By Butler L. T., Levit B.

Enable be a soft compact orientated manifold with out boundary, imbedded in a Euclidean area , and allow be a delicate map of right into a Riemannian manifold Λ. An unknown kingdom is saw through X = , the place > zero is a small parameter and is a white Gaussian noise. For a given gentle previous on and delicate estimators of the map we derive a secondorder asymptotic enlargement for the similar Bayesian probability. The calculation contains the geometry ofthe underlying areas and , specifically, the integration-by-parts formulation. utilizing this end result, a second-order minimax estimator of is located in line with the fashionable concept of harmonic maps and hypo-elliptic differential operators.

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1584.

Proposed by Murray S. Klamkin, University of Alberta. Determine the maximum value of the sum tan B C tan + λ + 2 2 tan C A tan + λ + 2 2 tan A B tan + λ 2 2 where A, B, C are the angles of a triangle and λ is a nonnegative constant. 37 of O. ) 1416. Proposed by Hidetosi Fukagawa, Aichi, Japan. In the figure, the unit square ABCD and the line l are fixed, and the unit square P QRS rotates with P and Q lying on l and AB respectively. X is the foot of the perpendicular from S to l. Find the position of point Q so that the length XY is a maximum.

Determine f (n) and find all RC-sets in Nn of cardinality f (n). 1096. Proposed by Murray S. Klamkin, University of Alberta. Determine the maximum and minimum values of S ≡ cos A B C A B C cos cos + sin sin sin , 4 4 4 4 4 4 where A, B, C are the angles of a triangle. ) 1098. Proposed by Jordi Dou, Barcelona, Spain. Characterize all trapezoids for which the circumscribed ellipse of minimal area is a circle. 1102. Proposed by George Tsintsifas, Thessaloniki, Greece. Let σn = A0 A1 . . An be an n-simplex in n-dimensional Euclidean space.