Higher-order term, elimination

If the term in brackets is expanded and the higher-order terms are eliminated, this expression simplifies to... 

The higher-order terms denoted as Gn(Z) are not known. We expect [3] that the magnitude of G11 does not exceeded by half the value of the leading logarithmic term (In3(Za) for the ns states and In2(Za) for the 2p states) at Z = 1 and use the estimation for GII( 1) for any Z. The purpose of this calculation is to eliminate any other sources of theoretical uncertainty and to study a value of C 2s by a comparison with experiment. 

Various mathematical transforms and their reverse, or "back" transforms, are often used in succession to "clean up" and filter out noise effects from experimental data, by first calculating the full transform, then eliminating the higher-order terms which are ascribed to noise, and finally computing the back transform. They are also very useful for solving differential equations. 

Note that, for those terms with oti = pj, the problem of eliminating higher-order terms is the same as that in the Birkhoff normal form for stable fixed points. Then, it is possible to eliminate higher-order terms as long as the set of frequencies (d n = 2,..., N) satisfies the nonresonance condition... 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

We should emphasize that we make no particular assumption as to what orbitals are employed. Some of the contributions to H discussed here show up in higher orders of MBPT if the Hartree-Fock reference is used. For example, we can drop all Hn Tl,T2) contributions from Eqs. (178) and (179), since they become fourth- and higher-order terms when the Hartree-Fock orbitals are employed (there is no first-order contribution to T in this case). We can also eliminate the [T2contribution from Eq. (170)... 

Eliminating any higher-order term) Now we generalize the method of the last exercise. Suppose we have managed to eliminate a number of higher-order terms, so that the system has been transformed into X - RX — X + a X" -I- ), where n > 3. Use the near-identity transformation x = X +... 

Experiments whose outcomes are listed in [65] have shown only a qualitative agreement with the theory [64], while quantitative differences, in particular, in the extent of drop deformation, appeared to be significant. An attempt to take into account the higher-order terms in the asymptotic expansion was made in [66], but it did not eliminate the discrepancy with experimental results. It should be noted that performing a successful experiment is a difficult task because one has to impose a strict requirement that drops should remain motionless (no sedimentation or lifting), for which it is necessary to make sure that the drops are very small, and the densities of the internal and external liquids should not differ by a lot. 

In a harmonic potential where co = 2jr x 100 kHz for a " Ca ion, the equilibrium distance, Az, equals 26.0 pm, while typically applied amplitudes in the experiments described in Section 10.4 are about 5 pm. In this case, it is necessary to consider the effect of higher-order terms in the Coulomb interaction for the BR mode and, in some cases, also for the COM mode for two different ions. This effect could be reduced, but not eliminated, if much smaller amplitudes could be detected by using a state-of-the-art imaging system with almost diffraction-limited resolution... 

In view of this situation, Watson [24] proposed what he called an structure. As mentioned above, he pointed out that 21-lo, which was referred to as / , was very close to the equilibrium moment of inertia 1. Watson called the stracture derived from a set of the "mass-dependence" (r i) stracture. In a number of examples he has shown that the stracture is indeed very close to the structnre except for some parameters involving hydrogen. A drawback of this method is that data for more isotopic species than are necessary for the r, method are needed for stracture determination. Nakata et al. [29-31] pointed out that there existed additivity relations for isotopic effects on the stracture parameters and employed this fact to eliminate higher-order terms in the expansion of moments of inertia in terms of isotope mass differences which were neglected in the original treatment of Watson. In this way Nakata et al. have expanded the applicability of the method. A similar approach was proposed by Harmony et al. [32-34]. They noticed that the ratio p = IJIq is not significantly dependent on isotopic species (but may be different for different inertial axes) and proposed to use [/ ] (2p l)[fo]i for the moment of inertia of the i-th isotopic species. The stracture thus obtained is called the stracture. 

With the aid of the above definitions, the solution of G from Flquation 116, on sinipli-fical ion and elimination of the higher-order terms, lakes the form... 

Also, the more accurate formula proposed by Maroulis [41] allows to eliminate the contribution of higher-order terms to the polarizability ... 

The third derivatives 9 AGcc/9o 9)39)/ = AG can be obtained differentiating three times Eq. (1.20) and eliminating higher-order terms according to the (2n- -l)... 

Letx = aiFrjIRuT. Using a power series expansion to describe e and eliminating the higher order terms for a numerically small values of x, 0 for small x,... 

These transformations, after elimination of terms that are appropriately higher order in capillary number, yield the following expressions (2JL) ... 

Here 2tt Aft = trm is imposed to eliminate the effect by the zero-order interaction of the RF field, otherwise a term containing Ix will remain and it will distort the longitudinal magnetization, transverse magnetization, as well as the BSPS of the 13C . For the on-resonance condition, 5 = 0, all the higher-order average Hamiltonians vanish since [7f(/ ), = 0 for arbitrary l and... 

The theoretical evaluation of kx and the higher order coefficients in Equation (42) requires more than merely retaining additional terms in Equation (40). Einstein s derivation of Equations (32) and (38) is based on the restriction that the dispersion is very dilute, and therefore simply retaining higher order coefficients in Equation (40) will not eliminate the above restriction. 

We now have to consider the question of retardation, that is, the elimination of t(). In the trace term, we have already neglected the retardation, which corresponds to neglecting higher-order concentration terms. As a result we introduced immediately the relevant Mdller operators in (3.43). However, the retardation terms coming from the first right-hand-side contribution of (3.43) would be of the same order as the trace term however, here the four-particle terms represent an approximation only (only certain contributions are taken into account). Therefore, we will neglect the mentioned retardation coming from f.23, and we write... 

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