View source for On 2D Inverse Problems/Y-Δ and star-mesh transforms


The effective conductivities b/w boundary nodes and the Dirichlet-to-Neumann map of a network are invariant under the following '''star-mesh transform''' and '''Y-Δ move''' (a special case of the star-mesh transform in which the center is an interior node and has the degree ''3''), illustrated by the following drawings from Wikipedia:

[[File:Star-mesh transform.svg|thumbnail| Star-mesh transform ]]

'''Exercise (**)'''. Let ''d'' be a diagonal entry of the Kirchhoff matrix ''K'' of a network ''G'', corresponding to an interior node. Use the Schur complement formula <math>\Lambda_G = K/C = (K/d)/(C/d)</math> for the Dirichlet-to-Neumann map to prove the invariance.

The series or parallel connection rules for replacing conductors follow from the invariance property of the Y-Δ move and can be viewed as its special cases, as also erasing an interior spike or a loop. 

[[File:Wye-delta-2.svg|thumbnail]]

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