Propagator definition

Equation (A 1,6.8). along with the definitions (Al.h.S) and (Al.6.6) constitute the central equation for the propagation of electromagnetic waves in free space. The fonn of section A 1,6.4 admits hamionic solutions of the fonn... 

In Chap. 5, p was defined as the fraction (or probability) of functional groups that had reacted at a certain point in the polymerization. According to the current definition provided by Eq. (6.66), p is the fraction (or probability) of propagation steps among the combined total of propagation and termination steps. The quantity 1 - p is therefore the fraction (or... 

The kinetic chain length has a slightly different definition in the presence of chain transfer. Instead of being simply the ratio Rp/R, it is redefined to be the rate of propagation relative to the rates of all other steps that compete with propagation specifically, termination and transfer (subscript tr) ... 

Fracture Mechanics. Linear elastic fracture mechanics (qv) (LEFM) can be appHed only to the propagation and fracture stages of fatigue failure. LEFM is based on a definition of the stress close to a crack tip in terms of a stress intensification factor K, for which the simplest general relationship is... 

Wa.ter Qua.litySta.nda.rds, The first step in water quahty standards is stream use classification. The individual states must decide what the uses of their water will be. The four categories, as defined by the EPA, are Class A, primary water contact recreation Class B, propagation of desirable aquatic life Class C, pubHc water suppHes prior to treatment and Class D, agricultural and industrial uses. States may vary the definition of these classes to meet their own needs. The second step is to develop water-quaHty criteria. This is the specific concentration of a pollutant that is allowable for the designated use. 

Definition of Dust E losion A dust explosion is the rapid combustion of a dust cloud. In a confined or nearly confined space, the explosion is characterized by relatively rapid development of pressure with a flame propagation and the evolution of large quantities of heat and reaction products. The required oxygen for this combustion is mostly supphed oy the combustion air. The condition necessaiy for a dust explosion is a simultaneous presence of a dust cloud of proper concentration in air that will support combustion and a suitable ignition source. 

Fundamental, laminar, and turbulent burning velocities describe three modes of flame propagation (see the Glossary for definitions). The fundamental burning velocity, S, is as its name implies, a fundamental property of a flammable mixture, and is a measure of how fast reactants are consumed and transformed into products of combustion. Fundamental burning velocity data for selected gases and vapors are listed in Appendix C of NFPA68 (1998). 

A deflagration or detonation flame arrester fails hy definition if any flame propagates from the unprotected to the protected side. Failures can result for a numher of reasons, some of which are listed helow ... 

A uniform definition of a failure and a method of classifying failures is essential if data from different sources are to be compared. The anatomy of a failure includes the initiating or root cause of a failure that is propagated by contributory causes and results in a failure mode—the effect by which a failure occurs or is observed. Modes include failure to operate, no output, failure to alarm on demand. The end result of a failure sequence is the failure effect, such as no fluid is pumped to the absorber, or a tank overflows. As discussed in Appendix A of IEEE Std. 500-1984, only the equipment failure mode is relevant for data that are needed in a CPQRA. The failure model used in this book is based upon those in the IEEE publication and IPRDS. ... 

Many emulsion polymerizations can be described by so-called zero-one kinetics. These systems are characterized by particle sizes that are sufficiently small dial entry of a radical into a particle already containing a propagating radical always causes instantaneous termination. Thus, a particle may contain either zero or one propagating radical. The value of n will usually be less than 0.4. In these systems, radical-radical termination is by definition not rate determining. Rates of polymerization are determined by the rates or particle entry and exit rather than by rates of initiation and termination. The main mechanism for exit is thought to be chain transfer to monomer. It follows that radical-radical termination, when it occurs in the particle phase, will usually be between a short species (one that lias just entered) and a long species. 

Whereas the propagation of conditions through the modeling relationships always produces consistent completions in the definition of the initial state, this may not be true with the user-driven specifications. It is possible that a valve specified by the user to be OPEN, is found by the propagation of other conditions to be CLOSED. To detect such potential inconsistencies, we have developed a dependency network (Steele, 1980) to keep track of the flow of computations during constraint propagation. The dependency network is built on the undirected graph that represents the... 

Propagation of the above states through the modeling relationships completes the definition of the initial state of the plant. The results are shown in Fig. 14a. (Note that valves V34 through V43, which are downstream of the mixers, are considered closed, and they do not affect the synthesis of the switchover procedure.) The desired goal state is operationally defined by the following conditions ... 

Mixer-2-input-flow-i = positive i = HCN, H2SO4, HCHO Propagating these values through the modeling equations, we can complete the definition of the goal state (see Fig. 14b). Comparing initial and... 

A better alternative would be to use the propagation of errors definition, which takes into consideration values of both and si when calculating the MDL. This would involve generating at least five calibration curves in order to obtain an accurate measurement of si and Sm ... 

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