Birth-death operators

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

Bose birth-death (creation-annihilation) operators a-, at, 6-, fet ... 

The solar body that is the fruit of the Great Work is a vehicle in which the powers of both the physical and astral bodies are fully manifested. The astral body takes on the materiality of the physical, while the physical body inherits the powers of the astral vehicle. The result is a body, freed from the Wheel of Birth and Death, able to materialize at will—physical enough to be touched, to eat, yet subtle enough to raise the vibrations of its atoms at will and so operate upon any level of the... 

The Lotka-Volterra equations written in the dimensionless parameters contain only several control parameters birth and death rates a, (3 and the ratio of diffusion coefficients k = Da/(Da + Z)B), 0 k < 1, i.e., Da = 2k, Db = 2(1 - k) whereas their sum is constant, DA + DB = 2. Lastly, it is also the space dimension d determining the functionals J[Z], equations (5.1.36) to (5.1.38), the Laplace operator (3.2.8) as well as the boundary condition (8.2.21) for the correlation functions of similar particles. Before discussing the results of the joint solution of the complete set of the kinetic equations, let us consider first the following statements. 

In an earlier study by Wolfe et al. (1985) of Air Force personnel involved in Operation Ranch Hand, a significant increase in the number of reported neonatal deaths (no additional details provided), as compared to a comparison group of Air Force military employees not stationed in Vietnam, was observed. The incidence of major defects, prematurity, learning disabilities, or infant deaths was not increased in the Ranch Hand personnel. A significant increase in the incidence of minor health effects such as birth marks, rashes, and neonatal jaundice was reported by the Ranch Hand veterans. It should be noted that the pregnancy outcomes were self-reported, and this finding was not corroborated by the follow-up study (Wolfe et al. 1995) which used birth certificates, medical records, and death certificates to assess possible relationships between paternal exposure to 2,3,7,8-TCDD and developmental effects in offspring. 

The batch reactor is perhaps the most natural of all chemical reactors. Most events in the natural world are time-varying batch events. They start from some initial conditions and proceed to change dynamically with time over some batch time. The whole cycle of animal and plant life is a batch operation. The conception, gestation, birth, growth, and death of an animal occur in a batch cycle. Our first exposure to a chemical reaction is probably in a high school chemistry class in which the reaction takes place batchwise in a test tube or flask. 

The stages that a physical process or a management system goes through as it proceeds from birth to death. These stages include conception, design, deployment, acquisition, operation, maintenance, decommissioning, and disposal. 

In this notation D/Dt represents a generalization of the substantial time derivative operator. The B — D terms are the net birth and death terms collectively representing the net rate of production of particles of state (x, r) at time t in an environment of state Y. 

The first two terms on the right represent the addition (birth) and loss (death) of drops of volume v due to breakage. The next two terms deal with formation and loss due to coalescence, and the last two terms represent droplet flow into and out of the vessel. The last two terms are eliminated for batch operation. If the system is noncoalescing, the middle two terms are eliminated. If the system is purely coalescing (no breakage), the first two terms are eliminated. Purely coalescing systems exist, at least initially, when the impeller speed is decreased. At steady state, eq. (12-53) can be written as... 

The chemical operator Pchem acting on P describes a birth and death process, while the diffusion operator Idiff describes the random walk taking place between and its neighbors t... 

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