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## Due June 7

### Problem 1

For $m\ge 2$, there is no Dirichlet eigenfunction on $V_m$ with respect to $\lambda_m=2$.

### Problem 2

1. The Dirichlet eigenfunctions on $V_m$ constructed in class, with respect to $\lambda_m=5$, are linearly independent.
2. If we add the three eigenfunctions chained from $p_1$ to $p_2$, from $p_2$ to $p_3$, and from $p_3$ to $p_1$, they are linearly dependent.

### Problem 3

Let $\mathcal E_m$ be a sequence of Dirichlet forms such that
$\displaystyle \mathcal E_m(u,v) = \sum_{j=1}^N \frac{1}{r_j}\mathcal E_{m-1}(u\circ f_j,v\circ f_j)$
and, given a function $u$ on $V_0$,
$\min\{\mathcal E_1(v,v): v|_{V_0} = u\} = \mathcal E_0(u,u)$,
then $\min\{\mathcal E_m(v,v): v|_{V_{m-1}} = u\} = \mathcal E_{m-1}(u,u)$ for every $m\ge1$ and any given function $u$ on $V_{m-1}$.

### Problem 4

Calculate all harmonic structures on the interval, self-similar with contractions $f_1(x) = \dfrac{1}{2}x$ and $f_2(x) = \dfrac{1}{2}x + \frac{1}{2}$ on the real line.

### Problem 5

Let $\mathcal E$ be a symmetric bilinear form on functions on the set $V$, and suppose $H$ is the symmetric operator that induces $\mathcal E$,
$\mathcal E(u,v) = -\langle u, Hv \rangle$.
The the following are equivalent:
1. $\mathcal E(u,u)\ge 0$, and $\mathcal E(u,u)=0$ if and only if $u$ is constant.
2. $H$ is nonpositive definite, and $Hu=0$ if and only if $u$ is constant.
(Hint: Use the spectral theorem.)