Further, using a first order mean-value approach, a probabilistic formulation is proposed.
We discuss first order mean field and second order Onsager truncations of the Plefka expansion of the Gibbs free energy.
Yet, only first order (mean value) and second order (pair-wise in the covariance matrix) suffice for a complete description of the Gaussian probability density.
Finally, a number of theorems is proven concerning the Plefka expansion, relating the first order mean field and the second order Onsager approximation to the Bethe approximation.
In such a way, in the z-scored values, data variance was taken into account twice – by means of first order (mean value) and second order (SD) statistics [34].
These processes are exactly analogous to the statistical method of weighting the (first order) mean of an experimental observation according to its (second order) standard error: an experimental 'prediction error' of high precision will compel a change in the null 'prediction'prediction
The second order mean velocity field is computed in the neighborhood of the stack plates.
In this article we consider the following fourth order mean field equation on smooth domain Ω⋐R4:Δ2u="ϱKeu∫ΩKeuin Ω,u= Δu= 0on ∂Ω, where ϱ∈R and 0<K∈C2.
These relationships serve as a first-order means to identify locations of potentially unstable areas in a landscape, thereby providing a basis to assess the potential impacts of future catchment disturbances.
Another result by van Benthem is that the type <1> quantifiers accepted by pushdown automata are precisely those for which the corresponding binary relation between numbers is definable (with first-order means) in additive arithmetic, i.e., in the model (N, +), where N = {0, 1, 2,…}.
('Monadic second-order' means that the logic is like first-order, except that we can also quantify over sets of elements but not over binary relations on elements, for example).
