A Poisson * Geometric Convolution Law for the Number of Components in Unlabelled Combinatorial Structures

1998 ◽  
Vol 7 (1) ◽  
pp. 89-110 ◽  
Author(s):  
HSIEN-KUEI HWANG
Keyword(s):  
Probability Measure ◽  
Finite Fields ◽  
Generating Function ◽  
Asymptotic Estimates ◽  
Combinatorial Structures ◽  
Number Of Components ◽  
Random Mapping ◽  
Uniform Probability ◽  
Arithmetical Semigroups ◽  
Precise Asymptotic

Given a class of combinatorial structures [Cscr ], we consider the quantity N(n, m), the number of multiset constructions [Pscr ] (of [Cscr ]) of size n having exactly m [Cscr ]-components. Under general analytic conditions on the generating function of [Cscr ], we derive precise asymptotic estimates for N(n, m), as n→∞ and m varies through all possible values (in general 1[les ]m[les ]n). In particular, we show that the number of [Cscr ]-components in a random (assuming a uniform probability measure) [Pscr ]-structure of size n obeys asymptotically a convolution law of the Poisson and the geometric distributions. Applications of the results include random mapping patterns, polynomials in finite fields, parameters in additive arithmetical semigroups, etc. This work develops the ‘additive’ counterpart of our previous work on the distribution of the number of prime factors of an integer [20].

scholarly journals Asymptotics of Smallest Component Sizes in Decomposable Combinatorial Structures of Alg-Log Type

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Li Dong ◽  
Zhicheng Gao ◽  
Daniel Panario ◽  
Bruce Richmond
Keyword(s):  
Generating Function ◽  
Combinatorial Structure ◽  
Asymptotic Estimates ◽  
Asymptotic Results ◽  
Combinatorial Structures ◽  
Combinatorial Objects ◽  
International Audience ◽  
Restricted Pattern

International audience A decomposable combinatorial structure consists of simpler objects called components which by thems elves cannot be further decomposed. We focus on the multi-set construction where the component generating function C(z) is of alg-log type, that is, C(z) behaves like c + d(1 -z/rho)(alpha) (ln1/1-z/rho)(beta) (1 + o(1)) when z is near the dominant singularity rho. We provide asymptotic results about the size of thes mallest components in random combinatorial structures for the cases 0 < alpha < 1 and any beta, and alpha < 0 and beta=0. The particular case alpha=0 and beta=1, the so-called exp-log class, has been treated in previous papers. We also provide similar asymptotic estimates for combinatorial objects with a restricted pattern, that is, when part of its factorization patterns is known. We extend our results to include certain type of integers partitions. partitions

Functionals of random mappings: exact and asymptotic results

1991 ◽  
Vol 23 (3) ◽  
pp. 437-455 ◽  
Author(s):  
P. J. Donnelly ◽  
W. J. Ewens ◽  
S. Padmadisastra
Keyword(s):  
Frequency Spectrum ◽  
Exact Results ◽  
Asymptotic Results ◽  
Number Of Components ◽  
Random Mapping ◽  
Random Mappings ◽  
Simulation Results ◽  
Central Tool

A random mapping partitions the set {1, 2, ···, m} into components, where i and j are in the same component if some functional iterate of i equals some functional iterate of j. We consider various functionals of these partitions and of samples from it, including the number of components of ‘small' size and of size O(m) as m → ∞the size of the largest component, the number of components, and various symmetric functionals of the normalized component sizes. In many cases exact results, while available, are uniformative, and we consider various approximations. Numerical and simulation results are also presented. A central tool for many calculations is the ‘frequency spectrum', both exact and asymptotic.

scholarly journals A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

2019 ◽  
Vol 7 ◽  
Author(s):  
DANIEL M. KANE ◽  
ROBERT C. RHOADES
Keyword(s):  
Asymptotic Behavior ◽  
Modular Form ◽  
Generating Function ◽  
Generating Functions ◽  
Asymptotic Properties ◽  
Bootstrap Percolation ◽  
Asymptotic Result ◽  
Integer Partitions ◽  
Mock Modular Form ◽  
Precise Asymptotic

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of$n$without$k$consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without$k$consecutive parts. Andrews showed that when$k=2$, the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For$k>2$, the asymptotic properties of the generating functions have proved more difficult to obtain. Using$q$-series identities and the$k=2$case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case$k=3$was given by Zagier.This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.

scholarly journals Moments of the weighted Cantor measures

2019 ◽  
Vol 52 (1) ◽  
pp. 256-273
Author(s):  
Steven N. Harding ◽  
Alexander W. N. Riasanovsky
Keyword(s):  
Probability Measure ◽  
Generating Function ◽  
Exponential Decay ◽  
Legendre Polynomials ◽  
Borel Probability Measure ◽  
Moment Generating Function ◽  
Unit Interval ◽  
Uniform Error ◽  
Natural Case ◽  
Borel Probability

AbstractBased on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying {\mu ^\alpha }(E) = \sum\nolimits_{n = 0}^{N - 1} {{\alpha _n}{\mu ^\alpha }(\varphi _n^{ - 1}(E))} where ϕn : x ↦ (x + n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L_{{\mu ^\alpha }}^2 [0, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xm dμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error ε in O((log log(1/ε)) · m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2, 1/2].

scholarly journals Some New Constructions of Authentication Codes with Arbitration and Multi-Receiver from Singular Symplectic Geometry

2011 ◽  
Vol 2011 ◽  
pp. 1-18
Author(s):  
You Gao ◽  
Huafeng Yu
Keyword(s):  
Probability Distribution ◽  
Finite Fields ◽  
Symplectic Geometry ◽  
Authentication Codes ◽  
Uniform Probability ◽  
Different Types ◽  
Uniform Probability Distribution ◽  
New Construction

A new construction of authentication codes with arbitration and multireceiver from singular symplectic geometry over finite fields is given. The parameters are computed. Assuming that the encoding rules are chosen according to a uniform probability distribution, the probabilities of success for different types of deception are also computed.

scholarly journals Croissance des boules et des géodésiques fermées dans les nilvariétés

1983 ◽  
Vol 3 (3) ◽  
pp. 415-445 ◽  
Author(s):  
Pierre Pansu
Keyword(s):  
Universal Cover ◽  
Asymptotic Estimates ◽  
Precise Asymptotic

AbstractIf (M, g) is a riemannian nilmanifold, the homothetic metrics εg˜ on the universal cover M converge in the sense of Gromov for small ε. In this convergence the volume of balls and the number of closed geodesies go to a limit, and precise asymptotic estimates are given for these numbers.

scholarly journals Reliability Analysis of k-out-of-n Systems of Components with Potentially Brittle Behavior by Universal Generating Function and Linear Programming

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Yi Chang ◽  
Yanshuo Mai ◽  
Linli Yi ◽  
Liuming Yu ◽  
Ying Chen ◽  
...  
Keyword(s):  
Linear Programming ◽  
Generating Function ◽  
Failure Probability ◽  
Number Of Components ◽  
Brittle Behavior ◽  
Extreme Load ◽  
The Individual ◽  
Initial Cracks ◽  
Joint Probabilities ◽  
Universal Generating Function

Due to initial cracks, careless construction, and extreme load conditions, components with brittle behavior may exist in a structural system. The presence of brittle behavior of components usually is accompanied by a low strength. However, existing methods for calculating the reliability of structures of components with brittle behavior are rather complicated or impossible. By means of decomposing the entire system into a set of subsystems, this paper proposed a method to estimate the bounds on failure probability of k-out-of-n system of components with potentially brittle behavior by using universal generating function (UGF) and linear programming (LP). Based on the individual component state probabilities and joint probabilities of the states of a small number of components, the proposed method can provide the bounds for the failure probability of a system with a large number of components. The accuracy and efficiency of the proposed method are investigated using numerical examples.

On a random mapping (T, Pj)

1984 ◽  
Vol 21 (1) ◽  
pp. 186-191 ◽  
Author(s):  
Jerzy Jaworski
Keyword(s):  
Fundamental Lemma ◽  
Number Of Components ◽  
Random Mapping ◽  
Finite Set

A random mapping (T,Pj) of a finite set V into itself is studied. We give a new proof of the fundamental lemma of [6]. Our method leads to the derivation of several results which cannot be deduced from [6]. In particular we determine the distribution of the number of components, cyclical points and ancestors of a given point.

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