Part IB, 2011, Paper 4

# Part IB, 2011, Paper 4

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Paper 4, Section I, E

commentLet $B[0,1]$ denote the set of bounded real-valued functions on $[0,1]$. A distance $d$ on $B[0,1]$ is defined by

$d(f, g)=\sup _{x \in[0,1]}|f(x)-g(x)| .$

Given that $(B[0,1], d)$ is a metric space, show that it is complete. Show that the subset $C[0,1] \subset B[0,1]$ of continuous functions is a closed set.

Paper 4, Section II, E

commentDefine a contraction mapping and state the contraction mapping theorem.

Let $(X, d)$ be a non-empty complete metric space and let $\phi: X \rightarrow X$ be a map. Set $\phi^{1}=\phi$ and $\phi^{n+1}=\phi \circ \phi^{n}$. Assume that for some integer $r \geqslant 1, \phi^{r}$ is a contraction mapping. Show that $\phi$ has a unique fixed point $y$ and that any $x \in X$ has the property that $\phi^{n}(x) \rightarrow y$ as $n \rightarrow \infty$.

Let $C[0,1]$ be the set of continuous real-valued functions on $[0,1]$ with the uniform norm. Suppose $T: C[0,1] \rightarrow C[0,1]$ is defined by

$T(f)(x)=\int_{0}^{x} f(t) d t$

for all $x \in[0,1]$ and $f \in C[0,1]$. Show that $T$ is not a contraction mapping but that $T^{2}$ is.

Paper 4, Section I, E

commentLet $f(z)$ be an analytic function in an open subset $U$ of the complex plane. Prove that $f$ has derivatives of all orders at any point $z$ in $U$. [You may assume Cauchy's integral formula provided it is clearly stated.]

Paper 4, Section II, D

commentState and prove the convolution theorem for Laplace transforms.

Use Laplace transforms to solve

$2 f^{\prime}(t)-\int_{0}^{t}(t-\tau)^{2} f(\tau) d \tau=4 t H(t)$

with $f(0)=0$, where $H(t)$ is the Heaviside function. You may assume that the Laplace transform, $\widehat{f}(s)$, of $f(t)$ exists for Re $s$ sufficiently large.

Paper 4, Section I, C

commentA plane electromagnetic wave in a vacuum has electric field

$\mathbf{E}=\left(E_{0} \sin k(z-c t), 0,0\right)$

What are the wavevector, polarization vector and speed of the wave? Using Maxwell's equations, find the magnetic field B. Assuming the scalar potential vanishes, find a possible vector potential $\mathbf{A}$ for this wave, and verify that it gives the correct $\mathbf{E}$ and $\mathbf{B}$.

Paper 4, Section II, D

commentShow that an irrotational incompressible flow can be determined from a velocity potential $\phi$ that satisfies $\nabla^{2} \phi=0$.

Given that the general solution of $\nabla^{2} \phi=0$ in plane polar coordinates is

$\phi=\sum_{n=-\infty}^{\infty}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right) r^{n}+c \log r+b \theta$

obtain the corresponding fluid velocity.

A two-dimensional irrotational incompressible fluid flows past the circular disc with boundary $r=a$. For large $r$, the flow is uniform and parallel to the $x$-axis $(x=r \cos \theta)$. Write down the boundary conditions for large $r$ and on $r=a$, and hence derive the velocity potential in the form

$\phi=U\left(r+\frac{a^{2}}{r}\right) \cos \theta+\frac{\kappa \theta}{2 \pi}$

where $\kappa$ is the circulation.

Show that the acceleration of the fluid at $r=a$ and $\theta=0$ is

$\frac{\kappa}{2 \pi a^{2}}\left(-\frac{\kappa}{2 \pi a} \mathbf{e}_{r}-2 U \mathbf{e}_{\theta}\right)$

Paper 4, Section II, F

commentSuppose that $P$ is a point on a Riemannian surface $S$. Explain the notion of geodesic polar co-ordinates on $S$ in a neighbourhood of $P$, and prove that if $C$ is a geodesic circle centred at $P$ of small positive radius, then the geodesics through $P$ meet $C$ at right angles.

Paper 4, Section I, F

commentA ring $R$ satisfies the descending chain condition (DCC) on ideals if, for every sequence $I_{1} \supseteq I_{2} \supseteq I_{3} \supseteq \ldots$ of ideals in $R$, there exists $n$ with $I_{n}=I_{n+1}=I_{n+2}=\ldots$ Show that $\mathbb{Z}$ does not satisfy the DCC on ideals.

Paper 4, Section II, F

commentState and prove the Hilbert Basis Theorem.

Is every ring Noetherian? Justify your answer.

Paper 4, Section I, G

comment(i) Let $V$ be a vector space over a field $F$, and $W_{1}, W_{2}$ subspaces of $V$. Define the subset $W_{1}+W_{2}$ of $V$, and show that $W_{1}+W_{2}$ and $W_{1} \cap W_{2}$ are subspaces of $V$.

(ii) When $W_{1}, W_{2}$ are finite-dimensional, state a formula for $\operatorname{dim}\left(W_{1}+W_{2}\right)$ in terms of $\operatorname{dim} W_{1}, \operatorname{dim} W_{2}$ and $\operatorname{dim}\left(W_{1} \cap W_{2}\right)$.

(iii) Let $V$ be the $\mathbb{R}$-vector space of all $n \times n$ matrices over $\mathbb{R}$. Let $S$ be the subspace of all symmetric matrices and $T$ the subspace of all upper triangular matrices (the matrices $\left(a_{i j}\right)$ such that $a_{i j}=0$ whenever $\left.i>j\right)$. Find $\operatorname{dim} S, \operatorname{dim} T, \operatorname{dim}(S \cap T)$ and $\operatorname{dim}(S+T)$. Briefly justify your answer.

Paper 4, Section II, G

commentLet $V$ be an $n$-dimensional $\mathbb{R}$-vector space and $f, g: V \rightarrow V$ linear transformations. Suppose $f$ is invertible and diagonalisable, and $f \circ g=t \cdot(g \circ f)$ for some real number $t>1$.

(i) Show that $g$ is nilpotent, i.e. some positive power of $g$ is 0 .

(ii) Suppose that there is a non-zero vector $v \in V$ with $f(v)=v$ and $g^{n-1}(v) \neq 0$. Determine the diagonal form of $f$.

Paper 4, Section I, H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain on a state space $S$, and let $p_{i j}(n)=\mathbb{P}\left(X_{n}=j \mid X_{0}=i\right)$.

(i) What does the term communicating class mean in terms of this chain?

(ii) Show that $p_{i i}(m+n) \geqslant p_{i j}(m) p_{j i}(n)$.

(iii) The period $d_{i}$ of a state $i$ is defined to be

$d_{i}=\operatorname{gcd}\left\{n \geqslant 1: p_{i i}(n)>0\right\}$

Show that if $i$ and $j$ are in the same communicating class and $p_{j j}(r)>0$, then $d_{i}$ divides $r$.

Paper 4, Section I, A

commentUse the method of characteristics to find a continuous solution $u(x, y)$ of the equation

$y \frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y}=0$

subject to the condition $u(0, y)=y^{4}$.

In which region of the plane is the solution uniquely determined?

Paper 4, Section II, A

commentLet $D$ be a two dimensional domain with boundary $\partial D$. Establish Green's second identity

$\int_{D}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d A=\int_{\partial D}\left(\phi \frac{\partial \psi}{\partial n}-\psi \frac{\partial \phi}{\partial n}\right) d s$

where $\frac{\partial}{\partial n}$ denotes the outward normal derivative on $\partial D$.

State the differential equation and boundary conditions which are satisfied by a Dirichlet Green's function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$ for the Laplace operator on the domain $D$, where $\mathbf{r}_{0}$ is a fixed point in the interior of $D$.

Suppose that $\nabla^{2} \psi=0$ on $D$. Show that

$\psi\left(\mathbf{r}_{0}\right)=\int_{\partial D} \psi(\mathbf{r}) \frac{\partial}{\partial n} G\left(\mathbf{r}, \mathbf{r}_{\mathbf{0}}\right) d s$

Consider Laplace's equation in the upper half plane,

$\nabla^{2} \psi(x, y)=0, \quad-\infty<x<\infty \quad \text { and } \quad y>0$

with boundary conditions $\psi(x, 0)=f(x)$ where $f(x) \rightarrow 0$ as $|x| \rightarrow \infty$, and $\psi(x, y) \rightarrow 0$ as $\sqrt{x^{2}+y^{2}} \rightarrow \infty$. Show that the solution is given by the integral formula

$\psi\left(x_{0}, y_{0}\right)=\frac{y_{0}}{\pi} \int_{-\infty}^{\infty} \frac{f(x)}{\left(x-x_{0}\right)^{2}+y_{0}^{2}} d x$

[ Hint: It might be useful to consider

$G\left(\mathbf{r}, \mathbf{r}_{0}\right)=\frac{1}{2 \pi}\left(\log \left|\mathbf{r}-\mathbf{r}_{0}\right|-\log \left|\mathbf{r}-\tilde{\mathbf{r}}_{0}\right|\right)$

for suitable $\tilde{\mathbf{r}}_{\mathbf{0}}$. You may assume $\nabla^{2} \log \left|\mathbf{r}-\mathbf{r}_{0}\right|=2 \pi \delta\left(\mathbf{r}-\mathbf{r}_{0}\right)$. ]

Paper 4, Section II, 13G

commentLet $X, Y$ be topological spaces and $X \times Y$ their product set. Let $p_{Y}: X \times Y \rightarrow Y$ be the projection map.

(i) Define the product topology on $X \times Y$. Prove that if a subset $Z \subset X \times Y$ is open then $p_{Y}(Z)$ is open in $Y$.

(ii) Give an example of $X, Y$ and a closed set $Z \subset X \times Y$ such that $p_{Y}(Z)$ is not closed.

(iii) When $X$ is compact, show that if a subset $Z \subset X \times Y$ is closed then $p_{Y}(Z)$ is closed

Paper 4, Section I, B

commentConsider the multistep method for numerical solution of the differential equation $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$ :

$\sum_{l=0}^{s} \rho_{l} \mathbf{y}_{n+l}=h \sum_{l=0}^{s} \sigma_{l} \mathbf{f}\left(t_{n+l}, \mathbf{y}_{n+l}\right), \quad n=0,1, \ldots$

What does it mean to say that the method is of order $p$, and that the method is convergent?

Show that the method is of order $p$ if

$\sum_{l=0}^{s} \rho_{l}=0, \quad \sum_{l=0}^{s} l^{k} \rho_{l}=k \sum_{l=0}^{s} l^{k-1} \sigma_{l}, \quad k=1,2, \ldots, p$

and give the conditions on $\rho(w)=\sum_{l=0}^{s} \rho_{l} w^{l}$ that ensure convergence.

Hence determine for what values of $\theta$ and the $\sigma_{i}$ the two-step method

$\mathbf{y}_{n+2}-(1-\theta) \mathbf{y}_{n+1}-\theta \mathbf{y}_{n}=h\left[\sigma_{0} \mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)+\sigma_{1} \mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)+\sigma_{2} \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)\right]$

is (a) convergent, and (b) of order 3 .

Paper 4, Section II, H

commentA company must ship coal from four mines, labelled $A, B, C, D$, to supply three factories, labelled $a, b, c$. The per unit transport cost, the outputs of the mines, and the requirements of the factories are given below.

\begin{tabular}{c|c|c|c|c|c} & $A$ & $B$ & $C$ & $D$ & \ \hline$a$ & 12 & 3 & 5 & 2 & 34 \ \hline$b$ & 4 & 11 & 2 & 6 & 21 \ \hline$c$ & 3 & 9 & 7 & 4 & 23 \ \hline & 20 & 32 & 15 & 11 & \end{tabular}

For instance, mine $B$ can produce 32 units of coal, factory a requires 34 units of coal, and it costs 3 units of money to ship one unit of coal from $B$ to $a$. What is the minimal cost of transporting coal from the mines to the factories?

Now suppose increased efficiency allows factory $b$ to reduce its requirement to $20.8$ units of coal, and as a consequence, mine $B$ reduces its output to $31.8$ units. By how much does the transport cost decrease?

Paper 4, Section $I$, C

commentConsider the 3-dimensional oscillator with Hamiltonian

$H=-\frac{\hbar^{2}}{2 m} \nabla^{2}+\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}+4 z^{2}\right)$

Find the ground state energy and the spacing between energy levels. Find the degeneracies of the lowest three energy levels.

[You may assume that the energy levels of the 1-dimensional harmonic oscillator with Hamiltonian

$H_{0}=-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+\frac{m \omega^{2}}{2} x^{2}$

$\left.\operatorname{are}\left(n+\frac{1}{2}\right) \hbar \omega, n=0,1,2, \ldots .\right]$

Paper 4, Section II, H

commentConsider independent random variables $X_{1}, \ldots, X_{n}$ with the $N\left(\mu_{X}, \sigma_{X}^{2}\right)$ distribution and $Y_{1}, \ldots, Y_{n}$ with the $N\left(\mu_{Y}, \sigma_{Y}^{2}\right)$ distribution, where the means $\mu_{X}, \mu_{Y}$ and variances $\sigma_{X}^{2}, \sigma_{Y}^{2}$ are unknown. Derive the generalised likelihood ratio test of size $\alpha$ of the null hypothesis $H_{0}: \sigma_{X}^{2}=\sigma_{Y}^{2}$ against the alternative $H_{1}: \sigma_{X}^{2} \neq \sigma_{Y}^{2}$. Express the critical region in terms of the statistic $T=\frac{S_{X X}}{S_{X X}+S_{Y Y}}$ and the quantiles of a beta distribution, where

$S_{X X}=\sum_{i=1}^{n} X_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} X_{i}\right)^{2} \text { and } S_{Y Y}=\sum_{i=1}^{n} Y_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} Y_{i}\right)^{2}$

[You may use the following fact: if $U \sim \Gamma(a, \lambda)$ and $V \sim \Gamma(b, \lambda)$ are independent, then $\left.\frac{U}{U+V} \sim \operatorname{Beta}(a, b) .\right]$

Paper 4 , Section II, D

commentDerive the Euler-Lagrange equation for the integral

$\int_{x_{0}}^{x_{1}} f\left(y, y^{\prime}, y^{\prime \prime}, x\right) d x$

where the endpoints are fixed, and $y(x)$ and $y^{\prime}(x)$ take given values at the endpoints.

Show that the only function $y(x)$ with $y(0)=1, y^{\prime}(0)=2$ and $y(x) \rightarrow 0$ as $x \rightarrow \infty$ for which the integral

$\int_{0}^{\infty}\left(y^{2}+\left(y^{\prime}\right)^{2}+\left(y^{\prime}+y^{\prime \prime}\right)^{2}\right) d x$

is stationary is $(3 x+1) e^{-x}$.