809,043 research outputs found
An asymptotic sampling formula for the coalescent with Recombination
Ewens sampling formula (ESF) is a one-parameter family of probability
distributions with a number of intriguing combinatorial connections. This
elegant closed-form formula first arose in biology as the stationary
probability distribution of a sample configuration at one locus under the
infinite-alleles model of mutation. Since its discovery in the early 1970s, the
ESF has been used in various biological applications, and has sparked several
interesting mathematical generalizations. In the population genetics community,
extending the underlying random-mating model to include recombination has
received much attention in the past, but no general closed-form sampling
formula is currently known even for the simplest extension, that is, a model
with two loci. In this paper, we show that it is possible to obtain useful
closed-form results in the case the population-scaled recombination rate
is large but not necessarily infinite. Specifically, we consider an asymptotic
expansion of the two-locus sampling formula in inverse powers of and
obtain closed-form expressions for the first few terms in the expansion. Our
asymptotic sampling formula applies to arbitrary sample sizes and
configurations.Comment: Published in at http://dx.doi.org/10.1214/09-AAP646 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Closed formula for the relative entropy of entanglement
The long-standing problem of finding a closed formula for the relative
entropy of entanglement (REE) for two qubits is addressed. A compact-form
solution to the inverse problem, which characterizes an entangled state for a
given closest separable state, is obtained. Analysis of the formula for a large
class of entangled states strongly suggests that a compact analytical solution
of the original problem, which corresponds to finding the closest separable
state for a given entangled state, can be given only in some special cases. A
few applications of the compact-form formula are given to show additivity of
the REE, to relate the REE with the Rains upper bound for distillable
entanglement, and to show that a Bell state does not have a unique closest
separable state.Comment: 7 pages, the title was modified as suggested by the PRA editor
Instability of semi-Riemannian closed geodesics
A celebrated result due to Poincar\'e affirms that a closed non-degenerate
minimizing geodesic on an oriented Riemannian surface is hyperbolic.
Starting from this classical theorem, our first main result is a general
instability criterion for timelike and spacelike closed semi-Riemannian
geodesics on a (non)oriented manifold. A key role is played by the spectral
index, a new topological invariant that we define through the spectral flow
(being the Morse index truly infinite) of a path of selfadjoint Fredholm
operators. A major step in the proof of this result is a em new spectral flow
formula. Bott's iteration formula, introduced by author in 1956, relates in a
clear way the Morse index of an iterated closed Riemannian geodesic and the
so-called -Morse indices. Our second result is a semi-Riemannian
generalization of the famous Bott-type iteration formula in the case of closed
(resp. timelike closed) Riemannian (resp. Lorentzian) geodesics. Our last
result is a strong instability result obtained by controlling the Morse index
of the geodesic and of all of its iterations.Comment: 33 pages, 2 figures. Fixed some typos and updated references. arXiv
admin note: text overlap with arXiv:1705.0917
Shadow world evaluation of the Yang-Mills measure
A new state-sum formula for the evaluation of the Yang-Mills measure in the
Kauffman bracket skein algebra of a closed surface is derived. The formula
extends the Kauffman bracket to diagrams that lie in surfaces other than the
plane. It also extends Turaev's shadow world invariant of links in a circle
bundle over a surface away from roots of unity. The limiting behavior of the
Yang-Mills measure when the complex parameter approaches -1 is studied. The
formula is applied to compute integrals of simple closed curves over the
character variety of the surface against Goldman's symplectic measure.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-17.abs.htm
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