. These models consist of a quasilinear parabolic evolution problem for the displacement of an elastic membrane coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate.

Local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are addressed. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account — the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem.

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