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For a given surface in three-dimensional Euclidean space, a reflectivity function on the surface, and a distribution of light sources, Lambertian rendering generates an image: a function on a square. Inverting Lambertian rendering is an ill-posed problem. We prove that by using partial motion derivatives of images of the surface, it is possible to make the inverse Lambertian rendering problem well-posed, provided the illumination distribution or the surface reflectivity are known. Although the inverse rendering equations are non-local and nonlinear, we show that under certain conditions they can be solved explicitly in a closed form. The general results are illustrated in the example of a sphere. If two surfaces are available, then the full inverse rendering problem is well-posed.

AMS classification scheme numbers: 45K05, 45G10

We consider flows on compact orientable two-dimensional manifolds all points of which are non-wandering. An inter-relation between so-called Conley-Lyapunov-Peixoto graphs of such flows and Cayley graphs of finite groups is clarified. We describe all finite groups that are `good candidates' to symmetry groups of the flows in question. An application of the above results to the integrability problem of Hamiltonian flows is suggested. We conclude with two questions which might be of interest to group theorists.

In this paper we extend Shishikura's result on the Hausdorff dimension of the boundary of the Mandelbrot set to higher-dimensional parameter spaces, for example the space of degree d polynomials, and show that some parameter subsets, including the boundary of the connectedness locus, have Hausdorff dimensions equal to the real dimension of the parameter space (which is four for cubic polynomials).

PACS number: 0545

Systems of ordinary differential equations modelling coupled cells with `wreath product' coupling have been the subject of recent research. For identical cells, such systems can have interesting symmetries. The basic existence theorem for Hopf bifurcation in the symmetric case is the equivariant Hopf theorem, which involves isotropy subgroups with a two-dimensional fixed-point subspace (called -axial). A classification theorem for -axial subgroups in wreath products has been presented by Dionne et al. However, their classification is incomplete: it omits some -axial subgroups in some cases. We provide a complete classification of the -axial subgroups in wreath products. We also classify the maximal isotropy subgroups for these groups.

Vortices or merons are the relevant topological solitons in a two-dimensional isotropic Heisenberg antiferromagnet immersed in a uniform magnetic field. The dynamics of such solitons is studied numerically within the discrete spin model as well as analytically within a continuum approximation based on a suitable extension of the nonlinear model. Vortex dynamics is affected rather profoundly by the applied field and acquires the characteristic features of the Hall effect of electrodynamics or the Magnus effect of fluid dynamics. In particular, a single vortex is always spontaneously pinned, two like vortices form a rotating bound state, and a vortex-antivortex pair undergoes Kelvin motion.

The time-dependent Ginzburg-Landau equations of superconductivity define a dynamical process when the applied magnetic field varies with time. Sufficient conditions (in terms of the time rate of change of the applied field) are given that, if satisfied, guarantee that this dynamical process is asymptotically autonomous. As time goes to infinity, the dynamical process asymptotically approaches a dynamical system whose attractor coincides with the omega-limit set of the dynamical process.

Asymptotically stable attractors supporting an invariant measure, for which the ergodic theorem holds almost everywhere with respect to Lebesgue measure, can be approximated by a space discretization procedure called Ulam's method. As an application of this result we propose the use of this method to approximate the `chaotic' attractors of flows in lower dimensions. A Monte Carlo implementation makes this feasible. The approximation method can be extended to attractors whose neighbourhoods contain positively invariant compact sets called blocks. Note that such attractors can fail to have open basins of attraction. When the attractor is uniquely ergodic, we also prove the weak convergence of the approximate measures constructed by the method and as an application, we show the weak convergence of Ulam's method for the logistic map at the Feigenbaum parameter value. More generally, using the work of Buescu and Stewart on transitive attractors of continuous maps, we prove weak convergence of the approximate measures and convergence of their supports to classes of Lyapounov stable attracting Cantor sets.

We develop a general, coordinate-free theory for the reduction of volume-preserving flows with a volume-preserving symmetry on three-manifolds. The reduced flow is generated by a one-degree-of-freedom Hamiltonian which is the generalization of the Bernoulli invariant from hydrodynamics. The reduction procedure also provides global coordinates for the study of symmetry-breaking perturbations. Our theory gives a unified geometric treatment of the integrability of three-dimensional, steady Euler flows and two-dimensional, unsteady Euler flows, as well as quasigeostrophic and magnetohydrodynamic flows.

This paper considers the fifth-order equation

for a certain class of nonlinearity f. A solitary-wave solution to this equation is a solution of the form , where c is a constant and r vanishes as . It is shown that the equation has at least one non-zero solitary-wave solution when c<0, and a countably infinite family of geometrically distinct solitary-wave solutions when c<0, .

For a fixed Hamiltonian H on the cotangent bundle of a compact manifold M and a fixed energy level e, we prove that the set of potentials on M such that the Hamiltonian flow of is Anosov, is the interior in the topology of the set of potentials such that the flow has no conjugate points.

A reduced periodic orbit is one which is periodic modulo a rigid motion. If such an orbit for the planar N-body problem is collision free then it represents a conjugacy class in the projective coloured braid group. Under a `strong force' assumption which excludes the original Newtonian potential we prove that in most conjugacy classes there is a collision-free reduced periodic solution to Newton's N-body equations. These are the classes that are `tied' in the sense of Gordon. We give explicit homological conditions which ensure that a class is tied. The method of proof is the direct method of the calculus of variations. For the three-body problem we obtain qualitative information regarding the shape of our solutions which leads to a partial symbolic dynamics.