## 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 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 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.

## 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 9/3/18: A numerical study of in-plane wave propagation in mooring cables

A numerical study of in-plane wave propagation in mooring cables

Biswajit Basu

Trinity College Dublin

Friday 9 March 2018

2.15pm, Blue Room, 4th floor, Main Building, DIT Kevin Street

Abstract:

This talk focuses on wave propagation in mooring cables. For cables in ocean applications, there are three sources of nonlinearities affecting the cable response: the hydrodynamic effect, geometric stiffening effect, and seabed effect. To this end, numerical solutions are pursued, where the partial differential equations describing the dynamics of the submerged mooring cable are formulated in Lagrangian coordinates and solved using space-time finite difference methods. Specifically, a two-dimensional cable is considered and the wave propagation is studied when the cable is subjected to harmonic excitation at the fairlead. The wave propagation with seabed effects is also compared to that in a fully suspended cable. The nonlinear mechanisms governing the propagation of waves are discussed based on the numerical results. Further, some results on hydrodynamic analysis using Lagrangian based smooth particle hydrodynamics (SPH) are presented. Comparisons are made with some recent results and the impact of nonlinear hydrodynamics on offshore applications is emphasized.

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