All seminars (unless otherwise stated) will take place on Mondays at 3.00 pm in Room 500 which is located on the 5th floor of the Mathematics Department. See Where to Find Us for further details. There will be tea afterwards in room 606.

If you require any more information on the Applied seminars please contact Professor Yaroslav Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

## 11 January 2010 - CANCELLED DUE TO WEATHER

Dr. M. Bruni - University of Portsmouth

##### Dark Side Dynamics: UDM models with fast transition

Abstract:

In this talk I first discuss some general properties of Unified Dark Matter (UDM) models. Using as an example models with an affine equation of state (EoS) I show how in general UDM models need to be extremely close to the standard LCDM model in order to be viable. The main problem is that the effective speed of sound - governing the Jeans length of the perturbations - needs to be extremely small, otherwise perturbations become incompatible with CMB and matter power spectra. In the second part of the talk I show how this problem can be avoided by UDM models with a fast transition. I will then introduce a phenomenological toy model and show that - for a wide range of parameter values - UDM models with a fast transition can fit the CMB and matter power spectra.

## 18 January 2010

Dr. M.Dafermos - University of Cambridge

##### The black hole stability problem

Abstract:

This talk will review our current understanding of waves on black hole backgrounds, and the intimate connection of this with the non-linear stability problem of black holes in general relativity.

## 25 January 2010

Prof. M. Tierz - Brandeis University, USA

##### On Random matrices and Schur polynomials

Abstract:

We study a Gaussian random matrix ensemble with hyperbolic interactions and show how to solve it with q-orthogonal polynomials. We also briefly discuss the relevance of the model in topological gauge theory and its connections with non-intersecting Brownian motion. The average of Schur polynomials over the random matrix ensemble will be studied in detail as it is deeply related to classical results in combinatorics, like the hook-content formula and the Jacobi-Trudi identity.

## 01 February 2010

Dr. D.Tseluiko - Loughborough University

##### Complex spatio-temporal dynamics on falling liquid films

Abstract:

We analyse the coherent-structure interaction and the formation of bound states in active dispersive dissipative nonlinear media using a falling viscous film as a prototype. The coherent structures in this case are localised pulses that dominate the evolution of the film. We study experimentally the interaction dynamics on a film coating a vertical fibre and show evidence for formation of bound states. A theoretical explanation is provided through a coherent-structures theory of a simple model for the flow.

## 08 February 2010

Dr. G. Vilensky - UCL

##### Interaction of nonlinear ultrasound waves with soft biological tissue

Abstract:

The work examines a new approach to modelling of nonlinear sound propagation in the media with continuously distributed relaxation times. The main feature of the proposed theory is that it avoids the use of a potentially infinite number of relaxation equations for a given set of distinct relaxation frequencies. Instead these are replaced by a single evolution equation of the Boltzmann type. The structure of its right-hand side depends on the adopted irreversible thermodynamic model and contains an appropriate linear combination of the time derivatives of the macroscopic thermodynamic parameters such as the density and the entropy. The rheological coefficients appearing before these functions are expressed in terms of the absorption coefficient as a function of the wave frequency. Since the dependency of absorption coefficient on the sound frequency is, normally, available from the related experimental observations, the above rheological coefficients can be recovered directly from the experimental data and causality considerations.

The other important feature of the model is that it does not place any restrictions on the character of the nonlinearity of the equation of state of the medium.The proposed systems of equations are considered from the standpoint of their application to ultrasound propagation in the soft tissue. In order to clarify the relative orders of magnitude of the experimental parameters occurring in the theoretical model, the use is made of the limiting case when the wave field is only slightly perturbed from the plane wave form, so that it is almost one-dimensional, and its transverse variations to the propagation direction are small. The nonlinearity is assumed weak and of the same order as the dissipation and the effects due to the transverse perturbations. In conventional nonlinear acoustics this theoretical model is known as the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation or parabolic approximation. In a wider physical context the same approach results in a range of other remarkable nonlinear wave equations, such as the Kadomtsev-Petviashvili equation.

The work extends KZK analysis to the case of the medium with continuously distributed relaxations. To the author's knowledge, this extension has not been reported in the literature. The resultant nonlinear wave equation is shown to contain an extra "memory" term which is a fractional derivative of the total pressure with respect to the retarded time. As a result, the extended KZK equation takes the form of the Volterra integral equation as is typically the case for physical systems with memory.

## 15 February 2010

READING WEEK - NO SEMINAR

## 22 February 2010

Prof. F. W. Hehl - Cologne and Columbia, MO

On the change in form of Maxwell's equations during the last 150 years - spotlights on the history of classical electrodynamics

Abstract:

Starting with Maxwell's equations for the electromagnetic field (1865), we first point out how Maxwell brought his system of equations into quaternionic form. Subsequently, we recognize that what we call

Maxwell's equation nowadays is a creation of Heaviside and Hertz. We touch the development of vector calculus (Hamilton, Grassmann, Gibbs, Foeppl) and of tensor calculus (Riemann, Christoffel, Ricci, Levi-Civita) both around 1900. Then we study the impact of special and of general relativity on Maxwell's equations. In particular we follow up the metric-free and topological version of Maxwell's equations via exterior differential forms and period integrals. Some alternative formulations via spinors, Clifford algebras, chains and cochains... are mentioned.

## 01 March 2010

Dr. Marco Bruni - University of Portsmouth

##### Dark Side Dynamics: UDM models with fast transition

Abstract:

In this talk I first discuss some general properties of Unified Dark Matter (UDM) models. Using as an example models with an affine equation of state (EoS) I show how in general UDM models need to be extremely close to the standard LCDM model in order to be viable. The main problem is that the effective speed of sound - governing the Jeans length of the perturbations - needs to be extremely small, otherwise perturbations become incompatible with CMB and matter power spectra. In the second part of the talk I show how this problem can be avoided by UDM models with a fast transition. I will then introduce a phenomenological toy model and show that - for a wide range of parameter values - UDM models with a fast transition can fit the CMB and matter power spectra.

## 08 March 2010

Prof. Evgeny Ferapontov - Loughborough

##### Soliton equations in 2+1 dimensions: Deformations of dispersionless limits

Abstract:

I will discuss a procedure to reconstruct dispersive terms for (2+1)D dispersionless integrable systems. It is based on the requirement that all hydrodynamic reductions of the dispersionless system are `inherited' by the corresponding dispersive counterpart.