Algebraic combinatorics by Christopher D. Godsil

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By Christopher D. Godsil

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Example text

K / D k 1 for k 1 a -string. 0 D We want to extend a function f 2 Vn n in such a way that the extended function nC1 belongs to f 2 VnC1 . From Eq. 6), we know that this is possible only if n and nC1 are in the same -string. ) belongs to Vn n for all n. So, the following problem arises: is such function f on S1 uniformly continuous and hence can be extended by continuity to S? When this is the case, we can consider the extended function fQ as an eigenfunction for the Laplacian on the whole gasket and define the corresponding eigenvalue as the limit of a suitably renormalized sequence f n g.

Consider the system S with the base b D 2 and digits f0; 1g. S / D Œ 23 ; 13 . 2). 2. Introduce a system S with the base b D 1 C i and digits f0; 1g. S / D ZŒi , which consists of numbers of the form a C i b, a; b 2 Z. These are called the Gaussian integers. S /, it is a fractal compact 50 3 Harmonic Functions on the Sierpi´nski Gasket Fig. 1 C F / : Here, as always, when an arithmetic operation is applied to a set, it means that it is applied to each element of the set. A picture of this set is shown in Fig.

Denote by V the space of smooth functions on with compact support restricted by some boundary conditions; see below. 9) P @vk 2 where d n x is the standard measure on Rn and jgrad vj2 D nkD1 j @x j . 11) @ where @ is the normal derivative and d n 1 y is a measure on the boundary @ . Suppose we restrict v by an appropriate boundary condition that forces the boundary integral in Eq. 11) to vanish. Then the operator  will be exactly the ratio of Q1 and Q0 . 13) In both cases,  is a nonnegative self-adjoint operator in L2 .

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