Probability distribution, structure invariants

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 RTD quantifies the number of fluid particles which spend different durations in a reactor and is dependent upon the distribution of axial velocities and the reactor length [3]. The impact of advection field structures such as vortices on the molecular transit time in a reactor are manifest in the RTD [6, 33], MRM measurement of the propagator of the motion provides the velocity probability distribution over the experimental observation time A. The residence time is a primary means of characterizing the mixing in reactor flow systems and is provided directly by the propagator if the velocity distribution is invariant with respect to the observation time. In this case an exact relationship between the propagator and the RTD, N(t), exists... 

Figure 9.1 The conditional probability distribution, P(4>hk). of the three-phase structure invariants, 4>hK having associated parameters Ahk with values of 0,1, 2,4, and 6. When A ss 0, all values of 4>jjk 3te equally likely, and no information useful for phase determination is available. However, the sum of the three phases for most invariants with A 6 is close to 0°, and an estimate of one phase can be made if the other two are known.

To do so, we first provide in Section 2 a brief overview of Markov Chains and Monte-Carlo simulations. Section 3 presents the structure of our Markov model of failures and replacement as well as its imder-lying assumptions. In Section 4, we run Monte-Carlo simulations of the model and generate probability distributions for the lifecycle cost and utility of the two considered architectures that serve as a basis for our comparative analysis. Important trends and invariants are identified and discussed. For example, changes in average lifecycle cost and utility resulting from fractionation are observed, as well as reductions in cost risk. We conclude this work in Section 5. 

The probability distribution of the maximum value (i.e. the largest extreme) is often approximated by one of the asymptotic extreme value distributions. Hence for structures subjected to a single time-varying action, a random process model is replaced by a random variable model and the principles and methods for time invariant models may he apphed. 

For a general structural system and limit state, similar to the case of barrier crossing at a constant level of a single-degree-of-freedom oscillator, Gueri and Rackwitz (1986) included R as basic variables, together with X, and reduced the dynamic problem to a time-invariant one with n + m basic variables therefore, the first-order reliability method can be used, where m is the number of response variables. The required information includes the explicit form of the performance function, G the probability distributions not only of system parameters, X, but also of response variables, R and even the joint PDF,, R(x, r). The method is very complicated because of the number of response variables and the difficulty of determining the distribution of response variables of interest. For some limit states, it may not be possible to identify R and determine the performance function in the explicit form. 

The time invariance in the spatial distribution of p is another manifestation of the self-similarity of the structures generated by time- and spatially periodic chaotic flows. Such invariant statistical properties can be demonstrated by computing the probability density H(log p). If the probability density function of the scaled variable (p/(p is computed according to eq. (3-13), the distributions are scaled automatically by the mean density. 

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