State sum models for quantum gravity

This paper reviews the construction of quantum field theory on a 4-dimensional spacetime by combinatorial methods, and discusses the recent developments in the direction of a combinatorial construction of quantum gravity.

Dynamic Link Inclusion in Online PDF Journals

Two complementary de facto standards for the publication of electronic documents are HTML on theWorldWideWeb and Adobe s PDF (Portable Document Format) language for use with Acrobat viewers. Both these formats provide support for hypertext features to be embedded within documents. We present a method, which allows links and other hypertext material to be kept in an abstract form in separate link databases. The links can then be interpreted or compiled at any stage and applied, in the correct format to some specific representation such as HTML or PDF. This approach is of great value in keeping hyperlinks relevant, up-to-date and in a form which is independent of the finally delivered electronic document format. Four models are discussed for allowing publishers to insert links into documents at a late stage. The techniques discussed have been implemented using a combination of Acrobat plug-ins, Web servers and Web browsers.

Dynamic instabilities in scalar neural field equations with space-dependent delays

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These non-local models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg-Landau equations describing a Turing-Hopf bifurcation with modulation group velocity of O(1). Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin-Feir instabilities.