# Part IB, 2016

### Jump to course

Paper 1, Section II, G

commentLet $(X, d)$ be a metric space.

(a) What does it mean to say that $\left(x_{n}\right)_{n}$ is a Cauchy sequence in $X$ ? Show that if $\left(x_{n}\right)_{n}$ is a Cauchy sequence, then it converges if it contains a convergent subsequence.

(b) Let $\left(x_{n}\right)_{n}$ be a Cauchy sequence in $X$.

(i) Show that for every $m \geqslant 1$, the sequence $\left(d\left(x_{m}, x_{n}\right)\right)_{n}$ converges to some $d_{m} \in \mathbb{R}$.

(ii) Show that $d_{m} \rightarrow 0$ as $m \rightarrow \infty$.

(iii) Let $\left(y_{n}\right)_{n}$ be a subsequence of $\left(x_{n}\right)_{n}$. If $\ell, m$ are such that $y_{\ell}=x_{m}$, show that $d\left(y_{\ell}, y_{n}\right) \rightarrow d_{m}$ as $n \rightarrow \infty$.

(iv) Show also that for every $m$ and $n$,

$d_{m}-d_{n} \leqslant d\left(x_{m}, x_{n}\right) \leqslant d_{m}+d_{n}$

(v) Deduce that $\left(x_{n}\right)_{n}$ has a subsequence $\left(y_{n}\right)_{n}$ such that for every $m$ and $n$,

$d\left(y_{m+1}, y_{m}\right) \leqslant \frac{1}{3} d\left(y_{m}, y_{m-1}\right)$

and

$d\left(y_{m+1}, y_{n+1}\right) \leqslant \frac{1}{2} d\left(y_{m}, y_{n}\right)$

(c) Suppose that every closed subset $Y$ of $X$ has the property that every contraction mapping $Y \rightarrow Y$ has a fixed point. Prove that $X$ is complete.

Paper 2, Section I, G

comment(a) What does it mean to say that the function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is differentiable at the point $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$ ? Show from your definition that if $f$ is differentiable at $x$, then $f$ is continuous at $x$.

(b) Suppose that there are functions $g_{j}: \mathbb{R} \rightarrow \mathbb{R}^{m}(1 \leqslant j \leqslant n)$ such that for every $x=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$,

$f(x)=\sum_{j=1}^{n} g_{j}\left(x_{j}\right) .$

Show that $f$ is differentiable at $x$ if and only if each $g_{j}$ is differentiable at $x_{j}$.

(c) Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by

$f(x, y)=|x|^{3 / 2}+|y|^{1 / 2}$

Determine at which points $(x, y) \in \mathbb{R}^{2}$ the function $f$ is differentiable.

Paper 2, Section II, G

comment(a) What is a norm on a real vector space?

(b) Let $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$ be the space of linear maps from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$. Show that

$\|A\|=\sup _{0 \neq x \in \mathbb{R}^{m}} \frac{\|A x\|}{\|x\|}, \quad A \in L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right),$

defines a norm on $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$, and that if $B \in L\left(\mathbb{R}^{\ell}, \mathbb{R}^{m}\right)$ then $\|A B\| \leqslant\|A\|\|B\|$.

(c) Let $M_{n}$ be the space of $n \times n$ real matrices, identified with $L\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right)$ in the usual way. Let $U \subset M_{n}$ be the subset

$U=\left\{X \in M_{n} \mid I-X \text { is invertible }\right\}$

Show that $U$ is an open subset of $M_{n}$ which contains the set $V=\left\{X \in M_{n} \mid\|X\|<1\right\}$.

(d) Let $f: U \rightarrow M_{n}$ be the map $f(X)=(I-X)^{-1}$. Show carefully that the series $\sum_{k=0}^{\infty} X^{k}$ converges on $V$ to $f(X)$. Hence or otherwise, show that $f$ is twice differentiable at 0 , and compute its first and second derivatives there.

Paper 3, Section I, G

comment(a) Let $X$ be a subset of $\mathbb{R}$. What does it mean to say that a sequence of functions $f_{n}: X \rightarrow \mathbb{R}(n \in \mathbb{N})$ is uniformly convergent?

(b) Which of the following sequences of functions are uniformly convergent? Justify your answers.

(i) $f_{n}:(0,1) \rightarrow \mathbb{R}, \quad f_{n}(x)=\frac{1-x^{n}}{1-x}$

(ii) $f_{n}:(0,1) \rightarrow \mathbb{R}, \quad f_{n}(x)=\sum_{k=1}^{n} \frac{1}{k^{2}} x^{k}$.

(iii) $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$, $\quad f_{n}(x)=x / n$.

(iv) $f_{n}:[0, \infty) \rightarrow \mathbb{R}, \quad f_{n}(x)=x e^{-n x}$.

Paper 3, Section II, G

commentLet $X$ be a metric space.

(a) What does it mean to say that a function $f: X \rightarrow \mathbb{R}$ is uniformly continuous? What does it mean to say that $f$ is Lipschitz? Show that if $f$ is Lipschitz then it is uniformly continuous. Show also that if $\left(x_{n}\right)_{n}$ is a Cauchy sequence in $X$, and $f$ is uniformly continuous, then the sequence $\left(f\left(x_{n}\right)\right)_{n}$ is convergent.

(b) Let $f: X \rightarrow \mathbb{R}$ be continuous, and $X$ be sequentially compact. Show that $f$ is uniformly continuous. Is $f$ necessarily Lipschitz? Justify your answer.

(c) Let $Y$ be a dense subset of $X$, and let $g: Y \rightarrow \mathbb{R}$ be a continuous function. Show that there exists at most one continuous function $f: X \rightarrow \mathbb{R}$ such that for all $y \in Y$, $f(y)=g(y)$. Prove that if $g$ is uniformly continuous, then such a function $f$ exists, and is uniformly continuous.

[A subset $Y \subset X$ is dense if for any nonempty open subset $U \subset X$, the intersection $U \cap Y$ is nonempty.]

Paper 4, Section I, G

comment(a) What does it mean to say that a mapping $f: X \rightarrow X$ from a metric space to itself is a contraction?

(b) State carefully the contraction mapping theorem.

(c) Let $\left(a_{1}, a_{2}, a_{3}\right) \in \mathbb{R}^{3}$. By considering the metric space $\left(\mathbb{R}^{3}, d\right)$ with

$d(x, y)=\sum_{i=1}^{3}\left|x_{i}-y_{i}\right|$

or otherwise, show that there exists a unique solution $\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}$ of the system of equations

$\begin{aligned} &x_{1}=a_{1}+\frac{1}{6}\left(\sin x_{2}+\sin x_{3}\right), \\ &x_{2}=a_{2}+\frac{1}{6}\left(\sin x_{1}+\sin x_{3}\right), \\ &x_{3}=a_{3}+\frac{1}{6}\left(\sin x_{1}+\sin x_{2}\right) . \end{aligned}$

Paper 4, Section II, G

comment(a) Let $V$ be a real vector space. What does it mean to say that two norms on $V$ are Lipschitz equivalent? Prove that every norm on $\mathbb{R}^{n}$ is Lipschitz equivalent to the Euclidean norm. Hence or otherwise, show that any linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ is continuous.

(b) Let $f: U \rightarrow V$ be a linear map between normed real vector spaces. We say that $f$ is bounded if there exists a constant $C$ such that for all $u \in U,\|f(u)\| \leqslant C\|u\|$. Show that $f$ is bounded if and only if $f$ is continuous.

(c) Let $\ell^{2}$ denote the space of sequences $\left(x_{n}\right)_{n \geqslant 1}$ of real numbers such that $\sum_{n \geqslant 1} x_{n}^{2}$ is convergent, with the norm $\left\|\left(x_{n}\right)_{n}\right\|=\left(\sum_{n \geqslant 1} x_{n}^{2}\right)^{1 / 2}$. Let $e_{m} \in \ell^{2}$ be the sequence $e_{m}=\left(x_{n}\right)_{n}$ with $x_{m}=1$ and $x_{n}=0$ if $n \neq m$. Let $w$ be the sequence $\left(2^{-n}\right)_{n}$. Show that the subset $\{w\} \cup\left\{e_{m} \mid m \geqslant 1\right\}$ is linearly independent. Let $V \subset \ell^{2}$ be the subspace it spans, and consider the linear map $f: V \rightarrow \mathbb{R}$ defined by

$f(w)=1, \quad f\left(e_{m}\right)=0 \quad \text { for all } m \geqslant 1 .$

Is $f$ continuous? Justify your answer.

Paper 3, Section II, G

comment(a) Prove Cauchy's theorem for a triangle.

(b) Write down an expression for the winding number $I(\gamma, a)$ of a closed, piecewise continuously differentiable curve $\gamma$ about a point $a \in \mathbb{C}$ which does not lie on $\gamma$.

(c) Let $U \subset \mathbb{C}$ be a domain, and $f: U \rightarrow \mathbb{C}$ a holomorphic function with no zeroes in $U$. Suppose that for infinitely many positive integers $k$ the function $f$ has a holomorphic $k$-th root. Show that there exists a holomorphic function $F: U \rightarrow \mathbb{C}$ such that $f=\exp F$.

Paper 4, Section I, G

commentState carefully Rouché's theorem. Use it to show that the function $z^{4}+3+e^{i z}$ has exactly one zero $z=z_{0}$ in the quadrant

$\{z \in \mathbb{C} \mid 0<\arg (z)<\pi / 2\}$

and that $\left|z_{0}\right| \leqslant \sqrt{2}$.

Paper 1, Section I, A

commentClassify the singularities of the following functions at both $z=0$ and at the point at infinity on the extended complex plane:

$\begin{aligned} f_{1}(z) &=\frac{e^{z}}{z \sin ^{2} z}, \\ f_{2}(z) &=\frac{1}{z^{2}(1-\cos z)}, \\ f_{3}(z) &=z^{2} \sin (1 / z) \end{aligned}$

Paper 1, Section II, A

commentLet $w=u+i v$ and let $z=x+i y$, for $u, v, x, y$ real.

(a) Let A be the map defined by $w=\sqrt{z}$, using the principal branch. Show that A maps the region to the left of the parabola $y^{2}=4(1-x)$ on the $z-$ plane, with the negative real axis $x \in(-\infty, 0]$ removed, into the vertical strip of the $w-$ plane between the lines $u=0$ and $u=1$.

(b) Let $\mathrm{B}$ be the map defined by $w=\tan ^{2}(z / 2)$. Show that $\mathrm{B}$ maps the vertical strip of the $z$-plane between the lines $x=0$ and $x=\pi / 2$ into the region inside the unit circle on the $w$-plane, with the part $u \in(-1,0]$ of the negative real axis removed.

(c) Using the results of parts (a) and (b), show that the map C, defined by $w=\tan ^{2}(\pi \sqrt{z} / 4)$, maps the region to the left of the parabola $y^{2}=4(1-x)$ on the $z$-plane, including the negative real axis, onto the unit disc on the $w$-plane.

Paper 2, Section II, A

commentLet $a=N+1 / 2$ for a positive integer $N$. Let $C_{N}$ be the anticlockwise contour defined by the square with its four vertices at $a \pm i a$ and $-a \pm i a$. Let

$I_{N}=\oint_{C_{N}} \frac{d z}{z^{2} \sin (\pi z)}$

Show that $1 / \sin (\pi z)$ is uniformly bounded on the contours $C_{N}$ as $N \rightarrow \infty$, and hence that $I_{N} \rightarrow 0$ as $N \rightarrow \infty$.

Using this result, establish that

$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}=\frac{\pi^{2}}{12}$

Paper 3, Section I, A

commentThe function $f(x)$ has Fourier transform

$\tilde{f}(k)=\int_{-\infty}^{\infty} f(x) e^{-i k x} d x=\frac{-2 k i}{p^{2}+k^{2}},$

where $p>0$ is a real constant. Using contour integration, calculate $f(x)$ for $x<0$. [Jordan's lemma and the residue theorem may be used without proof.]

Paper 4, Section II, A

comment(a) Show that the Laplace transform of the Heaviside step function $H(t-a)$ is

$\int_{0}^{\infty} H(t-a) e^{-p t} d t=\frac{e^{-a p}}{p}$

for $a>0$.

(b) Derive an expression for the Laplace transform of the second derivative of a function $f(t)$ in terms of the Laplace transform of $f(t)$ and the properties of $f(t)$ at $t=0$.

(c) A bar of length $L$ has its end at $x=L$ fixed. The bar is initially at rest and straight. The end at $x=0$ is given a small fixed transverse displacement of magnitude $a$ at $t=0^{+}$. You may assume that the transverse displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$, and so the tranverse displacement $y(x, t)$ is the solution to the problem:

$\begin{array}{cl} \frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}} & \text { for } 0<x<L \text { and } t> \\ y(x, 0)=\frac{\partial y}{\partial t}(x, 0)=0 & \text { for } 0<x<L, \\ y(0, t)=a ; y(L, t)=0 & \text { for } t>0 . \end{array}$

(i) Show that the Laplace transform $Y(x, p)$ of $y(x, t)$, defined as

$Y(x, p)=\int_{0}^{\infty} y(x, t) e^{-p t} d t$

is given by

$Y(x, p)=\frac{a \sinh \left[\frac{p}{c}(L-x)\right]}{p \sinh \left[\frac{p L}{c}\right]}$

(ii) By use of the binomial theorem or otherwise, express $y(x, t)$ as an infinite series.

(iii) Plot the transverse displacement of the midpoint of the bar $y(L / 2, t)$ against time.

Paper 1, Section II, D

comment(a) From the differential form of Maxwell's equations with $\mathbf{J}=\mathbf{0}, \mathbf{B}=\mathbf{0}$ and a time-independent electric field, derive the integral form of Gauss's law.

(b) Derive an expression for the electric field $\mathbf{E}$ around an infinitely long line charge lying along the $z$-axis with charge per unit length $\mu$. Find the electrostatic potential $\phi$ up to an arbitrary constant.

(c) Now consider the line charge with an ideal earthed conductor filling the region $x>d$. State the boundary conditions satisfied by $\phi$ and $\mathbf{E}$ on the surface of the conductor.

(d) Show that the same boundary conditions at $x=d$ are satisfied if the conductor is replaced by a second line charge at $x=2 d, y=0$ with charge per unit length $-\mu$.

(e) Hence or otherwise, returning to the setup in (c), calculate the force per unit length acting on the line charge.

(f) What is the charge per unit area $\sigma(y, z)$ on the surface of the conductor?

Paper 2, Section I, $6 \mathrm{D}$

comment(a) Derive the integral form of Ampère's law from the differential form of Maxwell's equations with a time-independent magnetic field, $\rho=0$ and $\mathbf{E}=\mathbf{0}$.

(b) Consider two perfectly-conducting concentric thin cylindrical shells of infinite length with axes along the $z$-axis and radii $a$ and $b(a<b)$. Current $I$ flows in the positive $z$-direction in each shell. Use Ampère's law to calculate the magnetic field in the three regions: (i) $r<a$, (ii) $a<r<b$ and (iii) $r>b$, where $r=\sqrt{x^{2}+y^{2}}$.

(c) If current $I$ now flows in the positive $z$-direction in the inner shell and in the negative $z$-direction in the outer shell, calculate the magnetic field in the same three regions.

Paper 2, Section II, D

comment(a) State the covariant form of Maxwell's equations and define all the quantities that appear in these expressions.

(b) Show that $\mathbf{E} \cdot \mathbf{B}$ is a Lorentz scalar (invariant under Lorentz transformations) and find another Lorentz scalar involving $\mathbf{E}$ and $\mathbf{B}$.

(c) In some inertial frame $S$ the electric and magnetic fields are respectively $\mathbf{E}=\left(0, E_{y}, E_{z}\right)$ and $\mathbf{B}=\left(0, B_{y}, B_{z}\right)$. Find the electric and magnetic fields, $\mathbf{E}^{\prime}=\left(0, E_{y}^{\prime}, E_{z}^{\prime}\right)$ and $\mathbf{B}^{\prime}=\left(0, B_{y}^{\prime}, B_{z}^{\prime}\right)$, in another inertial frame $S^{\prime}$ that is related to $S$ by the Lorentz transformation,

$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma v / c & 0 & 0 \\ -\gamma v / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

where $v$ is the velocity of $S^{\prime}$ in $S$ and $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$.

(d) Suppose that $\mathbf{E}=E_{0}(0,1,0)$ and $\mathbf{B}=\frac{E_{0}}{c}(0, \cos \theta, \sin \theta)$ where $0 \leqslant \theta \leqslant \pi / 2$, and $E_{0}$ is a real constant. An observer is moving in $S$ with velocity $v$ parallel to the $x$-axis. What must $v$ be for the electric and magnetic fields to appear to the observer to be parallel? Comment on the case $\theta=\pi / 2$.

Paper 3, Section II, D

comment(a) State Faraday's law of induction for a moving circuit in a time-dependent magnetic field and define all the terms that appear.

(b) Consider a rectangular circuit DEFG in the $z=0$ plane as shown in the diagram below. There are two rails parallel to the $x$-axis for $x>0$ starting at $\mathrm{D}$ at $(x, y)=(0,0)$ and $G$ at $(0, L)$. A battery provides an electromotive force $\mathcal{E}_{0}$ between $D$ and $G$ driving current in a positive sense around DEFG. The circuit is completed with a bar resistor of resistance $R$, length $L$ and mass $m$ that slides without friction on the rails; it connects $E$ at $(s(t), 0)$ and $F$ at $(s(t), L)$. The rest of the circuit has no resistance. The circuit is in a constant uniform magnetic field $B_{0}$ parallel to the $z$-axis.

[In parts (i)-(iv) you can neglect any magnetic field due to current flow.]

(i) Calculate the current in the bar and indicate its direction on a diagram of the circuit.

(ii) Find the force acting on the bar.

(iii) If the initial velocity and position of the bar are respectively $\dot{s}(0)=v_{0}>0$ and $s(0)=s_{0}>0$, calculate $\dot{s}(t)$ and $s(t)$ for $t>0$.

(iv) If $\mathcal{E}_{0}=0$, find the total energy dissipated in the circuit after $t=0$ and verify that total energy is conserved.

(v) Describe qualitatively the effect of the magnetic field caused by the induced current flowing in the circuit when $\mathcal{E}_{0}=0$.

Paper 4, Section I, D

comment(a) Starting from Maxwell's equations, show that in a vacuum,

$\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}=\mathbf{0} \quad \text { and } \quad \boldsymbol{\nabla} \cdot \mathbf{E}=0 \quad \text { where } \quad c=\sqrt{\frac{1}{\epsilon_{0} \mu_{0}}} .$

(b) Suppose that $\mathbf{E}=\frac{E_{0}}{\sqrt{2}}(1,1,0) \cos (k z-\omega t)$ where $E_{0}, k$ and $\omega$ are real constants.

(i) What are the wavevector and the polarisation? How is $\omega$ related to $k$ ?

(ii) Find the magnetic field $\mathbf{B}$.

(iii) Compute and interpret the time-averaged value of the Poynting vector, $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$.

Paper 1, Section I, C

commentConsider the flow field in cartesian coordinates $(x, y, z)$ given by

$\mathbf{u}=\left(-\frac{A y}{x^{2}+y^{2}}, \frac{A x}{x^{2}+y^{2}}, U(z)\right)$

where $A$ is a constant. Let $\mathcal{D}$ denote the whole of $\mathbb{R}^{3}$ excluding the $z$ axis.

(a) Determine the conditions on $A$ and $U(z)$ for the flow to be both incompressible and irrotational in $\mathcal{D}$.

(b) Calculate the circulation along any closed curve enclosing the $z$ axis.

Paper 1, Section II, $17 \mathrm{C}$

comment(a) For a velocity field $\mathbf{u}$, show that $\mathbf{u} \cdot \boldsymbol{\nabla} \mathbf{u}=\boldsymbol{\nabla}\left(\frac{1}{2} \mathbf{u}^{2}\right)-\mathbf{u} \times \boldsymbol{\omega}$, where $\boldsymbol{\omega}$ is the flow vorticity.

(b) For a scalar field $H(\mathbf{r})$, show that if $\mathbf{u} \cdot \nabla H=0$, then $H$ is constant along the flow streamlines.

(c) State the Euler equations satisfied by an inviscid fluid of constant density subject to conservative body forces.

(i) If the flow is irrotational, show that an exact first integral of the Euler equations may be obtained.

(ii) If the flow is not irrotational, show that although an exact first integral of the Euler equations may not be obtained, a similar quantity is constant along the flow streamlines provided the flow is steady.

(iii) If the flow is now in a frame rotating with steady angular velocity $\Omega \mathbf{e}_{z}$, establish that a similar quantity is constant along the flow streamlines with an extra term due to the centrifugal force when the flow is steady.

Paper 2, Section I, C

commentA steady, two-dimensional unidirectional flow of a fluid with dynamic viscosity $\mu$ is set up between two plates at $y=0$ and $y=h$. The plate at $y=0$ is stationary while the plate at $y=h$ moves with constant speed $U \mathbf{e}_{x}$. The fluid is also subject to a constant pressure gradient $-G \mathbf{e}_{x}$. You may assume that the fluid velocity $\mathbf{u}$ has the form $\mathbf{u}=u(y) \mathbf{e}_{x}$.

(a) State the equation satisfied by $u(y)$ and its boundary conditions.

(b) Calculate $u(y)$.

(c) Show that the value of $U$ may be chosen to lead to zero viscous shear stress acting on the bottom plate and calculate the resulting flow rate.

Paper 3, Section II, C

commentA layer of thickness $h_{1}$ of a fluid of density $\rho_{1}$ is located above a layer of thickness $h_{2}$ of a fluid of density $\rho_{2}>\rho_{1}$. The two-fluid system is bounded by two impenetrable surfaces at $y=h_{1}$ and $y=-h_{2}$ and is assumed to be two-dimensional (i.e. independent of $z$ ). The fluid is subsequently perturbed, and the interface between the two fluids is denoted $y=\eta(x, t)$.

(a) Assuming irrotational motion in each fluid, state the equations and boundary conditions satisfied by the flow potentials, $\varphi_{1}$ and $\varphi_{2}$.

(b) The interface is perturbed by small-amplitude waves of the form $\eta=\eta_{0} e^{i(k x-\omega t)}$, with $\eta_{0} k \ll 1$. State the equations and boundary conditions satisfied by the linearised system.

(c) Calculate the dispersion relation of the waves relating the frequency $\omega$ to the wavenumber $k$.

Paper 4, Section II, C

comment(a) Show that for an incompressible fluid, $\nabla \times \boldsymbol{\omega}=-\nabla^{2} \mathbf{u}$, where $\boldsymbol{\omega}$ is the flow vorticity,

(b) State the equation of motion for an inviscid flow of constant density in a rotating frame subject to gravity. Show that, on Earth, the local vertical component of the centrifugal force is small compared to gravity. Present a scaling argument to justify the linearisation of the Euler equations for sufficiently large rotation rates, and hence deduce the linearised version of the Euler equations in a rapidly rotating frame.

(c) Denoting the rotation rate of the frame as $\boldsymbol{\Omega}=\Omega \mathbf{e}_{z}$, show that the linearised Euler equations may be manipulated to obtain an equation for the velocity field $\mathbf{u}$ in the form

$\frac{\partial^{2} \nabla^{2} \mathbf{u}}{\partial t^{2}}+4 \Omega^{2} \frac{\partial^{2} \mathbf{u}}{\partial z^{2}}=\mathbf{0}$

(d) Assume that there exist solutions of the form $\mathbf{u}=\mathbf{U}_{0} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)]$. Show that $\omega=\pm 2 \Omega \cos \theta$ where the angle $\theta$ is to be determined.

Paper 1, Section I, F

comment(a) Describe the Poincaré disc model $D$ for the hyperbolic plane by giving the appropriate Riemannian metric.

(b) Let $a \in D$ be some point. Write down an isometry $f: D \rightarrow D$ with $f(a)=0$.

(c) Using the Poincaré disc model, calculate the distance from 0 to re $e^{i \theta}$ with $0 \leqslant r<1$

(d) Using the Poincaré disc model, calculate the area of a disc centred at a point $a \in D$ and of hyperbolic radius $\rho>0$.

Paper 2, Section II, F

comment(a) Let $A B C$ be a hyperbolic triangle, with the angle at $A$ at least $\pi / 2$. Show that the side $B C$ has maximal length amongst the three sides of $A B C$.

[You may use the hyperbolic cosine formula without proof. This states that if $a, b$ and $c$ are the lengths of $B C, A C$, and $A B$ respectively, and $\alpha, \beta$ and $\gamma$ are the angles of the triangle at $A, B$ and $C$ respectively, then

$\cosh a=\cosh b \cosh c-\sinh b \sinh c \cos \alpha .]$

(b) Given points $z_{1}, z_{2}$ in the hyperbolic plane, let $w$ be any point on the hyperbolic line segment joining $z_{1}$ to $z_{2}$, and let $w^{\prime}$ be any point not on the hyperbolic line passing through $z_{1}, z_{2}, w$. Show that

$\rho\left(w^{\prime}, w\right) \leqslant \max \left\{\rho\left(w^{\prime}, z_{1}\right), \rho\left(w^{\prime}, z_{2}\right)\right\}$

where $\rho$ denotes hyperbolic distance.

(c) The diameter of a hyperbolic triangle $\Delta$ is defined to be

$\sup \{\rho(P, Q) \mid P, Q \in \Delta\}$

Show that the diameter of $\Delta$ is equal to the length of its longest side.

Paper 3, Section I, $5 F$

comment(a) State Euler's formula for a triangulation of a sphere.

(b) A sphere is decomposed into hexagons and pentagons with precisely three edges at each vertex. Determine the number of pentagons.

Paper 3, Section II, F

comment(a) Define the cross-ratio $\left[z_{1}, z_{2}, z_{3}, z_{4}\right]$ of four distinct points $z_{1}, z_{2}, z_{3}, z_{4} \in \mathbb{C} \cup\{\infty\}$. Show that the cross-ratio is invariant under Möbius transformations. Express $\left[z_{2}, z_{1}, z_{3}, z_{4}\right]$ in terms of $\left[z_{1}, z_{2}, z_{3}, z_{4}\right]$.

(b) Show that $\left[z_{1}, z_{2}, z_{3}, z_{4}\right]$ is real if and only if $z_{1}, z_{2}, z_{3}, z_{4}$ lie on a line or circle in $\mathbb{C} \cup\{\infty\}$.

(c) Let $z_{1}, z_{2}, z_{3}, z_{4}$ lie on a circle in $\mathbb{C}$, given in anti-clockwise order as depicted.

Show that $\left[z_{1}, z_{2}, z_{3}, z_{4}\right]$ is a negative real number, and that $\left[z_{2}, z_{1}, z_{3}, z_{4}\right]$ is a positive real number greater than 1 . Show that $\left|\left[z_{1}, z_{2}, z_{3}, z_{4}\right]\right|+1=\left|\left[z_{2}, z_{1}, z_{3}, z_{4}\right]\right|$. Use this to deduce Ptolemy's relation on lengths of edges and diagonals of the inscribed 4-gon:

$\left|z_{1}-z_{3}\right|\left|z_{2}-z_{4}\right|=\left|z_{1}-z_{2}\right|\left|z_{3}-z_{4}\right|+\left|z_{2}-z_{3}\right|\left|z_{4}-z_{1}\right|$

Paper 4, Section II, F

commentLet $\alpha(s)=(f(s), g(s))$ be a simple curve in $\mathbb{R}^{2}$ parameterised by arc length with $f(s)>0$ for all $s$, and consider the surface of revolution $S$ in $\mathbb{R}^{3}$ defined by the parameterisation

$\sigma(u, v)=(f(u) \cos v, f(u) \sin v, g(u))$

(a) Calculate the first and second fundamental forms for $S$. Show that the Gaussian curvature of $S$ is given by

$K=-\frac{f^{\prime \prime}(u)}{f(u)}$

(b) Now take $f(s)=\cos s+2, g(s)=\sin s, 0 \leqslant s<2 \pi$. What is the integral of the Gaussian curvature over the surface of revolution $S$ determined by $f$ and $g$ ?

[You may use the Gauss-Bonnet theorem without proof.]

(c) Now suppose $S$ has constant curvature $K \equiv 1$, and suppose there are two points $P_{1}, P_{2} \in \mathbb{R}^{3}$ such that $S \cup\left\{P_{1}, P_{2}\right\}$ is a smooth closed embedded surface. Show that $S$ is a unit sphere, minus two antipodal points.

[Do not attempt to integrate an expression of the form $\sqrt{1-C^{2} \sin ^{2} u}$ when $C \neq 1$. Study the behaviour of the surface at the largest and smallest possible values of $u$.]

Paper 1, Section II, E

comment(a) Let $I$ be an ideal of a commutative ring $R$ and assume $I \subseteq \bigcup_{i=1}^{n} P_{i}$ where the $P_{i}$ are prime ideals. Show that $I \subseteq P_{i}$ for some $i$.

(b) Show that $\left(x^{2}+1\right)$ is a maximal ideal of $\mathbb{R}[x]$. Show that the quotient ring $\mathbb{R}[x] /\left(x^{2}+1\right)$ is isomorphic to $\mathbb{C} .$

(c) For $a, b \in \mathbb{R}$, let $I_{a, b}$ be the ideal $(x-a, y-b)$ in $\mathbb{R}[x, y]$. Show that $I_{a, b}$ is a maximal ideal. Find a maximal ideal $J$ of $\mathbb{R}[x, y]$ such that $J \neq I_{a, b}$ for any $a, b \in \mathbb{R}$. Justify your answers.

Paper 2, Section I, E

commentLet $R$ be an integral domain.

Define what is meant by the field of fractions $F$ of $R$. [You do not need to prove the existence of $F$.]

Suppose that $\phi: R \rightarrow K$ is an injective ring homomorphism from $R$ to a field $K$. Show that $\phi$ extends to an injective ring homomorphism $\Phi: F \rightarrow K$.

Give an example of $R$ and a ring homomorphism $\psi: R \rightarrow S$ from $R$ to a ring $S$ such that $\psi$ does not extend to a ring homomorphism $F \rightarrow S$.

Paper 2, Section II, E

comment(a) State Sylow's theorems and give the proof of the second theorem which concerns conjugate subgroups.

(b) Show that there is no simple group of order 351 .

(c) Let $k$ be the finite field $\mathbb{Z} /(31)$ and let $G L_{2}(k)$ be the multiplicative group of invertible $2 \times 2$ matrices over $k$. Show that every Sylow 3-subgroup of $G L_{2}(k)$ is abelian.

Paper 3, Section I, E

commentLet $G$ be a group of order $n$. Define what is meant by a permutation representation of $G$. Using such representations, show $G$ is isomorphic to a subgroup of the symmetric group $S_{n}$. Assuming $G$ is non-abelian simple, show $G$ is isomorphic to a subgroup of $A_{n}$. Give an example of a permutation representation of $S_{3}$ whose kernel is $A_{3}$.

Paper 3, Section II, E

comment(a) Define what is meant by an algebraic integer $\alpha$. Show that the ideal

$I=\{h \in \mathbb{Z}[x] \mid h(\alpha)=0\}$

in $\mathbb{Z}[x]$ is generated by a monic irreducible polynomial $f$. Show that $\mathbb{Z}[\alpha]$, considered as a $\mathbb{Z}$-module, is freely generated by $n$ elements where $n=\operatorname{deg} f$.

(b) Assume $\alpha \in \mathbb{C}$ satisfies $\alpha^{5}+2 \alpha+2=0$. Is it true that the ideal (5) in $\mathbb{Z}[\alpha]$ is a prime ideal? Is there a ring homomorphism $\mathbb{Z}[\alpha] \rightarrow \mathbb{Z}[\sqrt{-1}]$ ? Justify your answers.

(c) Show that the only unit elements of $\mathbb{Z}[\sqrt{-5}]$ are 1 and $-1$. Show that $\mathbb{Z}[\sqrt{-5}]$ is not a UFD.

Paper 4, Section I, $2 \mathrm{E}$

commentGive the statement and the proof of Eisenstein's criterion. Use this criterion to show $x^{p-1}+x^{p-2}+\cdots+1$ is irreducible in $\mathbb{Q}[x]$ where $p$ is a prime.

Paper 4, Section II, E

commentLet $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module.

(a) Show that every submodule of $M$ is finitely generated.

(b) Show that each maximal element of the set

$\mathcal{A}=\{\operatorname{Ann}(m) \mid 0 \neq m \in M\}$

is a prime ideal. [Here, maximal means maximal with respect to inclusion, and $\operatorname{Ann}(m)=\{r \in R \mid r m=0\} .]$

(c) Show that there is a chain of submodules

$0=M_{0} \subseteq M_{1} \subseteq \cdots \subseteq M_{l}=M$

such that for each $0<i \leqslant l$ the quotient $M_{i} / M_{i-1}$ is isomorphic to $R / P_{i}$ for some prime ideal $P_{i}$.

Paper 1, Section $I$, $1 F$

comment(a) Consider the linear transformation $\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by the matrix

$\left(\begin{array}{rrr} 5 & -6 & -6 \\ -1 & 4 & 2 \\ 3 & -6 & -4 \end{array}\right)$

Find a basis of $\mathbb{R}^{3}$ in which $\alpha$ is represented by a diagonal matrix.

(b) Give a list of $6 \times 6$ matrices such that any linear transformation $\beta: \mathbb{R}^{6} \rightarrow \mathbb{R}^{6}$ with characteristic polynomial

$(x-2)^{4}(x+7)^{2}$

and minimal polynomial

$(x-2)^{2}(x+7)$

is similar to one of the matrices on your list. No two distinct matrices on your list should be similar. [No proof is required.]

Paper 1, Section II, F

commentLet $M_{n, n}$ denote the vector space over $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Let $\operatorname{Tr}: M_{n, n} \rightarrow F$ denote the trace functional, i.e., if $A=\left(a_{i j}\right)_{1 \leqslant i, j \leqslant n} \in M_{n, n}$, then

$\operatorname{Tr}(A)=\sum_{i=1}^{n} a_{i i}$

(a) Show that Tr is a linear functional.

(b) Show that $\operatorname{Tr}(A B)=\operatorname{Tr}(B A)$ for $A, B \in M_{n, n}$.

(c) Show that $\operatorname{Tr}$ is unique in the following sense: If $f: M_{n, n} \rightarrow F$ is a linear functional such that $f(A B)=f(B A)$ for each $A, B \in M_{n, n}$, then $f$ is a scalar multiple of the trace functional. If, in addition, $f(I)=n$, then $f=$ Tr.

(d) Let $W \subseteq M_{n, n}$ be the subspace spanned by matrices $C$ of the form $C=A B-B A$ for $A, B \in M_{n, n}$. Show that $W$ is the kernel of Tr.

Paper 2, Section I, F

commentFind a linear change of coordinates such that the quadratic form

$2 x^{2}+8 x y-6 x z+y^{2}-4 y z+2 z^{2}$

takes the form

$\alpha x^{2}+\beta y^{2}+\gamma z^{2}$

for real numbers $\alpha, \beta$ and $\gamma$.

Paper 2, Section II, F

commentLet $M_{n, n}$ denote the vector space over a field $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Given $B \in M_{n, n}$, consider the two linear transformations $R_{B}, L_{B}: M_{n, n} \rightarrow$ $M_{n, n}$ defined by

$L_{B}(A)=B A, \quad R_{B}(A)=A B$

(a) Show that $\operatorname{det} L_{B}=(\operatorname{det} B)^{n}$.

[For parts (b) and (c), you may assume the analogous result $\operatorname{det} R_{B}=(\operatorname{det} B)^{n}$ without proof.]

(b) Now let $F=\mathbb{C}$. For $B \in M_{n, n}$, write $B^{*}$ for the conjugate transpose of $B$, i.e., $B^{*}:=\bar{B}^{T}$. For $B \in M_{n, n}$, define the linear transformation $M_{B}: M_{n, n} \rightarrow M_{n, n}$ by

$M_{B}(A)=B A B^{*}$

Show that $\operatorname{det} M_{B}=|\operatorname{det} B|^{2 n}$.

(c) Again let $F=\mathbb{C}$. Let $W \subseteq M_{n, n}$ be the set of Hermitian matrices. [Note that $W$ is not a vector space over $\mathbb{C}$ but only over $\mathbb{R} .]$ For $B \in M_{n, n}$ and $A \in W$, define $T_{B}(A)=B A B^{*}$. Show that $T_{B}$ is an $\mathbb{R}$-linear operator on $W$, and show that as such,

$\operatorname{det} T_{B}=|\operatorname{det} B|^{2 n}$

Paper 3, Section II, F

commentLet $\alpha: V \rightarrow V$ be a linear transformation defined on a finite dimensional inner product space $V$ over $\mathbb{C}$. Recall that $\alpha$ is normal if $\alpha$ and its adjoint $\alpha^{*}$ commute. Show that $\alpha$ being normal is equivalent to each of the following statements:

(i) $\alpha=\alpha_{1}+i \alpha_{2}$ where $\alpha_{1}, \alpha_{2}$ are self-adjoint operators and $\alpha_{1} \alpha_{2}=\alpha_{2} \alpha_{1}$;

(ii) there is an orthonormal basis for $V$ consisting of eigenvectors of $\alpha$;

(iii) there is a polynomial $g$ with complex coefficients such that $\alpha^{*}=g(\alpha)$.

Paper 4, Section I, F

commentFor which real numbers $x$ do the vectors

$(x, 1,1,1), \quad(1, x, 1,1), \quad(1,1, x, 1), \quad(1,1,1, x),$

not form a basis of $\mathbb{R}^{4}$ ? For each such value of $x$, what is the dimension of the subspace of $\mathbb{R}^{4}$ that they span? For each such value of $x$, provide a basis for the spanned subspace, and extend this basis to a basis of $\mathbb{R}^{4}$.

Paper 4, Section II, F

comment(a) Let $\alpha: V \rightarrow W$ be a linear transformation between finite dimensional vector spaces over a field $F=\mathbb{R}$ or $\mathbb{C}$.

Define the dual map of $\alpha$. Let $\delta$ be the dual map of $\alpha$. Given a subspace $U \subseteq V$, define the annihilator $U^{\circ}$ of $U$. Show that $(\operatorname{ker} \alpha)^{\circ}$ and the image of $\delta$ coincide. Conclude that the dimension of the image of $\alpha$ is equal to the dimension of the image of $\delta$. Show that $\operatorname{dim} \operatorname{ker}(\alpha)-\operatorname{dim} \operatorname{ker}(\delta)=\operatorname{dim} V-\operatorname{dim} W$.

(b) Now suppose in addition that $V, W$ are inner product spaces. Define the adjoint $\alpha^{*}$ of $\alpha$. Let $\beta: U \rightarrow V, \gamma: V \rightarrow W$ be linear transformations between finite dimensional inner product spaces. Suppose that the image of $\beta$ is equal to the kernel of $\gamma$. Then show that $\beta \beta^{*}+\gamma^{*} \gamma$ is an isomorphism.

Paper 1, Section II, H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk on the integers, starting at $X_{0}=0$.

(a) What does it mean to say that a Markov chain is irreducible? What does it mean to say that an irreducible Markov chain is recurrent? Show that $\left(X_{n}\right)_{n} \geqslant 0$ is irreducible and recurrent.

[Hint: You may find it helpful to use the limit

$\lim _{k \rightarrow \infty} \sqrt{k} 2^{-2 k}\left(\begin{array}{c} 2 k \\ k \end{array}\right)=\sqrt{\pi}$

You may also use without proof standard necessary and sufficient conditions for recurrence.]

(b) What does it mean to say that an irreducible Markov chain is positive recurrent? Determine, with proof, whether $\left(X_{n}\right)_{n \geqslant 0}$ is positive recurrent.

(c) Let

$T=\inf \left\{n \geqslant 1: X_{n}=0\right\}$

be the first time the chain returns to the origin. Compute $\mathbb{E}\left[s^{T}\right]$ for a fixed number $0<s<1$.

Paper 2, Section II, H

comment(a) Prove that every open communicating class of a Markov chain is transient. Prove that every finite transient communicating class is open. Give an example of a Markov chain with an infinite transient closed communicating class.

(b) Consider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ with state space $\{a, b, c, d\}$ and transition probabilities given by the matrix

$P=\left(\begin{array}{cccc} 1 / 3 & 0 & 1 / 3 & 1 / 3 \\ 0 & 1 / 4 & 0 & 3 / 4 \\ 1 / 2 & 1 / 2 & 0 & 0 \\ 0 & 2 / 3 & 0 & 1 / 3 \end{array}\right)$

(i) Compute $\mathbb{P}\left(X_{n}=b \mid X_{0}=d\right)$ for a fixed $n \geqslant 0$.

(ii) Compute $\mathbb{P}\left(X_{n}=c\right.$ for some $\left.n \geqslant 1 \mid X_{0}=a\right)$.

(iii) Show that $P^{n}$ converges as $n \rightarrow \infty$, and determine the limit.

[Results from lectures can be used without proof if stated carefully.]

Paper 3, Section I, H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain such that $X_{0}=i$. Prove that

$\sum_{n=0}^{\infty} \mathbb{P}_{i}\left(X_{n}=i\right)=\frac{1}{\mathbb{P}_{i}\left(X_{n} \neq i \text { for all } n \geqslant 1\right)}$

where $1 / 0=+\infty$. [You may use the strong Markov property without proof.]

Paper 4, Section I, H

commentConsider two boxes, labelled $\mathrm{A}$ and B. Initially, there are no balls in box $\mathrm{A}$ and $k$ balls in box B. Each minute later, one of the $k$ balls is chosen uniformly at random and is moved to the opposite box. Let $X_{n}$ denote the number of balls in box A at time $n$, so that $X_{0}=0$.

(a) Find the transition probabilities of the Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ and show that it is reversible in equilibrium.

(b) Find $\mathbb{E}(T)$, where $T=\inf \left\{n \geqslant 1: X_{n}=0\right\}$ is the next time that all $k$ balls are again in box $B$.

Paper 1, Section II, A

comment(a) Consider the general self-adjoint problem for $y(x)$ on $[a, b]$ :

$-\frac{d}{d x}\left[p(x) \frac{d}{d x} y\right]+q(x) y=\lambda w(x) y ; \quad y(a)=y(b)=0$

where $\lambda$ is the eigenvalue, and $w(x)>0$. Prove that eigenfunctions associated with distinct eigenvalues are orthogonal with respect to a particular inner product which you should define carefully.

(b) Consider the problem for $y(x)$ given by

$x y^{\prime \prime}+3 y^{\prime}+\left(\frac{1+\lambda}{x}\right) y=0 ; \quad y(1)=y(e)=0 .$

(i) Recast this problem into self-adjoint form.

(ii) Calculate the complete set of eigenfunctions and associated eigenvalues for this problem. [Hint: You may find it useful to make the substitution $\left.x=e^{s} .\right]$

(iii) Verify that the eigenfunctions associated with distinct eigenvalues are indeed orthogonal.

Paper 2, Section I, A

commentUse the method of characteristics to find $u(x, y)$ in the first quadrant $x \geqslant 0, y \geqslant 0$, where $u(x, y)$ satisfies

$\frac{\partial u}{\partial x}-2 x \frac{\partial u}{\partial y}=\cos x$

with boundary data $u(x, 0)=\cos x$.

Paper 2, Section II, A

commentConsider a bar of length $\pi$ with free ends, subject to longitudinal vibrations. You may assume that the longitudinal displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$ :

$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}$

for $x \in(0, \pi)$ and $t>0$ with boundary conditions:

$\frac{\partial y}{\partial x}(0, t)=\frac{\partial y}{\partial x}(\pi, t)=0$

for $t>0$. The bar is initially at rest so that

$\frac{\partial y}{\partial t}(x, 0)=0$

for $x \in(0, \pi)$, with a spatially varying initial longitudinal displacement given by

$y(x, 0)=b x$

for $x \in(0, \pi)$, where $b$ is a real constant.

(a) Using separation of variables, show that

$y(x, t)=\frac{b \pi}{2}-\frac{4 b}{\pi} \sum_{n=1}^{\infty} \frac{\cos [(2 n-1) x] \cos [(2 n-1) c t]}{(2 n-1)^{2}}$

(b) Determine a periodic function $P(x)$ such that this solution may be expressed as

$y(x, t)=\frac{1}{2}[P(x+c t)+P(x-c t)]$

Paper 3, Section $\mathbf{I}$, A

commentCalculate the Green's function $G(x ; \xi)$ given by the solution to

$\frac{d^{2} G}{d x^{2}}=\delta(x-\xi) ; \quad G(0 ; \xi)=\frac{d G}{d x}(1 ; \xi)=0$

where $\xi \in(0,1), x \in(0,1)$ and $\delta(x)$ is the Dirac $\delta$-function. Use this Green's function to calculate an explicit solution $y(x)$ to the boundary value problem

$\frac{d^{2} y}{d x^{2}}=x e^{-x}$

where $x \in(0,1)$, and $y(0)=y^{\prime}(1)=0$.

Paper 3, Section II, B

(a) Show that the Fourier transform of $f(x)=e^{-a^{2} x^{2}}$, for $a>0$, is

$\tilde{f}(k)=\frac{\sqrt{\pi}}{a} e^{-\frac{k^{2}}{4 a^{2}}},$

stating clearly any properties of the Fourier transform that you use.

[Hint: You may assume that $\int_{0}^{\infty} e^{-t^{2}} d t=\sqrt{\pi} / 2$.]

(b) Consider now the Cauchy problem for the diffusion equation in one space dimension, i.e. solving for $\theta(x, t)$ satisfying:

$\frac{\partial \theta}{\partial t}=D \frac{\partial^{2} \theta}{\partial x^{2}} \quad \text { with } \theta(x, 0)=g(x)$

where $D$ is a positive constant and $g(x)$ is specified. Consider the following property of a solution:

Property P: If the initial data $g(x)$