Inequalities on Infinite Networks

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Inequalities on networks have played important roles in the theory of netwoks. We study several famous inequalities on networks such as Wirtinger's inequality, Hardy's inequality, Poincare-Sobolev's inequality and the strong isoperimetric inequality, etc. These inequalities are closely related to the smallest eigenvalue of weighted discrete Laplacian. We discuss some relations between these inequalities and the potential-theorerteic magnitude of the ideal boundary of an infinite network.

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