User:DVD206/Effective conductivities

The effective conductivity between two separate nodes of an electrical network G without the boundary is equal to the ratio between the total current flowing between the nodes and the difference in potentials between the two nodes. The effective conductivities can be calculated in terms of the Schur complement of the Kirchhoff matrix of the network G.

Exercise 1 :

Suppose we know the effective conductivities between all pairs of nodes of a subset $\partial G$ of a network G. Then we can find the Dirichlet-to-Neumann map of $\{G,\partial G\}$. Also the converse is true.

Exercise 2 :

Suppose we know the effective conductivities between all pairs of nodes of a network G. Then we can find the individual conductivities of the edges of the network G.