10,954 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
Shape Outlier Detection and Visualization for Functional Data: the Outliergram
We propose a new method to visualize and detect shape outliers in samples of
curves. In functional data analysis we observe curves defined over a given real
interval and shape outliers are those curves that exhibit a different shape
from the rest of the sample. Whereas magnitude outliers, that is, curves that
exhibit atypically high or low values at some points or across the whole
interval, are in general easy to identify, shape outliers are often masked
among the rest of the curves and thus difficult to detect. In this article we
exploit the relation between two depths for functional data to help visualizing
curves in terms of shape and to develop an algorithm for shape outlier
detection. We illustrate the use of the visualization tool, the outliergram,
through several examples and asses the performance of the algorithm on a
simulation study. We apply them to the detection of outliers in a children
growth dataset in which the girls sample is contaminated with boys curves and
viceversa.Comment: 27 pages, 5 figure
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
Factorization of quadratic polynomials in the ring of formal power series over
We establish necessary and sufficient conditions for a quadratic polynomial
to be irreducible in the ring of formal power series with integer
coefficients. For and prime, we show that is reducible in if and only if it is reducible in , the
ring of polynomials over the -adic integers.Comment: 15 page
Dynamics on Grassmannians and resolvents of cone operators
The paper proves the existence and elucidates the structure of the asymptotic
expansion of the trace of the resolvent of a closed extension of a general
elliptic cone operator on a compact manifold with boundary as the spectral
parameter tends to infinity. The hypotheses involve only minimal conditions on
the symbols of the operator. The results combine previous investigations by the
authors on the subject with an analysis of the asymptotics of a family of
projections related to the domain. This entails a fairly detailed study of the
dynamics of a flow on the Grassmannian of domains.Comment: 34 page
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