The Algorithms

Perhaps the simplest and most obvious way to compute the opmtional space inertia matrix (or its inverse) is by the explicit inversion and multiplication of the Jacobian and joint space inertia matrices as shown in Equations 4.19 and 4.20. We will call this the Explicit Inversion/Multiplication Method. Although we will see that this is not the most efficient approach, it is in standard use today and may serve as a benchmark for new computational approaches. 

The scalar operations (multiplications, additions) required to compute A and its inverse using the Explicit Inversiori/Multiplication Method are shown in Thble 4.2. All operations are given for an V degree-of-freedom manipulator 

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The algorithm employed in the estimation process linearizes the constraint equations at each iterative step at current estimates of the true values for the variables and parameters. 

Application of the algorithm for analysis of vapor-liquid equilibrium data can be illustrated with the isobaric data of 0th-mer (1928) for the system acetone(1)-methanol(2). For simplicity, the van Laar equations are used here to express the activity coefficients. 

This basic approach can be developed into a formal algorithm known as the problem table algorithm. To jllustrate the algorithm, it can be developed using the data from Fig. 6.2 given in Table 6.2 for AT ,i = 10°C. 

A simple algorithm can be developed (see App. E) to target the minimum total number of shells (as a real, i.e., noninteger number) for a stream set based on the temperature distribution of the composite curves. The algorithm starts by dividing the composite... 

Solution Figure 16.16a shows the streeun grid with the CP tables for the above- and below-pinch designs. Following the algorithms in Fig. 16.15, the... 

The algorithm may calculate Qnmia and Qcmm to be unchanged. In this case, the designer knows that the match will not penalize the design in terms of increased utility usage. 

The algorithm may calculate an increase in Qnmin and Qcmin- This means that the match is transferring heat across the pinch or that there is some feature of the design that will cause cross-pinch heat transfer if the design was completed. If the match is not transferring heat across the pinch directly, then the increase in utility will result from the match being too big as a result of the tick-off heuristic. 

Various partitions, resulted from the different combinations of clustering parameters. The estimation of the number of classes and the selection of optimum clustering is based on separability criteria such as the one defined by the ratio of the minimum between clusters distance to the maximum of the average within-class distances. In that case the higher the criterion value the more separable the clustering. By plotting the criterion value vs. the number of classes and/or the algorithm parameters, the partitions which maximise the criterion value is identified and the number of classes is estimated. 

To implement the reconstruction of the initial image, using denoised and/or noisy data given by simulated projections The algorithm (1) and the Gibbs functional in the form (12) were used for the reconstruction. The coefficients a and P were optimized every time. 

Comparison of the measurements with the microdensitometers and the algorithms of calculation inclusive the filter function and the accuracy of measurement of all project partners. 

For the practical evaluation of the algorithm described previously it is integrated into the NDT Sean Manager system (DBA Systems Inc, Melbourne, FL, U.S.A). This system allows film digitisation, display, evaluation and archiveraent of images /3,4/ and was developed for the needs of computer based industrial NDT film inspection. A snapshot of the user interfaee for wall thickness evaluation is shown in fig. 3. 

Measure Wall Thickness This window is used for the dialog to calibrate the algorithm aceording formula (3) and for point wise measurements after calibration. The row Ideal indicates the nominal wall thickness used, IQI indicates the wall thickness values used for calibration and the detected optical density. Local can be used for noise reduction and compensation of geometric effects. 

The interpretated value of crack depth hi is calculated by means of the algorithm of solving inverse task, the parameters I or T have limit values correspondingly. 

The algorithm of calculating crack depth is realized in electropotential device Zond IGT-97 for measuring cracks depth. Its structure diagram is shown in Fig. 8 Using quasi-direct current is the device particular feature that made it possible to reduce its dimensions and weight. 

THE ALGORITHM FOR SIZING OF CRACKS WITH COMPLEX CROSS-SECTIONS... 

The algorithm for sizing of eraeks with complex cross-sections and unknown shapes based on the method was used in for sizing of cracks oriented perpendicularly to the applied field. This algoritlim is presented in Fig.3. In this paper, the same algorithm is applied readily to sizing of cracks with non-perpendicular orientation with respect to the applied field. 

The algorithm leads to computation of the width 2a, and the depths d, dj, dj, d, d , d at six equidistant points along the y -axis of the cross-section of a crack, as well as the surface density of charge m=4 ju c at the crack walls. In its formulation from Fig.3, the algoritlun is adapted to cracks with a constant width. 

The algorithm contains five minimisation procedures which are performed the same way as in the method " i.e. by minimisation of the RMS between the measured unidirectional distribution and the corresponding theoretical distribution of die z-component of the intensity of the leakage field. The aim of the first minimisation is to find initial approximations of the depth d, of the crack in the left half of its cross-section, die depth d in its right half, its half-width a, and the parameter c. The second minimisation gives approximations of d, and d and better approximations of a and c based on estimation of d,= d, and d,= d,j. Improved approximations of d] and d4 are determined by the third minimisation while fixing new estimations of d dj, dj, and dj. Computed final values dj , d/, a and c , whieh are designated by a subscript c , are provided by the fourth minimisation, based on improved estimations of d, dj, dj, and d . The fifth minimisation computes final values d, , d, dj, d while the already computed dj , d/, a and c are fixed. 

Fig.3 The algorithm for sizing of cracks with complex cross-sections and unknown shapes. The five minimisation procedures are numbered consecutively.

Table 1. Computed results from the algorithm for sizing of the cracks from Fig.4a, Fig.4b, and Fig.4c and orientation angles d)=0°, 0=30° and 0=45°, and true values of the eorresponding crack parameters.

The data from Table 2 show that the algorithm developed in allows sizing of different cracks with complex cross-sections and unknown shapes for orientation angles not exceeding 45°. It is seen that the width 2a and the parameter c (or the surface density of charge m=4 r // e at the crack walls) are determined with 100% accuracy for all of the Case Symbols studied. The errors in the computation of the depths dj and di are less than 4% while the errors in the computation of d, dj, d, and d are less than 20% independent of the shape of the investigated crack and its orientation angle O <45°. 

The initial classification model of dispersion properties of engineering materials was obtained The algorithm of its creation includes ... 

Further evidence of the efficacy of the algorithm for locating points of conical intersection is provided in Figure 3, which reports additional points on the 2E 2E intersection seam, detemiined by introducing the geometrical constraint. 

This completes the outline of FAMUSAMM. The algorithm has been implemented in the MD simulation program EGO VIII [48] in a sequential and a parallelized version the latter has been implemented and tested on a number of distributed memory parallel computers, e.g., IBM SP2, Cray T3E, Parsytec CC and ethernet-linked workstation clusters running PVM or MPI. 

Taking into account the hydration shell of the NA and the possibility of the water content changing we are forced to consider the water -I- nucleic acid as an open system. In the present study a phenomenological model taking into account the interdependence of hydration and the NA conformation transition processes is offered. In accordance with the algorithm described above we consider two types of the basic processes in the system and thus two time intervals the water adsorption and the conformational transitions of the NA, times of the conformational transitions being much more greater... 

The algorithm of Hao et al. [30] essentially minimizes X), eq. (8), at each step to obtain and then adds to it a rescaled velocity term to... 

Again, the algorithm allows at least ten times larger time step to be used than LFV for the same accuracy. 

Following the procedure defined by (23) the fourth order SISM for MD simulations written explicitly can be found In ref. [22]. In the fourth order SISM additional steps in the algorithm occur due to additional force evaluations. 

The algorithm was applied to the MD simulations of a box of water molecules. The three-center water model was used [23]. The initial positions were at the equilibrium therefore all displacements were zero. The initial velocities were... 

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