Kramers basis

This expression represents a simplification of the original expansion in the primitive Kramers pairs basis. Although there is not a reduction in the size of the one-particle basis, we need only consider half the matrix elements, and there is therefore a 50% reduction in the amount of work. The lack of reduction might be expected because the matrix elements are potentially spin-dependent. The X q operators are called Kramers single-replacement operators, and they define what we will call a Kramers basis. 

We have previously shown that in a Kramers basis we have the following relations ... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

Approaching the matter from an entirely different angle, a semiquantitative estimation of ascorbic acid or total iodine-reducing substances might provide a suitable basis. Delays in handling involve rather marked losses of ascorbic acid. Kramer and Mahoney (20) have observed a relationship between quality and the amount of iodine-reducible substances remaining in lima beans. 

Table 12-4 gives the characters, basis functions, and the case a, b, or c to which the irreducible representation of Hu and belong for the point T. The degeneracy in and is the usual Kramers spin degeneracy, which is removed in cases (2) and (4) because of the absence of 6 in these symmetry groups. 

Leonard, J. A., and Kramer, M. A., Radial basis (unction networks for classifying process faults. IEEE Control Syst. 11, pp. 31-38, (1991). 

Of the several approaches that draw upon this general description, radial basis function networks (RBFNs) (Leonard and Kramer, 1991) are probably the best-known. RBFNs are similar in architecture to back propagation networks (BPNs) in that they consist of an input layer, a single hidden layer, and an output layer. The hidden layer makes use of Gaussian basis functions that result in inputs projected on a hypersphere instead of a hyperplane. RBFNs therefore generate spherical clusters in the input data space, as illustrated in Fig. 12. These clusters are generally referred to as receptive fields. 

Kramer, et al.(262,327) and Knoll(233), on the basis of G, G response concluded that gels kept in continuous motion during the crosslink formation period have completely different 3-D networks than gels allowed to stand in a quiescent state prior to analysis on the rheometer. This observation confirmed the early comments of Conway(317) and has had important implications for fracturing gel research and modeling since crosslinked fracturing gels as applied in the field ordinarily do not experience a quiescent period, yet nearly all test data accumulated prior to 1986 was collected from samples that had experienced at least a momentary quiescent period. 

One problem encountered in solving Eq. (11.12) is the modeling of the prior distribution P x. It is assumed that this distribution is not known in advance and must be calculated from historical data. Several methods for estimating the density function of a set of variables are presented in the literature. Among these methods are histograms, orthogonal estimators, kernel estimators, and elliptical basis function (EBF) estimators (see Silverman, 1986 Scott, 1992 Johnston and Kramer, 1994 Chen et al., 1996). A wavelet-based density estimation technique has been developed by Safavi et al. (1997) as an alternative and superior method to other common density estimation techniques. Johnston and Kramer (1998) have proposed the recursive state... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

Kaissling K.-E., Klein U., de Kramer J. J., Keil T. A., Kanaujia S. and Hemberger J. (1985) Insect olfactory cells electrophysiological and biochemical studies. In Molecular Basis of Nerve Activity, eds J. P. Changeux and F. Hucho. Walter de Gruyter Co., Berlin. 

Kramer, A., Keitel, T., Winkler, K Stdcklein, W., Hohne, W and Schneider-Mergener, J. (1997) Molecular basis for the binding promiscuity of an anti-p24 (HIV-1) monoclonal antibody. Cell 91,799-809. 

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