DH Common Room

The Riemann zeta function involves, for Re s>1, the summation of the inverse s-th powers of the integers. A class of zeta-like functions is obtained if the s-th powers of integers which contain specified digits are omitted from the summation. The numerical summation of such series, their convergence properties and analytic continuation are considered in this lecture.

In this talk, a network topology is presented that allows both peer-to-peer and non-local traffic in a cellular based TDD-CDMA system known as opportunity driven multiple access (ODMA). The key to offering appropriate performance of peer-to-peer communication in such a system relies on the use of a routing algorithm which minimises interference. This talk will discuss the constraints and limitations on the capacity of such a system using a variety of routing techniques. A congestion based routing algorithm will be presented that attempts to minimize the overall power of the system as well as providing a measure of feasibility. This technique provides the lowest required transmit power in all circumstances, and the highest capacity in nearly all cases studied. All of the routing algorithms considered allocate TDD time slots on a first come first served basis according to a set of pre-defined rules. This fact is utilised to enable the development of a combined routing and resource allocation algorithm for TDD-CDMA relaying. A novel method of time slot allocation according to relaying requirements is then developed.

Two measures of assessing congestion are presented based on matrix norms. One is suitable for a current interior point solution, the other is more elegant but is not currently suitable for efficient minimisation and thus practical implementation.

Moving hydrodynamic boundaries (waves and bubbles, for example) produce acoustic signatures.

It is by now well known that one cannot HEAR the shape of a

drum: There are many known examples of isospectral yet not isometric "drums". Recently we discovered that the sequences of integers formed by counting the nodal domains of successive eigenfunctions encode geometrical information, which can also be used to resolve spectral ambiguities. I shall discuss these sequences and indicate how the information stored in the nodal sequences can be deciphered.