Vincent Coll

Meander Graphs and Frobenius Seaweed Lie Algebras II

We provide a recursive classification of meander graphs, showing that each meander is identified by a unique sequence of fundamental graph theoretic moves. This sequence is called the meander’s signature and can be used to construct arbitrarily large sets of meanders, Frobenius or otherwise, of any size and configuration. In certain special cases, the signature is used to produce an explicit formula for the index of seaweed Lie subalgebra of sl(n) in terms of elementary functions.

Two Generalizations of a Property of the Catenary

Abstract A well-known property of the catenary curve is that the ratio of the area under the curve to the arc length of the curve is independent of the interval over which these quantities are concurrently measured. We develop two higher-dimensional generalizations of this invariant ratio, and find that each invariant ratio identifies a class of hypersurfaces connected to classical objects from differential geometry.

Gabriel's Horn: A Revolutionary Tale

Summary We generalize hypersurfaces of revolution by allowing the profile curve to be dependent on more than one variable. We call these more general objects spherical arrays and we use them to introduce the reader to volume calculations, which are normally reserved for a more advanced course in differential geometry. The rich symmetry of the spherical arrays allow us to build objects with properties reminiscent of those for which Gabriels Horn became a cause célèbre.

Integer solutions to box optimization problems

Summary Two approaches contribute to the problem of finding integer side-lengths for open rectangular and polygonal boxes that optimize volume. One method adapts Pythagorean triples to an ellipse, while the other is based on counting factorizations.

Enomoto and Ota’s Conjecture Holds for Large Graphs

In 2000, Enomoto and Ota conjectured that if a graph G satisfies

A Characteristic Averaging Property of the Catenary

\sigma _{2}(G) \ge |G| + k - 1

The proper diameter of a graph

σ2(G)≥|G|+k-1, then for any set of k vertices