Linear propagator

Assuming linear propagation of nuclei, making appropriate substitutions into eqn. (1) and integrating, it may be shown that the total length of nuclei at time t, L(t), is given by... 

Applying equation (4.3.4) for the linear propagation of errors gives the covariance matrix Sy of the vector... 

In a real experiment with MatLab, the mean CSr was 79.8 ppm and the standard deviation 20.4ppm. The computed mean l/CSr was 0.0136ppm-1 and the standard deviation 0.0045 ppm -. These estimates are significantly biased relative to the linear propagation theory which would predict 0.0125 and 0.0031, respectively. This example shows that linear propagation should be applied with utmost care when a variable depends on another through a strongly non-linear relationship. <=... 

Now, we would like to attach a variance to the estimates of a and P that make c2 minimum. Given the complex and non-Unear nature of the gradient equations (5.4.67) and (5.4.68), we assume for simplicity that a and / are normally distributed and resort to linear propagation in order to retrieve an estimate for the covariance matrix of a and p. The covariance matrix Saj, of the vector (aT, ft) is approximated by... 

Branching steps need not necessarily occur rapidly because of the multiplication effect thus, their activation energies may be higher than those of the linear propagation reactions with which they compete [14],... 

Table 10.1 Linear propagation of lattice distortion (in terms of nnmber of sites r) calculated for silicate/hydrothermal solutions (liyama, 1974) and silicate/melt equilibria (Ottonello et al., 1978).

A quantitative description of the solidification process of a reactive mixture is based on the kinetic equations for that define the reaction parameters. One of these parameters is the "calorimetric" degree of conversion, which describes the linear propagation of polymeric chains ... 

The second limiting case arises when the rate of linear propagation of the reacting interface is so great that each particle decomposes as soon as it has a nucleus. For this case, Hume and Colvin give... 

A hnal operation that hnishes one solver step is to calculate the held in the frame moving with a an appropriate group velocity so that the pulse is kept in the center of the computational domain. This is achieved by applying a modihed linear propagator... 

Once again we follow the general procedure and approximate the linear propagator by its paraxial version ... 

M. Rodriguez, R. Bourayou, G. Mejean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eisloffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Woste, J.-P. Wolf, Kilometer-range non-linear propagation of femtosecond laser pulses, Physical Review E 69, 036607 (2004)... 

Scheme 22 Pseudo- linear propagation mode in the diyne polycyclotrimerization...

Polymer industries depend on the spontaneous polymerization of molecules into chains in response to an appropriate trigger. Polymerization reaction under STM was first observed by Grim et al., although the reaction itself was not induced with a current from the tunneling probe [37]. Recently, by using an STM tip, Okawa and Aono have successfully initiated and terminated the linear propagation of the chain polymerization of a diacetylene compound into a polydiacetylene compound at any chosen point with a spatial precision of about 1 nm [38]. 

The desirable situation for the occurrence and linear propagation of a detonation for homogeneous explosives exists in confined tubes or cylindrical shaped explosives, in which the system should not fall below the critical diameter (characteristic for every individual explosive) as it otherwise causes the wave front to be disturbed ( loss of energy to outside) and therefore the detonation velocity will be reduced. While for many secondary explosives one inch is a good tube diame-... 

Sonic waves comprise linear propagation of regions of compression and rarefaction, and are normally quite harmless. The systems normally adjust in-phase or with a lag time depending on the frequency, and the laws of thermodynamics are obeyed (Blandamer, 1973). However, at ultrasonic frequencies, if the power is increased, micro-cavita-... 

Figure 11 To the medium under investigation is rigidly attached the laboratory Cartesian reference system x, y, z, whereas to the molecule under consideration is attached the Cartesian system of axes x, Xj, Xj. The static electric field E, which polarizes the medium non-linearly, acts along the z-axis. The light wave which serves for analysing the induced non-linearity propagates along y...

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