Cosine, defined

Here W is the probability of finding the molecule on the level under examination, but (cos 6) is the mean value of the cosine defining angular momentum orientation for a separate particle on this level. 

The direction cosines define the orientation of the crystal lattice with respect to the condenser plates. The longitudinal susceptibility is... 

The two terms correspond to different polarization of phonons. The cosine term corresponds to displacements along the rotation axis or the direction tp = 0. The sine contribution arises from the phonons polarized along the line tp = The interaction (6.29) does not change the symmetry of the (p potential, and, in this respect, it is symmetric coupling, as defined in sections 2.3 and 2.5. Nonetheless, the role of the cosine and sine couplings is different. The former ( breathing modes ) just modulate the barrier (6.22), while the latter ( shaking modes ) displace the potential. 

The E-state indices may define chemical spaces that are relevant in similarity/ diversity search in chemical databases. This similarity search is based on atom-type E-state indices computed for the query molecule [55]. Each E-state index is converted to a z score, Z =(% -p )/0 , where is the ith E-state atomic index, p is its mean and O is its standard deviation in the entire database. The similarity was computed with the EucHdean distance and with the cosine index and the database used was the Pomona MedChem database, which contains 21000 chemicals. Tests performed for the antiinflamatory drug prednisone and the antimalarial dmg mefloquine as query molecules demonstrated that the chemicals space defined by E-state indices is efficient in identifying similar compounds from drug and drug-tike databases. 

The phase spectrum 0(n) is defined as 0(n) = arctan(A(n)/B(n)). One can prove that for a symmetrical peak the ratio of the real and imaginary coefficients is constant, which means that all cosine and sine functions are in phase. It is important to note that the Fourier coefficients A(n) and B(n) can be regenerated from the power spectrum P(n) using the phase information. Phase information can be applied to distinguish frequencies corresponding to the signal and noise, because the phases of the noise frequencies randomly oscillate. 

Furthermore, it is seen from Bq. (9) that any relation involving the cosine of an included angle may be written in terms of the scalar product of the vectors which define it. Finally, the reader is warned that a relation such as A B = A C does not imply that B = C, as... 

The average incident tangential momentum is muh while the average scattered tangential momentum is muf. If the gas molecule equilibrates with the surface and the scattered momentum is zero, we have Knudsen cosine scattering and complete accommodation of the incident gas molecule with the surface. On the other extreme, if specular reflection occurs, the incident momentum is retained upon scattering and mut = muf. The momentum accommodation coefficient, / is introduced to describe the type of scattering that does occur, and it is defined by... 

In powder EPR simulators we use the orientation of the static-field vector B with respect to the molecular xyz-axes system as the definition of molecular orientation. The orientation is defined in terms of the polar angles 0, direction cosines as defined previously in Equation 5.3. To solve Equation 8.17 we have to define the direction cosines k, of B, in terms of the direction cosines li of B. 

For powder EPR in perpendicular mode Bx lies in a plane perpendicular to B, and Bx samples all possible orientations in this plane. We define a vector v in this plane by means of an angle a between Bx and v. Then, the direction cosines of Bx are... 

The first cosine factor represents a wave like the originals with average wavelength and frequency and moving with velocity (uq + us2)/ k + k2). In the case of electromagnetic waves u = cAq and w2 = cfc2j so that v = c(ki + k2)/(ki + k2) = c, the original velocity. The composite wave however, has an amplitude that varies within a profile defined by the factor... 

As pointed out in the previous section, the Chebyshev operator can be viewed as a cosine propagator. By analogy, both the energy wave function and the spectrum can also be obtained using a spectral method. More specifically, the spectral density operator can be defined in terms of the conjugate Chebyshev order (k) and Chebyshev angle (0) 128 132... 

Other integrals of importance arc the sine and cosine integrals Ci(a ), Si (a ), which are defined by the equations... 

On the other hand, factor analysis involves other manipulations of the eigen vectors and aims to gain insight into the structure of a multidimensional data set. The use of this technique was first proposed in biological structure-activity relationship (i. e., SAR) and illustrated with an analysis of the activities of 21 di-phenylaminopropanol derivatives in 11 biological tests [116-119, 289]. This method has been more commonly used to determine the intrinsic dimensionality of certain experimentally determined chemical properties which are the number of fundamental factors required to account for the variance. One of the best FA techniques is the Q-mode, which is based on grouping a multivariate data set based on the data structure defined by the similarity between samples [1, 313-316]. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or covariance) relationships explored with R-mode factor analysis. The measure of similarity used is the cosine theta matrix, i. e., the matrix whose elements are the cosine of the angles between all sample pairs [1,313-316]. 

To illustrate the different complexities, CHARMM s energies arising from torsional terms depend only on the two central atoms, and there is only one cosine term. For MM2 and MM3, there are three cosine terms for each four-atom sequence that defines a torsion angle. MM2 treats lone pairs of electrons on hydroxyl and ether oxygen atoms as separate "atoms that also must be parameterized. Therefore, many more parameters must be used with the MM2 and MM3 programs than the CHARMM program. (The parameters for all the atomic sequences in glucose, and many other molecules, are furnished with MM2 and MM3.)... 

In this expression 1, m, n denote the direction cosines specifying the relative orientations of the principal axes in the monomer and dimer. (The expressions defining the values D and E are D = -3/2 Z and E = 1/2 (Y - X).) For the face-to-face structure proposed for [ZnTCP]2 (14) exciton theory predicts that dimerization should not affect the out-of-plane component (Z) of the tensor. The in-plane component, and therefore E, depends on the angle of rotation of one porphyrin plane relative to the other. According to the exciton model the observed reduction in E (cf. Table II) corresponds to an angle of rotation of about 23. This is reasonably close to the value predicted by molecular models (14). 

The factor r1 enters because the Cartesian spherical harmonics clmp are defined in terms of the direction cosines in a Cartesian coordinate system. The expressions for clmp are listed in appendix D. As an example, the c2mp functions have the form 3z2 — 1, xz, yz, (x2 — y2)/2 and xy, where x, y and z are the direction cosines of the radial vector from the origin to a point in space. 

The results of the integrations depend on the spectral density, which is defined as the cosine Fourier transform of the dynamical friction Eq. (8) ... 

The asymmetry parameter, which was defined in Section 3.4 as the average cosine of the scattering angle, depends, in general, on the polarization state of the incident light. However, the asymmetry parameter for a spherical particle is clearly independent of polarization and is given by... 

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