Roland K. W. Roeder — Page 2

arXiv2006

On Poincaré's Fourth and Fifth Examples of Limit Cycles at Infinity

Errors are found in example problems from Henri Poincaré's paper ``Mémoire sur les courbes définies par une équation différentielle.'' Examples four and five from chapter seven and examples one, two, and three from chapter nine do not have the limit cycles at infinity predicted by Poincaré. Instead they have fixed points at every point at infinity. In order to understand the errors made by Poincaré, examples four and five are studied at length. Replacement equations for the fourth and fifth examples are suggested based on the supposition that terms were omitted from Poincaré's equations.

math.DS↗

arXiv2006

Compact hyperbolic tetrahedra with non-obtuse dihedral angles

Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of $\mathbb{R}^E$ \cite{DIAZ}. If $C$ has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angles $A_C$ obtained by restricting to {\em non-obtuse} angles is a convex polytope. In this paper we explain why Andreev did not consider tetrahedra, the only polyhedra having fewer than five faces, by demonstrating that the space of dihedral angles of compact hyperbolic tetrahedra, after restricting to non-obtuse angles, is non-convex. Our proof provides a simple example of the ``method of continuity'', the technique used in classification theorems on polyhedra by Alexandrow \cite{ALEX}, Andreev \cite{AND}, and Rivin-Hodgson \cite{RH}.

math.GT↗