Stochastic process, 118

Wax N (ed) 1954 Selected Papers on Noise and Stochastic Processes (New York Dover)... 

Montroll E W and Shuler K E 1958 The application of the theory of stochastic processes to chemical kinetics Adv. Chem. Phys. 1 361-99... 

Oppenheim I, Shuler K E and Weiss G H 1977 Stochastic Processes in Chemicai Physics, The Master Equation (Cambridge, MA MIT Press)... 

Van Kampen N G 1981 Stochastic Processes in Physics and Chemistry (Amsterdam North-Holland)... 

Ultimately a plasmid is defined by its mode of DNA repHcation. DNA repHcation is initiated at a single, characteristic sequence, termed the origin. The origin sequence determines the copy number of the plasmid relative to the host chromosome and the host enzymes that are involved in plasmid repHcation. Two different plasmids that contain the same origin sequence are termed incompatible. This term does not refer to the active exclusion of one plasmid by another from the ceU but rather to a stochastic process by which the two plasmids are partitioned differentially into progeny ceUs. A ceU which contains two plasmids of the same incompatibiHty group segregates two clonal populations, each of which has one of the two plasmids in it. 

As discussed above, by changing the geometry of the lattice, it is possible to change the intrinsic nature of the stochastic process. On the other hand, Meng et al. [80] have shown that by adding a new reaction channel, namely... 

Diffusion is a stochastic process associated with the Brownian motion of atoms. For simplicity we assume a one-dimensional Brownian motion where a particle moves a lattice unit <2 in a short time period Td in a direction either forward or backward. After N timesteps the displacement of the particle from the starting point is... 

To see the connection between this stochastic process and a chemically reacting system, consider the first step of Scheme IX. Each (real) molecule of A has an equal and constant probability of reacting in time t. In the simulation, each position in the grid has an equal and constant probability (p) of being selected. For this first-order reaction, the chemical system is described by... 

Nearby the melting point, the formation or disappearance of a nucleus is actually a stochastic process. Thus, the nucleation process can be treated by using the method of the stochastic processes. This method has... 

Y. Beers, Introduction to the Theory of Error, Addison-Wesley Publishing Company, Cambridge, Mass., 1953. In this connection see also J. L. Doob, Stochastic Processes, John Wiley and Sons, New York, 1953 R. D. Evans, The Atomic Nucleus, McGraw-Hill Book Company, New York, 1955. 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

The complete specification of a random process requires us to have some way of writing down an infinite number of distribution functions. For practical reasons, this is an impossible task unless all the distribution functions can be specified by means of a rule that enables one to calculate any distribution function of interest in terms of a finite amount of prespecified information. The following examples will illustrate these ideas by showing howr some particular stochastic processes of interest can be defined. 

A. Blanc-Lapierre and B. Forfcet, Thhorie des Fonctions Aleatoires, Masson et Cie, Paris, 1953 J. L. Boob, Stochastic Process, John Wiley and Sons, Inc., New York, 1953 . Parzen, Stochastic Process, Holden-Bay, Inc., San Francisco, 1962. 

W. B. Davenport, Jr., and W. L. Boot, An Introduction to the Theory of Random Signal and Notae, McGraw-Hill Book Co., Hew York, 1958 J. L. Doob, Stochastic Process, John Wiley and Sons, Inn., Hew York, 1953. 

The literature of science is replete with models. This variety enables one to make some interesting observations. Thus, for example, one rarely regards models as unique or absolute, although, through the choice of a specific one (e.g., a differential equation), unique solutions to problems may be obtained. A model is formulated to serve a specific purpose. Some models may be suitable for generalization, others may not be. These generalizations are more profitably made as extrapolations for scientific purposes, and occasionally as useful philosophical observations. A model must be flexible to absorb new information, and, hence, stochastic processes have broader and richer applicability than deterministic models. 

The problems of operations research have stimulated new developments in several mathematical fields various aspects of game theory, stochastic processes, the calculus of variations, graph theory, and numerical analysis, to name a few. 

Whatever model is used to describe an operations research problem, be it a differential equation, a mathematical program, or a stochastic process, there is a natural tendency to seek a maximum or a minimum with a certain purpose in mind. Thus, one often finds optimization problems imbedded in the models of operations research. 

A stochastic process is a family of random variables X( depending on a parameter t (e.g., time). This definition may be extended to include... 

J. L. Doob, Stochastic Processes, John Wiley and Sons, Inc., New York, 1953. 

A Compound Distribution.—We now give an illustration of computations performed on a stochastic process. 

Steadiness of vacuum and one-particle states, 657 Steck B.,5Z8 Steck operator, 538 Steepest descents, method of, 62 Stochastic processes, 102,269 Strangeness quantum number, 516 Strategic saddle point, 309 Strategy, 308 mixed, 309... 

Rytov S. M. Principles of Statistical Radiophysics Vol. I Elements of Random Process Theory. Springer, Berlin. (1987) [Stochastic Processes. Moscow, Nauka (1976)]. 

Najera I, Richman DD, Olivares I, Rojas JM, Peinado MA, Perucho M, Najera R, Lopez GaHndez C (1994) Natural occurrence of drug resistance mutations in the reverse transcriptase of human immunodeficiency virus type 1 isolates. AIDS Res Hum Retroviruses 10 1479-1488 Nijhuis M, Boucher CAB, Schipper R Leitner T, Schuurman R, Albert J (1998) Stochastic processes strongly influence HIV-1 evolution during suboptimal protease inhibitor therapy. Proc Natl Acad Sci USA 95 14441-14446... 

The statistic models consider surface roughness as a stochastic process, and concern the averaged or statistic behavior of lubrication and contact. For instance, the average flow model, proposed by Patir and Cheng [2], combined with the Greenwood and Williamsons statistic model of asperity contact [3] has been one of widely accepted models for mixed lubrication in early times. 

X 10 N and the value obtained by the simplest form, 1.45 X 10 " N m n = 918 and a = 0.31 nm for Equation 21.1). These comparisons imphed that the measurements were consistent with the theoretical predictions. The deviation between the rupture length of 260.9 nm and the fitted-contour length indicated that the polymer chain was not fully stretched at the rupture event. The reason for this was that the rupture event was a stochastic process and was dependent on many factors such as pulling speed, bond strength, and temperature. The vahdity of the freely jointed (FJC) model (dashed fine) was also checked ... 

The principle of electrochemical noise experiments is to monitor, without perturbation, the spontaneous fluctuations of potential or current which occur at the electrode surface. The stochastic processes which give rise to the noise signals are related to the electrode kinetics which govern the corrosion rate of the system. Much can be learned about the corrosion of the coated substrate from these experiments. The technique of these measurements is discussed elsewhere (A). 

A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1994)... 

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