13,749 research outputs found
Weak Field Expansion of Gravity: Graphs, Matrices and Topology
We present some approaches to the perturbative analysis of the classical and
quantum gravity. First we introduce a graphical representation for a global
SO(n) tensor (\pl)^d h_\ab, which generally appears in the weak field
expansion around the flat space: g_\mn=\del_\mn+h_\mn. Making use of this
representation, we explain 1) Generating function of graphs (Feynman diagram
approach), 2) Adjacency matrix (Matrix approach), 3) Graphical classification
in terms of "topology indices" (Topology approach), 4) The Young tableau
(Symmetric group approach). We systematically construct the global SO(n)
invariants. How to show the independence and completeness of those invariants
is the main theme. We explain it taking simple examples of \pl\pl h-, {and}
(\pl\pl h)^2- invariants in the text. The results are applied to the analysis
of the independence of general invariants and (the leading order of) the Weyl
anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10
dimensions).Comment: 41pages, 26 figures, Latex, epsf.st
Volume of representation varieties
We introduce the notion of volume of the representation variety of a finitely
presented discrete group in a compact Lie group using the push-forward measure
associated to a map defined by a presentation of the discrete group. We show
that the volume thus defined is invariant under the Andrews-Curtis moves of the
generators and relators of the discrete group, and moreover, that it is
actually independent of the choice of presentation if the difference of the
number of generators and the number of relators remains the same. We then
calculate the volume of the representation variety of a surface group in an
arbitrary compact Lie group using the classical technique of Frobenius and
Schur on finite groups. Our formulas recover the results of Witten and Liu on
the symplectic volume and the Reidemeister torsion of the moduli space of flat
G-connections on a surface up to a constant factor when the Lie group G is
semisimple.Comment: 27 pages in AMS-LaTeX forma
Improved bounds and algorithms for graph cuts and network reliability
Karger (SIAM Journal on Computing, 1999) developed the first fully-polynomial
approximation scheme to estimate the probability that a graph becomes
disconnected, given that its edges are removed independently with probability
. This algorithm runs in time to obtain an
estimate within relative error .
We improve this run-time through algorithmic and graph-theoretic advances.
First, there is a certain key sub-problem encountered by Karger, for which a
generic estimation procedure is employed, we show that this has a special
structure for which a much more efficient algorithm can be used. Second, we
show better bounds on the number of edge cuts which are likely to fail. Here,
Karger's analysis uses a variety of bounds for various graph parameters, we
show that these bounds cannot be simultaneously tight. We describe a new graph
parameter, which simultaneously influences all the bounds used by Karger, and
obtain much tighter estimates of the cut structure of . These techniques
allow us to improve the runtime to , our results also
rigorously prove certain experimental observations of Karger & Tai (Proc.
ACM-SIAM Symposium on Discrete Algorithms, 1997). Our rigorous proofs are
motivated by certain non-rigorous differential-equation approximations which,
however, provably track the worst-case trajectories of the relevant parameters.
A key driver of Karger's approach (and other cut-related results) is a bound
on the number of small cuts: we improve these estimates when the min-cut size
is "small" and odd, augmenting, in part, a result of Bixby (Bulletin of the
AMS, 1974)
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