Markov chain theory probabilities

The parameters a = l/rij5 the number of which equals m(m — IX are reciprocal reactivity ratios (2.8) of binary copolymers. Markov chain theory allows one, without any trouble, to calculate at any m, all the necessary statistical characteristics of the copolymers, which are formed at given composition x of the monomer feed mixture. For instance, the instantaneous composition of the multicomponent copolymer is still determined by means of formulae (3.7) and (3.8), the sums which now contain m items. In the general case the problems of the calculation of the instantaneous values of sequence distribution and composition distribution of the Markov multicomponent copolymers were also solved [53, 6]. The availability of the simple algebraic expressions puts in question the expediency of the application of the Monte-Carlo method, which was used in the case of terpolymerization [85,99-103], for the calculations of the above statistical characteristics. Actually, the probability of any sequence MjMjWk. .. Mrl 4s of consecutive monomer units, selected randomly from a polymer chain is calculated by means of the elementary formula ... 

The process described above is thus repeated with constant time intervals. So, we have a discrete time t = nAr where n is the number of displacement steps. By the rules of probability balance and by the prescriptions of the Markov chain theory, the probability that shows a particle in position i after n motion steps and having a k-type motion is written as follows ... 

The basic elements of Markov-chain theory are the state space, the one-step transition probability matrix or the policy-making matrix and the initial state vector termed also the initial probability function In order to develop in the following a portion of the theory of Markov chains, some definitions are made and basic probability concepts are mentioned. 

Recent applications of the Markov chain theory in geology have introduced continuous-lag modelling of spatial variability (Carle and Fogg, 1997 Fogg et al., 1998). The continuous-lag approach extends the probability of state transitions recorded at fixed intervals (discrete-lag) to any desired interval by considering conditional rates of... 

MARKOV CHAIN THEORY DEFINITION OF THE PROBABILITY MATRIX... 

From the finite Markov chain theory, it follows that the atomic fiactions x and x ) of atoms A and B in the alloy can be calculated with the probabilities and the following equations... 

Calculations. The statistical analysis of the copolymer sequencing was performed in terms of Markov chain theory, using the 2nd order Markov chain model. The equations were numerically solved with respect to the conditional probabilities by least squares minimization of the deviations of calculated and measured pentad contributions. 

The instantaneous composition of a copolymer X formed at a monomer mixture composition x coincides, provided the ideal model is applicable, with stationary vector ji of matrix Q with the elements (8). The mathematical apparatus of the theory of Markov chains permits immediately one to wright out of the expression for the probability of any sequence P Uk in macromolecules formed at given x. This provides an exhaustive solution to the problem of sequence distribution for copolymers synthesized at initial conversions p l when the monomer mixture composition x has had no time to deviate noticeably from its initial value x°. As for the high-conversion copolymerization products they evidently represent a mixture of Markovian copolymers prepared at different times, i.e. under different concentrations of monomers in the reaction system. Consequently, in order to calculate the probability of a certain sequence Uk, it is necessary to average its instantaneous value P Uk over all conversions p preceding the conversion p up to which the synthesis was conducted. 

Because the dependence of probability P Uk on x should be established by means of the theory of Markov chains, in order to make such an averaging it is necessary to know how the monomer mixture composition drifts with conversion. This kind of information is available [2,27] from the solution of the following set of differential equations ... 

A general and precise description of stereoisomerism in polymers is suggested on the basis of the repetition theory which describes the distinct patterns along a line that can be obtained from a three-dimensional motif. The probability models for describing the" stero-sequence length in various possible cases of interest in stereoregular polymers are discussed. It is shown that for describing the stereosequence structure, the simplest probability model must involve a Markov chain with four probability parameters. 

The consistent kinetic analysis of the copolymerization with the simultaneous occurrence of the reactions (2.1) and (2.5) leads to the conclusion that the probabilities of the sequences of the monomer units M, and M2 in the macromolecules can not be described by a Markov chain of any finite order. Consequently, in this very case we deal with non-Markovian copolymers, the general theory for which is not yet available [6]. However, a comprehensive statistical description of the products of the complex-radical copolymerization within the framework of the Seiner-Litt model via the consideration of the certain auxiliary Markov chain was carried out [49, 59, 60]. 

For the penultimate model, we just have the triad conditional probabilities we started with, of course. What we re actually doing is using the theory of Markov chains, named after a Russian mathematician who studied the probability of mutually dependent events. In this general approach we would write (Equation 6-30) ... 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

Graph theory Describes topology of networks and subnetworks, based on quantification of the number of nodes (signaling components) and links between them. Dynamic properties of networks through Boolean analysis. Network analysis based on probabilities (Markov chain and Bayesian) to identify paths and relationships between different nodes in the network. (75-81)... 

Note that we have completed the sum over Q with the term Q = Q because its probability vanishes. The term in the bracket exhausts all possibilities for a move from the state (g, —1), thus it adds to one. Hence 77(g) is a solution of (66) and by the theory of Markov chains, it is the probability distribution of the stationary state. 

We now turn to a formal treatment of the sampling scheme through the theory of Markov chains. Let the points in F be numbered i = 1,2,3,... and consider a random walker on these points. We are concerned with the conditional probability that the walker is at point j at step (t +1) given that it was at point ko at step 0, k at step 1,..., kr-i at step t -1), and i at step t. We... 

Recall the well-known theorem of the standard theory of the ergodic Markov chains one can state the following (e.g. Feller [4]) In any finite irreducible, aperiodic Markov chain with the transition matrix P, the limit of the power matrices/ exists if r tends to infinity. This limit matrix has identical rows, its rows are the stationary probability vector of the Markov chain, y = [v,Vj,...,v,...,v ], that is v = v P, fiuthermore v, >0 ( = 1,...,R) and... 

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