We consider an anisotropic diffusion equation of the form ut = ∇∇(D(x)u) in two dimensions, which arises in various applications, including the modelling of wolf movement along seismic lines and the invasive spread of certain brain tumours along white matter neural fibre tracts. We consider a degenerate case, where the diffusion tensor D(x) has a zero-eigenvalue for certain values of x. Based on a regularisation procedure and various pointwise and integral a priori estimates, we establish the global existence of very weak solutions to the degenerate limit problem. Moreover, we show that in the large time limit these solutions approach profiles that exhibit a Dirac-type mass concentration phenomenon on the boundary of the region in which diffusion is degenerate, which is quite surprising for a linear diffusion equation. The results are illustrated by numerical examples.

In this note, we consider a nonlinear diffusion equation with a bistable reaction termarising in population dynamics. Given a rather general initial data, we investigate itsbehavior for small times as the reaction coefficient tends to infinity: we prove ageneration of interface property.

The oriented movement of biological cells or organisms in response to a chemical gradientis called chemotaxis. The most interesting situation related to self-organizationphenomenon takes place when the cells detect and response to a chemical which is secretedby themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) manyparticularized models have been proposed to describe the aggregation phase of thisprocess. Most of efforts were concentrated, so far, on mathematical models in which theformation of aggregate is interpreted as finite time blow-up of cell density. In recentlyproposed models cells are no more treated as point masses and their finite volume isaccounted for. Thus, arbitrary high cell densities are precluded in such description and athreshold value for cells density is a priori assumed. Different modelingapproaches based on this assumption lead to a class of quasilinear parabolic systems withstrong nonlinearities including degenerate or singular diffusion. We give a survey ofanalytical results on the existence and uniqueness of global-in-time solutions, theirconvergence to stationary states and on a possibility of reaching the density threshold bya solution. Unsolved problems are pointed as well.

Several problems of practical interest in robotics can be modelled as the convolution of functions on the Euclidean motion group. These include the evaluation of reachable positions and orientations at the distal end of a robot manipulator arm. A natural inverse problem arises when one wishes to design rather than to model manipulators. Namely, by considering a serial-chain robot arm as a concatenation of segments, we examine how statistics of known segments can be used to select, or design, the remainder of the structure so as to attain the desired statistical properties of the whole structure. This is then a deconvolution density estimation problem for the Euclidean motion group. We prove several results about the convergence of these deconvolution estimators to the true underlying density under certain smoothness assumptions. A practical implementation to the design of planar robot arms is demonstrated.

Proteus mirabilis are bacteria that make strikingly regular spatial-temporal patterns on agar surfaces.In this paper we investigate a mathematical model that has beenshown to display these structures when solved numerically. The model consists of an ordinarydifferential equation coupled with a partial differential equation involving a first-orderhyperbolic aging term together with nonlinear degenerate diffusion. The system is shown toadmit global weak solutions.

A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

We consider a Markov chain on the d-dimensional (d-allelî) non-negative lattice points with the sum of components being N, for which one-step transition consists of two stages—independent reproduction and random sampling. Convergence to a degenerate diffusion process when N → ∞ is proved. We show how difference among alleles in means and variances of offspring numbers affects the limit diffusion, giving a rigorous multi-allelic version of a result of Gillespie.