Heterofullerenes: Structure and property predictions,possible uses and synthesis proposals

Summary Substituting carbon atoms of fullerenes by heteroatoms and vacancies will lead to new and yet unknown spherical-shaped molecules termed hereafter as heterofullerenes. The enormous structural diversity of these molecules is investigated and their structural, electronic and thermochemical properties are predicted using semiempirical computations. Computational results for complexes with ions lead to the hypothesis that these molecules behave like microscopic Faraday cages in which the electrons concentrate on the outer side of the sphere. It is predicted that some of these heterofullerenes are structurally and electronically similar to phthalocyanines and related molecules but offer many additional advantages. Potential uses such as adding heterofullerenes to fullerene materials, as superior starting materials for the fabrication of diamonds, as catalysts in hydrogenation reactions, as components of materials dominated until now by phthalocyanines, etc., are discussed. Simple synthetic routes to these compounds that are based on minor alternations of existing methods for fullerene production are proposed. On the basis of the thermochemical calculations, we believe that the most promising possibility consists of using metal cyanide/graphite composite target rods instead of pure graphite rods as in a conventional fullerene synthesis.     

The Apollonius problem in four-dimensional Laguerre planes

The number of circles of a four-dimensional locally compact Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.     

An interior-point affine-scaling trust-region method for semismooth equations with box constraints

An algorithm for the solution of a semismooth system of equations with box constraints is described. The method is an affine-scaling trust-region method. All iterates generated by this method are strictly feasible. In this way, possible domain violations outside or on the boundary of the box are avoided. The method is shown to have strong global and local convergence properties under suitable assumptions, in particular, when the method is used with a special scaling matrix. Numerical results are presented for a number of problems arising from different areas.     

Reflection groups on Riemannian manifolds

We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds. Mathematics Subject Classification Primary 51F15, 53C20, 20F55, 22E40     

Solving non-linear portfolio optimization problems with the primal-dual interior point method

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.     

Half-regular and regular points in compact polygons

We investigate the derived structures of compact polygons at half-regular and at regular points. This enables us to give a geometric characterization of the real or complex split Cayley hexagon.