Comlab

We calculate an analytic value for the correlation coefficient between a geometric, or exponential, Brownian motion and its time-average, a novelty being our use of divided differences to elucidate formulae. This provides a simple approximation for the value of certain Asian options regarding them as exchange options. We also illustrate that the higher moments of the time-average can be expressed neatly as divided differences of the exponential function via the Hermite-Genocchi integral relation, as well as demonstrating that these expressions agree with those obtained by Oshanin and Yor when the drift term vanishes.

The formation of steady patterns in one space dimension is generically

governed, at small amplitude, by the Ginzburg-Landau equation.

But in systems with a conserved quantity, there is a large-scale neutral

mode that must be included in the asymptotic analysis for pattern

formation near onset. The usual Ginzburg-Landau equation for the amplitude

of the pattern is then coupled to an equation for the large-scale mode.

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These amplitude equations show that for certain parameters all regular

periodic patterns are unstable. Beyond the stability boundary, there

exist stable stationary solutions in the form of spatially modulated

patterns or localised patterns. Many more exotic localised states are

found for patterns in two dimensions.

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Applications of the theory include convection in a magnetic field,

providing an understanding of localised states seen in numerical

simulations.

The lattice Boltzmann equation has been used successfully used to simulate

nearly incompressible flows using an isothermal equation of state, but

much less work has been done to determine stable implementations for other

equations of state. The commonly used nine velocity lattice Boltzmann

equation supports three non-hydrodynamic or "ghost'' modes in addition to

the macroscopic density, momentum, and stress modes. The equilibrium value

of one non-hydrodynamic mode is not constrained by the continuum equations

at Navier-Stokes order in the Chapman-Enskog expansion. Instead, we show

that it must be chosen to eliminate a high wavenumber instability. For

general barotropic equations of state the resulting stable equilibria do

not coincide with a truncated expansion in Hermite polynomials, and need

not be positive or even sign-definite as one would expect from arguments

based on entropy extremisation. An alternative approach tries to suppress

the instability by enhancing the damping the non-hydrodynamic modes using

a collision operator with multiple relaxation times instead of the common

single relaxation time BGK collision operator. However, the resulting

scheme fails to converge to the correct incompressible limit if the

non-hydrodynamic relaxation times are fixed in lattice units. Instead we

show that they must scale with the Mach number in the same way as the

stress relaxation time.

The trapezoidal rule for numerical integration is remarkably accurate when

the integrand under consideration is smooth and periodic. In this

situation it is superior to more sophisticated methods like Simpson's rule

and even the Gauss-Legendre rule. In the first part of the talk we

discuss this phenomenon and give a few elementary examples. In the second

part of the talk we discuss the application of this idea to the numerical

evaluation of contour integrals in the complex plane.

Demonstrations involving numerical differentiation, the computation

of special functions, and the inversion of the Laplace transform will be

presented.

Plane Couette flow - the flow between two infinite parallel plates moving in opposite directions -

undergoes a discontinuous transition from laminar flow to turbulence as the Reynolds number is

increased. Due to its simplicity, this flow has long served as one of the canonical examples for understanding shear turbulence and the subcritical transition process typical of channel and pipe flows. Only recently was it discovered in very large aspect ratio experiments that this flow also exhibits remarkable pattern formation near transition. Steady, spatially periodic patterns of distinct regions of turbulent and laminar flow emerges spontaneously from uniform turbulence as the Reynolds number is decreased. The length scale of these patterns is more than an order of magnitude larger than the plate separation. It now appears that turbulent-laminar patterns are inevitable intermediate states on the route from turbulent to laminar flow in many shear flows. I will explain how we have overcome the difficulty of simulating these large scale patterns and show results from studies of three types of patterns: periodic, localized, and intermittent.

Sparse grids yield numerical solutions to PDEs with a

significantly reduced number of degrees of freedom. The relative

benefit increases with the dimensionality of the problem, which makes

multi-factor models for financial derivatives computationally tractable.

An outline of a convergence proof for the so called combination

technique will be given for a finite difference discretisation of the

heat equation, for which sharp error bounds can be shown.

Numerical examples demonstrate that by an adaptive (heuristic)

choice of the subspaces European and American options with up to thirty

(and most likely many more) independent variables can be priced with

high accuracy.

The computation of flows of compressible fluids will be

discussed, exploiting the symmetric form of the equations describing

compressible flow.

A cell is a wonderously complex object. In this talk I will

give an overview of some of the mathematical frameworks that are needed

in order to make progress to understanding the complex dynamics of a

cell. The talk will consist of a directed random walk through discrete

Markov processes, stochastic differential equations, anomalous diffusion

and fractional differential equations.