By Pierre H. Berard, G. Besson
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Additional resources for Spectral geometry: direct and inverse problems
B(Ci Cj ) ) the list of digital points of B (resp. B) that are located on the right (resp. left) side of each elementary part (Ck Ck+1 ) of (Ci Cj ) with i ≤ k < j. 2 Digital Circle and Circular Arc Definition 1 (Digital circle (Fig. b)). A digital contour C is a digital circle iﬀ there exists a Euclidean disk D(ω, r) that contains B but no points of ¯ B. Deﬁnition 2 is the analog of Deﬁnition 1 for parts of C. Definition 2 (Circular arc (Fig. c)). A part (Ci Cj ) of C (with i < j) is a circular arc iﬀ there exists a Euclidean disk D(ω, r) that contains B(Ci Cj ) but ¯(C C ) .
In recent years, good progress has been made in the development of new data assimilation methods but the fact remains that less than 10% of the total volume of incoming satellite data can be assimilated. The presence of clouds is one important obstacle and the eﬀort is now focusing on representing and includinging clouds in models of the atmospheric radiative transfer in the assimilation. There are questions associated with the estimation of the observations error statistics, taking into account that, for satellite instruments, data cannot be assumed to be independent as their error is spatially correlated.
In , it is shown how a Riemannian metric geometrically gives rise to a midpoint formation (out of which, in turn, the Levi-Civita aﬃne connection may be constructed, by the process given by the Theorem). Problem: Since a midpoint formation structure μ gives rise to an aﬃne connection λ by a geometric construction, and an aﬃne connection λ gives rise to a curvature r, likewise constructed geometrically, one gets by concatenation of these constructions a geometric construction of r out of μ. Is there a more direct geometric way of getting r from μ?