Markovic

Ross Kinderman and J. Laurie Snell Markov fields and their applications. Contemporary Mathematics v. 1. American Mathematical Society, 1980. 

A. Mohammad-Djafari M. Nikolova. Eddy current tomography using binary markov model. Signal Processing, 49 119-132, 1996. 

While the Smoliichowski equation is necessary for a Markov process, in general it is not sufficient, but known counter-examples are always non-Gaiissian as well. 

Using W2 = 17jP2, (A3.2.81 and (A3.2.9) may be used to satisfy the Smoluchowski equation, (A3.2.2). another necessary property for a stationary process. Thus u(t) is an example of a stationary Gaussian-Markov... 

Markov process [12], Denoting tire inverse of R.j by and using the definition... 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

Onsager s theory can also be used to detemiine the fomi of the flucUiations for the Boltzmaim equation [15]. Since hydrodynamics can be derived from the Boltzmaim equation as a contracted description, a contraction of the flucUiating Boltzmann equation detemiines fluctuations for hydrodynamics. In general, a contraction of the description creates a new description which is non-Markovian, i.e. has memory. The Markov... 

Green M S 1954 Markov random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids J. Chem. Phys. 22 398... 

To conclude this section it should be pointed out again that the friction coefficient has been considered to be frequency independent as implied in assuming a Markov process, and that zero-frequency friction as represented by solvent viscosity is an adequate parameter to describe the effect of friction on observed reaction rates. 

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

The Boltzmaim weight appears implicitly in the way the states are chosen. The fomi of the above equation is like a time average as calculated in MD. The MC method involves designing a stochastic algorithm for stepping from one state of the system to the next, generating a trajectory. This will take the fomi of a Markov chain, specified by transition probabilities which are independent of the prior history of the system. 

Markovic N and Billing G D 1997 Semi-classical treatment of chemical reactions extension to 3D wave packets Chem. Phys. 224 53... 

We have tacitly assumed that the rate constants depend only on the last unit of the chain. In such a situation, the copolymerization is called a Markov copolymerization of first order. The special case (i), r r- = 1, is a Markov copolymerization of order zero. If reactivity also depends on the penultimate unit of the chain, the polymerization is a Markov copolymerization of second order. 

Tidsweii i M, Lucas C A, Markovic N M and Ross P N 1995 Surface structure determination using anomaious x-ray scattering Underpotentiai deposition of copper on Pt(111) Phys. Rev. B 51 10 205-8... 

Markovic N M, Gasteiger H A and Ross P N 1995 Copper electrodeposition on Pt(111) in the presence of chloride and (bi)sulphate Rotating ring-Pt(111) disk electrode studies Langmuir 11 4098-108... 

P. Deuflhard, W. Huisinga, A. Fischer, Ch. Schiitte. Identification of Almost Invariant Aggregates in Nearly Uncoupled Markov Chains. Preprint, Preprint SC 98-03, Konrad Zuse Zentrum, Berlin (1998)... 

The equilibrium distribution of the system can be determined by considering the result c applying the transition matrix an infinite number of times. This limiting dishibution c the Markov chain is given by pij jt = lim, o p(l)fc -... 

Closely related to the transition matrix is the stochastic matrix, whose elements are labelle a . TTiis matrix gives the probability of choosing the two states m and n between whic the move is to be made. It is often known as the underlying matrix of the Markov chain, the probability of accepting a trial move from m to n is then the probability of makir a transition from m to n (7r, ) is given by multiplying the probability of choosing states... 

Fig. 10.22 Hidden Markov model used for protein sequence analysis, are match states (corresponding in this...

Mian, K Sjolander and D Haussler 1994. Hidden Markov Models in Computational Biology. Applications to Protein Modelling. Journal of Molecular Biology 235 1501-1531). 

Having built a hidden Markov model for a particular family of proteins, it can then b< used to search a database. A score is computed for each sequence in the database anc those sequences that score significantly more than other sequences of a similar length ar( identified. This was demonstrated for two key families of proteins, globins and kinases ii the original paper [Krogh et al. 1994]. For the kinases, 296 sequences with a Z score abov<... 

Eddy S R 1996. Hidden Markov Models. Current Opinion in Structural Biology 6 361-365. 

Rabiner L R 1989. A Tutorial on Hidden Markov Models and Selected Applications in Spe Recognition. Fhroceedings cf the IEEE TI-lSl-19rB. 

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