Dummy time variable

Here, u is a dummy (time ) variable. Equation (4.5.10] can be integrated when a second piece of information is available. We have discussed a few cases. One of these, the simplest, is that C (0,t) always remains zero. It is a somewhat academic case which would apply if the molecules arriving at the interface were Immediately absorbed by the upper phase (a liquid, in this case). Then the process is continually diffusion-controlled and... 

X = time constant of the exponential modifier t = centre of gravity (top) of the Gaussian t = dummy integration variable... 

In this case, b is a constant vector, but we can use an equation that will work even when the b vector is a function of time. Recall that x is a parameter (equation (2.12)). We will use b2 with the dummy integration variable tl (even though it is not needed) to illustrate the procedure. 

Note that in this equation t has become the fixed time of the observation of the strain and, for the purposes of integration, can be regarded as a constant. The stress history is accounted for in terms of the dummy integration variable s. The lower limit of integration is taken as -qo, because the complete stress history contributes to the observed strain. The upper limit is t, the time of observation of the strain, because stresses applied after t can have no effect on the observed strain. 

Here D is the diffusion coefficient, t is the time, t is a dummy integration variable. Using Equation (8), respective T(t) dependencies can be obtained, while the Equations (l)-(7) serve as boundary condition for the diffusion model. This set of equations yield a quasi-equilibrium diffusion model which means that at a given surface pressure the composition of the surface layer under dynamic conditions is equal to that in the equilibrium. Another regime of adsorption kinetics, called kinetic model, can also be described by assuming compositions of the adsorption layer that can differ from the equilibrium state. The deviation of the adsorption layer from the equilibrium composition is the result of the finite rate of the transition process between the adsorption states. In case of two adsorption states we have6... 

Bu = second virial coefficient, m3/mol, Equations 7, 10, and 13 Cij = pair direct correlation function Cij = spacial integral of c i times density, Equation 1 ft = component fugacity, KPa Hu = Henry s constant of solute i in solvent /, KPa K12 = binary parameter, Equation 12 N = number of components P = total pressure, KPa r = separation of molecular centers, meters R = universal gas constant, KJ/mol-K t = dummy integrating variable, Equations 3-6 and 19-23 T = absolute temperature, K T = characteristic temperature, K % = T/Tt ... 

The dummy time is introduced so that no confusion can arise as to what variables enter into the integration process. 

I estimated a version of equation (7.1) in which i denotes vaccine i (i = 1, 2,..., 13), and the continuous variable f was replaced by a set of year dummies. The model was estimated via weighted least squares, where the weight was equal to the market value (price times quantity) of that vaccine in thatyear. The coefficients on these year variables maybe considered values of a Center for Disease Control vaccine price index. Nominal FSS and Centers for Disease Control vaccine price indexes are compared in Figure 7.2... 

Equation (1.15) has been integrated numerically on a logarithmic scale to determine the dummy variable The logarithmic scale weighs more uniformly the relaxation times, which may cover several orders of magnitude. A change of variables, y = n (k = ln(T g/T ), transformed the distribution function given by... 

Results from these experimental runs were used as x, q data records to fit the parameters of six ANNs. In the experimental effort, a different feedforward ANN was used after each intermediate secondary measurement was obtained in the simulation-based effort, only one ANN accommodates all secondary measurements, and averaged dummy inputs are used for those secondary measurements not yet obtained. In addition in the experimental effort, a different ANN was used for final thickness and final void content predictions in the simulation-based effort, one ANN was used to predict both final thickness and final void content. The advantage of using one ANN to predict all values of q is that the parameters of only one ANN need be fitted. Fitting the parameters of an ANN for each variable in q is much more time-consuming. The disadvantage, however, is that the parameters A and abias are the same for each variable in q when just one ANN is used as an on-line model. When a different ANN is used for each variable in q, the parameters in A and abias are unique for each of those output variables, which results in increased on-line prediction accuracy. Similar speed-versus-accuracy arguments apply to the choice of one ANN for all secondary measurements versus an ANN for each secondary measurement. 

A two way fixed effects model Suppose the fixed effects model is modified to include a time specific dummy variable as well as an individual specific variable. Then, yit = a, + y, + P x + At every observation, the individual- and time-specific dummy variables sum to one, so there are some redundant coefficients. The discussion in Section 13.3.3 shows one way to remove the redundancy. Another useful way to do this is to include an overall constant and to drop one of the time specific and one of the time-dummy variables. The model is, thus, ylt = 5 + (a, - aj) + (y, - y,) + P x + e . (Note that the respective time or individual specific variable is zero when t or i equals one.) Ordinary least squares estimates of P can... 

Equations (89) and (90) define functions of time obtained by differentiation followed by substitution of x andy by l/jq and l/ 2,respectively,so that the dummy variable terms cancel in the derivative. , ... 

Fig. 6-13. Fit of the time series from regression with dummy variables...

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