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By W. T. Tutte

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D e 3 , a cube root of 1. ? S / for such a system? 3. Continued Fractions There is one more interesting numerical system related to the notion of continued fraction. Let k D fk1 ; k2 ; : : : g be a finite or infinite system of positive integers. 4) as n ! 1 if the sequence k is infinite. It is well known that the limit in question always exists. 0; 1/ is the value of the unique infinite continued fraction. 0; 1/, they can be values of two different finite continued fractions: k D fk1 ; : : : ; kn 1 ; 1g and k 0 D fk1 ; : : : ; kn 1 C 1g.

K/ ; 2 k+1 2 Œ0; 1, we have . / ˙ 1 1 D k a+ . kC1 2 / 2 . /C3 . 3) . n Corollary. l/ (here we have not only congruence but in fact equality, since in this case, 21 D 1). Proof of the theorem. Consider the triangular piece of the infinite gasket that is based on the segment Œk 1; k C 1. It is shown in Fig. 4. We denote the values of at the points k 1; k; k C1 by a ; a; aC respectively. Then the values bC ; b ; c in the remaining vertices shown in Fig. l/ is an integer when l < 2n . 42 3 Harmonic Functions on the Sierpi´nski Gasket The result is c D 5a 2a 3a C 2aC ; 5 bC D 2a 2aC ; b D 2a 2aC C 3a : 5 Consider now the functions g˙ W !

1 First Properties of Harmonic Functions We start with the following fact. 1. S1 / of all harmonic functions on S1 has dimension 3. S1 / are the values of this function at three boundary points. Proof. From linear algebra, we know that if a homogeneous system of linear equations in n variables has only the trivial solution, then the corresponding inhomogeneous system has a unique solution for any right-hand side. Sn / D 3 for all n 1. Therefore, every harmonic function on Sn has a unique harmonic extension to SnC1 , hence to S1 .

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Connectivity in Graphs by W. T. Tutte


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