Consistency of a moment set

The moments of an NDF represent some important physical properties of the underlying population of elements constituting the multiphase system under study. For this reason, they have to satisfy some simple rules. For instance, the positiveness of the density function over its support implies that the moment of order zero must be positive. Additionally, there are other simple, intuitive rules. For example, if the internal coordinate assumes only positive values (or in other words the moment is defined on a positive support) then the moment of order one (as well as all the other moments) must be positive. Another important property of the distribution is its variance (i.e. cr = m2 - m lmQ), which must be zero for a delta-function distribution, while it must be positive for polydisperse distributions. Accordingly, it has to be m2 m lmo. For higher-order moments the mathematical constraints are less intuitive and cannot be directly related to specific global properties of the multiphase systems. Fortunately, the theory of moments provides some interesting theorems that turn out to be very useful in determining whether a set of moments is invalid. 

Theorem 3.4 A moment set is said to be realizable if the Hankel-Hadamard determinants (Gautschi, 2004 Shohat G Tamarkin, 1943) are all non-negative  

Note that for k = 0 and / = 1 the Flankel-Fladamard determinant is mom2 - the validity 

Theorem 3.5 A necessary (but not sufficient) condition for validity is that of convexity of the function n(mk) with respect to k  

It is trivial to show that the convexity condition is equivalent to the positiveness of the Flankel-Hadamard determinants for k = 0,1 and I = 0,1,2, or in other words for the first four moments. For higher-order moments the equivalence is lost, since more stringent conditions are required by Theorem 3.4. However, it turns out that the convexity condition is useful for reasons that will be discussed below. 

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The concept of realizability or consistency of a moment set will be introduced in Section 3.1.4. 

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