Invariant probability

While there is, at present, no known CA analogue of a Froebenius-Perron construction, a systematic n -order approximation to the invariant probability distributions for CA systems is readily obtainable from the local structure theory (LST), developed by Gutowitz, et.al. [guto87a] LST is discussed in some detail in section 5.3. 

The spatial and temporal dimensions provide a convenient quantitative characterization of the various classes of large time behavior. The homogeneous final states of class cl CA, for example, are characterized by d l = dll = dmeas = dmeas = 0 such states are obviously analogous to limit point attractors in continuous systems. Similarly, the periodic final states of class c2 CA are analogous to limit cycles, although there does not typically exist a unique invariant probability measure on... 

Introducing these relations in Eq. (88) for the invariant probability, we obtain the escape-rate formula... 

On the other hand, the nonequilibrium steady states are constructed by weighting each phase-space trajectory with a probability which is different for their time reversals. As a consequence, the invariant probability distribution describing the nonequilibrium steady state at the microscopic phase-space level explicitly breaks the time-reversal symmetry. 

D.H. Kobe, Conventional and gauge-invariant probability amplitudes when electromagnetic potentials are turned on and off, Eur. J. Phys. 5 (1984) 172. 

The invariant probability density, which is associated to a surface of constant kinetic energy, can be written as a generalized function using the Dirac notation ... 

We are concerned with the determination of the functions L(/), h(/), and p l l ), from suitably designed population data to be determined from experiments. We assume that the cells are growing in a batch culture under balanced exponential growth conditions. Under these circumstances we have fid t) = where /(/) is the time-invariant probability dis-... 

An invariant probability measure Ip associated with this real zero can now be constructed. The stability of the trajectories... 

Considerable effort has been devoted to the asymptotic (in the sense of large size limit) properties of the invariant probability distribution, corresponding to the steady-state solution of the master equation. We briefly summarize some important results. 

It is useful to note that Theorem A.3 may be proven by looking at the ergodic properties of the backward recurrence time process A [Asmussen (2003), VII.2]. In particular one sees that the condition mx < oo is precisely the condition for A to be positive recurrent. This implies directly the existence of a unique invariant probability measure. One can write it explicitly ... 

In Fig. 2a, we compare the modulus of the normal component of the magnetic induction B (r) provided by the sensor and the one calculated by the model. Because of the excitation s shape, the magnetic induction B° (r) is rotation invariant. So, we only represent the field along a radii. It s obvious that the sensor does not give only the normal component B = but probably provides a combination, may be linear, of... 

Finally, and probably most importantly, the relations show that changes (of a nonhivial type) in the phase imply necessarily a change in the occupation number of the state components and vice versa. This means that for time-reversal-invariant situations, there is (at least) one partner state with which the phase-varying state communicates. 

As a consequence of this observation, the essential dynamics of the molecular process could as well be modelled by probabilities describing mean durations of stay within different conformations of the system. This idea is not new, cf. [10]. Even the phrase essential dynamics has already been coined in [2] it has been chosen for the reformulation of molecular motion in terms of its almost invariant degrees of freedom. But unlike the former approaches, which aim in the same direction, we herein advocate a different line of method we suggest to directly attack the computation of the conformations and their stability time spans, which means some global approach clearly differing from any kind of statistical analysis based on long term trajectories. 

The long term behavior of any system (3) is described by so-called invariant measures a probability measure /r is invariant, iff fi f B)) = ft(B) for all measurable subsets B C F. The associated invariant sets are defined by the property that B = f B). Throughout the paper we will restrict our attention to so-called SBR-measures (cf [16]), which are robust with respect to stochastic perturbations. Such measures are the only ones of physical interest. In view of the above considerations about modelling in terms of probabilities, the following interpretation will be crucial given an invariant measure n and a measurable set B C F, the value /r(B) may be understood as the probability of finding the system within B. 

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

After these preliminaries we are now ready for a mathematically precise definition of an almost invariant set. Let p M he any probability measure. Wc say that the set B is 5-almost invariant with respect to p if... 

As it turns out, there exists a relationship between those probabilities, by which sets are almost invariant, and associated eigenvalues X (cf. [6]). 

Lemma 3. Let p M be a probability measure and let X and Y be disjoint sets which are dx- resp. dy-almost invariant with respect to p. Moreover suppose that f (X) n Y = 0 and f Y) n A = 0. Then X UY is 6xuy-almost invariant with respect to p where... 

Fig. 9. Illustration of four almost invariant sets with respect to the probability measure i/4. The coloring is done according to the magnitude of the discrete density.

Type V isotherms of water on carbon display a considerable variety of detail, as may be gathered from the representative examples collected in Fig. 5.14. Hysteresis is invariably present, but in some cases there are well defined loops (Fig. 5.14(b). (t ), (capillary-condensed water. Extreme low-pressure hysteresis, as in Fig. 5.14(c) is very probably due to penetration effects of the kind discussed in Chapter 4. 

The residues not in the framework region form the loops between the p strands. These loops may vary in length and sequence among immunoglobulin chains of different classes but are constant within each class the sequence of the loops is invariant. The functions of these loops are not known, but they are probably involved in the effector functions of antibodies. When an antibody-antigen complex has been formed, signals are... 

The paramagnetic impurity which invariably accompanies Wilkinson s catalyst has proved difficult to identify. It is probably the air-stable, green, irans- (RhCl(CO)(PPh3)2j- see K. R. Dunbar and S. C Harfner, Innrg. Chem. 31, 36"76-9 (1992),... 

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