Action function

We seek the poles of the spectral function g(E) given by (3.7). In the WKB approximation the path integral in (3.7) is dominated by the classical trajectories which give an extremum to the action functional... 

Thus for Hamiltonians of finite dimension the effective action functional can be found by immediately integrating a system of ordinary differential equations. The simplest yet very important case is a bath of two-level systems. 

Vanadium pentoxide and mercuric oxide were used as catalysts for the hydrogen peroxide oxidation of bis(phenylthio)methane to its monooxide 17a31 (equation 5). From the synthetic point of view, it is interesting to note that vanadium pentoxide, in addition to its catalytic action, functions also as an indicator in this reaction. In the presence of hydrogen peroxide, the reaction mixture is orange while in the absence of hydrogen peroxide a pale yellow colour is observed. Thus, it is possible to perform the oxidation process as a titration ensuring that an excess of oxidant is never present. 

Mogenson, G, Jones, D and Yim, CY (1980) From motivation to action functional interface between the limbic system and the motor system. Progr. Neurobiol. 14 69-97. 

Protein kinases phosphorylate other proteins, and this action functions as a control switch for various cell activities. The switching functions make this family of proteins a very promising target for novel therapeutics. p38 mitogen-activated protein (MAP) kinase controls the production of growth factors and inflammatory cytokines, the molecules produced by the immune system that cause inflammation [54]. 

Division accdg to action (function is udamyi (impact), distantsionnyi ("distance", which corresponds to US "time"), mekba-nicheskii (mechanical) and dvoynogo deyst-viya ("double-action" which includes time and impact action)... 

We have shown elsewhere that the different bifurcation scenarios can be conveniently discussed in terms of area-preserving mappings generated by the action function [10]... 

S q(t)] has the form of an action functional for an auxiliary dynamical system with time-dependent Lagrangian L(q, q- 4>) ... 

The uncertainty principle requires that any extremum path should be spread, and the next step in our calculation is to find the prefactor in (3.54) by incorporating small fluctuations around the instanton solution, in the spirit of the usual steepest descent method. Following Callan and Coleman [1977], let us represent an arbitrary path in the form x t) = xins(T) + Sx(t) and expand the action functional up to quadratic terms in S (t), assuming that these deviations are small ... 

The relationships, rather similar in sense, for smooth dynamic systems were introduced in ref. 34 (p. 220 etc.) for studying the random perturbations via a method of action functionals. Close concepts can also be found in ref. 39. 

Mailman RB, Gay EA. Novel mechanisms of drag action functional selectivity at D-2 dopamine receptors (A lesson for drag discovery). Med Chem Res 2004 13 115-126. 

Soon after first publication Schrodinger s wave-mechanical model was extended by Madelung [39] on the basis of the obvious correspondence with the classical theory of hydrodynamics, already pointed out in Schrodinger s original papers [34] (II,p.l7). Writing the time dependence of k in terms of an action function... 

The xc potentials in (71) and (72) are formally given by functional derivatives of the xc part of the quantum mechanical action functionals ... 

Eqs. (101)-(103) constitute the KS scheme of time-dependent spin-density functional theory. With the xc action functional [ t. i] defined in analogy to Eq. (43), the spin-dependent xc potentials can be represented as functional derivatives ... 

Once the existence of u,o and A o is assumed, uniqueness follows from the above theorem. Up to this point the time-dependent HKS formalism is quite similar to the case without magnetic fields developed in Sect. 2. The variational representation of Dg,o and As,o, however, turns out to be much more complicated. Following Wacker, Kiimmel and Gross [60] the quantum mechanical action functional... 

Similarly, the action functional of noninteracting particles can be written as... 

Equation (225) was first obtained by Vignale [107] from invariance properties of the xc action functional aj/xc defined in Eq. (43). The derivation given here [108, 109] has the advantage of being independent of the stationary action principle. 

This should be cured by adding appropriate correlation terms to the xc part of the action functional, which is further backed by the observation that when combining the advantage of approximating by the optimized effective potential, together with a local density prescription of exchange and correlation in the xc kernels/ c and both singlet and the triplet spectrum are reproduced... 

At the risk of being redundant, we may state here the salient features of the TD-functional formalism. The first requirement is a variational principle, and for a time-dependent quantum description only a stationary action principle is available. With this a mapping theorem is established which turns the action functional into a functional of relevant physical quantities (which are the expectation values), and the condition of stationarity is now in terms of these variables instead of the entire density matrix. Thus the stationary property with respect to the density matrix now becomes one with respect to all the variables... 

This in turn implies the existence of an effective action functional W such that... 

A stationary principle involving an effective action functional determines the exact one-body Green function of the system... 

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