1,602 research outputs found
Computing the Girth of a Planar Graph in Linear Time
The girth of a graph is the minimum weight of all simple cycles of the graph.
We study the problem of determining the girth of an n-node unweighted
undirected planar graph. The first non-trivial algorithm for the problem, given
by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and
Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further
reduced the running time to O(n log n). In this paper, we solve the problem in
O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin
Quantum Stress Tensor Fluctuations and their Physical Effects
We summarize several aspects of recent work on quantum stress tensor
fluctuations and their role in driving fluctuations of the gravitational field.
The role of correlations and anticorrelations is emphasized. We begin with a
review of the properties of the stress tensor correlation function. We next
consider some illuminating examples of non-gravitational effects of stress
tensors fluctuations, specifically fluctuations of the Casimir force and
radiation pressure fluctuations. We next discuss passive fluctuations of
spacetime geometry and some of their operational signatures. These include
luminosity fluctuations, line broadening, and angular blurring of a source
viewed through a fluctuating gravitational field. Finally, we discuss the
possible role of quantum stress tensor fluctuations in the early universe,
especially in inflation. The fluctuations of the expansion of a congruence of
comoving geodesics grows during the inflationary era, due to non-cancellation
of anticorrelations that would have occurred in flat spacetime. This results in
subsequent non-Gaussian density perturbations and allows one to infer an upper
bound on the duration of inflation. This bound is consistent with adequate
inflation to solve the horizon and flatness problems.Comment: 15 pages, 1 figure; invited talk presented at the 3rd Mexican Meeting
on Experimental and Theoretical Physics, Mexico City, September 10-14, 200
On Kernel Formulas and Dispersionless Hirota Equations
We rederive dispersionless Hirota equations of the dispersionless Toda
hierarchy from the method of kernel formula provided by Carroll and Kodama. We
then apply the method to derive dispersionless Hirota equations of the extended
dispersionless BKP(EdBKP) hierarchy proposed by Takasaki. Moreover, we verify
associativity equations (WDVV equations) in the EdBKP hierarchy from
dispersionless Hirota equations and give a realization of associative algebra
with structure constants expressed in terms of residue formula.Comment: 30 pages, minor corrections, references adde
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