2,394 research outputs found
On the local structure of the Klein-Gordon field on curved spacetimes
This paper investigates wave-equations on spacetimes with a metric which is
locally analytic in the time. We use recent results in the theory of the
non-characteristic Cauchy problem to show that a solution to a wave-equation
vanishing in an open set vanishes in the ``envelope'' of this set, which may be
considerably larger and in the case of timelike tubes may even coincide with
the spacetime itself. We apply this result to the real scalar field on a
globally hyperbolic spacetime and show that the field algebra of an open set
and its envelope coincide. As an example there holds an analog of Borchers'
timelike tube theorem for such scalar fields and hence, algebras associated
with world lines can be explicitly given. Our result applies to cosmologically
relevant spacetimes.Comment: 17 pages, LaTeX, minor corrections, figures adde
TU Graz: Course: 707.000 Web Science and Web Technology: Lecture 2: Small World Problem
We will discuss several examples and research efforts related to the small world problem and set the ground for our discussion of network theory and social network analysis.
Readings: An Experimental Study of the Small World Problem, J. Travers and S. Milgram Sociometry 32 425-443 (1969) [Protected Access]
Optional: The Strength of Weak Ties, M.S. Granovetter The American Journal of Sociology 78 1360--1380 (1973) [Protected Access]
Optional: Worldwide Buzz: Planetary-Scale Views on an Instant-Messaging Network, J. Leskovec and E. Horvitz MSR-TR-2006-186. Microsoft Research, June 2007. [Web Link, the most recent and comprehensive study on the subject!]
Originally from: http://kmi.tugraz.at/staff/markus/courses/SS2008/707.000_web-science
Semi-Bounded Restrictions of Dirac Type Operators and the Unique Continuation Property
Let M be a connected Riemannian manifold and let D be a Dirac type operator
acting on smooth compactly supported sections in a Hermitian vector bundle over
M. Suppose D has a self-adjoint extension A in the Hilbert space of
square-integrable sections. We show that any -section contained in
a closed A-invariant subspace onto which the restriction of A is semi-bounded
has the unique continuation property: if vanishes on a non-empty open
subset of M, then it vanishes on all of M.Comment: 10 pages, LaTeX, minor corrections, reference adde
Explicit bounds on eigenfunctions and spectral functions on manifolds hyperbolic near a point
We derive explicit bounds for the remainder term in the local Weyl law for
locally hyperbolic manifolds, we also give the estimates of the derivative of
this remainder. We use these to obtain explicit bounds for the C^k-norms of the
L^2-normalised eigenfunctions in the case spectrum of the Laplacian is
discrete, e.g. for closed Riemannian manifolds. We also derive bounds for the
local heat trace. Our estimates are purely local and therefore also hold for
any manifold at points near which the metric is locally hyperbolic.Comment: 23 pages, 2 figure
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