scholarly journals Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Basile Morcrette ◽  
Hosam M. Mahmoud
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Probability Generating Function ◽  
Isomorphism Theorem ◽  
Systems Of Differential Equations ◽  
Exactly Solvable ◽  
Exact Distributions ◽  
International Audience ◽  
Weighted Probability

International audience This paper develops an analytic theory for the study of some Pólya urns with random rules. The idea is to extend the isomorphism theorem in Flajolet et al. (2006), which connects deterministic balanced urns to a differential system for the generating function. The methodology is based upon adaptation of operators and use of a weighted probability generating function. Systems of differential equations are developed, and when they can be solved, they lead to characterization of the exact distributions underlying the urn evolution. We give a few illustrative examples.

scholarly journals Strong Stability Preserving IMEX Methods for Partitioned Systems of Differential Equations

2021 ◽  
Author(s):  
Giuseppe Izzo ◽  
Zdzisław Jackiewicz
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Order Of Convergence ◽  
Large Range ◽  
Strong Stability ◽  
Strong Stability Preserving ◽  
Systems Of Differential Equations ◽  
Imex Methods ◽  
Stage Order

AbstractWe investigate strong stability preserving (SSP) implicit-explicit (IMEX) methods for partitioned systems of differential equations with stiff and nonstiff subsystems. Conditions for order p and stage order $$q=p$$ q = p are derived, and characterization of SSP IMEX methods is provided following the recent work by Spijker. Stability properties of these methods with respect to the decoupled linear system with a complex parameter, and a coupled linear system with real parameters are also investigated. Examples of methods up to the order $$p=4$$ p = 4 and stage order $$q=p$$ q = p are provided. Numerical examples on six partitioned test systems confirm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration, and they are also suitable for preserving the accuracy in the stiff limit or preserving the positivity of the numerical solution for large stepsizes.

scholarly journals The average position of the first maximum in a sample of geometric random variables

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Margaret Archibald ◽  
Arnold Knopfmacher
Keyword(s):  
Generating Function ◽  
Probability Generating Function ◽  
Random Variables ◽  
Expected Value ◽  
Factorial Moments ◽  
Maximum Value ◽  
Fourier Expansions ◽  
First Occurrence ◽  
International Audience ◽  
Average Position

International audience We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we compute the first two (factorial) moments. The asymptotic technique known as Rice's method then yields the main terms as well as the Fourier expansions of the fluctuating functions arising in the expected value and the variance.

scholarly journals Some exactly solvable models of urn process theory

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Philippe Flajolet ◽  
Philippe Dumas ◽  
Vincent Puyhaubert
Keyword(s):  
Large Deviation ◽  
Local Limit ◽  
Limit Laws ◽  
Coupon Collector ◽  
Exactly Solvable ◽  
Classical Models ◽  
International Audience ◽  
Contagion Model ◽  
Convergence Estimates

International audience We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector's problem, particle transfer (the Ehrenfest model), Friedman's "adverse campaign'' and Pólya's contagion model, as well as the OK Corral model (a basic case of Lanchester's theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail "semi-sacrificial'' urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of $2$-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., "autistic'' (generalized Pólya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable.

SOLUTION OF SOME SYSTEMS OF DIFFERENTIAL EQUATIONS IN MATHEMATICAL SYSTEM MAPLE

2013 ◽  
Vol 1 (05) ◽  
pp. 58-65
Author(s):  
Yunona Rinatovna Krakhmaleva ◽  
◽  
Gulzhan Kadyrkhanovna Dzhanabayeva ◽  
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations ◽  
Mathematical System

On decomposability of countable systems of differential equations

1993 ◽  
Vol 45 (10) ◽  
pp. 1598-1608
Author(s):  
A. M. Samoilenko ◽  
Yu. V. Teplinskii
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

scholarly journals Differential Games for an Infinite 2-Systems of Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1467
Author(s):  
Muminjon Tukhtasinov ◽  
Gafurjan Ibragimov ◽  
Sarvinoz Kuchkarova ◽  
Risman Mat Hasim
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Differential Games ◽  
Differential Game ◽  
Infinite System ◽  
The State ◽  
Geometric Constraints ◽  
Control Parameters ◽  
Systems Of Differential Equations ◽  
Pursuit And Evasion

A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to the origin of the Hilbert space l2 and the evader tries to prevent this. Differential game is completed if the state of the system reaches the origin of l2. The problem is to find a guaranteed pursuit and evasion times. We give an equation for the guaranteed pursuit time and propose an explicit strategy for the pursuer. Additionally, a guaranteed evasion time is found.

scholarly journals Period-Life of a Branching Process with Migration and Continuous Time

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 868
Author(s):  
Khrystyna Prysyazhnyk ◽  
Iryna Bazylevych ◽  
Ludmila Mitkova ◽  
Iryna Ivanochko
Keyword(s):  
Random Process ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Probability Generating Function ◽  
Time Interval ◽  
The Moment ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

Sign in / Sign up

Export Citation Format

Share Document