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## Due May 31st

### Problem 1

Let $u(x) = e^{\omega x}$ on $I=[0,1]$. Then, for each $m\ge 1$, $u|_{\mathcal P_m}$ is a discrete eigenfunction of $\Delta_m$ with eigenvalue
$\lambda_m = \dfrac{\omega^2}{4^m} + O(2^{-3m})$.
$\mathcal P_m$ is the dyadic partition $\{0, 1/2^m, \ldots, 1\}$ of $[0,1]$.

### Problem 2

Let $\phi(x) = 2 - \sqrt{4-x}$, for $x\in[0,4]$.
1. $\phi(x) = \dfrac{1}{4}x + O(x^2)$ as $x\to0$.
2. The sequence defined by, for given $\lambda_0\in[0,2]$, $\lambda_m = \phi(\lambda_{m-1})$ and $x_m = 4^m\lambda_m$ for $m\ge1$ satisfies
$x_m - x_{m-1} = O(2^{-m})$.
3. $x_m$ is Cauchy and hence converges.

### Problem 3

Let $\psi(x) = \dfrac{5 - \sqrt{25- 4x}}{2}$, for $x\in[0,4]$.
1. $\psi(x) = \dfrac{1}{5}x + O(x^2)$ as $x\to0$.
2. The sequence defined by, for given $\lambda_0\in[0,2]$, $\lambda_m = \psi(\lambda_{m-1})$ and $x_m = 5^m\lambda_m$ for $m\ge1$ satisfies
$x_m - x_{m-1} = O(5^{-m})$.
3. $x_m$ is Cauchy and hence converges.

### Problem 4

The sequence $u_m$ of functions resulting from the decimation process on the Sierpiński gasket constructs a uniformly continuous function on $V_*$.

### Problem 5

Calculate the eigenvalues and eigenvectors of the matrix
$\begin{pmatrix}4 & -1 & -1\\ -1 & 4 & -1 \\ -1 & - 1 & 4 \end{pmatrix}.$