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Non-Weyl Resonance Asymptotics for Quantum Graphs
We consider the resonances of a quantum graph that consists of a
compact part with one or more infinite leads attached to it. We discuss the
leading term of the asymptotics of the number of resonances of in
a disc of a large radius. We call a \emph{Weyl graph} if the
coefficient in front of this leading term coincides with the volume of the
compact part of . We give an explicit topological criterion for a
graph to be Weyl. In the final section we analyze a particular example in some
detail to explain how the transition from the Weyl to the non-Weyl case occurs.Comment: 29 pages, 2 figure
The Hecke Bicategory
We present an application of the program of groupoidification leading up to a
sketch of a categorification of the Hecke algebroid --- the category of
permutation representations of a finite group. As an immediate consequence, we
obtain a categorification of the Hecke algebra. We suggest an explicit
connection to new higher isomorphisms arising from incidence geometries, which
are solutions of the Zamolodchikov tetrahedron equation. This paper is
expository in style and is meant as a companion to Higher Dimensional Algebra
VII: Groupoidification and an exploration of structures arising in the work in
progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which
introduces the Hecke bicategory in detail.Comment: 27 pages, 11 .eps figures, Major revision of expositio
On Arithmetic Series involving the fractional part function
We present new relationships between the work of H. Davenport and A. I.
Popov. A new general formula involving the Von Mangoldt function is presented,
as well as a criteria for the Riemann Hypothesis
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