1. TANGENT SPHERES. Any four mutually tangent spheres determine six points of tangency. We say that a pair of tangencies {ti , t j } is opposite if the two spheres determining ti are distinct from the two spheres determining t j . Thus the six tangencies are naturally grouped into three opposite pairs, corresponding to the three ways of partitioning the four spheres into two pairs. Altshiller-C...

Vertex-colorings, edge-colorings and total-colorings of the Sierpiński gasket graphs Sn, the Sierpiński graphs S(n, k), graphs S (n, k), and graphs S(n, k) are considered. In particular, χ′′(Sn), χ (S(n, k)), χ(S(n, k)), χ(S(n, k)), χ(S(n, k)), and χ(S(n, k)) are determined.

We describe a new hybrid approach to modeling and animating animals. A pre–defined skin is modeled as a triangle mesh, such as may be purchased from a digital model vendor, or generated with a typical modeling program. This skin is then attached to the underlying bone, muscle, and tissue model. The rest shape of the skin is exactly as given in the surface model of the animal. Internal component...

Partee (1986) claimed without proof that the function BE is the only homomorphism that makes the Partee triangle commute. This paper shows that this claim is incorrect unless “homomorphism” is understood as “complete homomorphism.” It also shows that BE and A are the inverses of each other on certain natural assumptions.

Representation varieties of closed surface groups into SL(n,R) have been studied extensively by Hitchin and Labourie, and the dimension of a certain distinguished component of the variety was obtained by Hitchin using Higgs bundles. Here we determine the corresponding dimension for representations of triangle groups into SL(n,R), generalising some earlier work of Choi and Goldman in the case n ...

We investigate asphericity of the relative group presentation 〈G, t | atbtctdtet = 1〉 and prove it aspherical provided the subgroup of G generated by {ab−1, bc−1, cd−1, de−1} is neither finite cyclic nor a finite triangle group. We also prove a similar result for the closely related relative group presentation 〈G, s, t | αsβsγt = 1 = δtεtζs−1〉.

This paper introduces a new system for classifying scholarly journals in terms of their degree of 'application orientation'. The method extends earlier models and journals classification systems that were designed to tackle the crude duality

For a given triangle there are many points associated with the triangle that lie in its interior; examples include the incenter (which can be found by the intersection of the angle bisectors) and the centroid (which can be found by the intersection of the medians). Using this point one can naturally subdivide the triangle into either three or six “daughter” triangles. We can then repeat the sam...