Asymptotic properties

Because the pseudo-inverse filter is chosen from the class of additive filters, the regularization can be done without taking into account the noise, (n). At the end of this procedure the noise is transformed to the output of the pseudo-inverse filter (long dashed lines on Fig. 1). The regularization criteria F(a,a) has to fulfill the next conditions (i) leading to an additive filter algorithm, (ii) having the asymptotic property a, —> a, for K,M... 

This last representation is completely equivalent to the analytidty of t(ai) in Im 0 and the statement that a,t(a>) go to zero as u - oo. The analyticity property in turn is a direct consequence of the retarded or causal character of T(t), namely that it vanishes for t > 0. If t(ai) is analytic in the upper half plane, but instead of having the requisite asymptotic properties to allow the neglect of the contribution from the semicircle at infinity, behaves like a constant as o> — oo, we can apply Cauchy s integral to t(a,)j(o, — w0) where a>0 is some fixed point in the upper half plane within the contour. The result in this case, valid if t( - oo is... 

Because of the enormous range of difference approximations to an equation having similar asymptotic properties with respect to a grid step (the same order of accuracy or the number of necessary operations), their numerical realizations resulted in the appearance of different schemes for solving the basic problems in mathematical physics. 

Let us investigate the long-term asymptotic properties of this solution. Given that... 

It should be stressed that in this first group of works (1956-1970) there is nowhere any deviation from the Hamiltonian laws of dynamics. This feature was taken as a preliminary postulate of our work. Irreversibility appeared as an asymptotic property of the evolution of certain classes of systems. The term asymptotic refers to the large size of the system, as well as to the long time scale of observation. 

By means of a modification of the TFD method in the near nnclear region for the electron and energy densities, which introduces exact asymptotic properties, radial expectation values and the atomic density at the nucleus are evaluated, comparing fairly closely to the HF results, with a large improvement of the TF estimates. In addition to this, momentum expectation values can be estimated from semiclassical relations. 

Long-range asymptotic properties of KS orbitals were recalled in Sect. 4, as dictated by asymptotic properties of all potentials of KS equations. A new, simple and direct method was applied to obtain the asymptotic form of the exchange potential. For pure-state systems the known results were confirmed in Sects. 4.2 and 4.3, while for mixed-state systems a new, exact result was obtained in Sect. 5.6. 

If we use the fact that co is small and the asymptotic property Psir (r-> oo)... 

The statistical concept of consistency embodies the idea we can obtain as much accuracy and precision as desired by collecting enough data. Technically, there are different dehnitions of consistency recognized by mathematical statisticians. Consistency is one example of an asymptotic property. ... 

The fundamental asymptotic property which makes this formulation important can be summarized by the following assumption ... 

In order to compute a two step GLS estimate, we can use either the original variance estimates based on the separate least squares estimates or those obtained above in doing the LM test. Since both pairs are consistent, both FGLS estimators will have all of the desirable asymptotic properties. For our estimator, we... 

In some cases when detailed ab initio calculations of potential surfaces have become available, they have confirmed the major qualitative features of the surfaces deduced from experimental data and from preliminary data on correlation diagrams and the asymptotic properties of reactant, product, and intermediate states. One such case is the C+-H2 system,451 which has aroused considerable interst. Angular and energy distributions were determined experimentally for the CH+ + H products from the reactions of the C+(2P) ground state and the C+(4P) excited state with H2 (Table I, Jones et al.9b and references cited therein, and Jones et al.326). Chemiluminescence was also observed from this hydrogen-atom transfer reaction (Table IV).442,443 In addition, the reaction... 

Such a model must be formed from an undetermined number of atoms and allows the determination of asymptotic properties. This method, which is impossible for manual calculation, is, on the other hand, very well suited to mechanical means of calculation. In this way, chain problems may be suitably resolved using analysis (ref. 83) and machines. The resulting determination of asymptotic properties is satisfactory. 

Rigorous properties of the optimized effective potential (OEP) are derived. We present a detailed analysis of the asymptotic form of the OEP, going beyond the leading term. Furthermore, the asymptotic properties of the approximate OEP scheme of Krieger, Li and Iafrate [Phys. Lett. A 146, 256 (1990)] are analysed, showing that the leading asymptotic behavior is preserved by this approximation. 

---