Vector model summary

Contents 1. Introduction 176 2. Static NMR Spectra and the Description of Dynamic Exchange Processes 178 2.1. Simulation of static NMR spectra 178 2.2. Simulation of DNMR spectra with average density matrix method 180 3. Calculation of DNMR Spectra with the Kinetic Monte Carlo Method 182 3.1. Kinetic description of the exchange processes 183 3.2. Kinetic Monte Carlo simulation of DNMR spectra for uncoupled spin systems 188 3.3. Kinetic Monte Carlo simulation of coupled spin systems 196 3.4. The individual density matrix 198 3.5. Calculating the FID of a coupled spin system 200 3.6. Vector model and density matrix in case of dynamic processes 205 4. Summary 211 Acknowledgements 212 References 212... 

In summary, at each iteration of the estimation method we compute the model output, y(x kw), and the sensitivity coefficients, G for each data point i=l,...,N which are used to set up matrix A and vector b. Subsequent solution of the linear equation yields Akf f 1 and hence k[Pg.53]

In summary, the photon has been modeled as a doublet in rotation in the preferred frame E. Spin and energy have been obtained from a semiclassical analysis. Polarization corresponds to a fixed direction of vector L in E. In a nonpolarized photon vector L has a time-dependent direction. A particular case of nonpolarization is the ellipsoid, as in Hunter and Wadlinger [37]. Our Eq. (96) allows for the existence of multiphotons that vary in steps of half the ground-state photon energy such prediction differs of the prediction of Hunter and Wadlinger [37]. Photons in motion with respect to E will be considered elsewhere. The photon is the source of the electromagnetic field, as explained next. 

In summary, we have used the Rouse chain model to obtain the diffusion constant of the center of mass and the time-correlation function of the end-to-end vector, which reflects the rotational motion of the whole polymer molecule. Since N is proportional to the molecular weight M, and K is independent of molecular weight, Eqs. (3.41) and (3.62) indicate that Dq and Tr depend on the molecular weight, respectively, as... 

In summary, the system under study is extremely complicated. When the experiment is scaled down to suppress convection it no longer conforms to the necessary assumptions used to describe and model the system in the laboratory, nor does it any longer operate under pseudo-zeroth order conditions. This makes it impossible to use the scaling approach to perform convection-free experiments in the laboratory rather than in space. It has been shown, however, that reductions of four or five orders of magnitude in the g-vector are sufficient to achieve a convection free environment in this system. These conditions are readily achievable on the space shuttle and on space station, making them ideal locations for this research. 

In summary, the support vector machine (SVM) and partial least square (PLS) methods were used to develop quantitative structure activity relationship (QSAR) models to predict the inhibitory activity of nonpeptide HIV-1 protease inhibitors. Cenetic algorithm (CA) was employed to select variables that lead to the best-fitted models. A comparison between the obtained results using SVM with those of PLS revealed that the SVM model is much better than that of PLS. The root mean square errors of the training set and the test set for SVM model were calculated to be 0.2027, 0.2751, and the coefficients of determination (R2) are 0.9800, 0.9355 respectively. Furthermore, the obtained statistical parameter of leave-one-out cross-validation test (Q ) on SVM model was 0.9672, which proves the reliability of this model. Omar Deeb is thankful for Al-Quds University for financial support. 

Geometrically, PLS can be seen as the projection of the observation points in X space onto an A-dimensional hyperplane (Figure 2). The positions of the observation points on this hyperplane are given by the coordinates (scores) tg. These scores provide a good summary of X, similarly to principal components scores, and also serve as good predictors of the responses in Y. In this way the PLS scores constitute a compromise between the two objectives to model X and to model Y. The direction of the hyperplane is described by the loading matrix P, which has one vector pa per model dimension a. 

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