Results 1  10
of
12
The Complete Analysis of a Polynomial Factorization Algorithm Over Finite Fields
, 2001
"... This paper derives basic probabilistic properties of random polynomials over finite fields that are of interest in the study of polynomial factorization algorithms. We show that the main characteristics of random polynomial can be treated systematically by methods of "analytic combinatorics" based o ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
This paper derives basic probabilistic properties of random polynomials over finite fields that are of interest in the study of polynomial factorization algorithms. We show that the main characteristics of random polynomial can be treated systematically by methods of "analytic combinatorics" based on the combined use of generating functions and of singularity analysis. Our object of study is the classical factorization chain which is described in Fig. 1 and which, despite its simplicity, does not appear to have been totally analysed so far. In this paper, we provide a complete averagecase analysis.
Large Deviations for Integer Partitions
, 1998
"... We consider deviations from limit shape induced by uniformly distributed partitions (and strict partitions) of an integer n on the associated Young diagrams. We prove a full large deviation principle, of speed p n. The proof, based on projective limits, uses the representation of the uniform measure ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
We consider deviations from limit shape induced by uniformly distributed partitions (and strict partitions) of an integer n on the associated Young diagrams. We prove a full large deviation principle, of speed p n. The proof, based on projective limits, uses the representation of the uniform measure on partitions by means of suitably conditioned independent variables.
The PoissonDirichlet distribution and the scaleinvariant Poisson process
 COMBIN. PROBAB. COMPUT
, 1999
"... We show that the Poisson–Dirichlet distribution is the distribution of points in a scaleinvariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
We show that the Poisson–Dirichlet distribution is the distribution of points in a scaleinvariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that T � 1. Restricting both processes to (0,β] for 0 <β � 1, we give an explicit formula for the total variation distance between their distributions. Connections between various representations of the Poisson–Dirichlet process are discussed.
Cycle lengths in a permutation are typically Poisson
"... The set of cycle lengths of almost all permutations in Sn are “Poisson distributed”: we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain “ ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The set of cycle lengths of almost all permutations in Sn are “Poisson distributed”: we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain “normal order” (in the spirit of the ErdősTurán theorem). Our results were inspired by analogous questions about the size of the prime divisors of “typical ” integers. 1
Profiles of permutations
"... This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set S with asymptotic density σ and, on the other hand, permutations selected according to the Ewens distribution with parameter σ. In particular we ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set S with asymptotic density σ and, on the other hand, permutations selected according to the Ewens distribution with parameter σ. In particular we show that the asymptotic expected number of cycles of random permutations of [n] with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all 1 2 log n + O(1), and the variance is of the same order. Furthermore, we show that in permutations of [n] chosen from the Ewens distribution with parameter σ, the probability of a random element being in a cycle longer than γn approaches (1−γ) σ for large n. The same limit law holds for permutations with cycles carrying multiplicative weights with average σ. We draw parallels between the Ewens distribution and the asymptoticdensity case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions. 1
A PAIR OF DIFFERENCE DIFFERENTIAL EQUATIONS OF EULERCAUCHY TYPE
"... Abstract. We study two classes of linear difference differential equations analogous to EulerCauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory an ..."
Abstract
 Add to MetaCart
Abstract. We study two classes of linear difference differential equations analogous to EulerCauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general EulerCauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity. 1.