scholarly journals Asymptotics of Bivariate Analytic Functions with Algebraic Singularities

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Torin Greenwood
Keyword(s):  
Generating Functions ◽  
Analytic Functions ◽  
Broad Class ◽  
Multivariate Case ◽  
Univariate Case ◽  
International Audience ◽  
Rational Generating Functions ◽  
Algebraic Singularities

International audience In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to find asymptotic for- mulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions.

scholarly journals On a Class of Distributions Stable Under Random Summation

2012 ◽  
Vol 49 (02) ◽  
pp. 303-318 ◽  
Author(s):  
L. B. Klebanov ◽  
A. V. Kakosyan ◽  
S. T. Rachev ◽  
G. Temnov
Keyword(s):  
Normal Distribution ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Analytic Functions ◽  
Random Variable ◽  
Random Summation ◽  
The Stability ◽  
Rational Generating Functions ◽  
Family Of Distributions

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

scholarly journals Enumeration of minimal 3D polyominoes inscribed in a rectangular prism

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alain Goupil ◽  
Hugo Cloutier
Keyword(s):  
Geometric Structure ◽  
Generating Functions ◽  
Rectangular Prism ◽  
Nous Obtenons ◽  
Minimal Volume ◽  
Connected Sets ◽  
The Family ◽  
International Audience ◽  
Combinatorial Principles ◽  
Rational Generating Functions

International audience We consider the family of 3D minimal polyominoes inscribed in a rectanglar prism. These objects are polyominos and so they are connected sets of unitary cubic cells inscribed in a given rectangular prism of size $b\times k \times h$ and of minimal volume equal to $b+k+h-2$. They extend the concept of minimal 2D polyominoes inscribed in a rectangle studied in a previous work. Using their geometric structure and elementary combinatorial principles, we construct rational generating functions of minimal 3D polyominoes. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes. Nous considérons la famille des polyominos 3D de volume minimal inscrits dans un prisme rectangulaire. Ces objets sont des polyominos et sont donc des ensembles connexes de cubes unitaires. De plus ils sont inscrits dans un prisme rectangulaire de format $b\times k \times h$ donné et ont un volume minimal égal à $b+k+h-2$. Ces polyominos généralisent le concept de polyomino 2D étudié dans un travail précédent. Nous construisons des séries génératrices rationnelles de polyominos 3D minimaux et nous obtenons des formules exactes et des récurrences pour des sous-familles de ces polyominos.

scholarly journals On a Class of Distributions Stable Under Random Summation

2012 ◽  
Vol 49 (2) ◽  
pp. 303-318 ◽  
Author(s):  
L. B. Klebanov ◽  
A. V. Kakosyan ◽  
S. T. Rachev ◽  
G. Temnov
Keyword(s):  
Normal Distribution ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Analytic Functions ◽  
Random Variable ◽  
Random Summation ◽  
The Stability ◽  
Rational Generating Functions ◽  
Family Of Distributions

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

scholarly journals Counting with rational generating functions

2008 ◽  
Vol 43 (2) ◽  
pp. 75-91 ◽  
Author(s):  
Sven Verdoolaege ◽  
Kevin Woods
Keyword(s):  
Generating Functions ◽  
Rational Generating Functions

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

scholarly journals Finite Eulerian posets which are binomial or Sheffer

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Hoda Bidkhori
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
International Audience ◽  
Original Question ◽  
Eulerian Posets

International audience In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals. Nous étudions les ensembles partiellement ordonnés finis (EPO) qui sont soit binomiaux soit de type Sheffer (deux notions reliées aux séries génératrices et à la géométrie). Nos résultats sont les suivants: (1) nous déterminons la structure des EPO Euleriens et binomiaux; nous classifions ainsi les fonctions factorielles de tous ces EPO; (2) nous donnons une classification presque complète des fonctions factorielles des EPO Euleriens de type Sheffer; (3) dans la plupart de ces cas, nous déterminons complètement la structure des EPO Euleriens et Sheffer, ce qui est plus fort que classifier leurs fonctions factorielles. Nous étudions aussi les EPO Euleriens triangulaires. Cet article répond à des questions de R. Ehrenborg and M. Readdy. Il est aussi motivé par le travail de R. Stanley sur la reconnaissance du treillis booléen via l'étude des petits intervalles.

scholarly journals Univariate and Multivariate Outlier Identification for Skewed or Heavy-Tailed Distributions

2018 ◽  
Vol 18 (3) ◽  
pp. 517-532
Author(s):  
Vincenzo Verardi ◽  
Catherine Vermandele
Keyword(s):  
Computational Complexity ◽  
Multivariate Analyses ◽  
Outlier Identification ◽  
Multivariate Case ◽  
Univariate Case ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Identification Procedures

In univariate and in multivariate analyses, it is difficult to identify outliers in the case of skewed or heavy-tailed distributions. In this article, we propose simple univariate and multivariate outlier identification procedures that perform well with these types of distributions while keeping the computational complexity low. We describe the commands gboxplot (univariate case) and sdasym (multivariate case), which implement these procedures in Stata.

scholarly journals PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS

2015 ◽  
Vol 80 (2) ◽  
pp. 433-449 ◽  
Author(s):  
KEVIN WOODS
Keyword(s):  
Generating Functions ◽  
Counting Function ◽  
Order Theory ◽  
Characteristic Functions ◽  
Presburger Arithmetic ◽  
First Order ◽  
First Order Theory ◽  
Rational Generating Function ◽  
Rational Generating Functions

AbstractPresburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= (p1, . . . ,pn) are a subset of the free variables in a Presburger formula, we can define a counting functiong(p) to be the number of solutions to the formula, for a givenp. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

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