Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators

The operator $e^{-tA}$ and its trace are investigated in the case when $A$ is a non-self-adjoint elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter-ellipticity) we obtain a full asymptotic expansion in $t$ of the heat trace as $t\to 0^+$. As in the smooth compact case, the problem is reduced to the investigation of the resolvent $(A-\lambda)^{-1}$. The main step will consist in approximating this operator family by a parametrix to $A-\lambda$ using a suitable parameter-dependent calculus.Comment: 35 pages. Final version to appear in Math. Nachrichten. The paper has been improved. Section 4 has been rewritten and simplifie

Hecke operators on rational functions

We define Hecke operators U_m that sift out every m-th Taylor series coefficient of a rational function in one variable, defined over the reals. We prove several structure theorems concerning the eigenfunctions of these Hecke operators, including the pleasing fact that the point spectrum of the operator U_m is simply the set {+/- m^k, k in N} U {0}. It turns out that the simultaneous eigenfunctions of all of the Hecke operators involve Dirichlet characters mod L, giving rise to the result that any arithmetic function of m that is completely multiplicative and also satisfies a linear recurrence must be a Dirichlet character times a power of m. We also define the notions of level and weight for rational eigenfunctions, by analogy with modular forms, and we show the existence of some interesting finite-dimensional subspaces of rational eigenfunctions (of fixed weight and level), whose union gives all of the rational functions whose coefficients are quasi-polynomials.Comment: 35 pages, LaTe

On cyclic numbers and an extension of Midy's theorem

In this note we consider fractions of the form 1/m and their floating-point representation in various arithmetic bases. For instance, what is 1/7 in base 2005? And, what about 1/4? We give a simple algorithm to answer these questions. In addition, we discuss an extension of Midy's theorem whose proof relies on elementary modular arithmetic.Comment: 6 pages, aimed at undergraduate student

Adjoints of elliptic cone operators

We study the adjointness problem for the closed extensions of a general b-elliptic operator A in x^{-\nu}Diff^m_b(M;E), \nu>0, initially defined as an unbounded operator A:C_c^\infty(M;E)\subset x^\mu L^2_b(M;E)\to x^\mu L^2_b(M;E), \mu \in \R. The case where A is a symmetric semibounded operator is of particular interest, and we give a complete description of the domain of the Friedrichs extension of such an operator.Comment: 40 pages, LaTeX, preliminary versio

Final Focus System for a Muon Collider: A Test Model

The present scenario for a high luminosity 4 TeV on center of mass muon collider requires a beta function =3 mm at the interaction point. We discuss a test model of a basic layout which satisfies the requirements although it is not fully realistic.Comment: 9 pages, uses REVTEX macros. Submitted to the Proceedings of the Symposium on Physics Potential and Development of mu^+-mu^- Colliders, San Francisco, CA. Suppl. of the journal Nuclear Physics