10,163 research outputs found

    A generatingfunctionology approach to a problem of Wilf

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    Wilf posed the following problem: determine asymptotically as nn\to\infty 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 SS-restricted integer compositions

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    If SS is a cofinite set of positive integers, an "SS-restricted composition of nn" is a sequence of elements of SS, denoted λ=(λ1,λ2,...)\vec{\lambda}=(\lambda_1,\lambda_2,...), whose sum is nn. For uniform random SS-restricted compositions, the random variable B(λ)=iλi{\bf B}(\vec{\lambda})=\prod_i \lambda_i 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

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    For an ergodic Markov chain {X(t)}\{X(t)\} on N\Bbb N, with a stationary distribution π\pi, let Tn>0T_n>0 denote a hitting time for [n]c[n]^c, and let Xn=X(Tn)X_n=X(T_n). Around 2005 Guy Louchard popularized a conjecture that, for nn\to \infty, TnT_n is almost Geometric(pp), p=π([n]c)p=\pi([n]^c), XnX_n is almost stationarily distributed on [n]c[n]^c, and that XnX_n and TnT_n are almost independent, if p(n):=supip(i,[n]c)0p(n):=\sup_ip(i,[n]^c)\to 0 exponentially fast. For the chains with p(n)0p(n) \to 0 however slowly, and with supi,jp(i,)p(j,)TV<1\sup_{i,j}\,\|p(i,\cdot)-p(j,\cdot)\|_{TV}<1, we show that Louchard's conjecture is indeed true even for the hits of an arbitrary SnNS_n\subset\Bbb N with π(Sn)0\pi(S_n)\to 0. More precisely, a sequence of kk 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 ksupip(i,Sn)k\,\sup_ip(i,S_n), by a kk-long sequence of independent copies of (n,tn)(\ell_n,t_n), where n=Geometric(π(Sn))\ell_n= \text{Geometric}\,(\pi(S_n)), tnt_n is distributed stationarily on SnS_n, and n\ell_n is independent of tnt_n. 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 ν\nu, 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 μ=o(lnν)\mu=o(\ln\nu) and μ=o(ν1/2)\mu=o(\nu^{1/2}) 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

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    A random composition of nn appears when the points of a random closed set R~[0,1]\widetilde{\mathcal{R}}\subset[0,1] are used to separate into blocks nn points sampled from the uniform distribution. We study the number of parts KnK_n of this composition and other related functionals under the assumption that R~=ϕ(S)\widetilde{\mathcal{R}}=\phi(S_{\bullet}), where (St,t0)(S_t,t\geq0) is a subordinator and ϕ:[0,][0,1]\phi:[0,\infty]\to[0,1] is a diffeomorphism. We derive the asymptotics of KnK_n when the L\'{e}vy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function ϕ(x)=1ex\phi(x)=1-e^{-x}, we establish a connection between the asymptotics of KnK_n 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

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    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

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    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 GS(k)G_S(k) of the maximal extension of a global field kk that is unramified outside a finite set SS of places, as kk varies among a certain family of extensions of a fixed global field QQ. We prove a generalized version of the global Euler-Poincar\'{e} Characteristic, and define a group BS(k,A)B_S(k,A), for each finite simple GS(k)G_S(k)-module AA, to generalize the work of Koch about the pro-\ell completion of GS(k)G_S(k) to study the whole group GS(k)G_S(k). 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

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    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 qn/2nq^n/2^n. This bound is tight to within a 2+o(1)2 + o(1) factor for every nn as qq \to \infty, compared to previous bounds that were off by exponential factors in nn. (B) We give improved randomness extractors and "randomness mergers". Mergers are seeded functions that take as input Λ\Lambda (possibly correlated) random variables in {0,1}N\{0,1\}^N and a short random seed and output a single random variable in {0,1}N\{0,1\}^N that is statistically close to having entropy (1δ)N(1-\delta) \cdot N when one of the Λ\Lambda input variables is distributed uniformly. The seed we require is only (1/δ)logΛ(1/\delta)\cdot \log \Lambda-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 1o(1)1 - o(1) 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

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    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
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