Distribution higher order

In the same fashion as single variable distributions, higher-order moments can be considerd. The one of most interest is the second-order moment... 

The higher order distributions are defined analogously. 1 2 contains through the identity... 

In acidic solutions, equiUbtium is achieved more slowly. Polymerization of smaller species appears to occur sequentially a given polymer species first increases in size and then disappears, presumably because of its inclusion in higher order polymers. Depolymerization of siUcate species appears to be rapid, because crystalline Na2Si02 and Na2H2Si04 8H2O yield equivalent distributions of siUcate species in water upon dissolution. 

Many researchers have correlated the overall decomposition as an nxh. order reaction, with most paraffins following the first order and most olefins following a higher order. In general, isoparaffin rate constants are lower than normal paraffin rate constants. The rate constants are somewhat dependent on conversion due to inhibition effects that is, the rate constant often decreases with increasing conversion, and the order of conversion is not affected. This has been explained by considering the formation of aHyl radicals (38). To predict the product distribution, yields are often correlated as a function of conversion or other severity parameters (39). 

First-order sehemes use a uniform distribution aeross an element and seeond-order sehemes use a linear distribution aeross the element as shown in Fig. 9.16. Higher-order adveetion sehemes use more eomplex distributions aeross an element [29]. The distributions aeross the donor eell must be eon-strained to prevent numerieal oseillations. As an illustration, for seeond-order van Leer sehemes, the slope is limited using (9.15) and Fig. 9.17. The slope... 

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. 

Differentiation of Eq. (8) with respect to the position of a molecule gives a hierarchy of integro-differential equations, each of which relates a distribution function to the next higher order distribution function. Specifically,... 

Integral equations provide a satisfactory formalism for the study of homogeneous and inhomogeneous fluids. If the usual OZ equation is used, the best results are obtained from semiempirical closures such as the MV and DHH closures. However, this empirical element can be avoided by using integral equations that involve higher-order distribution functions, but at the cost of some computational complexity. 

J.E. Bjorkholm, C.V. Shank, Higher-order distributed feedback oscillators, Appl. Phys. Leu. 1972, 20. 306. 

Because much experimental work has been stimulated by the quasi-chemical theory, it is important to gain proper perspective by first describing the features of this theory.12 The term, quasichemical will be used to include the Bragg-Williams approximation as the zeroth-order theory, the Bethe or Guggenheim pair-distribution approximations as the first-order theory, and the subsequent elaborations by Yang,69 Li,28 or McGlashan31 as theories of higher order. 

Expressions for predicting monomer sequence distribution with higher order models and for monomer complex and other models have also been proposed. 

It was shown, in Eqs. (1-73), (1-74), (1-75), that a = 1, afy r> => 0, a = 0. As the zeroth approximation we shall assume that A mid /a are zero (their effects are negligibly small) if Eqs. (1-86) and (1-87) are multiplied by /a and A, respectively, we obtain the condition that og0 and -oSi are zero higher order equations would show that all the coefficients are zero. Thus, the coefficients are proportional to some power of /a (or A). The zero-order approximation to the distribution function is just the local maxwellian distribution... 

Although this lowest order approximation is used in determining the first order corrections to the distribution function, it is necessary to go to a higher order of approximation in determining the collision integral of Eq. (1-140). If we keep terms to first order in the small quantity m/M, the collision integral may be evaluated to give 28... 

But in order for a matrix to have a multiple root, it is necessary that its elements satisfy a certain algebraic relation to have a triple root they must satisfy two relations, and so forth for roots of higher order. Thus, if a matrix is considered as a point in 2-space, only those matrices that lie on a certain algebraic variety have multiple roots. Clearly, if the elements of a matrix are selected at random from any reasonable distribution, the probability that the matrix selected will have multiple roots is zero. Moreover, even if the matrix itself should have, the occurrence of any rounding errors would almost certainly throw the matrix off the variety and displace the roots away from one... 

The interpretation of the higher-order moments an is simplified if they are first centered about the first moment. To this end, we define the wth central moment pn of the distribution function or, equivalently,... 

The multidimensional theorem of averages can be used to calculate the higher-order joint distribution functions of derived sets of time functions, each of which is of the form... 

In accordance with the predictions that can be made on the basis of just the electric dipole approximation (see Section III.A) the observed dichroism is equal, but of opposite sign for the two enantiomers. This could be seen also in the valence shell ionization results for glycidol presented in Fig. 2. The added significance here is that a contribution to the angular distribution by higher order... 

When the transport equation for c is solved with a discretization scheme such as upwind, artificial diffusive fluxes are induced, effecting a smearing of the interface. When these diffusive fluxes are significant on the time-scale of the simulation, the information on the location of different fluid volumes is lost. The use of higher order discretization schemes is usually not sufficient to reduce the artificial smearing of the interface to a tolerable level. Hence special methods are used to guarantee that a physically reasonable distribution of the volume fraction field is maintained. 

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