Propagation equations

These equations are solved for (R ) and (M ) and substituted into the propagation equation. The rate of polymerization becomes... 

These two contributions can be evaluated separately in terms of the sensitivity coefficients of the result to the measured quantities by using the propagation equation of Kline and McClintock (1953) ... 

The main goal of the pn ocedure adopted to solve the propagator equation is to find the pole(s) of interested and the residues associated to it. Since G (E) has single isolated poles then the residue theorem can be applied to yield... 

Inclusion of this mass transfer term results in the propagation equation... 

Equations (8.11) and (8.12) are approximate expressions for propagating the estimate and the error covariance, and in the literature they are referred to as the extended Kalman filter (EKF) propagation equations (Jaswinski, 1970). Other methods for dealing with the same problem are discussed in Gelb (1974) and Anderson and Moore (1979). 

In case of a transverse wave, the propagation equation becomes ... 

Error propagation analysis is the estimation of error accumulation in a final result as a consequence of error in the individual components used to obtain the result. Given an equation explicitly expressing a result, the error propagation equation can be used to estimate the error in the result as a function of error in the other variables. 

Application of the Error Propagation Equation (Equation 11) gives ... 

This can be shown ly plication of the error propagation equation (Equation 11). If the error in the peak molecular weight M is represented by the error variance, s and we let... 

In thip appendix, a summary of the error propagation equations and objective functions used for standard characterization techniques are presented. These equations are Important for the evaluation of the errors associated with static measurements on the whole polymers and for the subsequent statistical comparison with the SEC estimates (see references 26 and 2J for a more detailed discussion of the equations). Among the models most widely used to correlate measured variables and polymer properties is the truncated power series model... 

The wave vector of a homogeneous wave may be written k = (k + zk")e, where k and k" are nonnegative and is a real unit vector in the direction of propagation. Equation (2.45) requires that... 

Our objective is to design an optical device that will change the polarization from horizontal to vertical linear polarization - a rotation of the Stokes vector 5i,52,53 from // = (1,0,0 to V = (0,1,0 - and to do so independently of the wavelength. For this purpose, we require a propagation equation for the Stokes vector, obtained from Eq. (5.14) and the definitions (5.16) in much the same way that Feynman et al. [9] convert the two-state TDSE into a torque equation for combinations of products of probability amplitudes see Appendix 5.B. The equations... 

Rather than using a medium that has a continually varying angle cp, we consider the rotation of cp to take place in discrete steps. To describe these changes, it is useful to replace the torque form of the propagation equation, (5.18), with one better suited to use in numerical simulation. 

To justify the basic propagation equation (5.14) or (5.19), it is essential that there be no reflections from interfaces. This means that either the individual layers be coated for broadband antireflection or that the incremental change of optical properties p. cp across an interface is small. 

In these appendices, we provide connections between the present Stokes-vector approach to optical design and alternatives. To write the propagation equation in a form appropriate to the two-state TDSE we define the dimensionless slowly varying parameters... 

Then the propagation equation is recognizable as that of the two-state atom. 

The basic differences between spherical and cylindrical symmetry are in the propagation equations for the water and expln products, the equations of state and the shock front conditions remaining unchanged. Thus, even for acoustic waves, pressure for cylindrical waves varies as r-1/2 F(t—r/c0) where F is an undetermined function, as compared with r 1 F(t—r/c0), valid at any distance for acoustic spherical waves. The development of a finite amplitude theory will not therefore be as simply related to the actual state of affairs, and errors incurred in approximations used will be larger than for spherical waves... 

These results, combined with a large number of field measurements, have shown that the propagation equation can typically be expressed as ... 

Bestian and Clauss proposed that the polymerization occured with isomerization on a cationic alkyltitanium species or one of its associated forms. Propagation by anionic and cationic species accounts for their results more easily. Most of the oligomer low molecular weight product was from anionic type propagation (Equation 8). However, the 7.8% of the dimer and the 30% of the trimer fractions were produced by cationic propagation of the n-butyl group (Equation 9). 

The propagation equation for elastic disturbances is obtained by adding the inertia force to the body force. Wc get then... 

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