Assigned equations

Since the Lindemann ratio dija 0.1 is empirically roughly the same for all substances, one expects the g value, as measured by sound attenuation, to be correlated with the glass transition temperature. Note that this relationship is independent of the details of the bead assignment. Equation (20), if rewritten as Tg di/a) g, is almost obvious, given the interaction of the form in Eq. (17) The typical lattice displacement, at Tg, is roughly A< ) j- dija. On the other hand, the typical structural excitations have the energy of about Tg, at the glass transition. 

In general for a set of nonlinear equations the necessary condition for determinancy is that there exists at least one admissible set of output variables for the equations (C7, S4). We can think of an output variable as that variable for which a given equation is solved either by an iteration process or by an elimination process. The set of all such assigned pairs of variables and equations is called the output set. Clearly an admissible output set must satisfy the conditions (i) each equation has exactly one output variable, (ii) each variable appears as the output variable of exactly one equation, (iii) each output variable must occur in the assigned equations in such a manner that it can be solved for uniquely. Such an output set (circled entities) is illustrated in terms of an occurrence matrix in Fig. 4. Algorithms for finding output sets have been published by Steward (S4) and Gupta et al. (G8). 

In this case we assign Equation 2 to fs (determinable), leaving two unassigned equations, 1 and 3. Variables fo and fi are still not determinable, but now Equation 1 is not assigned and contains only measured variables thus, this is a redundant equation and the associated variables are also redundant or overmeasured. 

Finally, let us consider fi measured instead of /3. The new occurrence matrix is in Table 3. Now, we can assign all the equations by assigning Equations 1, 2, and 3 to /3, /s, and fi, respectively. In this case, all the unmeasured process variables are determinable from the available information however, there are no redundant measured variables. 

A-X =f 7 Assigned equation for the estimation of unmeasured variable 7 it belongs to the output... 

First Step. From the output set assignment we obtain equations that are not assigned. This set E of not assigned equations is composed of three subsets of equations. 

Subset E2 of not assigned equations which contain unmeasured but determinable parameters (i.e. they have been assigned as output to equations not included in set E) ... 

Cyclic 1,1-enediamines 8 (n = 2) react with maleic anhydride to give products to which y-lactam fused heterocyclic structure 132 had been assigned (equation 48)126. 

Summarizing, we place a probability function for the position steps the position with a stochastic process. While in the deterministic case a fixed increment is added, in the stochastic case the generating function is multiplied by an increment. The assigned equation for the probability generating function to the stochastic equation X (t + At) = KAt + X (t) is therefore... 

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