Part IB, 2014, Paper 1

# Part IB, 2014, Paper 1

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Paper 1, Section II, F

commentDefine what it means for two norms on a real vector space $V$ to be Lipschitz equivalent. Show that if two norms on $V$ are Lipschitz equivalent and $F \subset V$, then $F$ is closed in one norm if and only if $F$ is closed in the other norm.

Show that if $V$ is finite-dimensional, then any two norms on $V$ are Lipschitz equivalent.

Show that $\|f\|_{1}=\int_{0}^{1}|f(x)| d x$ is a norm on the space $C[0,1]$ of continuous realvalued functions on $[0,1]$. Is the set $S=\{f \in C[0,1]: f(1 / 2)=0\}$ closed in the norm $\|\cdot\| 1$ ?

Determine whether or not the norm $\|\cdot\|_{1}$ is Lipschitz equivalent to the uniform $\operatorname{norm}\|\cdot\|_{\infty}$ on $C[0,1]$.

[You may assume the Bolzano-Weierstrass theorem for sequences in $\mathbb{R}^{n}$.]

Paper 1, Section I, B

commentLet $f(z)$ be an analytic/holomorphic function defined on an open set $D$, and let $z_{0} \in D$ be a point such that $f^{\prime}\left(z_{0}\right) \neq 0$. Show that the transformation $w=f(z)$ preserves the angle between smooth curves intersecting at $z_{0}$. Find such a transformation $w=f(z)$ that maps the second quadrant of the unit disc (i.e. $|z|<1, \pi / 2<\arg (z)<\pi)$ to the region in the first quadrant of the complex plane where $|w|>1$ (i.e. the region in the first quadrant outside the unit circle).

Paper 1, Section II, B

commentBy choice of a suitable contour show that for $a>b>0$

$\int_{0}^{2 \pi} \frac{\sin ^{2} \theta d \theta}{a+b \cos \theta}=\frac{2 \pi}{b^{2}}\left[a-\sqrt{a^{2}-b^{2}}\right]$

Hence evaluate

$\int_{0}^{1} \frac{\left(1-x^{2}\right)^{1 / 2} x^{2} d x}{1+x^{2}}$

using the substitution $x=\cos (\theta / 2)$.

Paper 1, Section II, A

commentThe region $z<0$ is occupied by an ideal earthed conductor and a point charge $q$ with mass $m$ is held above it at $(0,0, d)$.

(i) What are the boundary conditions satisfied by the electric field $\mathbf{E}$ on the surface of the conductor?

(ii) Consider now a system without the conductor mentioned above. A point charge $q$ with mass $m$ is held at $(0,0, d)$, and one of charge $-q$ is held at $(0,0,-d)$. Show that the boundary condition on $\mathbf{E}$ at $z=0$ is identical to the answer to (i). Explain why this represents the electric field due to the charge at $(0,0, d)$ under the influence of the conducting boundary.

(iii) The original point charge in (i) is released with zero initial velocity. Find the time taken for the point charge to reach the plane (ignoring gravity).

[You may assume that the force on the point charge is equal to $m d^{2} \mathbf{x} / d t^{2}$, where $\mathbf{x}$ is the position vector of the charge, and $t$ is time.]

Paper 1, Section I, B

commentConstant density viscous fluid with dynamic viscosity $\mu$ flows in a two-dimensional horizontal channel of depth $h$. There is a constant pressure gradient $G>0$ in the horizontal $x$-direction. The upper horizontal boundary at $y=h$ is driven at constant horizontal speed $U>0$, with the lower boundary being held at rest. Show that the steady fluid velocity $u$ in the $x$-direction is

$u=\frac{-G}{2 \mu} y(h-y)+\frac{U y}{h}$

Show that it is possible to have $d u / d y<0$ at some point in the flow for sufficiently large pressure gradient. Derive a relationship between $G$ and $U$ so that there is no net volume flux along the channel. For the flow with no net volume flux, sketch the velocity profile.

Paper 1, Section II, B

commentConsider the purely two-dimensional steady flow of an inviscid incompressible constant density fluid in the absence of body forces. For velocity $\mathbf{u}$, the vorticity is $\boldsymbol{\nabla} \times \mathbf{u}=\boldsymbol{\omega}=(0,0, \omega)$. Show that

$\mathbf{u} \times \boldsymbol{\omega}=\boldsymbol{\nabla}\left[\frac{p}{\rho}+\frac{1}{2}|\mathbf{u}|^{2}\right]$

where $p$ is the pressure and $\rho$ is the fluid density. Hence show that, if $\omega$ is a constant in both space and time,

$\frac{1}{2}|\mathbf{u}|^{2}+\omega \psi+\frac{p}{\rho}=C,$

where $C$ is a constant and $\psi$ is the streamfunction. Here, $\psi$ is defined by $\mathbf{u}=\boldsymbol{\nabla} \times \boldsymbol{\Psi}$, where $\boldsymbol{\Psi}=(0,0, \psi)$.

Fluid in the annular region $a<r<2 a$ has constant (in both space and time) vorticity $\omega$. The streamlines are concentric circles, with the fluid speed zero on $r=2 a$ and $V>0$ on $r=a$. Calculate the velocity field, and hence show that

$\omega=\frac{-2 V}{3 a}$

Deduce that the pressure difference between the outer and inner edges of the annular region is

$\Delta p=\left(\frac{15-16 \ln 2}{18}\right) \rho V^{2}$

[Hint: Note that in cylindrical polar coordinates $(r, \phi, z)$, the curl of a vector field $\mathbf{A}(r, \phi)=[a(r, \phi), b(r, \phi), c(r, \phi)]$ is

$\boldsymbol{\nabla} \times \mathbf{A}=\left[\frac{1}{r} \frac{\partial c}{\partial \phi},-\frac{\partial c}{\partial r}, \frac{1}{r}\left(\frac{\partial(r b)}{\partial r}-\frac{\partial a}{\partial \phi}\right)\right]$

Paper 1, Section I, F

commentDetermine the second fundamental form of a surface in $\mathbb{R}^{3}$ defined by the parametrisation

$\sigma(u, v)=((a+b \cos u) \cos v,(a+b \cos u) \sin v, b \sin u)$

for $0<u<2 \pi, 0<v<2 \pi$, with some fixed $a>b>0$. Show that the Gaussian curvature $K(u, v)$ of this surface takes both positive and negative values.

Paper 1, Section II, E

commentLet $G$ be a finite group and $p$ a prime divisor of the order of $G$. Give the definition of a Sylow $p$-subgroup of $G$, and state Sylow's theorems.

Let $p$ and $q$ be distinct primes. Prove that a group of order $p^{2} q$ is not simple.

Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $P$ a Sylow $p$-subgroup of H. Let $N_{G}(P)$ denote the normaliser of $P$ in $G$. Prove that if $g \in G$ then there exist $k \in N_{G}(P)$ and $h \in H$ such that $g=k h$.

Paper 1, Section I, G

commentState and prove the Steinitz Exchange Lemma. Use it to prove that, in a finitedimensional vector space: any two bases have the same size, and every linearly independent set extends to a basis.

Let $e_{1}, \ldots, e_{n}$ be the standard basis for $\mathbb{R}^{n}$. Is $e_{1}+e_{2}, e_{2}+e_{3}, e_{3}+e_{1}$ a basis for $\mathbb{R}^{3} ?$ Is $e_{1}+e_{2}, e_{2}+e_{3}, e_{3}+e_{4}, e_{4}+e_{1}$ a basis for $\mathbb{R}^{4} ?$ Justify your answers.

Paper 1, Section II, G

commentLet $V$ be an $n$-dimensional real vector space, and let $T$ be an endomorphism of $V$. We say that $T$ acts on a subspace $W$ if $T(W) \subset W$.

(i) For any $x \in V$, show that $T$ acts on the linear span of $\left\{x, T(x), T^{2}(x), \ldots, T^{n-1}(x)\right\}$.

(ii) If $\left\{x, T(x), T^{2}(x), \ldots, T^{n-1}(x)\right\}$ spans $V$, show directly (i.e. without using the CayleyHamilton Theorem) that $T$ satisfies its own characteristic equation.

(iii) Suppose that $T$ acts on a subspace $W$ with $W \neq\{0\}$ and $W \neq V$. Let $e_{1}, \ldots, e_{k}$ be a basis for $W$, and extend to a basis $e_{1}, \ldots, e_{n}$ for $V$. Describe the matrix of $T$ with respect to this basis.

(iv) Using (i), (ii) and (iii) and induction, give a proof of the Cayley-Hamilton Theorem.

[Simple properties of determinants may be assumed without proof.]

Paper 1, Section II, 20H

commentConsider a homogeneous Markov chain $\left(X_{n}: n \geqslant 0\right)$ with state space $S$ and transition $\operatorname{matrix} P=\left(p_{i, j}: i, j \in S\right)$. For a state $i$, define the terms aperiodic, positive recurrent and ergodic.

Let $S=\{0,1,2, \ldots\}$ and suppose that for $i \geqslant 1$ we have $p_{i, i-1}=1$ and

$p_{0,0}=0, p_{0, j}=p q^{j-1}, j=1,2, \ldots,$

where $p=1-q \in(0,1)$. Show that this Markov chain is irreducible.

Let $T_{0}=\inf \left\{n \geqslant 1: X_{n}=0\right\}$ be the first passage time to 0 . Find $\mathbb{P}\left(T_{0}=n \mid X_{0}=0\right)$ and show that state 0 is ergodic.

Find the invariant distribution $\pi$ for this Markov chain. Write down:

(i) the mean recurrence time for state $i, i \geqslant 1$;

(ii) $\lim _{n \rightarrow \infty} \mathbb{P}\left(X_{n} \neq 0 \mid X_{0}=0\right)$.

[Results from the course may be quoted without proof, provided they are clearly stated.]

Paper 1, Section II, D

comment(a) Legendre's differential equation may be written

$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+n(n+1) y=0, \quad y(1)=1$

Show that for non-negative integer $n$, this equation has a solution $P_{n}(x)$ that is a polynomial of degree $n$. Find $P_{0}, P_{1}$ and $P_{2}$ explicitly.

(b) Laplace's equation in spherical coordinates for an axisymmetric function $U(r, \theta)$ (i.e. no $\phi$ dependence) is given by

$\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial U}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial U}{\partial \theta}\right)=0$

Use separation of variables to find the general solution for $U(r, \theta)$.

Find the solution $U(r, \theta)$ that satisfies the boundary conditions

$\begin{aligned} &U(r, \theta) \rightarrow v_{0} r \cos \theta \quad \text { as } r \rightarrow \infty \\ &\frac{\partial U}{\partial r}=0 \quad \text { at } r=r_{0} \end{aligned}$

where $v_{0}$ and $r_{0}$ are constants.

Paper 1, Section II, E

commentDefine what it means for a topological space to be compact. Define what it means for a topological space to be Hausdorff.

Prove that a compact subspace of a Hausdorff space is closed. Hence prove that if $C_{1}$ and $C_{2}$ are compact subspaces of a Hausdorff space $X$ then $C_{1} \cap C_{2}$ is compact.

A subset $U$ of $\mathbb{R}$ is open in the cocountable topology if $U$ is empty or its complement in $\mathbb{R}$ is countable. Is $\mathbb{R}$ Hausdorff in the cocountable topology? Which subsets of $\mathbb{R}$ are compact in the cocountable topology?

Paper 1, Section I, $\mathbf{6 C}$

comment(i) A general multistep method for the numerical approximation to the scalar differential equation $y^{\prime}=f(t, y)$ is given by

$\sum_{\ell=0}^{s} \rho_{\ell} y_{n+\ell}=h \sum_{\ell=0}^{s} \sigma_{\ell} f_{n+\ell}, \quad n=0,1, \ldots$

where $f_{n+\ell}=f\left(t_{n+\ell}, y_{n+\ell}\right)$. Show that this method is of order $p \geqslant 1$ if and only if

$\rho\left(\mathrm{e}^{z}\right)-z \sigma\left(\mathrm{e}^{z}\right)=\mathcal{O}\left(z^{p+1}\right) \quad \text { as } \quad z \rightarrow 0$

where

$\rho(w)=\sum_{\ell=0}^{s} \rho_{\ell} w^{\ell} \quad \text { and } \quad \sigma(w)=\sum_{\ell=0}^{s} \sigma_{\ell} w^{\ell}$

(ii) A particular three-step implicit method is given by

$y_{n+3}+(a-1) y_{n+1}-a y_{n}=h\left(f_{n+3}+\sum_{\ell=0}^{2} \sigma_{\ell} f_{n+\ell}\right)$

where the $\sigma_{\ell}$ are chosen to make the method third order. [The $\sigma_{\ell}$ need not be found.] For what values of $a$ is the method convergent?

Paper 1, Section II, C

commentDefine a Householder transformation $\mathrm{H}$ and show that it is an orthogonal matrix. Briefly explain how these transformations can be used for QR factorisation of an $m \times n$ matrix.

Using Householder transformations, find a QR factorisation of

$A=\left[\begin{array}{rrr} 2 & 5 & 4 \\ 2 & 5 & 1 \\ -2 & 1 & 5 \\ 2 & -1 & 16 \end{array}\right]$

Using this factorisation, find the value of $\lambda$ for which

$A x=\left[\begin{array}{c} 1+\lambda \\ 2 \\ 3 \\ 4 \end{array}\right]$

has a unique solution $x \in \mathbb{R}^{3}$.

Paper 1, Section I, 8H

commentState and prove the Lagrangian sufficiency theorem.

Use the Lagrangian sufficiency theorem to find the minimum of $2 x_{1}^{2}+2 x_{2}^{2}+x_{3}^{2}$ subject to $x_{1}+x_{2}+x_{3}=1$ (where $x_{1}, x_{2}$ and $x_{3}$ are real).

Paper 1, Section II, A

commentConsider a particle confined in a one-dimensional infinite potential well: $V(x)=\infty$ for $|x| \geqslant a$ and $V(x)=0$ for $|x|<a$. The normalised stationary states are

$\psi_{n}(x)= \begin{cases}\alpha_{n} \sin \left(\frac{\pi n(x+a)}{2 a}\right) & \text { for }|x|<a \\ 0 & \text { for }|x| \geqslant a\end{cases}$

where $n=1,2, \ldots$.

(i) Determine the $\alpha_{n}$ and the stationary states' energies $E_{n}$.

(ii) A state is prepared within this potential well: $\psi(x) \propto x$ for $0<x<a$, but $\psi(x)=0$ for $x \leqslant 0$ or $x \geqslant a$. Find an explicit expansion of $\psi(x)$ in terms of $\psi_{n}(x) .$

(iii) If the energy of the state is then immediately measured, show that the probability that it is greater than $\frac{\hbar^{2} \pi^{2}}{m a^{2}}$ is

$\sum_{n=0}^{4} \frac{b_{n}}{\pi^{n}}$

where the $b_{n}$ are integers which you should find.

(iv) By considering the normalisation condition for $\psi(x)$ in terms of the expansion in $\psi_{n}(x)$, show that

$\frac{\pi^{2}}{3}=\sum_{p=1}^{\infty} \frac{A}{p^{2}}+\frac{B}{(2 p-1)^{2}}\left(1+\frac{C(-1)^{p}}{(2 p-1) \pi}\right)^{2}$

where $A, B$ and $C$ are integers which you should find.

Paper 1, Section I, $\mathbf{7 H} \quad$

commentConsider an estimator $\hat{\theta}$ of an unknown parameter $\theta$, and assume that $\mathbb{E}_{\theta}\left(\hat{\theta}^{2}\right)<\infty$ for all $\theta$. Define the bias and mean squared error of $\hat{\theta}$.

Show that the mean squared error of $\hat{\theta}$ is the sum of its variance and the square of its bias.

Suppose that $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables with mean $\theta$ and variance $\theta^{2}$, and consider estimators of $\theta$ of the form $k \bar{X}$ where $\bar{X}=\frac{1}{n} \sum_{i=1}^{n} X_{i}$.

(i) Find the value of $k$ that gives an unbiased estimator, and show that the mean squared error of this unbiased estimator is $\theta^{2} / n$.

(ii) Find the range of values of $k$ for which the mean squared error of $k \bar{X}$ is smaller $\operatorname{than} \theta^{2} / n$.

Paper 1, Section II, H

commentSuppose that $X_{1}, X_{2}$, and $X_{3}$ are independent identically distributed Poisson random variables with expectation $\theta$, so that

$\mathbb{P}\left(X_{i}=x\right)=\frac{e^{-\theta} \theta^{x}}{x !} \quad x=0,1, \ldots$

and consider testing $H_{0}: \theta=1$ against $H_{1}: \theta=\theta_{1}$, where $\theta_{1}$ is a known value greater than 1. Show that the test with critical region $\left\{\left(x_{1}, x_{2}, x_{3}\right): \sum_{i=1}^{3} x_{i}>5\right\}$ is a likelihood ratio test of $H_{0}$ against $H_{1}$. What is the size of this test? Write down an expression for its power.

A scientist counts the number of bird territories in $n$ randomly selected sections of a large park. Let $Y_{i}$ be the number of bird territories in the $i$ th section, and suppose that $Y_{1}, \ldots, Y_{n}$ are independent Poisson random variables with expectations $\theta_{1}, \ldots, \theta_{n}$ respectively. Let $a_{i}$ be the area of the $i$ th section. Suppose that $n=2 m$, $a_{1}=\cdots=a_{m}=a(>0)$ and $a_{m+1}=\cdots=a_{2 m}=2 a$. Derive the generalised likelihood ratio $\Lambda$ for testing

$H_{0}: \theta_{i}=\lambda a_{i} \text { against } H_{1}: \theta_{i}= \begin{cases}\lambda_{1} & i=1, \ldots, m \\ \lambda_{2} & i=m+1, \ldots, 2 m\end{cases}$

What should the scientist conclude about the number of bird territories if $2 \log _{e}(\Lambda)$ is $15.67 ?$

[Hint: Let $F_{\theta}(x)$ be $\mathbb{P}(W \leqslant x)$ where $W$ has a Poisson distribution with expectation $\theta$. Then

$\left.F_{1}(3)=0.998, \quad F_{3}(5)=0.916, \quad F_{3}(6)=0.966, \quad F_{5}(3)=0.433 .\right]$

Paper 1, Section I, C

commentDefine the Legendre transform $f^{*}(\mathbf{p})$ of a function $f(\mathbf{x})$ where $\mathbf{x} \in \mathbb{R}^{n}$.

Show that for $g(\mathbf{x})=\lambda f\left(\mathbf{x}-\mathbf{x}_{0}\right)-\mu$,

$g^{*}(\mathbf{p})=\lambda f^{*}\left(\frac{\mathbf{p}}{\lambda}\right)+\mathbf{p}^{\mathbf{T}} \mathbf{x}_{0}+\mu$

Show that for $f(\mathbf{x})=\frac{1}{2} \mathbf{x}^{\mathbf{T}} \mathbf{A} \mathbf{x}$ where $\mathbf{A}$ is a real, symmetric, invertible matrix with positive eigenvalues,

$f^{*}(\mathbf{p})=\frac{1}{2} \mathbf{p}^{\mathbf{T}} \mathbf{A}^{-\mathbf{1}} \mathbf{p}$