Deviation vector

Four samples from a Polynesian island gave the lead isotope compositions given in Table 4.3. Calculate the mean and standard deviation vectors, the covariance matrix and the correlation coefficient between the two isotope ratios. 

At small values of the deviation vector q this reduces to the product of three functions of sinc (x) type. It is shown for one component in Figure 4.3. 

Figure 4.3 The scattered intensity as a function of one component of the deviation vector...

The deviation vector q, with origin at the end of the reciprocal lattice vector h, has two components, q y horizontal, positive rightwards going, and q vertical, positive upwards going. For the symmetric reflection, these components are related to the deviations of specimen ( ) and analyser ( ) from their zero... 

The procedure of developing a semi-empirical parameterization can be generally formalized in terms of Eq. (2) as follows. A set of experimental energies 5(C QF5) corresponding to different chemical compositions C, molecular geometries Q, and electronic states with specific values of S and T is given. In the case when a response to an external field is to be reproduced the latter can be included into the coordinate set Q. Developing a parameterization means to find certain (sub)set of parameters w which minimizes the norm of the deviation vector with the components... 

The determinant of S is often called the generalized sample variance. It is proportional to the square of the volume generated by the p deviation vectors, x - x. ... 

The total sample variance is the sum of the diagonal elements of the sample variance-covariance matrix, S. Total variance =. S 2,, +. S 2 +...+ s2pp. Geometrically, the total sample variance is the sum of the squared lengths of the p deviation vectors, xx. 

The function in Example 4.4 can be used to autoscale a data matrix. The function determines the size of the argument, its mean vector, and its standard deviation vector. On the last fine, a MATLAB programming trick is used to extend the mean vector and standard deviation vector into matrices having the same number of rows as the original argument prior to subtraction and division. The expression ones < r, i) creates an r x 1 column vector of ones. When used as an index in the statement mn (ones(r,1), ), it instructs MATLAB to replicate the mean vector r times to give a matrix having the dimensions r x c. 

The behaviour of the system in the vicinity of the solution (10) depends on the deviation vector (f). 

The stability that we mentioned before refers to the evolution of the deviation vector ( ) = x (t) — x(l) between the perturbed solution x (t) and the periodic orbit x(t) at the same time t. If ( ) is bounded, then the periodic orbit is stable. In this case two particles, one on the periodic orbit x t) and the other on the perturbed orbit x (t), that start close to each other at t = 0, would always stay close. A necessary condition is that all the eigenvalues of the monodromy matrix be on the unit circle in the complex plane. However, in a Hamiltonian system this condition is not enough for stability, because there is only one eigenvector corresponding to the double unit eigenvalue and consequently a secular term always appears in the general solution, as can be seen from Equation (49). We remark that this secular term appears if the vector of initial deviation (0) = a (0) — s(0) has a component along the direction /2(C)). 

Therefore, the control quantities vector r(t) is used instead of the status quantities. In the course of the control process, the real values of the controlled quantities are measured and compared with the required values z(t). In accordance with the found deviations, individual controllers intervene into the process in compliance with the design requirements. The control deviations vector is marked e(t). The shown equation system (1) is reformed to ... 

In order to reconstruct the image information, an adequate spectral width is needed for the probe beam to satisfy the Bragg condition at every grating component. In this section, we will theoretically estimate such a spectral width. When we rewrite Eq. (9) using the deviation vectors Sea, the difference between jUi and jUo is given by... 

The first approach is to replace each polymer atom coordinate x, by the sum of its original position (x,) and a deviation vector. 

Below we leave off each atom subscript i from the deviation vector in order to simplify the notation. Differential changes in (x, — (x,)) and (/, u , VY are related by a Jacobian... 

The second approach is to express the deviation vector between x,- and (x,) in spherical coordinates,... 

Owing to the different bonding environment around an APB, atoms near the APB are expected to deviate from ideal lattice positions, as a result of which both the direction and the magnitude of the APB displacement vector p differ from that for an ideal APB. In general, the deviation vector 2p should vary as a function of the position relative to the APB. In most cases, however, a simplification is made in which the APB still. separates two rigidly displaced crystals but with a displacement vector in the form... 

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