Past

As an alternative to floating-point, several workers have proposed the use

of a logarithmic number system, in which a real number is represented as a

fixed-point logarithm. Multiplication and division therefore proceed in

minimal time with no rounding error. However, the system can only offer an

overall advantage if addition and subtraction can be performed with speed

and accuracy at least equal to that of floating-point, but this has

hitherto been difficult to achieve. We will present a number of original

techniques by which this has now been accomplished. We will then

demonstrate by means of simulations that the logarithmic system offers

around twofold improvements in speed and accuracy, and finally will

describe a new European collaborative project which aims to develop a

logarithmic microprocessor during the next three years.

We show that Sorensen's (1992) implicitly restarted Arnoldi method

(IRAM) (including its block extension) is non-stationary simultaneous

iteration in disguise. By using the geometric convergence theory for

non-stationary simultaneous iteration due to Watkins and Elsner (1991)

we prove that an implicitly restarted Arnoldi method can achieve a

super-linear rate of convergence to the dominant invariant subspace of

a matrix. We conclude with some numerical results the demonstrate the

efficiency of IRAM.

It has been known for some while now that every radial basis function

in common usage for multi-dimensional interpolation has associated with

it a uniquely defined Hilbert space, in which the radial basis function

is the `minimal norm interpolant'. This space is usually constructed by

utilising the positive definite nature of the radial function, but such

constructions turn out to be difficult to handle. We will present a

direct way of constructing the spaces, and show how to prove extension

theorems in such spaces. These extension theorems are at the heart of

improved error estimates in the $L_p$-norm.

We will examine how the various notions of partition regularity change as we change the ambient space. A typical question would be as follows. We say that the system of equations $Ax=b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax=b$. Rado proved that the system $Ax=b$ is partition regular if and only if it has a constant solution. What happens if the integers are replaced by the rationals, or the reals, or a more general ring? 

No previous knowledge of partition regularity is assumed. This is based on joint work with Paul Russell and joint work with Ben Barber, Neil Hindman and Dona Strauss.