9,119 research outputs found
Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives
We consider the problem of finding commuting self-adjoint extensions of the
partial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domain
C_c^\infty(\Omega) where the self-adjointness is defined relative to
L^2(\Omega), and \Omega is a given open subset of R^d. The measure on \Omega is
Lebesgue measure on R^d restricted to \Omega. The problem originates with I.E.
Segal and B. Fuglede, and is difficult in general. In this paper, we provide a
representation-theoretic answer in the special case when \Omega=I\times\Omega_2
and I is an open interval. We then apply the results to the case when \Omega is
a d-cube, I^d, and we describe possible subsets \Lambda of R^d such that
{e^(i2\pi\lambda \dot x) restricted to I^d:\lambda\in\Lambda} is an orthonormal
basis in L^2(I^d).Comment: LaTeX2e amsart class, 18 pages, 2 figures; PACS numbers 02.20.Km,
02.30.Nw, 02.30.Tb, 02.60.-x, 03.65.-w, 03.65.Bz, 03.65.Db, 61.12.Bt,
61.44.B
The Measure of a Measurement
While finite non-commutative operator systems lie at the foundation of
quantum measurement, they are also tools for understanding geometric iterations
as used in the theory of iterated function systems (IFSs) and in wavelet
analysis. Key is a certain splitting of the total Hilbert space and its
recursive iterations to further iterated subdivisions. This paper explores some
implications for associated probability measures (in the classical sense of
measure theory), specifically their fractal components.
We identify a fractal scale in a family of Borel probability measures
on the unit interval which arises independently in quantum information
theory and in wavelet analysis. The scales we find satisfy and , some . We identify these
scales by considering the asymptotic properties of
where are dyadic subintervals, and .Comment: 18 pages, 3 figures, and reference
Spectral asymptotics of periodic elliptic operators
We demonstrate that the structure of complex second-order strongly elliptic
operators on with coefficients invariant under translation by
can be analyzed through decomposition in terms of versions ,
, of with -periodic boundary conditions acting on
where . If the semigroup generated by
has a H\"older continuous integral kernel satisfying Gaussian bounds then the
semigroups generated by the have kernels with similar properties
and extends to a function on which is
analytic with respect to the trace norm. The sequence of semigroups
obtained by rescaling the coefficients of by converges in
trace norm to the semigroup generated by the homogenization
of . These convergence properties allow asymptotic analysis of
the spectrum of .Comment: 27 pages, LaTeX article styl
Harmonic analysis of iterated function systems with overlap
In this paper we extend previous work on IFSs without overlap. Our method
involves systems of operators generalizing the more familiar Cuntz relations
from operator algebra theory, and from subband filter operators in signal
processing.Comment: 37 page
A statistical study of the global structure of the ring current
[1] In this paper we derive the average configuration of the ring current as a function of the state of the magnetosphere as indicated by the Dst index. We sort magnetic field data from the Combined Release and Radiation Effects Satellite (CRRES) by spatial location and by the Dst index in order to produce magnetic field maps. From these maps we calculate local current systems by taking the curl of the magnetic field. We find both the westward (outer) and the eastward (inner) components of the ring current. We find that the ring current intensity varies linearly with Dst as expected and that the ring current is asymmetric for all Dst values. The azimuthal peak of the ring current is located in the afternoon sector for quiet conditions and near midnight for disturbed conditions. The ring current also moves closer to the Earth during disturbed conditions. We attempt to recreate the Dst index by integrating the magnetic perturbations caused by the ring current. We find that we need to multiply our computed disturbance by a factor of 1.88 ± 0.27 and add an offset of 3.84 ± 4.33 nT in order to get optimal agreement with Dst. When taking into account a tail current contribution of roughly 25%, this agrees well with our expectation of a factor of 1.3 to 1.5 based on a partially conducting Earth. The offset that we have to add does not agree well with an expected offset of approximately 20 nT based on solar wind pressure
Construction of Parseval wavelets from redundant filter systems
We consider wavelets in L^2(R^d) which have generalized multiresolutions.
This means that the initial resolution subspace V_0 in L^2(R^d) is not singly
generated. As a result, the representation of the integer lattice Z^d
restricted to V_0 has a nontrivial multiplicity function. We show how the
corresponding analysis and synthesis for these wavelets can be understood in
terms of unitary-matrix-valued functions on a torus acting on a certain vector
bundle. Specifically, we show how the wavelet functions on R^d can be
constructed directly from the generalized wavelet filters.Comment: 34 pages, AMS-LaTeX ("amsproc" document class) v2 changes minor typos
in Sections 1 and 4, v3 adds a number of references on GMRA theory and
wavelet multiplicity analysis; v4 adds material on pages 2, 3, 5 and 10, and
two more reference
Essential selfadjointness of the graph-Laplacian
We study the operator theory associated with such infinite graphs as
occur in electrical networks, in fractals, in statistical mechanics, and even
in internet search engines. Our emphasis is on the determination of spectral
data for a natural Laplace operator associated with the graph in question. This
operator will depend not only on , but also on a prescribed
positive real valued function defined on the edges in . In electrical
network models, this function will determine a conductance number for each
edge. We show that the corresponding Laplace operator is automatically
essential selfadjoint. By this we mean that is defined on the dense
subspace (of all the real valued functions on the set of vertices
with finite support) in the Hilbert space . The
conclusion is that the closure of the operator is selfadjoint in
, and so in particular that it has a unique spectral resolution,
determined by a projection valued measure on the Borel subsets of the infinite
half-line. We prove that generically our graph Laplace operator
will have continuous spectrum. For a given infinite graph
with conductance function , we set up a system of finite graphs with
periodic boundary conditions such the finite spectra, for an ascending family
of finite graphs, will have the Laplace operator for as its limit.Comment: 50 pages with TOC and figure
Confirming EIS Clusters. Optical and Infrared Imaging
Clusters of galaxies are important targets in observationally cosmology, as
they can be used both to study the evolution of the galaxies themselves and to
constrain cosmological parameters. Here we report on the first results of a
major effort to build up a sample of distant galaxy clusters to form the basis
for further studies within those fields. We search for simultaneous
overdensities in color and space to obtain supporting evidence for the reality
of the clusters. We find a confirmation rate for EIS clusters of 66%,
suggesting that a total of about 80 clusters with z>=0.6 are within reach using
the EIS cluster candidates.Comment: 4 pages, 2 figures, to appear in the proceedings of the IGRAP
International Conference 1999 on 'Clustering at high Redshift
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