31,265 research outputs found
Variational properties of the discrete Hilbert-Einstein functional
This is a survey on rigidity and geometrization results obtained with the
help of the discrete Hilbert-Einstein functional, written for the proceedings
of the "Discrete Curvature" colloquium in Luminy.Comment: 10 page
On Fermat's principle for causal curves in time oriented Finsler spacetimes
In this work, a version of Fermat's principle for causal curves with the same
energy in time orientable Finsler spacetimes is proved. We calculate the
secondvariation of the {\it time arrival functional} along a geodesic in terms
of the index form associated with the Finsler spacetime Lagrangian. Then the
character of the critical points of the time arrival functional is investigated
and a Morse index theorem in the context of Finsler spacetime is presented.Comment: 20 pages, minor corrections, references adde
On the Minimum Ropelength of Knots and Links
The ropelength of a knot is the quotient of its length and its thickness, the
radius of the largest embedded normal tube around the knot. We prove existence
and regularity for ropelength minimizers in any knot or link type; these are
curves, but need not be smoother. We improve the lower bound for the
ropelength of a nontrivial knot, and establish new ropelength bounds for small
knots and links, including some which are sharp.Comment: 29 pages, 14 figures; New version has minor additions and
corrections; new section on asymptotic growth of ropelength; several new
reference
On the exit time from a cone for random walks with drift
We compute the exponential decay of the probability that a given
multi-dimensional random walk stays in a convex cone up to time , as
goes to infinity. We show that the latter equals the minimum, on the dual cone,
of the Laplace transform of the random walk increments. As an example, our
results find applications in the counting of walks in orthants, a classical
domain in enumerative combinatorics.Comment: 21 pages, 2 figures, to appear in Revista Matem\'atica Iberoamerican
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