Past Differential Equations and Applications Seminar

The price we pay for this simplicity is that the change of variable

spinor --> coframe

makes the Dirac equation nonlinear. The morale of the talk is that, in our opinion, it is more natural to view the Dirac equation as a nonlinear equation for the unknown coframe rather than a linear equation for the unknown spinor.

 I will discuss the dynamics of synaptically coupled model neurons that undergo a form of accommodation in the presence of sustained activity. The basic model is an integral equation for synaptic activity that depends upon the non-local network connectivity, synaptic response, and firing rate of a single neuron. A phenomenological model of accommodation is examined whereby the firing rate is taken to be a simple state-dependent threshold function. As in the case without threshold accommodation classical Mexican-Hat connectivity is shown to allow for the existence of spatially localised states (bumps). Importantly an analysis of bump stability (in both one and two spatial dimensions) using recent Evans function techniques shows that bumps may undergo instabilities leading to the emergence of both breathers and travelling waves. Numerical simulations show that bifurcations in this model have the same generic properties as those seen in many other dissipative systems that support localised structures, and in particular those of coupled cubic complex Ginzburg-Landau equations, and three component reaction diffusion equations. Interestingly, travelling pulses in this model truly have a discrete character in the sense that they scatter as auto-solitons. /notices/events/abstracts/differential-equations/ht06/Coombes.shtml    

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.