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Normalization of the diffusive filters that represent the inhomogeneous covariance operators of variational assimilation, using asymptotic expansions and techniques of non-euclidean geometry; Riemannian geometry and the generic parametrix expansion method; Part II
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    Riemannian geometry and the generic parametrix expansion methodalgorithms
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    This note is the second part of a study which develops the geometrical ideas that underly the 'parametrix expansion' asymptotic method for estimating the shape of the impulse response function of the diffusion equation in a smoothly curved non-Euclidean space. This part of the study develops the algebraic tools required to treat the general n-dimensional case in Riemannian geometry. Notations are developed both to facilitate the hand calculations for the low-order expansion terms, and to enable the bulk of the algebraic manipulations to be automated, should it become desirable to carry out the asymptotic expansion to a higher order. The general procedure is illustrated by exhibiting, in moderate detail, the algebraic operations required to obtain the first two coefficients for the amplitude adjustment quotient in the n-dimensional case, and for the important special cases of two and three dimensions where the curvature terms upon which the formulas depend take on simpler forms.

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