Functions stationary

Another popular and convenient way to study the quantum dynamics of a vibrational system is wave packet propagation (Sepulveda and Grossmann, 1996). According to the ideas of Ehrenfest the center of these non-stationary functions follows during a certain time classical paths, thus representing a natural way of establishing the quantum-classical correspondence. In our case the dynamics of wave packets can be calculated quite easily by projection of the initial function into the basis set formed by the stationary eigen-... 

Among the most detailed and accurate investigations of positronium formation in the Ore gap are those of Humberston (1982, 1984) and Brown and Humberston (1984, 1985), who used an extension of the Kohn variational method described previously, see section 3.2, to two open channels. The single-channel Kohn functional, equation (3.37), is now replaced by the following stationary functional for the K-matrix ... 

The AYRSSM allows us to estimate the dependence of pollution level in the AYRS estuary as a function of anthropogenic activity. Suppose that the intensity of sources of heavy metals is such that their concentration in the water near Angarsk, Irkutsk, Krasnoyarsk, Bratsk, and Ust-Ilimsk can be described by a stationary function, supporting heavy metal concentrations at level h at each of these cities. Computer experiments show that there is a stable correlation between h, the heavy metal concentration in the AYRS estuary, and the water flow rate fi. An increase in h of 10% leads to a rise of pollution input to the Kara Sea by 2.5%. An increase in h of 1% leads to a rise of pollution input to the Kara Sea by 0.7%. These results are correct only when variances in values fi and h are close to their average estimates. Near their critical values the estimates are unstable and more detailed models are required. 

These two coupled equations involve the rates of ionization, recombination, and reverse transfer to the excited state. However, reverse transfer does not affect the shape of the distribution when ionization is under kinetic control. In such a case v exp(- t/x/j), p 0, and there is nothing to transfer back from p to v. Kinetic ionization creates a distribution identical in form to that of W/(r), regardless of the rate of reverse transfer. Hence, fo(r) may be affected only if ionization is under diffusional control. The maximal effect is expected to be at to = oo when there are stationary functions of distance ns(r) = limv, o.vv(.v) and ms r) = lim o p(.v), with a large dip in ns(r) near the contact and a hump in ms(r) at the same place (ns + ms = 1). 

Combining these equations, all three expressions for tan r] give the same result for the stationary functional,... 

For degenerate states more than one state with eigenvalue E take the form (9). Both are stationary, but a complex superposition, although again a stationary function, does not have the form (9) and could describe particles in motion [48]. 

In practice, m, is usually not a strictly stationary function, that is, one whose statistical properties are independent of time. Rather, the flow may change with time. However, we still wish to define a mean velocity this is done by defining... 

To find out if a stationary function is a minimum or a maximum is more complicated for more than one variable. There are special cases, e.g., two-dimensional inflexion points, saddle points. The second total derivative for /(x, y) is d /(x, y). It reads as... 

The second-order perturbation expressions that Stacey derived for ratios of linear and bilinear functionals are not unique to the variational approach. In this section these expressions were obtained with a straightforward perturbation approach (50). Moreover, Seki (63) has shown that the Usachev-Gandini approach can be used to derive a GPT expression for the static reactivity for perturbations that do not retain criticality. This does not detract from the value of the variational approach. The latter provides stationary functionals for all kinds of integral parameters from which one can obtain exact expressions for these parameters [e.g., Eq. (140) for the static reactivity] as well as a consistent derivation of the equations for the distribution functions and of constraints imposed on them. 

The purpose in using bilinear functionals for many applications is to improve the accuracy of the calculation relative to the accuracy that can be achieved with linear functionals. Bilinear functionals are used, for example, to construct stationary functionals for some integral parameter in which first-order errors in the value of the distribution functions cause only second-order errors in the parameter. There may be problems in which second-order errors in the value of the stationary functional (due to the neglect of spectral effects), exceed first-order errors in the value of the same parameter if calculated from a linear functional formulation. In other words, spectral effects may impair the advantage of bilinear formulations. 

Krishnakumar, K. (1989). Microgenetic algorithms for stationary and non-stationary function optimization. In G. Rodriguez (Ed),Proceedings of SPIEIntelligentControlandAdaptive Systems, Vol. 1196, (pp. 289-296). 

The derivatives of the excitation energies [Pg.56]

Stavropoulos, C. N., and S. D. Fassois. Non-Stationary Functional Series Modeling and Analysis of Hardware Reliability Series A Comparative Study Using Rail Vehicle Interfailure Times. Reliability Engineering and System Safety 68, no. 2 (2000) 169-183. 

Poulimenos AG, Fassois SD (2009a) Asymptotic analysis of non-stationary functional series TARMA estimators. In Proceedings of the 15th symposium on system identification, Saint-Malo... 

The iso-stationary function is obtained by inserting these extremal coordinates into Eq. 16. Since the edge distortions were transformed according to the irreps of S5, the iso-stationary function naturally decomposes into three independent terms, one for each irrep. 

The subscripts 0 and t of the spatial variables indicate the time at which they are to be taken. Actually the expressions given in Eq. 28 are statistically stationary functions, so that only the time interval matters rather than the absolute time. The brackets stand for an ensemble average over all spin systems in the sample. 

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