We define an abelian group, the conformal cobordism group of hyperbolic structures, which classifies the hyperbolic structures according to whether it bounds a (higher dimensional) conformally flat structure in a conformally invariant way. We then construct a homomorphism from this group to the circle group, using the eta invariant. The homomorphism can be highly nontrivial. It remains an inter...

The official slogan of predicativism is what Russell called the “vicious circle principle”: no class can contain elements which are definable only in terms of the class itself. Another way to say this is that we do not accept any entity which cannot be defined without referring to a class to which it belongs. It is hard to make this principle precise, but the underlying sentiment is that the el...

We consider the problem of locating a spherical circle with respect to existing facilities on a sphere, such that the sum of distances between the circle and the facilities is minimized, or such that the maximum distance is minimized. The problem properties are analyzed, and we give solution procedures. When the circle to be located is restricted to be a great circle, some simplifications are p...

The Descartes circle theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or “bends”) bi = 1 ri satisfy the relation (b1 + b2 + b3 + b4) 2 = 2(b1 + b 2 2 + b 2 3 + b 2 4). We show that similar relations hold involving the centers of the four circles in such a configuration, coordinatized as complex numbers, yielding a complex Des...

We give a complete and explicit characterization of the connected graphs which admit a continuous 3-to-1 map onto the circle, and of the connected graphs which admit a continuous 2-to-l map onto the circle. This generalizes earlier work of Heath and Hilton who considered the mappings of trees onto the circle.