All seminars (unless otherwise stated) will take place on **Mondays at 3.00pm in ****Bentham SB01 Seminar Room 3** which is located in Bentham House. Please click here for a map. There will be tea afterwards in Maths Room 606, which is located on the 6th floor of the Mathematics Department. See How to Find Us for further details. If you require any more information on the Applied seminars please contact Dr Ilia Kamotski e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937.

## 14 January 2013

Euan Spence - University of Bath

##### Is the Helmholtz equation really sign-indefinite?

Abstract:

The Helmholtz equation is arguably the most basic model of linear wave propagation, and so has been the subject of vast amounts of research. The standard variational formulations of the Helmholtz equation are sign-indefinite (i.e. not coercive), and this indefiniteness impacts both the analysis and the numerical analysis of the Helmholtz equation.

In the literature, one often finds this sign-indefiniteness attributed to the Helmholtz equation itself, with papers often including phrases like "the Helmholtz equation is highly indefinite". In this talk I will present novel, sign-definite formulations of the Helmholtz equation, and thus argue that, whereas the standard variational formulations of the Helmholtz equation are sign-indefinite, this indefiniteness is *not* a feature of the Helmholtz equation itself, only its standard formulations.

This talk is based on joint work with Simon Chandler-Wilde (Reading), Ivan Graham (Bath), Ilia Kamotski (UCL), Andrea Moiola (Reading), and Valery Smyshlayev (UCL).

## 21 January 2013

Richard Craster - Imperial College London

##### Title: The asymptotics of waveguides and their connection to homogenisation theory

Abstract:

This is a talk about the connection between asymptotic methods for waveguides, that identify trapped modes, with those of microstructured media. In the latter case a high frequency homogenization is developed and applied to topical problems from Physics.

## 28 January 2013

Julius Kaplunov - Keele University

##### Title: Asymptotic analysis of initial value problems for thin elastic plates

Abstract:

3D initial value problems for thin elastic plates are subject to asymptotic analysis. Arbitrary variation of initial data through the thickness is assumed. Hierarchy of iteration procedures is established. The initial conditions for various 2D plate models are derived.

## 04 February 2013

Marco Marletta - University of Cardiff

##### Approximating spectra of self-adjoint and dissipative operators

Abstract:

We consider various methods for approximating the spectra of self-adjoint or dissipative differential operators, including projection methods, Glazman decomposition and a new abstract version of the complex barrier method.

This is joint work with Rob Scheichl and Michael Strauss.

## 11 February 2013

**READING WEEK - NO SEMINAR**

## 18 February 2013

Simon Chandler-Wilde - Reading

##### High frequency acoustic scattering by screens: computation and analysis

Abstract:

We address, in the study of acoustic scattering by 2D and 3D planar screens, three inter-related and challenging questions. Each of these questions focuses particularly on the formulation of these problems as boundary integral equations. The first question is, roughly, does it make sense to consider scattering by screens which occupy arbitrary open sets in the plane, and do different screens cause the same scattering if the open sets they occupy have the same closure? This question is intimately related to rather deep properties of fractional Sobolev spaces on general open sets, and the capacity and Haussdorf dimension of their boundary. The second question is, roughly, that, in answering the first question, can we understand explicitly and quantitatively the influence of the frequency of the incident time harmonic wave? It turns out that we can, that the problems have variational formations with sesquilinear forms which are bounded and coercive on fractional Sobolev spaces, and that we can determine explicitly how continuity and coercivity constants depend on the frequency. The third question is: can we design computational methods, adapted to the highly oscillatory solution behaviour at high frequency, which have computational cost essentially independent of the frequency? The answer here is that in 2D we can provably achieve solutions to any desired accuracy using a number of degrees of freedom which provably grows only logarithmically with the frequency, and that it looks promising that some extension to 3D is possible. This is joint work with Dave Hewett, Steve Langdon, and Ashley Twigger, all at Reading.

## 25 February 2013

Dr. Hugo Touchette - QMUL

##### Title: An introduction to large deviation theory and some of its recent applications

Abstract:

The theory of large deviations, initiated by Cramer (1930s) and developed by Donsker and Varadhan (1970s), is concerned with the exponential decay of probabilities of fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance and engineering, as they yield valuable information about the large fluctuations of random systems around their most probable state or trajectory. In the context of physics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This talk will explore this and other connections between large deviation theory and statistical physics to show that much of the structure of statistical physics can be expressed in the language of large deviations. Many applications related to equilibrium and nonequilibrium systems will be discussed to illustrate this point.

## 04 March 2013

Prof. Colin Rogers - University of New South Wales

##### Title: On q-Gaussian Gasdynamic and Magnetogasdynamic Systems. Integrable Hamiltonian Reductions.

Abstract:

Integrable substructure in gasdynamic and magnetogasdynamics systems is investigated via a general elliptic vortex ansatz. Certain universal and Hamiltonian aspects of the admitted representations are uncovered. Thermodynamically consistent relations are obtained for which the systems with a q-Gaussian density distribution admit reduction to integrable systems of Dyson-type in 3+1-dimensions and to integrable systems of Ermakov-Ray-Reid type in 2+1-dimensions. In the latter case, in the conducting context, the Ermakov components of this nonlinear system describe the time-evolution of the semi-axes of the elliptic cylinder within which the magnetogasdynamics motion is confined.

## 11 March 2013

Gregory Vilensky - UCL

##### Biochemical hydrodynamics of protein solutions with high pressure relaxation kinetics

Abstract:

The work aims to formulate a rational approach to problems of biochemical hydrodynamics of protein - solvent systems. The common feature of such problems is the existence of the myriad of simultaneous biochemical reactions with a broadband distribution of reaction rates. Normally, the kinetic events can be initiated by the disturbance of the system from the state of thermodynamic equilibrium either via the perturbation of the pressure, the temperature or the chemical content of the solution. This work considers the case of pressure induced relaxation kinetics.

Typically, biochemical studies do not consider the effects of interaction between hydrodynamic motions and kinetics. Being primarily concerned with calculations of the reaction rates and other important characteristics of the involved biochemical processes, they represent the fluid state by means of effective steadystate values. However, when the relaxation times of the involved microscopic kinetics occupy a broadband interval, hydrodynamic and kinetic time scales may become comparable. In this situation the hydrodynamic state cannot be treated as passive to the kinetic changes. Neither can the effects of kinetics on the hydrodynamics be schematically represented by a small number of relaxation equations, since this would contradict the broadband nature of the involved kinetics. Consequently, one enters into a new area of hydrodynamics in which modelling of the interaction between the broadband protein kinetics and nonequilibrium fluid motion is required. This work illustrates the distinctive features of this new class of problems by considering the practically important example of pressure initiated fast relaxation kinetics in protein solutions.

Owing to the broadband distribution of the reaction rates, the approach to modelling based on a large number of ordinary differential equations of reaction kinetics is impractical as it results in a stiff system of governing equations. An alternative method is to use continuous distributions of relaxation times, in order to account for the kinetic effects. This approach results in a compact set of equations similar to the conventional Navier-Stokes-Fourier system, but with additional terms responsible for the production of entropy owing to the biochemical changes of the medium's structure.

The work analyzes one such methodology from the point of view of novel hydrodynamic effects caused by pressure induced fast protein kinetics which would not be observed, if conventional equations of fluid mechanics were used..

## 18 March 2013

Alexander Movchan - Liverpool University

##### Title: Meso-scale approximations to solutions of boundary value problems in domains with multiple perforations

Abstract:

Asymptotic approximations are derived for solutions of mixed boundary value problems for the Laplacian in a three-dimensional domain with many perforations of arbitrary shape, with the Neumann boundary conditions being prescribed on the surfaces of small voids. The only assumption made on the geometry is that the diameter of a void is assumed to be smaller compared to the distance to the nearest neighbor. The asymptotic approximation, obtained here, involves a linear combination of dipole fields constructed for individual voids, with the coefficients, which are determined by solving a linear algebraic system. The energy estimate is obtained for the remainder term of the asymptotic approximation. Examples and numerical simulations, based on the meso-scale approximations, are discussed.

Generalisations to problems of elasticity in composite media are considered as well. This talk is based on the joint work with Vladimir Maz'ya and Michael Nieves.