Stationarity method

Slightly overestimated Pi as compared with the pressure calculated on the basis of the one-centered approach may be conditioned by (i) the approximation related to the application of the partial stationarity method by concentrations of chain carriers (ii) the consumption of the initial reagents when reaching the first explosion limit. 

State vector, specification of, 493 Stationarity property of probability density functions, 136 Stationary methods, 60 Statistical independence, 148 Statistical matrix, 419 including description of "mixtures, 423... 

In an attempt to address these questions, a modern method of statistical physics was recently applied by Varotsos et al. (2007) to C02 observations made at Mauna Loa, Hawaii. The necessity to employ a modern method of C02 data analysis stems from the fact that most atmospheric quantities obey non-linear laws, which usually generate non-stationarities. These non-stationarities often conceal existing correlations between the examined time series and therefore, instead of applying the conventional Fourier spectral analysis to atmospheric time series, new analytical techniques capable of eliminating non-stationarities in the data should be utilized (Hu et al., 2001 Chen et al., 2002 Grytsai et al., 2005). 

Measurements of the environmental parameters in the monitoring regime provide sets of series of quantitative characteristics for the system of data processing, which cannot be analyzed because of their stationarity. There are many ways to overcome time dependence and thereby remove the contradiction between the applicability of statistical methods and the level of observational data stationarity. One such way consists in partitioning a series of noise-loaded measurements into quasi-stationary parts (Borodin et al., 1996 Krapivin el al., 2004). 

If however 10-2(1032) and 103(1034) are so small that they are comparable in magnitude with the reciprocal "time of reaction, which is of the same order of magnitude as wh the method of stationarity fails and we are compelled to use the much more difficult general method. Here we may determine x3 by chemical or physical analysis so that, in some cases at least, the reactions 1 —> 3 and 3 — 0 may be investigated separately. 

Radical polymerizations are almost always considered as kinetically stationary. However, the stationarity conditions are not always fulfilled. Living polymerizations with rapid initiation are stationary, but the character of the medium should not significantly change during polymerization in order to prevent shifts in the equilibria between ion pairs and free ions. All other polymerizations are non-stationary even, to some extent, living polymerizations with slow initiation. It is usually very difficult to define initiation and termination rates so as to permit exact kinetic analysis. When the concentration of active centres cannot be directly determined, indirect methods must be applied, and sometimes even just a trial search for best agreement with experiment. 

The second possibility is to use a gradient code, if this is available for the chosen method. The third method is the simplest, namely to evaluate E2 as an expectation value. This method is equivalent to the other two, if Eq satisfies a stationarity condition, like the Brillouin condition of Hartree-Fock theory. For non-stationary approaches, like MP2 or CC, the methods based on differentiations of the expectation value are more reliable. This is related to the validity or non-validity of the Hellmann-Feynman theorem. 

This approximation resolves the computational difficulty encountered in the direct exact formulation that requires repeated computations of the solution of linear simultaneous algebraic equations and determinants of the matrices with huge dimensions. The efficiency in the approximated expansion is gained by the appreciation that the conditioning information can be truncated within one period of the system only. For linear systems, the expressions for the reduced-order likelihood function p(yi, yj, - - -, yNp W, C) and the conditional PDFs p(.yn 0, yn-Np, yn-Np+1, , y -i, C) are available since they are Gaussian and the correlation functions are known in closed forms regardless of the stationarity of the response. For stationary response, the method is very efficient in the sense that evaluation of all the conditional PDFs p(ynW, yn-Np,yn-Np+i,, y -i, C) requires the inverse and determinant of two relatively small matrices only. 

So-called false-time-step relaxation is used to achieve stationarity. The semi-implicit method, which considers the pressure-Hnk of the pressure correction equation and the Reynolds equations, is the SIMPLEST algorithm. The sets of algebraic equations for each variable are solved iteratively by means of the ADI technique. An example of the simulated flow field is illustrated in Fig. 3. Good agreement can then be achieved between measured flow details and the simulation results for vessels and impellers of different geometry [1]. 

For non-asjrmptotic values of p and L the non-linear stationarity equations equations could be solved numerically [20] using a standard iterative method [25]. We fotmd that for a given set of parameters there is a chmn length (which depends on the strength of the disorder) such that for 0 < T < Tc there is only a replica sjunmetric solution. This is the case when the variational p irameters satisfy Xc = 1 and sq = Si. For L > Lc there is still a replica symmetric solution but we also find an additional replica symmetry breaking solution. So in this regime we find an additional solution such that 0 < Xc < 1... 

This method to derive approximate solutions for kinetic equations is called the principle of quasi-stationarity (first introduced by Max Eodenstein, see box), and is very helpful for the kinetic evaluation of complex reaction systems as the mathematical treatment of the kinetic systems becomes simpler. (Note that for the example discussed above the Eodenstein principle is not needed, as the exact equations (4.3.38)-(4.3.40) can be derived quite easily this example was just chosen to show this principle.)... 

While this equation is quite simple in form, it hides the fact that the universal functional F[p] is not available in explicit form. Numerous schemes have been formulated for a direct optimization of the density based on these stationarity conditions, but these methods have not really been competitive. The most efficient approach has been to invoke a quasi-independent-particle approximation, formulated in the Kohn-Sham equations. 

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