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A Bayesian Approach to the Estimation of Maps between by Butler L. T., Levit B.

By Butler L. T., Levit B.

Enable be a soft compact orientated manifold with no boundary, imbedded in a Euclidean house , and permit be a gentle map of right into a Riemannian manifold Λ. An unknown nation is saw through X = , the place > zero is a small parameter and is a white Gaussian noise. For a given gentle past on and gentle estimators of the map we derive a secondorder asymptotic growth for the comparable Bayesian threat. The calculation includes the geometry ofthe underlying areas and , particularly, the integration-by-parts formulation. utilizing this consequence, a second-order minimax estimator of is located in response to the fashionable idea of harmonic maps and hypo-elliptic differential operators.

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1 0 , l l . The solution of Eq. (33) read now w = w1+ i w2= l6e - ( K / 2 + 3 A ) + [$ 4 (1) V a a a e 3 ~= + ~ ( 1 ) v a ~ ab ( 2 7 ) v 1b m=O, - vaF:;7)vb + i v " F p ], (40) 28 and [the flux components were introduced in (35)]. In addition, one obtains a differential equation for the spinor with the non-trivial torsion components as introduced in Eq. F;i7)vb . 21, (43) To make the set of equations complete, we have to give the differential equations obeyed by the vector field u,which is straightforward if we use the differential equation for the spinor recall w& = 'p&Uc.

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