The affine deformation space of a rank two Schottky group: a picture gallery

The Margulis invariant of an affine hyperbolic element measures the signed Lorentzian displacement along the unique closed geodesic in its class. Given a group of affine transformations of Minkowski spacetime whose linear part is Schottky, the Margulis invariant is a useful tool in determining properness of its action. This paper, based on a presentation given at CGG IV Oostende 2005, describes the affine deformation space of a rank two Schottky group. Several illustrations are included, contrasting the difference between the case of a three-holed sphere and that of a one-holed torus. Paper based on a presentation given at CGG IV Oostende 2005. Attendance of the conference and the subsequent writing of this paper was made possible by an NSERC Discovery Grant.     

Flat Chains in Banach Spaces

We extend the notion of a flat chain with coefficients in a normed abelian group from Euclidean space to an arbitrary Banach space and prove a compactness result. We also remove the condition that a flat chain with arbitrary coefficients have finite mass in order for its support to exist. This research was part of the author’s Ph. D. dissertation at Stanford University.     

Lorentz spacetimes of constant curvature

This paper is unpublished work of Geoffrey Mess written in 1990, which gives a classification of flat and anti-de Sitter domains of dependence in 2+1 dimensions.      

Surfaces in $${\mathbb{S}^2\times\mathbb{R}}$$ with a canonical principal direction

We show a way to choose nice coordinates on a surface in and use this to study minimal surfaces. We show that only open parts of cylinders over a geodesic in are both minimal and flat. We also show that the condition that the projection of the direction tangent to onto the tangent space of the surface is a principal direction, is equivalent to the condition that the surface is normally flat in . We present classification theorems under the extra assumption of minimality or flatness. J. Fastenakels is a research assistant of the Research Foundation—Flanders (FWO). J. Van der Veken is a postdoctoral researcher supported by the Research Foundation—Flanders (FWO). This work was partially supported by project G.0432.07 of the Research Foundation—Flanders (FWO).     

Surfaces with high contact number and their characterization

The notion of contact number of a Euclidean submanifold was introduced in an earlier article (Proc. Edinb. Math. Soc. 47:69–100, 2004) as the highest order of contact of geodesics and normal sections on the submanifold. It was proved in (Proc. Edinb. Math. Soc. 47:69–100, 2004) that the contact number relates closely with the notions of isotropic submanifolds and holomorphic curves. One important problem concerning contact number is to construct Euclidean submanifolds with high contact number. The purpose of this article is thus to construct Euclidean surfaces with high contact number and to provide simple geometric characterization of such surfaces. Mathematics Subject Classification (2000) Primary 53C40, 53A10 Secondary 53B25, 53C42     

Convolution with measures on flat curves in low dimensions

We prove Lp→Lq convolution estimates for the affine arclength measure on certain flat curves in Rd when d∈{2,3,4}. For d=2,3, we also establish certain related Lorentz space estimates.     

Exact number of solutions of stationary reaction-diffusion equations

We study both existence and the exact number of positive solutions of the problem

     

A new lead Mo(IV) phosphate with a tunnel structure: Pb2Mo2O(PO4)2P2O7

A molybdenum (IV) phosphate containing lead, Pb₂Mo₂(PO₄)2P₂O₇, has been synthesized for the first time. It crystallizes in the space group C2/c with a=14.098(1) Å, b=14.187(2) Å, c=6.5592(4) Å and β=102.08(1)°. Its original tunnel structure, built up of Mo₂O₁₁ bioctahedra, P₂O₇ and PO₄ phosphate groups can be described from the assemblage of [Mo₄P₄O₂₄]_∞ ribbons interconnected through monophosphate groups. The stereoactivity of the 6s² lone pair of Pb²⁺, which is surrounded by nine oxygen atoms, is discussed.     

On flatness properties of cyclic S-posets

In this paper, we discuss flatness properties of cyclic S-posets. As applications, some partially ordered monoids are characterized and some results on S-acts can be also obtained. Research supported by the National Natural Science Foundation of China (No.10626012).     

Flat Möbius strips of given isotopy type in <Emphasis Type="Italic">R</Emphasis><Superscript>3</Superscript> whose centerlines are geodesics or lines of curvature

We construct flat Möbius strips of arbitrary isotopy types, whose centerlines are geodesics or lines of curvature.