Generator matrices

In addition, factorization of the partition function provides a vehicle for the efficient estimation of many other properties all that is required is that the property for a single chain conformation be expressible in factorial form similar to that for the partition function. If a conformation-dependent property/can be computed for a given (fixed) chain conformation by a product of generator matrices F > one for each bond,... 

Today generator matrices F are known for many properties,10 among them the population of different conformers, the relative stability of macromolecular diastereoisomers, the mean-square end-to-end distance, the radius of gyration, the molecular dipole moment, the molecular optical anisotropy (and, with it, the stress-optical coefficient, the Kerr effect, depolarized light scattering, and the... 

To generate matrices of numerical data which can be analysed statistically. 

An example of the application of the RIS model is mentioned in passing. It involves the work of Mattice and Galiatsatos on the dimensionless ratios of various moments of conformation-dependent physical properties of polymers. -55 Xhese ratios play an important role in the evaluation of the behavior of polymers. The properties were calculated through the use of generator matrices as just shown. 

V. Galiatsatos and W. L. Mattice, ]. Comput. Chem., 11, 396 (1990). Efficient Formulation of the Large Generator Matrices Required for Computation of the Higher Moments, and Mixed Moments, of Conformation-Dependent Properties. of Chain Molecules with Independent Bonds. 

Note that we can use L steps of the Lanczos method to generate matrices Qr, and Ti, L < N, and to introduce a natural approximation to vector d as... 

The same procedure may be repeated with all the elements of the group O to verify that (a) the set of e-orbitals does not mix with the set of ta-orbitals each set mixes only within itself, and (b) the set of e-orbitals and the set of ta-orbitals each generates matrices which are irreducible representations corresponding to e and ta respectively of the group 0. 

The above expression can be written in a much more compact form by introducing generator matrices A,-, the serial multiplication of which generates the unwieldy sum in equation (4.19). Thus... 

While a lack of cell adhesion will not promote cell survival or proliferation, there are several physiological processes that depend upon cell detachment from the ECM. Examples of this necessary de-adhesion can be found in normal development and tissue homeostasis, as anti-adhesive molecules cause different levels of de-adhesion ranging from complete cell detachment to the localized detachment required for cell migration. Investigation of these anti-adhesive molecules is important, as their use in biomaterials may provide a means to generate matrices that are conducive to cell migration by mimicking the adhesive/anti-adhesive nature of the native ECM. 

The higher even moments of the unperturbed dimensions, r P)Q and s p)q, p > 1, are accessible through an appropriate expansion in the dimensions of the generator matrices used for the simpler cases where p = 1 [11]. Dimensionless ratios formed from appropriate combinations of these even moments provide information about the shape of the distribution functions. Thus r ) measures the average value of r, r )dl(r )o measures the width of the distributirMi for and (r )o/ r )l measures the skewness of this distribution function. All flexible homopolymers wiU approach the Gaussian limit of = 5/3 as n 00, but nar-... 

Generator matrices are easily formulated for the computation of the mean square dipole moment, (p. )o. using the analogy between formulating r as a sum of bond vectors and... 

Flory reviewed in 1969 the development and applications of the rotational isomeric state scheme calculations, which allow, by matrix algebra, the statistical mechanical averaging over the rotational states of chain properties which may be expressed as a vector or tensor quantity associated with the chain bonds, and estimations of the probabilities of chosen conformational sequences. The methods were generalized and schemes for reducing the dimensions of certain generator matrices were presented in 1974, when comparisons were also made with an alternative Fourier expansion method, currently in use for atactic polypropylene. These techniques have greatly contributed to an understanding... 

A matrix product formed in this way can be used to generate matrices for the operator products we need when considering compound operations. For example, in the NH3 case of Figure 4.6, a rotation can be generated by applying the C3 matrix twice. This can be done by applying the C3 matrix once and then using the intermediate column vector in a second application ... 

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