Harmonic Algebraic Curves and Noncrossing Partitions

Motivated by Gauss's first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that we call a basketball, consisting of a pair of noncrossing matchings satisfying one additional constraint. We prove that every noncrossing matching arises from some harmonic curve, and deduce from this that every basketball arises from some polynomial.     

One cannot hear orbifold isotropy type

We show that the isotropy types of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. Indeed, we construct arbitrarily large families of mutually isospectral orbifolds with different isotropy types. Finally, we show that the corresponding singular strata of two isospectral orbifolds may not be homeomorphic. Received: 6 October 2005     

Every group has a terminating transfinite automorphism tower

Iteratively taking the automorphism group of any group leads, transfinitely, to a fixed point.

     

A techno-economic assessment of the pretreatment and fractionation steps of a biomass-to-ethanol process

Applied Biochemistry and Biotechnology - It is generally recognized that the front-end (pretreatment, fractionation, enzymatic hydrolysis) steps of a lignocellulose-to-ethanol process are both...     

Existence of magnetic charge

A status report is presented on the existence of quarks carrying the Dirac unit of magnetic chargeg = (137/2)e. The Paschen-Back effect in dyonium is discussed. From the dyonium model, Akers predicted the existence of a new meson at 1814 MeV withI ^G(J^PC) = 0⁺(0^–+). Experimental evidence now confirms the existence of the meson resonance.     

On characterizing collections arising from N-gons in the plane

Consider a labeled convex n-gon in the plane and a point x in general position in the interior. We associate with this a collection (x) of 3-element sets, namely those 3-element subsets of the vertices which contain x in their interior. We investigate the problem of characterizing those collections of subsets which can be represented in this matter. We give a complete characterization of this and a related problem.School of Mathematics, University of Minnesota, Minneapolis, MN 55455. This work was supported in part by NSF grant DMS-8700995.Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England. Work done while Hill Visiting Professor at the University of Minnesota.