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

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

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 $L^2$-section $\phi$ contained in a closed A-invariant subspace onto which the restriction of A is semi-bounded has the unique continuation property: if $\phi$ 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