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We show in two dimensions that measuring Dirichlet data for the conductivityequation on an open subset of the boundary and, roughly speaking, Neumann datain slightly larger set than the complement uniquely determines the conductivityon a simply connected domain. The proof is reduced to show a similar result forthe Schr\"odinger equation. Using Carleman estimates with degenerate weights weconstruct appropriate complex geometrical optics solutions to prove theresults.

Partial Data for the Calderon Problem in Two Dimensions