UCL

Much of the mathematical modelling of urban systems revolves around the use spatial interaction models, derived from information theory and entropy-maximisation techniques and embedded in dynamic difference equations. When framed in the context of a retail system, the

dynamics of centre growth poses an interesting mathematical problem, with bifurcations and phase changes, which may be analysed analytically. In this contribution, we present some analysis of the continuous retail model and corresponding discrete version, which yields insights into the effect of space on the system, and an understanding of why certain retail centers are more successful than others. This class of models turns out to have wide reaching applications: from trade and migration flows to the spread of riots and the prediction of archeological sites of interest, examples of which we explore in more detail during the talk.

We develop the idea of using Monte Carlo sampling of random portfolios to solve portfolio investment problems. We explore the need for more general optimization tools, and consider the means by which constrained random portfolios may be generated. DeVroye’s approach to sampling the interior of a simplex (a collection of non-negative random variables adding to unity) is already available for interior solutions of simple fully-invested long-only systems, and we extend this to treat, lower bound constraints, bounded short positions and to sample non-interior points by the method of Face-Edge-Vertex-biased sampling. A practical scheme for long-only and bounded short problems is developed and tested. Non-convex and disconnected regions can be treated by applying rejection for other constraints. The advantage of Monte Carlo methods is that they may be extended to risk functions that are more complicated functions of the return distribution, without explicit gradients, and that the underlying return distribution may be modeled parametrically or empirically based on general distributions. The optimization of expected utility, Omega, Sortino ratios may be handled in a similar manner to quadratic risk, VaR and CVaR, irrespective of whether a reduction to LP or QP form is available. Robustification is also possible, and a Monte Carlo approach allows the possibility of relaxing the general maxi-min approach to one of varying degrees of conservatism. Grid computing technology is an excellent platform for the development of such computations due to the intrinsically parallel nature of the computation. Good comparisons with established results in Mean-Variance and CVaR optimization are obtained, and we give some applications to Omega and expected Utility optimization. Extensions to deploy Sobol and Niederreiter quasi-random methods for random weights are also proposed. The method proposed is a two-stage process. First we have an initial global search which produces a good feasible solution for any number of assets with any risk function and return distribution. This solution is already close to optimal in lower dimensions based on an investigation of several test problems. Further precision, and solutions in 10-100 dimensions, are obtained by invoking a second stage in which the solution is iterated based on Monte-Carlo simulation based on a series of contracting hypercubes.

The process industries are one of the UK's major sectors and include

petrochemicals, pharmaceuticals, water, energy and the food industry,

amongst others. The design of a processing plant is a difficult task. This

is due to both the need to cater for multiple criteria (such as economics,

environmental and safety) and the use highly complex nonlinear models to

describe the behaviour of individual unit operations in the process. Early

in the design stages, an engineer may wish to use automated design tools to

generate conceptual plant designs which have potentially positive attributes

with respect to the main criteria. Such automated tools typically rely on

optimization for solving large mixed integer nonlinear programming models.

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This talk presents an overview of some of the work done in the Computer

Aided Process Engineering group at UCL. Primary emphasis will be given to

recent developments in hybrid optimization methods, including the use of

graphical interfaces based on problem specific visualization techniques to

allow the engineer to interact with embedded optimization procedures. Case

studies from petrochemical and water industries will be presented to

demonstrate the complexities involved and illustrate the potential benefits

of hybrid approaches.

How many triangles must a graph of density d contain? This old question due to Erdos was recently answered by Razborov, after many decades of progress by numerous authors.

We will consider the analogous question for tripartite graphs. Given a tripartite graph with prescribed edges densities between each

pair of classes how many triangles must it contain?

GIBSON BUILDING COMMON ROOM 2ND FLOOR

(Coffee and Cakes in Gibson Meeting Room - opposite common room)

The effects of electric fields on nonlinear free surface flows are investigated. Both inviscid and Stokes flows are considered.

Fully nonlinear solutions are computed by boundary integral equation methods and weakly nonlinear solutions are obtained by using long wave asymptotics and lubrication theory. Effects of electric fields on the stability of the flows are discussed. In addition applications to coating flows are presented.