10,647 research outputs found
Euclidean algorithms are Gaussian
This study provides new results about the probabilistic behaviour of a class
of Euclidean algorithms: the asymptotic distribution of a whole class of
cost-parameters associated to these algorithms is normal. For the cost
corresponding to the number of steps Hensley already has proved a Local Limit
Theorem; we give a new proof, and extend his result to other euclidean
algorithms and to a large class of digit costs, obtaining a faster, optimal,
rate of convergence. The paper is based on the dynamical systems methodology,
and the main tool is the transfer operator. In particular, we use recent
results of Dolgopyat.Comment: fourth revised version - 2 figures - the strict convexity condition
used has been clarifie
Gaussian Behavior of Quadratic Irrationals
We study the probabilistic behaviour of the continued fraction expansion of a
quadratic irrational number, when weighted by some "additive" cost. We prove
asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal
with the underlying dynamical system associated with the Gauss map, and its
weighted periodic trajectories. We work with analytic combinatorics methods,
and mainly with bivariate Dirichlet generating functions; we use various tools,
from number theory (the Landau Theorem), from probability (the Quasi-Powers
Theorem), or from dynamical systems: our main object of study is the (weighted)
transfer operator, that we relate with the generating functions of interest.
The present paper exhibits a strong parallelism with the methods which have
been previously introduced by Baladi and Vall\'ee in the study of rational
trajectories. However, the present study is more involved and uses a deeper
functional analysis framework.Comment: 39 pages In this second version, we have added an annex that provides
a detailed study of the trace of the weighted transfer operator. We have also
corrected an error that appeared in the computation of the norm of the
operator when acting in the Banach space of analytic functions defined in the
paper. Also, we give a simpler proof for Theorem
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache
Status of Average-x from Lattice QCD
As algorithms and computing power have advanced, lattice QCD has become a
precision technique for many QCD observables. However, the calculation of
nucleon matrix elements remains an open challenge. I summarize the status of
the lattice effort by examining one observable that has come to represent this
challenge, average-x: the fraction of the nucleon's momentum carried by its
quark constituents. Recent results confirm a long standing tendency to
overshoot the experimentally measured value. Understanding this puzzle is
essential to not only the lattice calculation of nucleon properties but also
the broader effort to determine hadron structure from QCD.Comment: proceedings for 3rd International Workshop on Nucleon Structure at
Large Bjorken
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