Flat regular polytopes

At the centre of the theory of abstract regular polytopes lies the amalgamation problem: given two regularn-polytopesP ₁ andP ₂, when does there exist a regular (n+1)-polytopeP whose facets are isomorphic toP ₁ and whose vertex-figures are isomorphic toP ₂? The most general circumstances known hitherto which lead to a positive answer involve flat polytopes, which are such that each vertex lies in each facet. The object of this paper is to describe an analogous but wider class of constructions, which generalize the previous results.     

Equivelar polyhedral manifolds inE 3

An equivelar polyhedral 2-manifold in the classM _p,q is one embedded inE ³ in which every face is a convexp-gon and every vertex isq-valent. Various constructions for equivelar manifolds are described, and it is shown that, in certain classesM _p,q, there is a manifold of given genusg≧2 for all but finitely manyg.     

Braiding of the attractor and the failure of iterative algorithms

Research partially supported by NSF Grant 8120790     

Positron trapping at dislocations in metals

The trapping rate of positrons at dislocations in metals, and its temperature dependence, are calculated. Two different trapping processes, with the excess energy absorbed in either electron-hole pair formation or by phonon creation, are considered and the former is found to be the most important. An extension of the theory to include depletion of the positron density around the dislocations in a diffusion approximation is included. The trapping is found to be transition limited if the temperature is low or the trap potential shallow. At room temperature diffusion is important for deep traps.     

Effects of an intense electric field on metal surface geometry

The effects of a strong positive electric field on the surface geometry of a single crystal, smooth metal surface are discussed. Explicit density functional calculations are presented for the field induced relaxation and evaporation of the Al(111) surface layer of atoms. The results are then explained in terms of a simple model and approximately extended to other metal surfaces. Our model determines the critical field for surface evaporation if there is no reconstruction. Finally, we will briefly mention the possibility of field induced reconstruction and its role in determining the critical field for stripping the surface layer of atoms.     

Tris-[2.21]- and tris-[2.2.2]-σ-homobenzenes

Tris-[2.2.1]- and tris-[2.2.2]-σ-homobenzenes are synthesised by reaction of 1,1′-bicyclobutenyl with appropriate dienophiles and hydrogenation of the adducts.     

On a problem of klee concerning convex polytopes

A non-simpliciald-polytope is shown to have strictly fewerk-faces ([(d−1)/2]≦k≦d−1) then some simpliciald-polytope with the same number of vertices; actual numerical bounds are given. This provides a strong affirmative answer to a problem of Klee.     

Polytopes with centrally symmetric facets

A new and conceptually simpler proof is given of the theorem of A. D. Aleksandrov and G. C. Shephard, that ad-polytope (d≧3), all of whose facets are centrally symmetric, is itself centrally symmetric.     

Braid groups and Hodge theory

This paper gives an account of the unitary representations of the braid group that arise via the Hodge theory of cyclic branched coverings of ${\mathbb{P}^1}$ , highlighting their connections with ergodic theory, complex reflection groups, moduli spaces of 1-forms and open problems in surface topology.