Notional average

This final method is based on the approximate methods for calculating compressible flow in a pipe, as described in Sections 6.8, 6.9 and Appendix 2, and in a valve, as described in Section 9.11.2. The major approximation made is that the specific volume of the gas in large stretches of pipework can be represented adequately by a notional average specific volume. The benefit of ASVAM is that it can be programmed to avoid implicit loops, and hence the associated problems of convergence in the main equations, while maintaining a very reasonable accuracy. 

Having derived the ratio of supply vessel to pipe outlet pressures, p /p), we may estimate the notional average specific volume throughout pipe and valve from equation (6.62), namely ... 

In order to get a better notion of the scaling relationship between gyration radius and average chain length for different density regimes, it is convenient... 

Thus, in order to reproduce the effect of an experimentally existing activation barrier for the scission/recombination process, one may introduce into the MC simulation the notion of frequency , lo, with which, every so many MC steps, an attempt for scission and/or recombination is undertaken. Clearly, as uj is reduced to zero, the average lifetime of the chains, which is proportional by detailed balance to Tbreak) will grow to infinity until the limit of conventional dead polymers is reached. In a computer experiment Lo can be easily controlled and various transport properties such as mean-square displacements (MSQ) and diffusion constants, which essentially depend on Tbreak) can be studied. 

The notion of the distribution function of a random variable is also useful in connection with problems where it is not possible or convenient to subject the underlying function X(t) to direct measurements, but where certain derived time functions of the form Y(t) = [X(t)] are available for observation. The theorem of averages then tdls us what averages of X(t) it is possible to calculate when all that is known is the distribution function of . The answer is quite simple if / denotes (almost) apy real-valuqd function of a real variable, then all X averages of the form... 

In the frame of the present review, we discussed different approaches for description of an overdamped Brownian motion based on the notion of integral relaxation time. As we have demonstrated, these approaches allow one to analytically derive exact time characteristics of one-dimensional Brownian diffusion for the case of time constant drift and diffusion coefficients in arbitrary potentials and for arbitrary noise intensity. The advantage of the use of integral relaxation times is that on one hand they may be calculated for a wide variety of desirable characteristics, such as transition probabilities, correlation functions, and different averages, and, on the other hand, they are naturally accessible from experiments. 

All notional particles have equal weight, and estimated statistical quantities are found using cell averages. 

The idea behind measures of location and central tendency is contained within the notion of the average. There are predominantly three summary statistics that are commonly used for describing this aspect of a set of data the arithmetic mean - normally shortened to the mean, the mode and the median. 

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