6,336 research outputs found
Feynman loops and three-dimensional quantum gravity
This paper explores the idea that within the framework of three-dimensional
quantum gravity one can extend the notion of Feynman diagram to include the
coupling of the particles in the diagram with quantum gravity. The paper
concentrates on the non-trivial part of the gravitational response, which is to
the large momenta propagating around a closed loop. By taking a limiting case
one can give a simple geometric description of this gravitational response.
This is calculated in detail for the example of a closed Feynman loop in the
form of a trefoil knot. The results show that when the magnitude of the
momentum passes a certain threshold value, non-trivial gravitational
configurations of the knot play an important role.
The calculations also provide some new information about a limit of the
coloured Jones polynomial which may be of independent mathematical interest.Comment: approx 14 pages. v2: minor descriptive changes and added refs. v3:
minor correction
Projection methods in conic optimization
There exist efficient algorithms to project a point onto the intersection of
a convex cone and an affine subspace. Those conic projections are in turn the
work-horse of a range of algorithms in conic optimization, having a variety of
applications in science, finance and engineering. This chapter reviews some of
these algorithms, emphasizing the so-called regularization algorithms for
linear conic optimization, and applications in polynomial optimization. This is
a presentation of the material of several recent research articles; we aim here
at clarifying the ideas, presenting them in a general framework, and pointing
out important techniques
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
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