scholarly journals Sampling parts of random integer partitions: a probabilistic and asymptotic analysis

2015 ◽  
Vol 25 (1) ◽  
pp. 79-95
Author(s):  
Ljuben Mutafchiev
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Asymptotic Analysis ◽  
Integer Partitions ◽  
Random Partition ◽  
Limiting Distributions ◽  
Sampling Plans ◽  
Random Integer ◽  
Sampling Procedures ◽  
The Relationship

Abstract Let λ be a partition of the positive integer n, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of λ at random. They obtained limiting distributions of the multiplicity μn = μn(λ) of the randomly-chosen part as n → ∞. The asymptotic behavior of the part size σn = σn(λ), under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of (μn; σn), as n → ∞, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for (μn; σn). Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of μn and σn.

scholarly journals A phase transition in the distribution of the length of integer partitions

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Dimbinaina Ralaivaosaona
Keyword(s):  
Phase Transition ◽  
Positive Integer ◽  
Gaussian Distribution ◽  
Gumbel Distribution ◽  
Limiting Distribution ◽  
Integer Partitions ◽  
Random Partition ◽  
Uniform Probability ◽  
International Audience

International audience We assign a uniform probability to the set consisting of partitions of a positive integer $n$ such that the multiplicity of each summand is less than a given number $d$ and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erdős and Lehner published in 1941 that the distributions of the length in random restricted $(d=2)$ and random unrestricted $(d \geq n+1)$ partitions behave very differently. In this paper we show that as the bound $d$ increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case.

scholarly journals On the enumeration and asymptotic growth of free quasigroup words

2020 ◽  
Vol 30 (07) ◽  
pp. 1465-1483
Author(s):  
Jonathan D. H. Smith ◽  
Stefanie G. Wang
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Numerical Data ◽  
Recursive Formula ◽  
Catalan Numbers ◽  
Euclidean Algorithm ◽  
Asymptotic Growth ◽  
Number Of Generators ◽  
The Relationship ◽  
Standard Techniques

This paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of this paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators.

scholarly journals Estimating the Dog Population, Responsible Pet Ownership, and Intestinal Parasitism in Dogs in Quito, Ecuador

2021 ◽  
Author(s):  
Colon Jaime Grijalva ◽  
Heather S. Walden ◽  
P. Cynda Crawford ◽  
Julie K. Levy ◽  
William E. Pine ◽  
...  
Keyword(s):  
Intestinal Parasites ◽  
Household Wealth ◽  
Gastrointestinal Parasites ◽  
Capital City ◽  
Pet Ownership ◽  
Positive Interaction ◽  
Stray Dogs ◽  
Household Factors ◽  
Sampling Procedures ◽  
The Relationship

Abstract In 2011, authorities of Quito, the capital city of Ecuador, approved an ordinance to promote public health and animal welfare through responsible pet ownership promotion. The population of dogs was not known, and the relationships between dog abundance, socio-economic factors, prevalence of zoonotic gastrointestinal parasites, and pet ownership responsibility had not been investigated. The objectives of this study were (i) to estimate the human:dog (HD) ratio, (ii) to examine the relationship between household factors and responsible pet ownership and (iii) to estimate the prevalence of households with one or more dogs infected with intestinal parasites in Quito, Ecuador. Space-based random sampling procedures were used for estimation of HD ratios in stray dogs and confined owned dogs. The relationship between household factors and a responsible pet ownership index was examined using logistic regression. Dog fecal samples were tested for intestinal parasites. Among stray dogs, the observed HD ratio was 58:1. Among dogs kept indoors, the observed HD ratio was 3,5:1. A positive interaction effect between number of dogs in study households and household living conditions (a proxy for household wealth) on responsible pet ownership was observed, which we discuss in this report. Prevalence of households with dogs infected with intestinal parasites was 28% (95% CI = 21-37). Ancylostoma spp. was the most frequent intestinal parasite in study dogs kept indoors. This study provides new information that can be used by policy makers to formulate, implement, and evaluate public policies and education programs aimed at enhancing pet ownership responsibility in Ecuador.

A General Asymptotic Scheme for the Analysis of Partition Statistics

2014 ◽  
Vol 23 (6) ◽  
pp. 1057-1086 ◽  
Author(s):  
PETER J. GRABNER ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Generating Function ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Higher Moments ◽  
Integer Partitions ◽  
Classical Singularity ◽  
Random Integer ◽  
Partition Statistics ◽  
New Statistics

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

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.

An asymptotic analysis for Hamilton–Jacobi equations with large Hamiltonian drift terms

2017 ◽  
Vol 0 (0) ◽  
Author(s):  
Taiga Kumagai
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Asymptotic Analysis ◽  
Open Subset ◽  
Dimensional Euclidean Space ◽  
Asymptotic Behavior Of Solutions ◽  
Two Dimensional ◽  
Jacobi Equations ◽  
Drift Terms

AbstractWe investigate the asymptotic behavior of solutions of Hamilton–Jacobi equations with large Hamiltonian drift terms in an open subset of the two-dimensional Euclidean space. The drift is given by

scholarly journals The Small-Péclet-Number Approximation for Radiative Zones and its Application to Shear Layer Instabilities

2004 ◽  
Vol 215 ◽  
pp. 346-347
Author(s):  
F. Lignières
Keyword(s):  
Asymptotic Behavior ◽  
Thermal Diffusivity ◽  
Asymptotic Analysis ◽  
Shear Layer ◽  
Peclet Number ◽  
Earth Atmosphere ◽  
Hydrodynamic Equations ◽  
The Earth ◽  
Shear Instabilities ◽  
Thermal Diffusivities

Thermal diffusivities within stars are higher by several orders of magnitude than in the earth atmosphere. This particularity of the stellar fluid has subtle effects on the dynamics of stably stratified radiative zones as it contributes to favour some flows and supress others. An asymptotic analysis of hydrodynamic equations for large values of the thermal diffusivity is presented and this results in a simplification of the dynamical effects of the thermal diffusivity. This asymptotic behavior proved to be very useful to understand shear instabilities of Kelvin-Helmholtz type.

scholarly journals The role of the range of dispersal in a nonlocal Fisher-KPP equation: an asymptotic analysis

2020 ◽  
pp. 2050032
Author(s):  
Julien Brasseur
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Nonlocal Problem ◽  
Type Equation ◽  
Independent Interest ◽  
Kpp Equation ◽  
Open Question

In this paper, we study the asymptotic behavior as [Formula: see text] of solutions [Formula: see text] to the nonlocal stationary Fisher-KPP type equation [Formula: see text] where [Formula: see text] and [Formula: see text]. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution [Formula: see text] and that [Formula: see text] as [Formula: see text] where [Formula: see text]. This generalizes the previously known results and answers an open question raised by Berestycki et al. Our method of proof is also of independent interest as it shows how to reduce this nonlocal problem to a local one. The sharpness of our assumptions is also briefly discussed.

Microbiological Guidelines and Sampling Plans for Dried Infant Cereals and Powdered Infant Formula from a Canadian National Microbiological Survey

1980 ◽  
Vol 43 (8) ◽  
pp. 613-616 ◽  
Author(s):  
D. L. COLLINS-THOMPSON ◽  
K. F. WEISS ◽  
G. W. RIEDEL ◽  
S. CHARBONNEAU
Keyword(s):  
Test Organism ◽  
Rejection Rate ◽  
Fecal Coliforms ◽  
Powdered Infant Formula ◽  
Sampling Plans ◽  
Infant Foods ◽  
Hygienic Practice ◽  
Infant Formulae ◽  
Sampling Procedures

The microbiological safety and quality of 130 lots of domestic and imported dried infant cereals and powdered infant formulae were determined using aerobic colony count (ACC), aerobic sporeformers, confirmed and fecal coliforms, Escherichia coli, Staphylococcus aureus, Bacillus cereus, Clostridium perfringens (spores), hemolytic streptococci and yeast and molds. Based on the analytical results and sampling procedures of this survey, no health hazards were found. The results were also used to develop a three-class acceptance plan for ACC (n = 5, c = 2, m = 103/g, M = 104/g), confirmed coliforms (n = 5, c = 1, m = < 1.8/g, M = 20/g) and Salmonella (n = 20, c = 0, m = 0, M = 0). These plans were influenced by the proposed Codex Code of Hygienic Practice for Foods for Infants and Children. Canadian-produced infant foods had an estimate lot rejection rate based on the three-class acceptance plan of 1.1 to 20.3%, based on cereal type and test organism.

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