10,163 research outputs found
A generatingfunctionology approach to a problem of Wilf
Wilf posed the following problem: determine asymptotically as
the probability that a randomly chosen part size in a randomly chosen
composition of n has multiplicity m. One solution of this problem was given by
Hitczenko and Savage. In this paper, we study this question using the
techniques of generating functions and singularity analysis.Comment: 12 page
Part-products of -restricted integer compositions
If is a cofinite set of positive integers, an "-restricted composition
of " is a sequence of elements of , denoted
, whose sum is . For uniform random
-restricted compositions, the random variable is asymptotically lognormal. The proof is
based upon a combinatorial technique for decomposing a composition into a
sequence of smaller compositions.Comment: 18 page
Tight Markov chains and random compositions
For an ergodic Markov chain on , with a stationary
distribution , let denote a hitting time for , and let
. Around 2005 Guy Louchard popularized a conjecture that, for , is almost Geometric(), , is almost
stationarily distributed on , and that and are almost
independent, if exponentially fast. For the
chains with however slowly, and with
, we show that Louchard's
conjecture is indeed true even for the hits of an arbitrary
with . More precisely, a sequence of consecutive hit
locations paired with the time elapsed since a previous hit (for the first hit,
since the starting moment) is approximated, within a total variation distance
of order , by a -long sequence of independent copies of
, where , is
distributed stationarily on , and is independent of . The
two conditions are easily met by the Markov chains that arose in Louchard's
studies as likely sharp approximations of two random compositions of a large
integer , a column-convex animal (cca) composition and a Carlitz (C)
composition. We show that this approximation is indeed very sharp for most of
the parts of the random compositions. Combining the two approximations in a
tandem, we are able to determine the limiting distributions of
and largest parts of the random cca composition and the
random C-composition, respectively. (Submitted to Annals of Probability in
August, 2009.
Asymptotic laws for compositions derived from transformed subordinators
A random composition of appears when the points of a random closed set
are used to separate into blocks
points sampled from the uniform distribution. We study the number of parts
of this composition and other related functionals under the assumption
that , where is a
subordinator and is a diffeomorphism. We derive the
asymptotics of when the L\'{e}vy measure of the subordinator is regularly
varying at 0 with positive index. Specializing to the case of exponential
function , we establish a connection between the asymptotics
of and the exponential functional of the subordinator.Comment: Published at http://dx.doi.org/10.1214/009117905000000639 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Las Vegas algorithm to solve the elliptic curve discrete logarithm problem
In this paper, we describe a new Las Vegas algorithm to solve the elliptic
curve discrete logarithm problem. The algorithm depends on a property of the
group of rational points of an elliptic curve and is thus not a generic
algorithm. The algorithm that we describe has some similarities with the most
powerful index-calculus algorithm for the discrete logarithm problem over a
finite field
Presentations of Galois groups of maximal extensions with restricted ramification
Motivated by the work of Lubotzky, we use Galois cohomology to study the
difference between the number of generators and the minimal number of relations
in a presentation of the Galois group of the maximal extension of a
global field that is unramified outside a finite set of places, as
varies among a certain family of extensions of a fixed global field . We
prove a generalized version of the global Euler-Poincar\'{e} Characteristic,
and define a group , for each finite simple -module , to
generalize the work of Koch about the pro- completion of to
study the whole group . In the setting of the nonabelian Cohen-Lenstra
heuristics, we prove that the objects studied by the Liu--Wood--Zureick-Brown
conjecture are always achievable by the random group that is constructed in the
definition the probability measure in the conjecture.Comment: Comments are welcom
Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers
We extend the "method of multiplicities" to get the following results, of
interest in combinatorics and randomness extraction. (A) We show that every
Kakeya set (a set of points that contains a line in every direction) in
\F_q^n must be of size at least . This bound is tight to within a factor for every as , compared to previous bounds
that were off by exponential factors in . (B) We give improved randomness
extractors and "randomness mergers". Mergers are seeded functions that take as
input (possibly correlated) random variables in and a
short random seed and output a single random variable in that is
statistically close to having entropy when one of the
input variables is distributed uniformly. The seed we require is only
-bits long, which significantly improves upon
previous construction of mergers. (C) Using our new mergers, we show how to
construct randomness extractors that use logarithmic length seeds while
extracting fraction of the min-entropy of the source.
The "method of multiplicities", as used in prior work, analyzed subsets of
vector spaces over finite fields by constructing somewhat low degree
interpolating polynomials that vanish on every point in the subset {\em with
high multiplicity}. The typical use of this method involved showing that the
interpolating polynomial also vanished on some points outside the subset, and
then used simple bounds on the number of zeroes to complete the analysis. Our
augmentation to this technique is that we prove, under appropriate conditions,
that the interpolating polynomial vanishes {\em with high multiplicity} outside
the set. This novelty leads to significantly tighter analyses.Comment: 26 pages, now includes extractors with sublinear entropy los
Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths
We introduce and characterize central probability distributions on Littelmann
paths. Next we establish a law of large numbers and a central limit theorem for
the generalized Pitmann transform. We then study harmonic functions on
multiplicative graphs defined from the tensor powers of finite-dimensional Lie
algebras representations. Finally, we show there exists an inverse of the
generalized Pitman transform defined almost surely on the set of infinite paths
remaining in the Weyl chamber and explain how it can be computed.Comment: 27 pages, minor corrections and a simpler definition of the Pitman
invers
- …