### Some Potential MMath Projects for 03/04

Ergodic Pumping (taken): Simulate the following caricature of what I
believe to be
the operation of myosin in muscle. 3 types of particle on an interval [0,1]:
1 piston of mass M eg 10, 1 driver of mass 1, N thermalisers of two masses
mu1, mu2, eg 0.1,0.2, temperature T of order 1. All bounce off the ends of
the interval. The thermalisers bounce
off the driver and piston if relative speed is less than a threshold of
order sqrt(T/mu), else pass through. The driver bounces off the piston.
Thermalisers of the same mass pass through each other, thermalisers of
different masses bounce off each other if relative speed is less than
a threshold of order sqrt(T/mu), else pass through each other.
The piston is subject to an external force -F. The initial configuration is
D=eps/2, P=eps small, with zero velocities, and the thermalisers uniformly
distributed with velocities from a Maxwellian at temperature T, i.e. normal
with variance T/mu.
Bayesian function fitting: If a function x: R to R is believed to be of the
form x(t) = Re sum_j a_j e^{iw_jt} for some number J of terms and some w_j,
a_j, with some distribution, and you are given y_n = x(t_n+d_n) + e_n for
some t_n with errors d_n and e_n from some distribution, then find the
posterior distribution for J, the a_j and the w_j. Test it on some simple
cases, eg J=1 or 2.

Configurations of particles: In many problems, eg celestial
mechanics, protein modelling, one is interested in configurations of N
particles in d-dimensional (usually d=3) Euclidean space modulo isometries.
Lagrange showed that the space of such configurations is isomorphic to the
space of nonnegative quadratic forms in (N-1) variables (think of them as
neutral
charge distributions on the N particles) of rank at most d. See whether this
representation can be used to do something useful, eg find minimum energy
configurations, fit protein structure to NMR data...

Triple Linkage: Engineering have built me a realisation of a mechanical
system with three disks connected by three rods that I proved to be chaotic
in a good sense. Test this experimentally. For example, in theory the
three angles should carry out a Brownian motion (on scales large compared
to 2pi).
Greg King (Engineering) could probably provide assistance with setting up
observation apparatus.

Alloy structures: Study numerically which configurations of two atoms with
the same crystal structure but different natural bond lengths can take on
a crystal lattice (e.g. Si, Ge). Start with a toy model, eg triangluar
lattice in 2D.

Turbine blades: Get information, eg from Rolls Royce, about the numbers
N and M of
rotors and stators they use in jet engines or gas turbines and think what
would be optimal from various points of view. For example, they should not be
the same, else you'd get strong oscillation in the torque (and sound
generation) at frequency
Nw (w=rotation frequency), but probably they should not be too different
in order to generate torque. Probably they should be chosen close but
coprime.

Robot walking: Simulate a model of a bipedal robot walking down a slope
and search for periodic orbits and calculate their stability.