Trend parameters

In some selected cases the g value and peak width can be related to maturity/thermal stress. One such case, where a column of generally homogeneous organic material was present and gave smoothly trending parameters, is shown in Table III. 

In many random vibration problems, the mean is constant and is thus easily estimated and removed from the signal (sample-mean adjusted signal). The case of a time-dependent mean (also referred to as deterministic trend) may be treated via proper techniques, such as curve fitting, high-pass filtering, or special parametric models (such as integrated models with a deterministic trend parameter (Box et al. 1994)). 

When K lor K = 1.07 as shown in Fig. 2.5 an interpretation in terms of a society at the crossroads between a liberal and a totalitarian state can be made. The reason for this is the higher willingness of the individual to adapt to the prevailing opinion. As can be seen in Figs. 2.3 b and c and Fig. 2.5 the value iTc = 1 is the critical value of the trend parameter k, where for (5 = 0 the potential (2.129) changes from the monostable to the bistable form when all the possible consequences for bifurcation etc., as discussed in Sect. 2.3, can be observed. 

In reality there may exist a feedback between the socio-configuration and the trend parameters in which case the latter also become dynamic variables. This general problem will also be discussed in Chap. 3 and in later models. Here the problem is simplified by assigning a certain assumed time dependence... 

In general they depend on the socio-configuration n and on trend parameters K (e.g. the adaptation and preference parameters of Chap. 2). The latter are considered as constants here, but may also become dynamic variables in an extended treatment (see, for instance. Chap. 5). 

If the trend parameters k are constants with time, the drift coefficients Ki(x) and the fluctuation coefficients Qij (x) become independent of time. Fokker-Planck equations of the form (3.32) with time independent coefficients are treated in textbooks on statistical physics [1.51-53]. 

The more general case, as already indicated in Sect. 3.1, consists of an interaction between socio-configuration x t) e , situation vector y( ) e and the trend parameters K t) = iti( ),. ..,KT(t) e X where the trend parameters K(t) are now represented as elements of a T-dimensional trend-space When thinking in terms of Langevin equations the coupled set of equations for the variables... 

Similarly, the time derivatives of the trend parameters K(t) (which determine the explicit form of the transition probabilities) describe the psychological and political reactions of trends to the changing situation and will also consist of a superposition of competing influences expressed in partial rates ... 

Here, the transition probabilities in general can not only depend on the socioconfiguration w and on trend parameters k but also on the material situation y. The first term of the rhs of (3.100) conserves the number... 

If the grossvariables are expected to (approximately) obey a closed subdynamics, it is self-consistently plausible that the transition probabilities p i and pj° only depend on Uai via the gross variables Gq. This assumption amounts to a specialized choice of the trend parameters... 

In (4.34) the several trends described by nif, and are automatically superimposed on each other. All trend parameters are assumed to remain constant in the time interval in which the migration process is to be investigated. In principle, however, it is also possible to take into account a time dependence for all trend parameters (Sect. 3.3). 

Some sets of reasonable trend parameters will now be chosen and the resulting paths of the mean values jc(r), y (r) graphed and discussed (Fig. 4.2a-8a). In one case of particular interest the variances cr (r), (r) and Oyy (t) will also... 

Figure 4.3. (a) x-y diagram of fluxlines of mean values using trend parameters — Tty — 0, K -Ky- 0.5 and = 1.0 leading to two stable foci... 

In Fig. 4.7 a, b strong asymmetrical external sympathy parameters and internal sympathy parameters are shown to lead to a phase transition. Figure 4.7 a shows mean value fluxhnes which approach the focus at x = y = 0 extremely slowly ( critical slowing down ). This is due to the fact that the focus becomes unstable for the values assumed for the trend parameters. The stationary dis-... 

In accordance with their substantial implications the trend parameters v, S, a and are named as follows ... 

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