scholarly journals A symbolic method to compute the probability distribution of the number of pattern occurences in random texts generated by stochastic 0L-systems

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Cedric Loi ◽  
Paul-Henry Cournède ◽  
Jean Françon
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Branching Process ◽  
Specific Pattern ◽  
Multitype Branching Process ◽  
Transformation Rules ◽  
Symbolic Method ◽  
International Audience ◽  
Pattern Occurrences ◽  
Random Text

International audience The analysis of pattern occurrences has numerous applications, in particular in biology. In this article, a symbolic method is proposed to compute the distribution associated to the number of occurences of a specific pattern in a random text generated by a stochastic 0L-system. To that purpose, a semiring structure is set for combinatorial classes composed of weighted words. This algebraic structure relies on new union and concatenation operators which, under some assumptions, are admissible constructions. Decomposing the combinatorial classes of interest by using these binary operators enables the direct translation of specifications into a set of functional equations relating generating functions thanks to transformation rules. The article ends with two examples. The first one deals with unary patterns and the connection with multitype branching process is established. The second one is about a pattern composed of two letters and underlines the importance of writing a proper specification.

scholarly journals A bijection between planar constellations and some colored Lagrangian trees

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Cedric Chauve
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Formula One ◽  
Lagrange Formula ◽  
International Audience ◽  
Planar Maps ◽  
New Formula

International audience Constellations are colored planar maps that generalize different families of maps (planar maps, bipartite planar maps, bi-Eulerian planar maps, planar cacti, ...) and are strongly related to factorizations of permutations. They were recently studied by Bousquet-Mélou and Schaeffer who describe a correspondence between these maps and a family of trees, called Eulerian trees. In this paper, we derive from their result a relationship between planar constellations and another family of trees, called stellar trees. This correspondence generalizes a well known result for planar cacti, and shows that planar constellations are colored Lagrangian objects (that is objects that can be enumerated by the Good-Lagrange formula). We then deduce from this result a new formula for the number of planar constellations having a given face distribution, different from the formula one can derive from the results of Bousquet-Mélou and Schaeffer, along with systems of functional equations for the generating functions of bipartite and bi-Eulerian planar maps enumerated according to the partition of faces and vertices.

scholarly journals Branching processes in random environment die slowly

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Vladimir Vatutin ◽  
Andreas Kyprianou
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Generating Functions ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Limit Distributions ◽  
Conditional Limit Theorems ◽  
International Audience ◽  
Conditional Limit

International audience Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.

scholarly journals Multitype linear fractional branching processes

2006 ◽  
Vol 43 (04) ◽  
pp. 1091-1106 ◽  
Author(s):  
A. Joffe ◽  
G. Letac
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Branching Processes ◽  
Main Tool ◽  
Linear Functions ◽  
Multitype Branching Process ◽  
The Mean

We complete a paper written by Edward Pollak in 1974 on a multitype branching process the generating functions of whose birth law are fractional linear functions with the same denominator. The main tool is a parameterization of these functions adapted using the mean matrix M and an element w of the first quadrant. We use this opportunity to give a self-contained presentation of Pollak's theory.

scholarly journals Generating trees for partitions and permutations with no k-nestings

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sophie Burrill ◽  
Sergi Elizalde ◽  
Marni Mishna ◽  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Young Tableaux ◽  
Set Partitions ◽  
Lattice Walks ◽  
Generating Trees ◽  
Generating Tree ◽  
International Audience ◽  
Tree Approach

International audience We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter. Nous décrivons une approche, basée sur l'utilisation d'arbres de génération, pour énumération et la génération exhaustive de partitions et permutations sans k-emboîtement. Contrairement aux travaux antérieurs qui reposent sur un lien entre ces objets, tableaux de Young et familles de chemins dans des treillis, notre approche traite directement partitions et diagrammes de permutations. Nous fournissons des équations fonctionnelles explicites pour les séries génératrices, avec k en tant que paramètre.

scholarly journals Multitype linear fractional branching processes

2006 ◽  
Vol 43 (4) ◽  
pp. 1091-1106 ◽  
Author(s):  
A. Joffe ◽  
G. Letac
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Branching Processes ◽  
Main Tool ◽  
Linear Functions ◽  
Multitype Branching Process ◽  
The Mean

We complete a paper written by Edward Pollak in 1974 on a multitype branching process the generating functions of whose birth law are fractional linear functions with the same denominator. The main tool is a parameterization of these functions adapted using the mean matrix M and an element w of the first quadrant. We use this opportunity to give a self-contained presentation of Pollak's theory.

A Symbolic Method to Analyse Patterns in Plant Structure

2014 ◽  
Vol 23 (5) ◽  
pp. 842-860
Author(s):  
C. LOI ◽  
P.-H. COURNÈDE ◽  
J. FRANÇON
Keyword(s):  
Plant Growth ◽  
Stochastic Models ◽  
Branching Process ◽  
Growth Dynamics ◽  
Complex Structures ◽  
Plant Structure ◽  
Multitype Branching Process ◽  
Symbolic Method ◽  
L Systems ◽  
Stochastic Distribution

Formal grammars such as L-systems have long been used to describe plant growth dynamics. In this article, they are used for a new purpose. The aim is to build a symbolic method that enables the computation of the stochastic distribution associated with the number of complex structures in plants whose organogenesis is driven by a multitype branching process. For that purpose, a new combinatorial framework is set in which plant structure is coded by a Dyck word. Moreover, organogenesis is represented by stochastic F0L-systems. In doing so, the problem is equivalent to determining the distribution of patterns in random words generated by a stochastic F0L-system. This method finds interesting applications in the parameter identification of stochastic models of plant development.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

scholarly journals Demographic Dynamics in Multitype Populations with Migrations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 246
Author(s):  
Manuel Molina-Fernández ◽  
Manuel Mota-Medina
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Branching Process ◽  
Probability Model ◽  
Biological Systems ◽  
Research Work ◽  
General Setting ◽  
Process Theory ◽  
Multitype Branching Process ◽  
Demographic Dynamics

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

Branching processes and functional differential equations determining steady-size distributions in cell populations

1995 ◽  
Vol 32 (01) ◽  
pp. 1-10
Author(s):  
Ziad Taib
Keyword(s):  
Differential Equation ◽  
Branching Process ◽  
Branching Processes ◽  
Functional Differential Equations ◽  
Cell Populations ◽  
Size Distributions ◽  
Multitype Branching Process ◽  
Functional Differential ◽  
Intuitive Interpretation ◽  
Infinite Sum

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

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.

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