Inner loop

In the inner loops of MD and MC programs, we consider an atom and loop over all atoms j to calculate the... 

As an example of how the approximate thermodynamic-property equations are handled in the inner loop, consider the calculation of K values. The approximate models for nearly ideal hquid solutions are the following empirical Clausius-Clapeyron form of the K value in terms of a base or reference component, b, and the definition of the relative volatility, Ot. 

Values of A and B for the base component are back-calculated for each stage in the outer loop from a suitable iC-value correlation (e.g. the SRK equation, which is also used to compute the iC values of the other components on each of the other stages so that values of Ot) can be computed). The values of A, B, and Ot are passed from the outer loop to the inner loop, where they are used to formulate the phase equihb-ria equation ... 

A computer solution was obtained as follows. The only initial assumptions are a condenser outlet temperature of 65 F and a bottoms-prodiict temperature of 165 F, The bubble-point temperature of the feed is computed as 123,5 F, In the initiahzation procedure, the constants A and B in (13-106) for inner-loop calcu-... 

In the inner-loop calculation sequence, component flow rates are computed from the MESH equations by the tridiagonal matrix method. The resulting bottoms-product flow rate deviates somewhat from the specified value of 50 lb mol/h. However, by modifying the component stripping factors with a base stripping factor, S, in (13-109) of 1,1863, the error in the bottoms flow rate is reduced to 0,73 percent. 

The initial inside-loop error from the solution of the normalized energy-balance equations, is found to be only 0,04624, This is reduced to 0,000401 after two iterations through the inner loop. 

At this point in the inside-out method, the revised column profiles of temperature and phase compositions are used in the outer loop with the complex SRK thermodynamic models to compute updates of the approximate K and H constants. Then only one inner-loop iteration is required to obtain satisfactory convergence of the energy equations. The K and H constants are again updated in the outer loop. After one inner-loop iteration, the approximate K and H constants are found to be sufficiently close to the SRK values that overall convergence is achieved. Thus, a total of only 3 outer-loop iterations and 4 inner-loop iterations are required. 

The system element dynamic equations can now be combined in the block diagram shown in Figure 4.31. Using equation (4.4), the inner-loop transfer function is... 

In this example, the inner loop is solved first using feedback. The controller and integrator are cascaded together (numpl, denpl) and then series is used to find the forward-path transfer function (numfp, denfp ). Feedback is then used again to obtain the closed-loop transfer function. 

The inner face of the activated receptor binds to the C-terminus of the G-protein a subunit (see Figure 7.3). Inner loop 3 (ic3) between transmembrane helices 5 and 6 of the receptor plays a critical role in this interaction. Note, however, that although the a subunit bears the primary binding site for the receptor, attachment of the py-dnner to the a subunit is essential for this interaction to occur. 

We now reduce the block diagram. The first step is to close the inner loop so the system becomes a standard feedback loop (Fig. 10.2b). With hindsight, the result should be intuitively obvious. For now, we take the slow route. Using the lower case letter locations in Fig. 10.2a, we write down the algebraic equations... 

With the choice of x = 0.5 s, but without the inner loop nor the secondary controller, the closed-loop equation is... 

A Routh-Hurwitz analysis can confirm that. The key point is that with cascade control, the system becomes more stable and allows us to use a larger proportional gain in the primary controller. The main reason is the much faster response (smaller time constant) of the actuator in the inner loop.2... 

This generic procedure affords the powder EPR absorption spectrum, which should be differentiated to get the powder EPR spectrum. Note that the whole procedure consists of three nested loops with the computation of an exponential (Equation 4.8) within the inner loop. Coded in a higher language (C, FORTRAN95) and run on a standard PC, this program will generate the EPR spectrum of a simple S = 1/2 or an effective S = 1/2 system in a split second (of the order of 10 ms or less). It is, however, useful to think about ways to make it as fast as possible, because extending... 

A more general case than (1) is that in which fgut is specified but N is not. This amounts to a two-dimensional search in which the procedure and criteria in case (1) constitute an inner loop in an outer-loop search for the appropriate value of N. Since N is a small integer, this usually entails only a small number of outer-loop iterations. 

To solve the problem a sequential quadratic programming code was used in the outer loop of calculations. Inner loops were used to evaluate the physical properties. Forward-finite differences with a step size of h = 10 7 were used as substitute for the derivatives. Equilibrium data were taken from Holland (1963). The results shown in Table E12.1B were essentially the same as those obtained by Sargent and Gaminibandara. 

In double CV, the CV strategy is applied in an outer loop (outer CV) to split all data into test sets and calibration sets, and in an inner loop (inner CV) to split the calibration set into training sets and validation sets (Figure 4.6). The inner loop is used to optimize the complexity of the model (for instance, the optimum number of PLS components as shown in Figure 4.5) the outer loop gives predicted values yjEST for all n objects, and from these data a reasonable estimation of the prediction performance for new cases can be derived (for instance, the SEPtest). It is important... 

The number of segments in the outer and inner loop (. 0ut and sin, respectively) may be different. Each loop of the outer CV results in an optimum complexity (for instance, optimum number of PLS components, Aopt)- In general, these Sout values are different for a final model the median of these values or the most frequent value can be chosen (a smaller complexity would avoid overfitting, a larger complexity would result in a more detailed model but with the risk of overfitting). A final model can be created from all n objects applying the final optimum complexity the prediction performance of this model has been estimated already by double CV. This strategy is especially useful for PLS and PCR. 

Basically, two types of approaches are developed here iterative (optimization-based) approaches like the one by Sippl et al. [101] and direct approaches like the one by Kabsch [102, 103], based on Lagrange multipliers. Unfortunately, the much expedient direct methods may fail to produce a sufficiently accurate solution on some degenerate cases. Redington [104] suggested a hybrid method with an improved version of the iterative approach, which requires the computation of only two 3x3 matrix multiplications in the inner loop of the optimization. 

On the other hand, the optimal control problem with a discretized control profile can be treated as a nonlinear program. The earliest studies come under the heading of control vector parameterization (Rosenbrock and Storey, 1966), with a representation of U t) as a polynomial or piecewise constant function. Here the mode is solved repeatedly in an inner loop while parameters representing V t) are updated on the outside. While hill climbing algorithms were used initially, recent efficient and sophisticated optimization methods require techniques for accurate gradient calculation from the DAE model. 

Shown in the letter o that has no inner loops and is relatively wide. This person will be honest and blunt when asked her opinion. If the o is open, then she will volunteer her frank opinion without being asked. 

Circles within circle letters on me right hand side. The larger me inner loop is, me more secrets this person wm withhold from omers. If the inner loop is huge, this person will try to avoid giving you a complete answer. 

Inner loops on the left side of the circle letters. This person is d... 

Second-order Monte Carlo analysis consists of 2 loops, the inner loop representing variability and the outer loop representing parameter uncertainty. To conduct an analysis, the following steps are required (also see Figure 7.1) ... 

Specify the number of inner and outer loop simulations for the 2nd-order Monte Carlo analysis. In the 1st outer loop simulation, values for the parameters with uncertainty (either constants or random variables) are randomly selected from the outer loop distributions. These values are then used to specify the inner loop constants and random variable distributions. The analysis then proceeds for the number of simulations specified by the analyst for the inner loop. This is analogous to a Ist-order Monte Carlo analysis. The analysis then proceeds to the 2nd outer loop simulation and the process is repeated. When the number of outer loop simulations reaches the value specified by the analyst, the analysis is complete. The result is a distribution of distributions, a meta-distribution that expresses uncertainty both from uncertainty and from variability (Figure 7.1). 

There are some issues associated with 2nd-order Monte Carlo analysis. Computational time can be a problem because the necessary number of replicates is squared with 2nd-order Monte Carlo analyses (i.e., number of inner loop simulations times number of outer loop simulations). In practice, specifying variability and uncertainty with random variables is a difficult exercise because the analyst is essentially trying to quantify what he or she does not know or only partially understands. 

Inputs for a 2nd-order Monte Carlo analysis to estimate exposure of Carolina wrens to a hypothetical pesticide (random variables are included in the inner loop of the Monte Carlo analysis, while random variables with uncertainty are included in both the inner and outer loops of the Monte Carlo analysis)... 

D Monte Carlo A kind of nested Monte Carlo simulation in which distributions representing both incertitude and variability are combined together. Typically, the outer loop selects random values for the parameters specifying the distributions used in an inner loop to represent variability. 

Fig. 23, Integrated density and step-height-dependent model parameter extraction approach. The outer loop finds the planarization length that best captures the density dependence, while the inner loop find the step-height model parameters that best explain up and down area polish data [48].

This is an inner loop used in the same spirit as in traditional Julia set computations. No complex numbers are required for the computation. Hold three of the coefficients constant and examine the plane detemined by the remaining two. This code runs in a manner similar to other fractal-generating codes in which color indicates divergence rate. rmu is a quaternion constant. 

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