Probabilistic theory

Unlike simple random variables that have no space or time dependence, the statistics of the random velocity field in homogeneous turbulence can be described at many different levels of complexity. For example, a probabilistic theory could be formulated in terms of the set of functions U(x, t) (x, t) e R3 x R However, from a CFD modeling perspective, such a theory would be of little practical use. Thus, we will consider only one-point and two-point formulations that describe a homogeneous turbulent flow by the velocity statistics at one or two fixed points in space and/or time. 

Fukuda, H. and Chou, T. W A probabilistic theory of the strength of short-fibre composites with variable fibre length and orientation,. /. Mater. Sci., 17, 1003 (1982). 

In Equation 3a, we take ej to be random, with zero mean and known distribution, in order to apply the probabilistic theory of hypothesis testing. Selection of the operator and the nature of ej are governed by (our perceptions of) the structure of Equation 2. Assumptions concerning and e are crucial. In the best of circumstances is linear (in the xj and B) and e is normal, independent and unbiased. Then,... 

Thus in the study of the most general problems in kinetic theory one was faced with two results that seemed to be completely irreconcilable with each other. It was plausible to put the blame for this on some internal contradiction in the foundations of the mechanico-probabilistic theory. (Cf. Sections 7 and 16.)... 

In accordance with the probabilistic nature of the biological activity concept, the most relevant methods for prediction of activity are those based on probabilistic theory and mathematical statistics, and the purpose of prediction is the complete relationship Pr(Doze) = P D). 

Hauptman, H. A new method in the probabilistic theory of the structure invariants. Acta Cryst. A31, 680-687 (1975). 

Suppes, P. (1970), A Probabilistic Theory of Causality, Amsterdam North Holland. 

This relation can be reformulated adopting one out of several possible modeling approaches. The conventional continuum mechanical approach for rewriting the interfacial momentum transfer terms for dispersed flows was outlined in sect 3.4.3. Hence, an alternative approach for calculating the interfacial momentum transfer terms based on kinetic or probabilistic theories, as proposed by Simonin and co-workers, is examined in this section. 

L. Devroye, L. Gyorfi and G. Lugosi, A Probabilistic Theory of Pattern Recognition Springer, Berlin (1997). 

Lin, Y. K. Probabilistic Theory of Structural Dynamics. Robert E. Krieger Publishing Company, Malabar, FL, 1976. 

The probabilistic theory of one-centered model for a chain process is described in [2]. Below its summary is given. 

Elementary probabilistic theory for non-branching chain reactions. An active intermediate being generated in an non-branching chain reaction has two chances (i) enter into a chain propagation reaction with a probability valued or (ii) to be lost with a probability value P, such that a + fi =. ... 

Elementary probabilistic theory of branching chain reactions. If branching occurs in a chain reaction then expression (1.1) for the average chain length includes the branching probability (<, which acts to decrease the chain breaking probability... 

LIN, Y.K. "Probabilistic Theory of Structural Dynamics," Robert E. Krleger Publ.Comp., 1967. 

Lin, Y.K. Probabilistic Theory of Structural Dgnemics, reprint Robert E. Krieger Publ. Company, 1976... 

The two probabilistic theories that will be mentioned in this book are conditional probabihty and Bayesian Belief Networks (BBN). 

Lin YK (1967) Probabilistic theory of structural dynamics. McGraw-Hill, New York Melchers RE (1999) Structural reliability analysis and prediction, 2nd edn. Wiley, Chichester Nigam NC (1983) Introduction to random vibrations. MIT Press, Cambridge... 

Lin YK (1976) Probabilistic theory of structural dynamics. Krieger, Huntington... 

The probabilistic approach is, in fact, a kind of numerical approach, where a probabilistic theory has been elaborated to eliminate the assumption of an infinite... 

---