scholarly journals Generating Functions of Stochastic L-Systems and Application to Models of Plant Development

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Cedric Loi ◽  
Paul Henry Cournède
Keyword(s):  
Generating Functions ◽  
Probability Distributions ◽  
Branching Processes ◽  
Plant Structure ◽  
L System ◽  
International Audience ◽  
Multitype Branching Processes ◽  
L Systems ◽  
Architectural Development ◽  
New Framework

International audience If the interest of stochastic L-systems for plant growth simulation and visualization is broadly acknowledged, their full mathematical potential has not been taken advantage of. In this article, we show how to link stochastic L-systems to multitype branching processes, in order to characterize the probability distributions and moments of the numbers of organs in plant structure. Plant architectural development can be seen as the combination of two subprocesses driving the bud population dynamics, branching and differentiation. By writing the stochastic L-system associated to each subprocess, we get the generating function associated to the whole system by compounding the associated generating functions. The modelling of stochastic branching is classical, but to model differentiation, we introduce a new framework based on multivariate phase-type random vectors.

Generalizations of the elementary renewal theorem to distributions defined by concave recurrence relations

1977 ◽  
Vol 14 (3) ◽  
pp. 516-526
Author(s):  
W. Reh
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Branching Processes ◽  
General Procedure ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Probability Generating Functions ◽  
Dependent Probability

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.

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 $nt$.

GENERATING FUNCTIONS FOR STOCHASTIC CONTEXT-FREE GRAMMARS

1990 ◽  
Vol 04 (04) ◽  
pp. 553-572 ◽  
Author(s):  
MICHAEL G. THOMASON
Keyword(s):  
Generating Functions ◽  
Pattern Analysis ◽  
Branching Processes ◽  
Single Type ◽  
Probability Generating Functions ◽  
Syntactic Pattern ◽  
Multitype Branching Processes ◽  
Stochastic Context Free Grammars ◽  
Context Free ◽  
Context Free Grammars

Stochastic context-free grammars are an important tool in syntactic pattern analysis and other applications as well. This paper discusses major results in single-type and multitype branching processes used to study a grammar’s stochastic derivations. Probability generating functions are well-established as a tool in this area and are used extensively here.

Generalizations of the elementary renewal theorem to distributions defined by concave recurrence relations

1977 ◽  
Vol 14 (03) ◽  
pp. 516-526
Author(s):  
W. Reh
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Branching Processes ◽  
General Procedure ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Probability Generating Functions ◽  
Dependent Probability

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.

Survival probabilities and extinction times for some multitype branching processes

1974 ◽  
Vol 6 (03) ◽  
pp. 446-462 ◽  
Author(s):  
Edward Pollak
Keyword(s):  
Generating Functions ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Single Individual ◽  
Survival Probabilities ◽  
Time To Extinction ◽  
Special Cases ◽  
Multitype Branching Processes ◽  
Mean Time To Extinction ◽  
Mean Time

This paper deals with the computation of survival probabilities and extinction times for multitype positively regular branching processes. If all of the generating functions of the offspring distributions are of the linear fractional form and have the same denominator, explicit expressions may be obtained for all of their iterates. It is then possible to obtain formulae for survival probabilities and bounds on the mean time to extinction, given extinction, of a line descended from a single individual. If there are two types and the offspring distributions are bivariate Poisson, their generating functions may be bounded by linear fractional generating functions. It is then possible to compute upper and lower bounds on mean times to extinction, given extinction, and this is done for some special cases.

Survival probabilities and extinction times for some multitype branching processes

1974 ◽  
Vol 6 (3) ◽  
pp. 446-462 ◽  
Author(s):  
Edward Pollak
Keyword(s):  
Generating Functions ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Single Individual ◽  
Survival Probabilities ◽  
Time To Extinction ◽  
Special Cases ◽  
Multitype Branching Processes ◽  
Mean Time To Extinction ◽  
Mean Time

This paper deals with the computation of survival probabilities and extinction times for multitype positively regular branching processes. If all of the generating functions of the offspring distributions are of the linear fractional form and have the same denominator, explicit expressions may be obtained for all of their iterates. It is then possible to obtain formulae for survival probabilities and bounds on the mean time to extinction, given extinction, of a line descended from a single individual. If there are two types and the offspring distributions are bivariate Poisson, their generating functions may be bounded by linear fractional generating functions. It is then possible to compute upper and lower bounds on mean times to extinction, given extinction, and this is done for some special cases.

Optimal Land Use Analysis Using Lindenmayer Grammars

2013 ◽  
Vol 8 (2) ◽  
pp. 163-175
Author(s):  
Urszula Żukowska ◽  
Grażyna Kalewska
Keyword(s):  
Land Use ◽  
Urban Development ◽  
Computer Program ◽  
L System ◽  
L Systems ◽  
The City

In today's world, when it is so important to use every piece of land for a particular purpose, both economically and ecologically, identifying optimal land use is a key issue. For this reason, an analysis of the optimal land use in a section of the city of Olsztyn, using the L-system Urban Development computer program, was chosen as the aim of this paper. The program uses the theories of L-systems and the cartographic method to obtain results in the form of sequences of productions or maps. For this reason, the first chapters outline both theories, i.e. the cartographic method to identify optimal land use and Lindenmayer grammars (called L-systems). An analysis based on a fragment of the map of Olsztyn was then carried out. Two functions were selected for the analysis: agricultural and forest-industrial. The results are presented as maps and sequences in individual steps.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

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