Stochastic population processes in the theory of radiative transfer

1970 ◽  
Vol 7 (02) ◽  
pp. 272-290
Author(s):  
P. J. Brockwell
Keyword(s):  
Radiative Transfer ◽  
Transition Probability ◽  
Collision Integral ◽  
Time Interval ◽  
Reciprocity Principle ◽  
Initial State ◽  
First Passage ◽  
Generating Functional ◽  
Generalized State

Summary Starting from a characterization of radiative transfer in terms of a collision rate λ and a single-collision transition probability Ψ, we study the distribution of the generalized state ζ(t) of a radiation particle at time t conditional on a specified initial state at time t = 0. The generalized state is a vector consisting of the state ω(t) at time t and the states ω 1, ω 2, …, ω n of the particle immediately after the collisions it experiences in the time interval (0, t]. The variable ζ(t) takes values in a population space and can be studied conveniently with the aid of a certain generating functional G. The first-collision integral equation and the backward integro-differential equation for G are derived. Simultaneous consideration of the first-collision and last-collision equations lead to a generalized reciprocity principle for G. First-passage problems are also considered. Finally a number of illustrative examples are given.

Stochastic population processes in the theory of radiative transfer

1970 ◽  
Vol 7 (2) ◽  
pp. 272-290 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Radiative Transfer ◽  
Transition Probability ◽  
Collision Integral ◽  
Time Interval ◽  
Reciprocity Principle ◽  
Initial State ◽  
First Passage ◽  
Generating Functional ◽  
Generalized State

SummaryStarting from a characterization of radiative transfer in terms of a collision rate λ and a single-collision transition probability Ψ, we study the distribution of the generalized state ζ(t) of a radiation particle at time t conditional on a specified initial state at time t= 0. The generalized state is a vector consisting of the state ω(t) at time t and the states ω1, ω2, …, ωn of the particle immediately after the collisions it experiences in the time interval (0, t]. The variable ζ(t) takes values in a population space and can be studied conveniently with the aid of a certain generating functional G. The first-collision integral equation and the backward integro-differential equation for G are derived. Simultaneous consideration of the first-collision and last-collision equations lead to a generalized reciprocity principle for G. First-passage problems are also considered. Finally a number of illustrative examples are given.

scholarly journals Probability distributions of COVID-19 tweet posted trends use a nonhomogeneous Poisson process

2020 ◽  
Vol 1 (4) ◽  
pp. 229-238
Author(s):  
Devi Munandar ◽  
Sudradjat Supian ◽  
Subiyanto Subiyanto
Keyword(s):  
Social Media ◽  
Poisson Process ◽  
Exponential Distribution ◽  
Probability Distributions ◽  
Nonhomogeneous Poisson Process ◽  
Time Interval ◽  
Initial State ◽  
Time Intervals ◽  
Time Parameters ◽  
Parameter Values

The influence of social media in disseminating information, especially during the COVID-19 pandemic, can be observed with time interval, so that the probability of number of tweets discussed by netizens on social media can be observed. The nonhomogeneous Poisson process (NHPP) is a Poisson process dependent on time parameters and the exponential distribution having unequal parameter values and, independently of each other. The probability of no occurrence an event in the initial state is one and the probability of an event in initial state is zero. Using of non-homogeneous Poisson in this paper aims to predict and count the number of tweet posts with the keyword coronavirus, COVID-19 with set time intervals every day. Posting of tweets from one time each day to the next do not affect each other and the number of tweets is not the same. The dataset used in this study is crawling of COVID-19 tweets three times a day with duration of 20 minutes each crawled for 13 days or 39 time intervals. The result of this study obtained predictions and calculated for the probability of the number of tweets for the tendency of netizens to post on the situation of the COVID-19 pandemic.

scholarly journals Stochastic Comparative Statics in Markov Decision Processes

2021 ◽  
Author(s):  
Bar Light
Keyword(s):  
Markov Decision Processes ◽  
Dynamic Pricing ◽  
Optimization Problems ◽  
Transition Probability ◽  
Random Variable ◽  
Comparative Statics ◽  
Decision Processes ◽  
Optimal Decision ◽  
Initial State ◽  
Markov Decision

In multiperiod stochastic optimization problems, the future optimal decision is a random variable whose distribution depends on the parameters of the optimization problem. I analyze how the expected value of this random variable changes as a function of the dynamic optimization parameters in the context of Markov decision processes. I call this analysis stochastic comparative statics. I derive both comparative statics results and stochastic comparative statics results showing how the current and future optimal decisions change in response to changes in the single-period payoff function, the discount factor, the initial state of the system, and the transition probability function. I apply my results to various models from the economics and operations research literature, including investment theory, dynamic pricing models, controlled random walks, and comparisons of stationary distributions.

Theoretical study of Zeno and anti-Zeno effects on photodissociation dynamics: A model approach

2016 ◽  
Vol 15 (08) ◽  
pp. 1650070 ◽  
Author(s):  
Bikram Nath ◽  
Chandan Kumar Mondal
Keyword(s):  
Repeated Measurements ◽  
Time Interval ◽  
Mathematical Tool ◽  
Population Transfer ◽  
Vibrational States ◽  
Initial State ◽  
Zeno Effect ◽  
Dissociation Dynamics ◽  
Vibrational Population

Zeno and anti-Zeno effects in the evolution of the multi-photonic dissociation dynamics of the diatomic molecule HBr[Formula: see text] owing to repeated measurements demand if the system in the initial state have been studied. The effects have been calculated numerically for the case of vibrational population transfer and dissociation dynamics of HBr[Formula: see text] taking it as a model. We use time-dependent Fourier grid Hamiltonian (TDFGH) method as a mathematical tool in presence of intense radiation field as perturbation. The effects have been explored through a probable mechanism of population transfer from the ground vibrational state to the different upper vibrational states which ultimately go to the dissociation continuum. The results show significant differences in the mechanism of population transfer and the significant role of time interval of measurement ([Formula: see text] in Zeno and anti-Zeno effects. In case of survival probability of ground vibrational states, there is Zeno effect when the frequency of the laser to which the molecule is submitted is near the vibrational [Formula: see text] to [Formula: see text] resonance, while there is anti-Zeno effect if it is far from this resonance.

The first-passage density of the Brownian motion process to a curved boundary

1992 ◽  
Vol 29 (02) ◽  
pp. 291-304 ◽  
Author(s):  
J. Durbin ◽  
D. Williams
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Sample Path ◽  
Practical Method ◽  
Time Interval ◽  
Brownian Motion Process ◽  
First Passage ◽  
Curved Boundary ◽  
Multiple Integrals ◽  
High Degree

An expression for the first-passage density of Brownian motion to a curved boundary is expanded as a series of multiple integrals. Bounds are given for the error due to truncation of the series when the boundary is wholly concave or wholly convex. Extensions to the Brownian bridge and to continuous Gauss–Markov processes are given. The series provides a practical method for calculating the probability that a sample path crosses the boundary in a specified time-interval to a high degree of accuracy. A numerical example is given.

Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (04) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn , where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and π n for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of π n to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r 0 > 1 with , then the exact convergence rate of π n to π is characterized by r 0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between π n and π based on the moments of A.

Final characterization of InLambda delay standards for supplementary time interval comparison

2018 ◽  
Author(s):  
Albin Czubla ◽  
Piotr Szterk ◽  
Roman Osmyk ◽  
Borut Pinter ◽  
Rado Lapuh ◽  
...  
Keyword(s):  
Time Interval ◽  
Interval Comparison

scholarly journals Wave length and celerity downstream from a hydraulic jump

RBRH ◽  
2019 ◽  
Vol 24 ◽  
Author(s):  
Ana Paula Gomes ◽  
Eduardo Pivatto Marzec ◽  
Luiz Augusto Magalhães Endres
Keyword(s):  
Dispersion Equation ◽  
Hydraulic Jump ◽  
Wave Length ◽  
Vortex Formation ◽  
Wave Period ◽  
Time Interval ◽  
Significant Wave ◽  
Wave Characteristics ◽  
Wave Celerity

ABSTRACT The wave period, i. e., the time interval which corresponds to a complete oscillation, is an important parameter of wave characterization. It allows the estimation of other important wave characteristics such as the length and celerity. This study aims at describing the results of a relationship among the significant, mean, and peak periods of waves generated downstream from a hydraulic jump. The frequency of vortex formation in the roller region within the hydraulic jump was used. Besides those relationships, wave lengths were also determined by the dispersion equation by considering the wave-current overlapping effect in order to identify the wave celerity. Estimated results of wave celerity were compared to their experimental results. Our findings allowed us to identify that the significant wave period was the most representative period for the characterization of a wave downstream from a hydraulic jump.

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