Perturbation index

Next, we shall consider how the nuclear kinetic energy is taken into consideration perturbatively. The natural perturbation index k is chosen to be... 

DAEs can be classified by their index. We defined in Sec. 2.6.3 the index of a linear DAE by the index of nilpotency of the corresponding matrix pencil. For the general, nonlinear case there are many different index definitions, which might lead to different classifications in special cases. Here we will introduce the so-called perturbation index which allows to categorize the equations by means of their sensitivity under perturbations, [HLR89]. 

Next, we show that multibody system equations have the perturbation index 3. Let p, V, X denote the solutions of the linear unperturbed system (5.1.3) and p, v, X the solution of the corresponding perturbed system... 

A sharper bound is given in [Arn95] where it has been shown, that S3 affects only the perturbations in A but not those in p and v. That motivates to distinguish also the index of variables A has perturbation index-3, while p and v have perturbation index-2. 

The differentiation index is defined as the number of differentiations required to obtain an ODE in all unknowns. Thus, the equations of motion with the constraint given on position level are an index-3 system conformal to their perturbation index. Analogously, the velocity constrained equations are an index-2 problem and the acceleration constrained equations an index-1 system. 

We consider in this book exclusively DAEs of the form (5.1.2) with a regular matrix Ei y). For these systems the differentiation index and the perturbation index are the same, [HLR89]. 

If the index of refraction of a thin material were modulated in Heu of its absorption, the resultant transmittance function for a gra ting prepared as in the absorption case is given by equation 9 where n is the average index of the thin film. An is the amphtude of the index perturbation, and T is the thickness of the film. 

The perturbation of the PMD symmetry is accompanied by a decrease in the charge alternation and by the appearance of bond alternation from one end group to another. The bond alternation ampHtude has been revealed to be proportional to the asymmetry degree, which can be calculated as the difference of topological indexes = 4>gj — 4>gg. The effect is maximum if Tgj > 45° and Tgg < 45°. If A4>j2 = 90°, the ideal polyene state is... 

The first-order energy involves only the perturbation operator and the unperturbed wavefunction. In an HF-LCAO treatment, the integrals would be over the LCAOs, and this implies a four-index transformation to integrals over the basis functions. 

The refractive index of a medium is the ratio of the speed of light in a vacuum to its speed in the medium, and is the square root of the relative permittivity of the medium at that frequency. When measured with visible light, the refractive index is related to the electronic polarizability of the medium. Solvents with high refractive indexes, such as aromatic solvents, should be capable of strong dispersion interactions. Unlike the other measures described here, the refractive index is a property of the pure liquid without the perturbation generated by the addition of a probe species. 

The index 0 indicates that the derivatives are taken for zero perturbation strengths. The equations for the first-order amplitudes and multipliers are obtained as ... 

The coupling of two angular momenta with equal magnimdes, as in the previous model of the perturbed H atom, is of special interest in leading to a monodromy matrix with index n = 2. 

P. R. Longaker and M. M. Litvak, Perturbation of the refractive index of absorbing media by a pulsed laser beam. J. Appl. Phys. 40(10), 4033-4041 (1969). 

The second contribution to the E-state index comes from the interactions between an atom i and all other atoms in the molecular graph. The perturbation on the intrinsic state value / of atom i due to another atomj depends on the difference between the corresponding intrinsic state values, (f- /,), and on the graph distance between atoms i andj. The overall perturbation on the intrinsic state value I of atom i is ... 

Finally, the E-state index S of atom i is the sum of the intrinsic state and of the perturbation term ... 

The next lowest unperturbed energy level however, is four-fold degenerate and, consequently, degenerate perturbation theory must be used to determine its perturbation corrections. For simplicity of notation, in the quantities and we drop the index n, which has the value... 

The index n refers to the order of perturbation. When all of these equations are combined one gets... 

At concentrations greater than about 0.01 M, refractive index changes and the perturbing effect of solute molecules or ions on the charge distribution of their neighbours both affect the value of s. Positive or negative deviations may result. 

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