Random function second-order moment

The second order moment can be expressed either in terms of the covariance or the variogram. The covariance of the random function at points and is defined to be... 

The random function is said to be weakly or second order stationary when its first two moments are invariant under simultaneous translation by h. That is, for every x and h ... 

Thus, if the assumption of second order stationarity holds, then statistical inferences about the first two moments become possible since each pair of observations that are separated by a distance h can be considered a different realization of the random function. 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

The present study shows that It is possible to evaluate the variability of statically determinate and statically indeterminate structures due to spatial variation of elastic properties without resort to finite element analysis. If a Green s function formulation is used, the mean square statistics of the indeterminate forces are obtained in a simple Integral form which is evaluated by numerical methods in negligible computer time. It was shown that the response variability problem becomes a problem Involving only few random variables, even if the material property is considered to constitute stochastic fields. The response variability was estimated using two methods, the First-Order Second Moment method, and the Monte Carlo simulation technique. 

C, within the one-phase channel. The temperature is chosen at the hydrophile-lipophile balance (HLB) temperature for a salinity of 0.49%. The SANS data taken with an oil-water pontrast are analyzed by using a random-wave model with an appropriate spectral function. The spectral function is an inverse eighth-order polynomial in wave number k, containing three length scales 1/a, 1/b, and 1/c, and has finite second and fourth moments. This three-... 

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