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Let be the Beurling algebra with weight on the unit circle and, for a closed set , let . We prove that, for , there exists a closed set of measure zero such that the quotient algebra is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras and the algebra of absolutely continuous functions on , we characterize the closed sets for which the restriction algebras and are generated by their idempotents.
In the shape from shading problem of computer vision one
attempts to recover the three-dimensional shape of an object or
landscape from the shading on a single image. Under the
assumptions that the surface is dusty, distant, and illuminated
only from above, the problem reduces to that of solving the
eikonal equation |Du|=f on a domain in . Despite
various existence and uniqueness theorems for smooth solutions,
we show that this problem is unstable, which is catastrophic for
general numerical algorithms.
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