2,560 research outputs found
Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators
The operator and its trace are investigated in the case when 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 of the heat trace as . As
in the smooth compact case, the problem is reduced to the investigation of the
resolvent . The main step will consist in approximating this
operator family by a parametrix to 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
Resolvents of cone pseudodifferential operators, asymptotic expansions and applications
We study the structure and asymptotic behavior of the resolvent of elliptic
cone pseudodifferential operators acting on weighted Sobolev spaces over a
compact manifold with boundary. We obtain an asymptotic expansion of the
resolvent as the spectral parameter tends to infinity, and use it to derive
corresponding heat trace and zeta function expansions as well as an analytic
index formula.Comment: 30 pages, 5 figure
On the Noncommutative Residue and the Heat Trace Expansion on Conic Manifolds
Given a cone pseudodifferential operator we give a full asymptotic
expansion as of the trace \Tr Pe^{-tA}, where is an elliptic
cone differential operator for which the resolvent exists on a suitable region
of the complex plane. Our expansion contains and new
terms whose coefficients are given explicitly by means of residue traces. Cone
operators are contained in some natural algebras of pseudodifferential
operators on which unique trace functionals can be defined. As a consequence of
our explicit heat trace expansion, we recover all these trace functionals.Comment: 15 page
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