On hearing the shape of a drum

This lecture was filmed under the auspices of the Committee on Educational Media of the Mathematical Association of America, and an expanded version of the script has been published in The American Mathematical Monthly.

Kac actually discusses only the case of a single component membrane (C=1) with a convex polygonal perimeter and one or more convex polygonal holes. A he observes the result (1.7 b) follows formally by letting the polygons approach smooth curves while the extension to C>1 is obvious from his analysis.

This follows by iterating the defining equation Twn=μnwn.

It is easily shown that the trace Tr{A}=∑j−1NAjj of a matrix A is invariant under the similarity transform A′=SAS−1. On choosing S to diagonalize A′ the A′ii becomes the eigenvalues, thereby proving the result.

Notice that in the case of doubled or trebled edge sites bα includes only those bonds crossed as the boundary passes the site.

There are essentially only three configurations.

It may be shown generally for two-dimensional walks that rs≈Dqs/(s−1) as s→∞ where D is a constant. Consequently the first term in (5.5) is approximately DNe-qz∑s(qz)s/(s+1)! which for large z approximates. DNe−qz(e−qz−1)/qz≃DNh2/dτ=D′|Ω|/τ as required by (5.4).

It is necessary here to consider strong embeddings in which points placed on adjacent lattice sites must always be incident on an edge corresponding to the lattice bond.

Uhlenbeck, 1962, I

Domb, 1960, Advances in Phys., 9, 315, 10.1080/00018736000101189

Collatz, 1957, Abh. Math. Sem. Univ. Hamburg, 21, 63, 10.1007/BF02941924

Hoover, 1962, J. Chem. Phys., 36, 3141, 10.1063/1.1732443

Sykes, 1966, Lattice Constant Systems in Crystal Statistics, J. Math. Phys., 7, 10.1063/1.1705066

Berge, 1962