Nontrivial generalization

Equation (36) nontrivially generalizes the weU-known scaling fi(r) = Xr which Fock [172] used in 1930 to prove the virial theorem. 2 0 is a constant... 

Within the frameworks of this formalism to account for consistently nonlinear phenomenon complex nature is a success, such as memory effects and spatial correlations. In addition the earlier known solutions are not only reproduced, but their nontrivial generalization is given. Another important feature is connected with fractal structures self-similarity using. Unlike the traditional methods of system description on the basis of averaging different procedures, when microscopic level erasing occurs, in fractal conception medium self-affine structure and thus within the frameworks of this conception system micro and macroscopic description levels are united. Exactly such method is important for complex multicomponent systems, discovered far from thermodynamic equilibrium state [35], which are polymers [12], The authors of Refs. [31, 32] are attempted two indicated trends combination. 

As one should expect from the terminology, the earliest application of the q-centroid, strictly speaking, the r-centroid method was made in diatomic spectroscopy [131], although in a completely different physical sense. The q-centroid approximation presents a nontrivial generalization of the r-centroid approach of diatomic spectroscopy to the case of the nonradiative decay of polyatomic molecules. The importance of studying the q-dependence of the nonadiabatic coupling element was emphasized by Freed and Gelbart [132] and Freed and lin [27]. 

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

Except for simple cases, it is generally a nontrivial task to compute the Lyapunov exponents of a flow. In trying to estimate A(x(0)) in equation 4.59, for example, the exponentially increasing norm, V t), may lead to computer overflow problems. 

The metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

The main problem is to find the free energy of the real interface with nonlocal energetic and entropic effects. For a general multicomponent interface the minimization of the nonlocal HS-B2-functional is a nontrivial numerical problem. Fortunately, the variational nature of the problem lends itself to a stepwise solution where simple para-metrization of the density profiles through the interface upon integration of the functional yields the free energy as a function of the parameters. In fact, if we take the profile to be a step function as in the case of local free energy then with local entropy we get the result... 

Determination of accurate models for fm and fsm is nontrivial (Drew and Passman, 1999), and no consensus exists on the exact forms needed to describe particular flows. Nevertheless, it is generally acknowledged that the momentum-exchange model must include drag terms with forms similar to... 

To define a feature extraction procedure it is necessary to consider that the output signal of a chemical sensor follows the variation of the concentration of gases at which it is exposed with a certain dynamics. The nontrivial handling of gas samples complicates the investigation of the dynamics of the sensor response. Generally, sensor response models based on the assumption of a very rapid concentration transition from two steady states results in exponential behaviour. 

For importance sampling in the lattice simulation, one can use the leading part of the determinant, [real, positive]. This proposal provides a nontrivial check on analytic results at asymptotic density and can be used to extrapolate to intermediate density. Furthermore, it can be applied to condensed matter systems like High-Tc superconductors, which in general suffers from a sign problem. 

Molecules, in general, have some nontrivial symmetry which simplifies mathematical analysis of the vibrational spectrum. Even when this is not the case, the number of atoms is often sufficiently small that brute force numerical solution using a digital computer provides the information wanted. Of course, crystals have translational symmetry between unit cells, and other elements of symmetry within a unit cell. For such a periodic structure the Hamiltonian matrix has a recurrent pattern, so the problem of calculating its eigenvectors and eigenvalues can be reduced to one associated with a much smaller matrix (i.e. much smaller than 3N X 3N where N is the number of atoms in the crystal). 

It should be noted, however, that the limit 0 is only a formal procedure, which does not necessarily lead to a unique or correct semiclassical limit. In the case of the mapping formulation, this is because of the following reasons (i) For a given molecule, the frequencies f)mi(x) will in general also depend in a nontrivial way on h. (ii) A slowly varying term may as well be included in the stationary phase treatment [147]. (iii) As indicated by the term resulting from the commutator = 8 , the effective action constant ... 

Equation (12.17) represents the required boundary condition. It should be emphasized that it is essentially nonlocal both in space and time. In general, the numerical implementation of the operator in the right hand side of Eq. (12.17) is a nontrivial task. 

Several general implications of CE/MS hyphenation have to be taken into consideration. As discussed, both electrical circuits of CE and ESI have to be maintained. This goal is best accomplished if the CE capillary exit is kept at ground potential, as normally done in CE. This is nontrivial and places severe demands on the MS entrance configuration. These aspects are discussed in more detail in Ref. 14. 

The nontrivial transformation rule of Eq. (2.231) for the Ito drift coefficient (or the drift velocity) is sometimes referred to as the Ito formula. Note that Eq. (2.166) is a special case of the Ito formula, as applied to a transformation from generalized coordinates to Cartesian bead coordinates. The method used above to derive Eq. (2.166) thus constitutes a poor person s derivation of the Ito formula, which is readily generalized to obtain the general transformation formula of Eq. (2.231). 

By setting the origin of the coordinate system at the intersection of the two mirror reflection lines, it is easy to see that only Eq. (E.3) of the four corrugation functions is invariant under the mirror reflection operation. The fourfold rotational symmetry further requires n = m, and a = To the lowest nontrivial corrugation component, the general form of the corrugation function is... 

When it comes to the analysis of similar approaches stemming from the DFT the numerous attempts to cope with the multiplet states must be mentioned [113,154]. In these papers attempts are made to construct symmetry dependent functionals capable to distinguish different multiplet states in a general direction proposed by [99,100]. It turns out, however, that the result [113] is demonstrated for the lower multiplets of the C atom which are all Roothaan terms. It is not clear whether this methodology is going to work when applied to the d-shell multiplets which may be either non-Roothaan ones or even nontrivially correlated multiple terms. 

There may still be a nontrivial choice as to which bridge the poor shall sleep under. The point is quite general the opportunity set rarely reduces to literally one physical option. 

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