Performance, Markovian

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

Under current treatment of statistical method a set of the states of the Markovian stochastic process describing the ensemble of macromolecules with labeled units can be not only discrete but also continuous. So, for instance, when the description of the products of living anionic copolymerization is performed within the framework of a terminal model the role of the label characterizing the state of a monomeric unit is played by the moment when this unit forms in the course of a macroradical growth [25]. 

The above-mentioned statistical characteristics of the chemical structure of heteropolymers are easy to calculate, provided they are Markovian. Performing these calculations, one may neglect finiteness of macromolecules equating to zero elements va0 of transition matrix Q. Under such an approach vector X of a copolymer composition whose components are X = P(M,) and X2 = P(M2) coincides with stationary vector n of matrix Q. The latter is, by definition, the left eigenvector of this matrix corresponding to its largest eigenvalue A,i, which equals unity. Components of the stationary vector... 

Thus, as can be inferred from the foregoing, the calculation of any statistical characteristics of the chemical structure of Markovian copolymers is rather easy to perform. The methods of statistical chemistry [1,3] can reveal the conditions for obtaining a copolymer under which the sequence distribution in macromolecules will be describable by a Markov chain as well as to establish the dependence of elements vap of transition matrix Q of this chain on the kinetic and stoichiometric parameters of a reaction system. It has been rigorously proved [ 1,3] that Markovian copolymers are formed in such reaction systems where the Flory principle can be applied for the description of macromolecular reactions. According to this fundamental principle, the reactivity of a reactive center in a polymer molecule is believed to be independent of its configuration as well as of the location of this center inside a macromolecule. 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

The Fourier transforms were performed in the standard way. No smoothing nor filtering was employed. Subtraction of the data from the least squares fit removes the constant or linear term characterizing a Markovian process. Fourier transform of the differences from the linear fit suppresses the enhancement of both the power and amplitude spectra at low frequencies. 

IV. MARKOVIAN MODELING OF ELECTROLYZER (ELECTROCHEMICAL REACTOR) PERFORMANCE... 

This equation can also be used for obtaining the master equation in the Markovian limit. In that case, the exponential decays faster than the state p evolves p(s) may therefore be replaced with pit) and taken out of the integral. Assuming still Re(f n) < 0, the integral may be performed explicitly and the master equation is... 

The same is true for re-re and si-si options. The statistical modeling of experimental pentad distribution can be performed by considering a first-order Markovian model. The insertion probability for a si... 

We argued and illustrated [13,14] that it is much easier to compute the above matrix than to perform the complete simulations from reactants to products. Arguments in support of the expected speedup are presented in Section 13.2.2. This matrix is finally used in a probabilistic non-Markovian framewoik to obtain the overall kinetics of the system. 

We start performing the above Markovian limit that gives ... 

More, within the potential form (5.397) the x and coordinates are separated and coupled, allowing the averages operations being performed firstly on the coordinates of the coupled x, electron, while replacing in the final result the referential electronic x influences. However, the present Markovian ELF picture is summarized by the following analytical steps (Putz, 2005) ... 

German, R. 2000. Performance Analysis of Communication Systems - Modeling with Non-Markovian Stochastic Petri Nets. New York John Chichester Wiley Sons. 

In order to rank performance of catalysts in copol)unerization with respect to comonomer incorporation and comonomer sequence distribution, copolymerization parameters have proven to be very useful. They are determined by means of nuclear magnetic resonance spectroscopy (NMR copolymer sequence analysis) taking into account the Markovian statistics of chain growth, as reviewed by Randall [4], Galimberti and coworkers [5] described the analysis of EPM prepared by means of... 

---