Mode equations

The total population dose D is the sum over the doses from M. ladure modes (equation 1.4-5). The probability of occuirence of the ith failure mode in the limit of large N is equation 1.4-6. The population dose per year per plant, and hence the plant risk, is given by equation 1.4-7 which says that risk is the expected consequence in the same. sense that an insurance premium is the expected consequence of the awards... 

Derivative mode (D) output is proportional to the rate of change of the input error, as can be seen from the three-mode equation. 

Formation of the intermediate organomercury peroxide 56 was rationalised in terms of an allylic mercuration providing an unsaturated hydroperoxide 55 that can cyclise by the favoured 54) 5-exo mode (equation 42 X = 02CCF3). However, this was not the main reaction pathway and the yields (2.7% and 0.6% of 2,m-10-dibromo-8,9-dioxabicyclo[5.2.1]decanes 54 were an order of magnitude lower than those of the 2,6-dibromides 53 obtained from 1,4-cyclooctadiene. 

Here for simplicity it is assumed that there is only one promoting mode. Equation (3.85) should be compared with Eq. (3.73). In Eq. (3.85) both vibronic and spin-orbit coupling are involved in W-,. /. Due to the rapid progress in ab initio calculations, it has now become possible to evaluate Wi >/ by using potential energy surfaces information obtained from ab initio calculations. 

The corresponding reaction with PlnSnM proceeds with complete inversion regardless of addition mode (equation 21). In this case, radical reactions appear unimportant. Displacement of the tosylate by a trimethyltin cuprate was also found to take place by inversion. 

Mode Equations to Determine the Effective Refractive Indices... 

From the three-layer mode equation, (15.3), the values cutoff and the penetration depth of the evanescent wave in the cover media, dv c, can be found for the three-layer structure as ... 

For an exhaustive analysis of the MCLW structure, the field distribution in the layers should be considered, similar to the analysis of the three-layer structure in the previous section. The field description of the three-layered structure in (15.1) can be expanded and applied for the four-layer structure to derive a mode equation derived in the same manner as the three-layer mode equation (15.1 and 15.2), e.g. by adding the electromagnetic fields in layer M to the solution ansatz ij/ in (15.1) ... 

This results in a mode equation quite similar to the three-layer mode equation, where only the term for the phase shift at the film/substrate boundary is interchanged with a new phase shift for the film/metal/substrate boundary ... 

Generally applicable mathematical representations of the dilational behavior of propellants have not been developed, as might be expected however, Fishman and Rinde (29) have derived empirical expressions for the formulations which they studied. These relationships give reasonable description of uniaxial behavior over wide ranges of strain, time, and temperature for several testing modes. Equation 1 is representative of one of the generalized expressions for the polyurethane and polybutadiene formulations studied. 

Several cyclofunctionalization reactions of alkynic alcohols are synthetically useful. Metal ion-promoted cyclofunctionalization of ris-2-propargylcyclopentanol systems proceeds by the 5-exo mode (equation 77 and Table 23).197 Protiodemetallation or reductive demetallation provides the cyclic enol ether in high yields. This method has been used by Noyori in the synthesis of prostacyclin (PGh).197b,197c Reactions with catalytic amounts of mercury(II) or palladium(II) salts gave the endocyclic enol ether as the major product.197 -198 A related cyclization with Ag2C03 has been reported by Chuche.191 Schwartz... 

Evaporated PDA(12-8) film was used as a nonlinear optical medium in a layered guided wave directional coupler. The directional coupling phenomenon happens in two adjacent waveguide by periodical energy transfer. The theory of linear directional coupler was exactly established [11]. It can be reduced to coupled mode equations ... 

In this study we suppose nonlinear organic material shows optical Kerr effect as n = n0+n2lEl2 and n2 = X<3)/(2n0). Moreover for simplification, we suppose the waveguides allow single mode propagations and TE polarization. After appropriate handling we get the following nonlinear coupled mode equations [ 12] ... 

For the solution Eq. 1 to be non-trivial, the following mode equation must be fulfilled [16,23] ... 

The sensitivity of the sensor is usually defined by the change in Nm per change in tic, i.e., dNm/dtic. An expression for this for, say, TE polarized fight can be derived by using perturbational analysis of the mode equation (Eq.2) [16] ... 

Fig. 1. Symmetric double-well potential U-(Q) for a pseudo-JT molecule with two nondegenerate electronic terms coupled to one low-symmetry mode [equation (9)]. The curve corresponds to strong coupling case with k = 4 and a relatively large energy gap, A = 12 (both in units of hcS). The dashed line represents the twofold degenerate ground-state energy level subject to a tunneling splitting.

The halocyclization of unsaturated benzyl sulfides is influenced by the degree of unsaturation. Thus alkenyl sulfides yield tetrahydrothiopyrans through (y-endo-trig and (y-exo-trig modes (Equation 141), whereas alkynyl sulfides afford 2-methylenetetrahydrothiopyrans by a 6-exo-dig cyclization (Equation 142) <1995JOC6468>. 

To obtain relation (3.46), one has to take into account that motion of the particles of the chain ought to be considered to be coherent. Now, it is not difficult to pass from equation (3.46) to the normal-mode equation... 

Now we can insert expressions for the modal fields and normalization constant into coupled mode equation (13.15)... 

To see how these harmonic solvent modes translate into vibrational friction, (43-46) consider how the correlation function for the solvent force on the frozen mode [Equation (12)], behaves at short times (47). The solvent modes themselves, q (t), are the displacements of the liquid along the 37V-dimensional eigenvectors e of each mode. Literally, if the 3 A-dimensional vector giving the position of every atom in the liquid at time t is R(t), the displacement from the time zero configuration R(0) is the sum... 

Figure 6 Simulations of the 2D Raman spectrum for the anharmonically (fully) coupled v and v4 modes [Equation (20)]. (a) The coupled response, (b) The g44 coupled response.

Flere again, the mechanism is proposed to follow the description of the overall process. The dissociation of a dimer (22) in one of two modes (equation 94) will scramble the... 

From the mode equation and equation (38), the waveguide sensitivity for both cover refractive index and the surface adlayer changes can be derived (three-layer planar waveguide). The derivation is given in detail in [7] and here only the results... 

Mode/ equations. The mathematical model requires eight concentration-independent coefficients a to fcf, k, and k2l. From these it calculates the five A coefficients with eqns 11.8 to 11.12 from these, the rates of aldehyde and alcohol with eqns 11.6 and 11.7 and finally the rates of olefin, paraffin, H2, and CO with eqns 11.2, 11.5, 11.10, 11.11, 11.3, and 11.4, respectively. Alternatively, eqns 11.8 to 11.12 can be used to replace the A coefficients in eqns 11.6 and 11.7 in order to obtain explicit rate equations for aldehyde and alcohol in terms of the phenomenological coefficients. However, the resulting rate equations are more cumbersome. 

The relationship between geometric mean and mode [equation (2.93)] takes the form ... 

Cyclic vinyloxiranes react with organocuprates with inversion of configuration (anti) and in a vinyl-ogous mode (equation 46), not by the expected Sn2 reaction that simple oxiranes are known to undergo. This subject has recently been summarize. ... 

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