Part II, 2008, Paper 4

Algebraic Topology

• 4.II.21F

  Algebraic Topology | Part II, 2008

  Let $X$ and $Y$ be topological spaces.

  (i) Show that a map $f: X \rightarrow Y$ is a homotopy equivalence if there exist maps $g, h: Y \rightarrow X$ such that $f g \simeq 1_{Y}$ and $h f \simeq 1_{X}$. More generally, show that a map $f: X \rightarrow Y$ is a homotopy equivalence if there exist maps $g, h: Y \rightarrow X$ such that $f g$ and $h f$ are homotopy equivalences.

  (ii) Suppose that $\tilde{X}$ and $\tilde{Y}$ are universal covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. Using path-lifting properties, show that if $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$.

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Applications of Quantum Mechanics

• 4.II.33E

  Applications of Quantum Mechanics | Part II, 2008

  Explain why the allowed energies of electrons in a three-dimensional crystal lie in energy bands. What quantum numbers can be used to classify the electron energy eigenstates?

  Describe the effect on the energy level structure of adding a small density of impurity atoms randomly to the crystal.

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Applied Probability

• 4.II.26I

  Applied Probability | Part II, 2008

  On a hot summer night, opening my window brings some relief. This attracts hordes of mosquitoes who manage to negotiate a dense window net. But, luckily, I have a mosquito trapping device in my room.

  Assume the mosquitoes arrive in a Poisson process at rate $\lambda$; afterwards they wander around for independent and identically distributed random times with a finite mean $\mathbb{E} S$, where $S$ denotes the random wandering time of a mosquito, and finally are trapped by the device.

  (a) Identify a mathematical model, which was introduced in the course, for the number of mosquitoes present in the room at times $t \geqslant 0$.

  (b) Calculate the distribution of $Q(t)$ in terms of $\lambda$ and the tail probabilities $\mathbb{P}(S>x)$ of the wandering time $S$, where $Q(t)$ is the number of mosquitoes in the room at time $t>0$ (assuming that at the initial time, $Q(0)=0$ ).

  (c) Write down the distribution for $Q^{\mathrm{E}}$, the number of mosquitoes in the room in equilibrium, in terms of $\lambda$ and $\mathbb{E} S$.

  (d) Instead of waiting for the number of mosquitoes to reach equilibrium, I close the window at time $t>0$. For $v \geqslant 0$ let $X(t+v)$ be the number of mosquitoes left at time $t+v$, i.e. $v$ time units after closing the window. Calculate the distribution of $X(t+v)$.

  (e) Let $V(t)$ be the time needed to trap all mosquitoes in the room after closing the window at time $t>0$. By considering the event $\{X(t+v) \geqslant 1\}$, or otherwise, compute $\mathbb{P}[V(t)>v]$.

  (f) Now suppose that the time $t$ at which I shut the window is very large, so that I can assume that the number of mosquitoes in the room has the distribution of $Q^{E}$. Let $V^{E}$ be the further time needed to trap all mosquitoes in the room. Show that

  $$\mathbb{P}\left[V^{E}>v\right]=1-\exp \left(-\lambda \mathbb{E}\left[(S-v)_{+}\right]\right),$$

  where $x_{+} \equiv \max (x, 0)$.

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Asymptotic Methods

• 4.II $. 31 \mathrm{~A}$

  Asymptotic Methods | Part II, 2008

  The Bessel equation of order $n$ is

  $$z^{2} y^{\prime \prime}+z y^{\prime}+\left(z^{2}-n^{2}\right) y=0 .$$

  Here, $n$ is taken to be an integer, with $n \geqslant 0$. The transformation $w(z)=z^{\frac{1}{2}} y(z)$ converts (1) to the form

  $$w^{\prime \prime}+q(z) w=0$$

  where

  $$q(z)=1-\frac{\left(n^{2}-\frac{1}{4}\right)}{z^{2}}$$

  Find two linearly independent solutions of the form

  $$w=e^{s z} \sum_{k=0}^{\infty} c_{k} z^{\rho-k}$$

  where $c_{k}$ are constants, with $c_{0} \neq 0$, and $s$ and $\rho$ are to be determined. Find recurrence relationships for the $c_{k}$.

  Find the first two terms of two linearly independent Liouville-Green solutions of (2) for $w(z)$ valid in a neighbourhood of $z=\infty$. Relate these solutions to those of the form (3).

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Classical Dynamics

• 4.I.9B

  Classical Dynamics | Part II, 2008

  (a) Show that the principal moments of inertia for an infinitesimally thin uniform rectangular sheet of mass $M$ with sides of length $a$ and $b$ (with $b<a$ ) about its centre of mass are $I_{1}=M b^{2} / 12, I_{2}=M a^{2} / 12$ and $I_{3}=M\left(a^{2}+b^{2}\right) / 12$.

  (b) Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of the sheet as viewed in the body frame are

  $$\begin{aligned} &I_{1} \frac{d \omega_{1}}{d t}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \frac{d \omega_{2}}{d t}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \end{aligned}$$

  and

  $$I_{3} \frac{d \omega_{3}}{d t}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} .$$

  A possible solution of these equations is such that the sheet rotates with $\omega_{1}=\omega_{3}=0$, and $\omega_{2}=\Omega=$ constant.

  By linearizing, find the equations governing small motions in the neighbourhood of this solution that have $\left(\omega_{1}, \omega_{3}\right) \neq 0$. Use these to show that there are solutions corresponding to instability such that $\omega_{1}$ and $\omega_{3}$ are both proportional to exp $(\beta \Omega t)$, with $\beta=\sqrt{\left(a^{2}-b^{2}\right) /\left(a^{2}+b^{2}\right)} .$

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• 4.II.15B

  Classical Dynamics | Part II, 2008

  (a) A Hamiltonian system with $n$ degrees of freedom has Hamiltonian $H=H(\mathbf{p}, \mathbf{q})$, where the coordinates $\mathbf{q}=\left(q_{1}, q_{2}, q_{3}, \ldots, q_{n}\right)$ and the momenta $\mathbf{p}=\left(p_{1}, p_{2}, p_{3}, \ldots, p_{n}\right)$ respectively.

  Show from Hamilton's equations that when $H$ does not depend on time explicitly, for any function $F=F(\mathbf{p}, \mathbf{q})$,

  $$\frac{d F}{d t}=[F, H],$$

  where $[F, H]$ denotes the Poisson bracket.

  For a system of $N$ interacting vortices

  $$H(\mathbf{p}, \mathbf{q})=-\frac{\kappa}{4} \sum_{\substack{i=1 \\ N}}^{N} \sum_{\substack{j=1 \\ j \neq i}}^{N} \ln \left[\left(p_{i}-p_{j}\right)^{2}+\left(q_{i}-q_{j}\right)^{2}\right]$$

  where $\kappa$ is a constant. Show that the quantity defined by

  $$F=\sum_{j=1}^{N}\left(q_{j}^{2}+p_{j}^{2}\right)$$

  is a constant of the motion.

  (b) The action for a Hamiltonian system with one degree of freedom with $H=H(p, q)$ for which the motion is periodic is

  $$I=\frac{1}{2 \pi} \oint p(H, q) d q .$$

  Show without assuming any specific form for $H$ that the period of the motion $T$ is given by

  $$\frac{2 \pi}{T}=\frac{d H}{d I}$$

  Suppose now that the system has a parameter that is allowed to vary slowly with time. Explain briefly what is meant by the statement that the action is an adiabatic invariant. Suppose that when this parameter is fixed, $H=0$ when $I=0$. Deduce that, if $T$ decreases on an orbit with any $I$ when the parameter is slowly varied, then $H$ increases.

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Coding and Cryptography

• 4.I.4G

  Coding and Cryptography | Part II, 2008

  What is a binary cyclic code of length $N$ ? What is the generator polynomial for such a cyclic code? Prove that the generator polynomial is a factor of $X^{N}-1$ over the field $\mathbb{F}_{2}$.

  Find all the binary cyclic codes of length 5 .

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Cosmology

• 4.I.10E

  Cosmology | Part II, 2008

  The Friedmann and Raychaudhuri equations are respectively

  $$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3} \rho-\frac{k c^{2}}{a^{2}} \quad \text { and } \quad \frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+\frac{3 P}{c^{2}}\right) \text {, }$$

  where $\rho$ is the mass density, $P$ is the pressure, $k$ is the curvature and $\dot{a} \equiv d a / d t$ with $t$ the cosmic time. Using conformal time $\tau$ (defined by $d \tau=d t / a$ ) and the equation of state $P=w \rho c^{2}$, show that these can be rewritten as

  $$\frac{k c^{2}}{\mathcal{H}^{2}}=\Omega-1 \quad \text { and } \quad 2 \frac{d \mathcal{H}}{d \tau}=-(3 w+1)\left(\mathcal{H}^{2}+k c^{2}\right)$$

  where $\mathcal{H}=a^{-1} d a / d \tau$ and the relative density is $\Omega \equiv \rho / \rho_{\text {crit }}=8 \pi G \rho a^{2} /\left(3 \mathcal{H}^{2}\right)$.

  Use these relations to derive the following evolution equation for $\Omega$

  $$\frac{d \Omega}{d \tau}=(3 w+1) \mathcal{H} \Omega(\Omega-1)$$

  For both $w=0$ and $w=-1$, plot the qualitative evolution of $\Omega$ as a function of $\tau$ in an expanding universe $\mathcal{H}>0$ (i.e. include curves initially with $\Omega>1$ and $\Omega<1$ ).

  Hence, or otherwise, briefly describe the flatness problem of the standard cosmology and how it can be solved by inflation.

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Differential Geometry

• 4.II $. 24 \mathrm{H} \quad$

  Differential Geometry | Part II, 2008

  Let $S \subset \mathbb{R}^{3}$ be a surface.

  (a) In the case where $S$ is compact, define the Euler characteristic $\chi$ and genus $g$ of $S$.

  (b) Define the notion of geodesic curvature $k_{g}$ for regular curves $\gamma: I \rightarrow S$. When is $k_{g}=0$ ? State the Global Gauss-Bonnet Theorem (including boundary term).

  (c) Let $S=\mathbb{S}^{2}$ (the standard 2-sphere), and suppose $\gamma \subset \mathbb{S}^{2}$ is a simple closed regular curve such that $\mathbb{S}^{2} \backslash \gamma$ is the union of two distinct connected components with equal areas. Can $\gamma$ have everywhere strictly positive or everywhere strictly negative geodesic curvature?

  (d) Prove or disprove the following statement: if $S$ is connected with Gaussian curvature $K=1$ identically, then $S$ is a subset of a sphere of radius 1 .

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Dynamical Systems

• 4.I.7A

  Dynamical Systems | Part II, 2008

  Let $F: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. State when $F$ is chaotic according to Glendinning's definition.

  Prove that if $F$ has a 3 -cycle then $F^{2}$ has a horseshoe.

  [You may assume the Intermediate Value Theorem.]

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• 4.II.14A

  Dynamical Systems | Part II, 2008

  Explain the difference between a hyperbolic and a nonhyperbolic fixed point $\mathbf{x}_{0}$ for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$.

  Consider the system in $\mathbb{R}^{2}$, where $\mu$ is a real parameter,

  $$\begin{aligned} &\dot{x}=x\left(\mu-x+y^{2}\right) \\ &\dot{y}=y\left(1-x-y^{2}\right) \end{aligned}$$

  Show that the fixed point $(\mu, 0)$ has a bifurcation when $\mu=1$, while the fixed points $(0, \pm 1)$ have a bifurcation when $\mu=-1$.

  [The fixed point at $(0,-1)$ should not be considered further.]

  Analyse each of the bifurcations at $(\mu, 0)$ and $(0,1)$ in turn as follows. Make a change of variable of the form $\mathbf{X}=\mathbf{x}-\mathbf{x}_{0}(\mu), \nu=\mu-\mu_{0}$. Identify the (non-extended) stable and centre linear subspaces at the bifurcation in terms of $X$ and $Y$. By finding the leading-order approximation to the extended centre manifold, construct the evolution equation on the extended centre manifold, and determine the type of bifurcation. Sketch the local bifurcation diagram, showing which fixed points are stable.

  [Hint: the leading-order approximation to the extended centre manifold of the bifurcation at $(0,1)$ is $Y=a X$ for some coefficient a.]

  Show that there is another fixed point in $y>0$ for $\mu<1$, and that this fixed point connects the two bifurcations.

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Electrodynamics

• 4.II $. 35 \mathrm{D} \quad$

  Electrodynamics | Part II, 2008

  The Maxwell field tensor is given by

  $$F^{a b}=\left(\begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & -B_{x} \\ E_{z} & -B_{y} & B_{x} & 0 \end{array}\right)$$

  A general 4-velocity is written as $U^{a}=\gamma(1, \mathbf{v})$, where $\gamma=\left(1-|\mathbf{v}|^{2}\right)^{-1 / 2}$, and $c=1$. A general 4-current density is written as $J^{a}=(\rho, \mathbf{j})$, where $\rho$ is the charge density and $\mathbf{j}$ is the 3 -current density. Show that

  $$F^{a b} U_{b}=\gamma(\mathbf{E} \cdot \mathbf{v}, \mathbf{E}+\mathbf{v} \times \mathbf{B})$$

  In the rest frame of a conducting medium, Ohm's law states that $\mathbf{j}=\sigma \mathbf{E}$ where $\sigma$ is the conductivity. Show that the relativistic generalization to a frame in which the medium moves with uniform velocity $\mathbf{v}$ is

  $$J^{a}-\left(J^{b} U_{b}\right) U^{a}=\sigma F^{a b} U_{b}$$

  Show that this implies

  $$\mathbf{j}=\rho \mathbf{v}+\sigma \gamma(\mathbf{E}+\mathbf{v} \times \mathbf{B}-(\mathbf{v} \cdot \mathbf{E}) \mathbf{v})$$

  Simplify this formula, given that the charge density vanishes in the rest frame of the medium

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Fluid Dynamics II

• 4.II.37A

  Fluid Dynamics II | Part II, 2008

  Viscous incompressible fluid of uniform density is extruded axisymmetrically from a thin circular slit of small radius centred at the origin and lying in the plane $z=0$ in cylindrical polar coordinates $r, \theta, z$. There is no external radial pressure gradient. It is assumed that the fluid forms a thin boundary layer, close to and symmetric about the plane $z=0$. The layer has thickness $\delta(r) \ll r$. The $r$-component of the steady Navier-Stokes equations may be approximated by

  $$u_{r} \frac{\partial u_{r}}{\partial r}+u_{z} \frac{\partial u_{r}}{\partial z}=\nu \frac{\partial^{2} u_{r}}{\partial z^{2}}$$

  (i) Prove that the quantity (proportional to the flux of radial momentum)

  $$\mathcal{F}=\int_{-\infty}^{\infty} u_{r}^{2} r d z$$

  is independent of $r$.

  (ii) Show, by balancing terms in the momentum equation and assuming constancy of $\mathcal{F}$, that there is a similarity solution of the form

  $$u_{r}=-\frac{1}{r} \frac{\partial \Psi}{\partial z}, \quad u_{z}=\frac{1}{r} \frac{\partial \Psi}{\partial r}, \quad \Psi=-A \delta(r) f(\eta), \quad \eta=\frac{z}{\delta(r)}, \quad \delta(r)=C r$$

  where $A, C$ are constants. Show that for suitable choices of $A$ and $C$ the equation for $f$ takes the form

  $$\begin{gathered} -f^{\prime 2}-f f^{\prime \prime}=f^{\prime \prime \prime} ; \\ f=f^{\prime \prime}=0 \text { at } \eta=0 ; \quad f^{\prime} \rightarrow 0 \text { as } \eta \rightarrow \infty ; \\ \int_{-\infty}^{\infty} f_{\eta}^{2} d \eta=1 . \end{gathered}$$

  (iii) Give an inequality connecting $\mathcal{F}$ and $\nu$ that ensures that the boundary layer approximation $(\delta \ll r)$ is valid. Solve the equation to give a complete solution to the problem for $u_{r}$ when this inequality holds.

  [Hint: $\left.\int_{-\infty}^{\infty} \operatorname{sech}^{4} x d x=4 / 3 .\right]$

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Further Complex Methods

• 4.I.8C

  Further Complex Methods | Part II, 2008

  The Hilbert transform $\hat{f}$ of a function $f$ is defined by

  $$\hat{f}(t)=\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \frac{f(\tau)}{t-\tau} d \tau,$$

  where $\mathcal{P}$ denotes the Cauchy principal value.

  Show that the Hilbert transform of $\frac{\sin t}{t}$ is $\frac{1-\cos t}{t} .$

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Galois Theory

• 4.II.18H

  Galois Theory | Part II, 2008

  Let $L=\mathbf{C}(z)$ be the function field in one variable, $n>0$ an integer, and $\zeta_{n}=e^{2 \pi i / n}$.

  Define $\sigma, \tau: L \rightarrow L$ by the formulae

  $$(\sigma f)(z)=f\left(\zeta_{n} z\right), \quad(\tau f)(z)=f(1 / z),$$

  and let $G=\langle\sigma, \tau\rangle$ be the group generated by $\sigma$ and $\tau$.

  (i) Find $w \in \mathbf{C}(z)$ such that $L^{G}=\mathbf{C}(w)$.

  [You must justify your answer, stating clearly any theorems you use.]

  (ii) Suppose $n$ is an odd prime. Determine the subgroups of $G$ and the corresponding intermediate subfields $M$, with $\mathbf{C}(w) \subseteq M \subseteq L$.

  State which intermediate subfields $M$ are Galois extensions of $\mathbf{C}(w)$, and for these extensions determine the Galois group.

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General Relativity

• 4.II.36E

  General Relativity | Part II, 2008

  A solution of the Einstein equations is given by the metric

  $$\mathrm{d} s^{2}=-\left(1-\frac{2 M}{r}\right) \mathrm{d} t^{2}+\frac{1}{\left(1-\frac{2 M}{r}\right)} \mathrm{d} r^{2}+r^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}\right)$$

  For an incoming light ray, with constant $\theta, \phi$, show that

  $$t=v-r-2 M \log \left|\frac{r}{2 M}-1\right|,$$

  for some fixed $v$ and find a similar solution for an outgoing light ray. For the outgoing case, assuming $r>2 M$, show that in the far past $\frac{r}{2 M}-1 \propto \exp \left(\frac{t}{2 M}\right)$ and in the far future $r \sim t$.

  Obtain the transformed metric after the change of variables $(t, r, \theta, \phi) \rightarrow(v, r, \theta, \phi)$. With coordinates $\hat{t}=v-r, r$ sketch, for fixed $\theta, \phi$, the trajectories followed by light rays. What is the significance of the line $r=2 M$ ?

  Show that, whatever path an observer with initial $r=r_{0}<2 M$ takes, he must reach $r=0$ in a finite proper time.

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Geometry of Group Actions

• 4.I $3 \mathrm{G}$

  Geometry of Group Actions | Part II, 2008

  Define the hyperbolic metric (in the sense of metric spaces) on the 3 -ball.

  Given a finite set in hyperbolic 3 -space, show there is at least one closed ball of minimal radius containing that set.

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• 4.II.12G

  Geometry of Group Actions | Part II, 2008

  What does it mean for a subgroup $G$ of the Möbius group to be discrete?

  Show that a discrete group necessarily acts properly discontinuously in hyperbolic 3-space.

  [You may assume that a discrete subgroup of a matrix group is a closed subset.]

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Graph Theory

• 4.II.17F

  Graph Theory | Part II, 2008

  For $s \geqslant 2$, let $R(s)$ be the least integer $n$ such that for every 2 -colouring of the edges of $K_{n}$ there is a monochromatic $K_{s}$. Prove that $R(s)$ exists.

  For any $k \geqslant 1$ and $s_{1}, \ldots, s_{k} \geqslant 2$, define the Ramsey number $R_{k}\left(s_{1}, \ldots, s_{k}\right)$, and prove that it exists.

  Show that, whenever the positive integers are partitioned into finitely many classes, some class contains $x, y, z$ with $x+y=z$.

  [Hint: given a finite colouring of the positive integers, induce a colouring of the pairs of positive integers by giving the pair $i j(i<j)$ the colour of $j-i$.]

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Linear Analysis

• 4.II.22F

  Linear Analysis | Part II, 2008

  Let $H$ be a Hilbert space. Show that if $V$ is a closed subspace of $H$ then any $f \in H$ can be written as $f=v+w$ with $v \in V$ and $w \perp V$.

  Suppose $U: H \rightarrow H$ is unitary (that is to say $U U^{*}=U^{*} U=I$ ). Let

  $$A_{n} f=\frac{1}{n} \sum_{k=0}^{n-1} U^{k} f$$

  and consider

  $$X=\{g-U g: g \in H\}$$

  (i) Show that $U$ is an isometry and $\left\|A_{n}\right\| \leqslant 1$.

  (ii) Show that $X$ is a subspace of $H$ and $A_{n} f \rightarrow 0$ as $n \rightarrow \infty$ whenever $f \in X$.

  (iii) Let $V$ be the closure of $X$. Show that $A_{n} v \rightarrow 0$ as $n \rightarrow \infty$ whenever $v \in V$.

  (iv) Show that, if $w \perp X$, then $U w=w$. Deduce that, if $w \perp V$, then $U w=w$.

  (v) If $f \in H$ show that there is a $w \in H$ such that $A_{n} f \rightarrow w$ as $n \rightarrow \infty$.

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Logic and Set Theory

• 4.II.16G

  Logic and Set Theory | Part II, 2008

  Prove Hartog's Lemma that for any set $x$ there is an ordinal $\alpha$ which cannot be mapped injectively into $x$.

  Now assume the Axiom of Choice. Prove Zorn's Lemma and the Well-ordering Principle.

  [If you appeal to a fixed point theorem then you should state it clearly.]

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Mathematical Biology

• 4.I.6B

  Mathematical Biology | Part II, 2008

  A semi-infinite elastic filament lies along the positive $x$-axis in a viscous fluid. When it is perturbed slightly to the shape $y=h(x, t)$, it evolves according to

  $$\zeta h_{t}=-A h_{x x x x}$$

  where $\zeta$ characterises the viscous drag and $A$ the bending stiffness. Motion is forced by boundary conditions

  $$h=h_{0} \cos (\omega t) \quad \text { and } \quad h_{x x}=0 \quad \text { at } \quad x=0, \quad \text { while } \quad h \rightarrow 0 \quad \text { as } \quad x \rightarrow \infty$$

  Use dimensional analysis to find the characteristic length $\ell(\omega)$ of the disturbance. Show that the steady oscillating solution takes the form

  $$h(x, t)=h_{0} \operatorname{Re}\left[e^{i \omega t} F(\eta)\right] \quad \text { with } \quad \eta=x / \ell,$$

  finding the ordinary differential equation for $F$.

  Find two solutions for $F$ which decay as $x \rightarrow \infty$. Without solving explicitly for the amplitudes, show that $h(x, t)$ is the superposition of two travelling waves which decay with increasing $x$, one with crests moving to the left and one to the right. Which dominates?

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Number Fields

• 4.II.20G

  Number Fields | Part II, 2008

  (a) Explain what is meant by an integral basis of an algebraic number field. Specify such a basis for the quadratic field $k=\mathbb{Q}(\sqrt{2})$.

  (b) Let $K=\mathbb{Q}(\alpha)$ with $\alpha=\sqrt[4]{2}$, a fourth root of 2 . Write an element $\theta$ of $K$ as

  $$\theta=a+b \alpha+c \alpha^{2}+d \alpha^{3}$$

  with $a, b, c, d \in \mathbb{Q}$. By computing the relative traces $T_{K / k}(\theta)$ and $T_{K / k}(\alpha \theta)$, show that if $\theta$ is an algebraic integer of $K$, then $2 a, 2 b, 2 c$ and $4 d$ are rational integers. By further computing the relative norm $N_{K / k}(\theta)$, show that

  $$a^{2}+2 c^{2}-4 b d \text { and } 2 a c-b^{2}-2 d^{2}$$

  are rational integers. Deduce that $1, \alpha, \alpha^{2}, \alpha^{3}$ is an integral basis of $K$.

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Number Theory

• 4.I.1H

  Number Theory | Part II, 2008

  Let $p$ be an odd prime number. Assuming that the multiplicative group of $\mathbb{Z} / p \mathbb{Z}$ is cyclic, prove that the multiplicative group of units of $\mathbb{Z} / p^{n} \mathbb{Z}$ is cyclic for all $n \geqslant 1$.

  Find an integer $a$ such that its residue class in $\mathbb{Z} / 11^{n} \mathbb{Z}$ generates the multiplicative group of units for all $n \geqslant 1$.

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• 4.II.11H

  Number Theory | Part II, 2008

  Let $N>1$ be an integer, which is not a square, and let $p_{k} / q_{k}(k=1,2, \ldots)$ be the convergents to $\sqrt{N}$. Prove that

  $$\left|p_{k}^{2}-q_{k}^{2} N\right|<2 \sqrt{N} \quad(k=1,2, \ldots)$$

  Explain briefly how this result can be used to generate a factor base $B$, and a set of $B$-numbers which may lead to a factorization of $N$.

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Numerical Analysis

• 4.II.39C

  Numerical Analysis | Part II, 2008

  Let $A \in \mathbb{R}^{n \times n}$ be a real matrix with $n$ linearly independent eigenvectors. When calculating eigenvalues of $A$, the sequence $\mathbf{x}^{(k)}, k=0,1,2, \ldots$, is generated by power method $\mathbf{x}^{(k+1)}=A \mathbf{x}^{(k)} /\left\|A \mathbf{x}^{(k)}\right\|$, where $\mathbf{x}^{(0)}$ is a real nonzero vector.

  (a) Describe the asymptotic properties of the sequence $\mathbf{x}^{(k)}$, both in the case where the eigenvalues $\lambda_{i}$ of $A$ satisfy $\left|\lambda_{i}\right|<\left|\lambda_{n}\right|, i=1, \ldots, n-1$, and in the case where $\left|\lambda_{i}\right|<\left|\lambda_{n-1}\right|=\left|\lambda_{n}\right|, i=1, \ldots, n-2$. In the latter case explain how the (possibly complexvalued) eigenvalues $\lambda_{n-1}, \lambda_{n}$ and their corresponding eigenvectors can be determined.

  (b) Let $n=3$, and suppose that, for a large $k$, we obtain the vectors

  $$\mathbf{y}_{k}=\mathbf{x}_{k}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right], \quad \mathbf{y}_{k+1}=A \mathbf{x}_{k}=\left[\begin{array}{l} 2 \\ 3 \\ 4 \end{array}\right], \quad \mathbf{y}_{k+2}=A^{2} \mathbf{x}_{k}=\left[\begin{array}{l} 2 \\ 4 \\ 6 \end{array}\right]$$

  Find two eigenvalues of $A$ and their corresponding eigenvectors.

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Optimization and Control

• 4.II.29I

  Optimization and Control | Part II, 2008

  State Pontryagin's maximum principle for the controllable dynamical system with state-space $\mathbb{R}^{+}$, given by

  $$\dot{x}_{t}=b\left(t, x_{t}, u_{t}\right), \quad t \geqslant 0,$$

  where the running costs are given by $c\left(t, x_{t}, u_{t}\right)$, up to an unconstrained terminal time $\tau$ when the state first reaches 0 , and there is a terminal cost $C(\tau)$.

  A company pays a variable price $p(t)$ per unit time for electrical power, agreed in advance, which depends on the time of day. The company takes on a job at time $t=0$, which requires a total amount $E$ of electrical energy, but can be processed at a variable level of power consumption $u(t) \in[0,1]$. If the job is completed by time $\tau$, then the company will receive a reward $R(\tau)$. Thus, it is desired to minimize

  $$\int_{0}^{\tau} u(t) p(t) d t-R(\tau)$$

  subject to

  $$\int_{0}^{\tau} u(t) d t=E, \quad u(t) \in[0,1],$$

  with $\tau>0$ unconstrained. Take as state variable the energy $x_{t}$ still needed at time $t$ to complete the job. Use Pontryagin's maximum principle to show that the optimal control is to process the job on full power or not at all, according as the price $p(t)$ lies below or above a certain threshold value $p^{*}$.

  Show further that, if $\tau^{*}$ is the completion time for the optimal control, then

  $$p^{*}+\dot{R}\left(\tau^{*}\right)=p\left(\tau^{*}\right)$$

  Consider a case in which $p$ is periodic, with period one day, where day 1 corresponds to the time interval $[0,2]$, and $p(t)=(t-1)^{2}$ during day 1 . Suppose also that $R(\tau)=1 /(1+\tau)$ and $E=1 / 2$. Determine the total energy cost and the reward associated with the threshold $p^{*}=1 / 4$.

  Hence, show that any threshold low enough to carry processing over into day 2 is suboptimal.

  Show carefully that the optimal price threshold is given by $p^{*}=1 / 4$.

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Partial Differential Equations

• 4.II.30C

  Partial Differential Equations | Part II, 2008

  (i) Define the Fourier transform $\hat{f}=\mathcal{F}(f)$ of a Schwartz function $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$, and also of a tempered distribution $u \in \mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$.

  (ii) From your definition, compute the Fourier transform of the distribution $W_{t} \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$ given by

  $$W_{t}(\psi)=<W_{t}, \psi>=\frac{1}{4 \pi t} \int_{\|y\|=t} \psi(y) d \Sigma(y)$$

  for every Schwartz function $\psi \in \mathcal{S}\left(\mathbb{R}^{3}\right)$. Here $d \Sigma(y)=t^{2} d \Omega(y)$ is the integration element on the sphere of radius $t$.

  Hence deduce the formula of Kirchoff for the solution of the initial value problem for the wave equation in three space dimensions,

  $$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0$$

  with initial data $u(0, x)=0$ and $\frac{\partial u}{\partial t}(0, x)=g(x), x \in \mathbb{R}^{3}$, where $g \in \mathcal{S}\left(\mathbb{R}^{3}\right)$. Explain briefly why the formula is also valid for arbitrary smooth $g \in C^{\infty}\left(\mathbb{R}^{3}\right)$.

  (iii) Show that any $C^{2}$ solution of the initial value problem in (ii) is given by the formula derived in (ii) (uniqueness).

  (iv) Show that any two $C^{2}$ solutions of the initial value problem for

  $$\frac{\partial^{2} u}{\partial t^{2}}+\frac{\partial u}{\partial t}-\Delta u=0$$

  with the same initial data as in (ii), also agree for any $t>0$.

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Principles of Quantum Mechanics

• 4.II.32D

  Principles of Quantum Mechanics | Part II, 2008

  Define the interaction picture for a quantum mechanical system with Schrödinger picture Hamiltonian $H_{0}+V(t)$ and explain why either picture gives the same physical predictions. Derive an equation of motion for interaction picture states and use this to show that the probability of a transition from a state $|n\rangle$ at time zero to a state $|m\rangle$ at time $t$ is

  $$P(t)=\frac{1}{\hbar^{2}}\left|\int_{0}^{t} e^{i\left(E_{m}-E_{n}\right) t^{\prime} / \hbar}\left\langle m\left|V\left(t^{\prime}\right)\right| n\right\rangle d t^{\prime}\right|^{2}$$

  correct to second order in $V$, where the initial and final states are orthogonal eigenstates of $H_{0}$ with eigenvalues $E_{n}$ and $E_{m}$.

  Consider a perturbed harmonic oscillator:

  $$H_{0}=\hbar \omega\left(a^{\dagger} a+\frac{1}{2}\right), \quad V(t)=\hbar \lambda\left(a e^{i \nu t}+a^{\dagger} e^{-i \nu t}\right)$$

  with $a$ and $a^{\dagger}$ annihilation and creation operators (all usual properties may be assumed). Working to order $\lambda^{2}$, find the probability for a transition from an initial state with $E_{n}=\hbar \omega\left(n+\frac{1}{2}\right)$ to a final state with $E_{m}=\hbar \omega\left(m+\frac{1}{2}\right)$ after time $t$.

  Suppose $t$ becomes large and perturbation theory still applies. Explain why the rate $P(t) / t$ for each allowed transition is sharply peaked, as a function of $\nu$, around $\nu=\omega$.

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Principles of Statistics

• 4.II.27I

  Principles of Statistics | Part II, 2008

  Define sufficient statistic, and state the factorisation criterion for determining whether a statistic is sufficient. Show that a Bayesian posterior distribution depends on the data only through the value of a sufficient statistic.

  Given the value $\mu$ of an unknown parameter $M$, observables $X_{1}, \ldots, X_{n}$ are independent and identically distributed with distribution $\mathcal{N}(\mu, 1)$. Show that the statistic $\bar{X}:=n^{-1} \sum_{i=1}^{n} X_{i}$ is sufficient for $\mathrm{M}$.

  If the prior distribution is $\mathrm{M} \sim \mathcal{N}\left(0, \tau^{2}\right)$, determine the posterior distribution of $\mathrm{M}$ and the predictive distribution of $\bar{X}$.

  In fact, there are two hypotheses as to the value of M. Under hypothesis $H_{0}$, $\mathrm{M}$ takes the known value 0 ; under $H_{1}, \mathrm{M}$ is unknown, with prior distribution $\mathcal{N}\left(0, \tau^{2}\right)$. Explain why the Bayes factor for choosing between $H_{0}$ and $H_{1}$ depends only on $\bar{X}$, and determine its value for data $X_{1}=x_{1}, \ldots, X_{n}=x_{n}$.

  The frequentist $5 \%$-level test of $H_{0}$ against $H_{1}$ rejects $H_{0}$ when $|\bar{X}| \geqslant 1.96 / \sqrt{n}$. What is the Bayes factor for the critical case $|\bar{x}|=1.96 / \sqrt{n}$ ? How does this behave as $n \rightarrow \infty$ ? Comment on the similarities or differences in behaviour between the frequentist and Bayesian tests.

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Probability and Measure

• 4.II $. 25 \mathrm{~J}$

  Probability and Measure | Part II, 2008

  (i) A stepfunction is any function $s$ on $\mathbb{R}$ which can be written in the form

  $$s(x)=\sum_{k=1}^{n} c_{k} I_{\left(a_{k}, b_{k}\right]}(x), \quad x \in \mathbb{R},$$

  where $a_{k}, b_{k}, c_{k}$ are real numbers, with $a_{k}<b_{k}$ for all $k$. Show that the set of all stepfunctions is dense in $L^{1}(\mathbb{R}, \mathcal{B}, \mu)$. Here, $\mathcal{B}$ denotes the Borel $\sigma$-algebra, and $\mu$ denotes Lebesgue measure.

  [You may use without proof the fact that, for any Borel set $B$ of finite measure, and any $\varepsilon>0$, there exists a finite union of intervals $A$ such that $\mu(A \triangle B)<\varepsilon$.]

  (ii) Show that the Fourier transform

  $$\hat{s}(t)=\int_{\mathbb{R}} s(x) e^{i t x} d x$$

  of a stepfunction has the property that $\hat{s}(t) \rightarrow 0$ as $|t| \rightarrow \infty$.

  (iii) Deduce that the Fourier transform of any integrable function has the same property.

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Representation Theory

• 4.II.19G

  Representation Theory | Part II, 2008

  (a) Let $A$ be a normal subgroup of a finite group $G$, and let $V$ be an irreducible representation of $G$. Show that either $V$ restricted to $A$ is isotypic (a sum of copies of one irreducible representation of $A)$, or else $V$ is induced from an irreducible representation of some proper subgroup of $G$.

  (b) Using (a), show that every (complex) irreducible representation of a $p$-group is induced from a 1-dimensional representation of some subgroup.

  [You may assume that a nonabelian $p$-group $G$ has an abelian normal subgroup $A$ which is not contained in the centre of $G$.]

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Riemann Surfaces

• 4.II.23H

  Riemann Surfaces | Part II, 2008

  Let $\Lambda$ be a lattice in $\mathbb{C}$ generated by 1 and $\tau$, where $\operatorname{Im} \tau>0$. The Weierstrass function $\wp$ is the unique meromorphic $\Lambda$-periodic function on $\mathbb{C}$, such that the only poles of $\wp$ are at points of $\Lambda$ and $\wp(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$.

  Show that $\wp$ is an even function. Find all the zeroes of $\wp^{\prime}$.

  Suppose that $a$ is a complex number such that $2 a \notin \Lambda$. Show that the function

  $$h(z)=(\wp(z-a)-\wp(z+a))(\wp(z)-\wp(a))^{2}-\wp^{\prime}(z) \wp^{\prime}(a)$$

  has no poles in $\mathbb{C} \backslash \Lambda$. By considering the Laurent expansion of $h$ at $z=0$, or otherwise, deduce that $h$ is constant.

  [General properties of meromorphic doubly-periodic functions may be used without proof if accurately stated.]

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Statistical Modelling

• 4.I $. 5 \mathrm{~J} \quad$

  Statistical Modelling | Part II, 2008

  A long-term agricultural experiment had $n=90$ grassland plots, each $25 \mathrm{~m} \times 25 \mathrm{~m}$, differing in biomass, soil pH, and species richness (the count of species in the whole plot). While it was well-known that species richness declines with increasing biomass, it was not known how this relationship depends on soil pH. In the experiment, there were 30 plots of "low pH", 30 of "medium pH" and 30 of "high pH". Three lines of the data are reproduced here as an aid.

  Briefly explain the commands below. That is, explain the models being fitted.

  Let $H_{1}, H_{2}$ and $H_{3}$ denote the hypotheses represented by the three models and fits. Based on the output of the code below, what hypotheses are being tested, and which of the models seems to give the best fit to the data? Why?

  

  Finally, what is the value obtained by the following command?

  

  $$\begin{aligned} & >\operatorname{grass}[\mathrm{c}(1,31,61),] \\ & \text { pH Biomass Species } \\ & 1 \text { high } 0.4692972 \quad 30 \\ & 31 \text { mid } 0.1757627 \quad 29 \\ & 61 \text { low } 0.1008479 \quad 18 \\ & >\text { fit1 <- glm(Species Biomass, family = poisson) } \\ & >\text { fit2 <- glm(Species } ~ \mathrm{pH}+\text { Biomass, family }=\text { poisson) } \\ & >\text { fit3 <- glm(Species } \mathrm{pH} * \text { Biomass, family }=\text { poisson) } \end{aligned}$$

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• 4.II.13J

  Statistical Modelling | Part II, 2008

  Consider the following generalized linear model for responses $y_{1}, \ldots, y_{n}$ as a function of explanatory variables $x_{1}, \ldots, x_{n}$, where $x_{i}=\left(x_{i 1}, \ldots, x_{i p}\right)^{\top}$ for $i=1, \ldots, n$. The responses are modelled as observed values of independent random variables $Y_{1}, \ldots, Y_{n}$, with

  $$Y_{i} \sim \operatorname{ED}\left(\mu_{i}, \sigma_{i}^{2}\right), \quad g\left(\mu_{i}\right)=x_{i}^{\top} \beta, \quad \sigma_{i}^{2}=\sigma^{2} a_{i},$$

  Here, $g$ is a given link function, $\beta$ and $\sigma^{2}$ are unknown parameters, and the $a_{i}$ are treated as known.

  [Hint: recall that we write $Y \sim E D\left(\mu, \sigma^{2}\right)$ to mean that $Y$ has density function of the form

  $$f\left(y ; \mu, \sigma^{2}\right)=a\left(\sigma^{2}, y\right) \exp \left\{\frac{1}{\sigma^{2}}[\theta(\mu) y-K(\theta(\mu))]\right\}$$

  for given functions a and $\theta .]$

  [ You may use without proof the facts that, for such a random variable $Y$,

  $$\left.E(Y)=K^{\prime}(\theta(\mu)), \quad \operatorname{var}(Y)=\sigma^{2} K^{\prime \prime}(\theta(\mu)) \equiv \sigma^{2} V(\mu) .\right]$$

  Show that the score vector and Fisher information matrix have entries:

  $$U_{j}(\beta)=\sum_{i=1}^{n} \frac{\left(y_{i}-\mu_{i}\right) x_{i j}}{\sigma_{i}^{2} V\left(\mu_{i}\right) g^{\prime}\left(\mu_{i}\right)}, \quad j=1, \ldots, p$$

  and

  $$i_{j k}(\beta)=\sum_{i=1}^{n} \frac{x_{i j} x_{i k}}{\sigma_{i}^{2} V\left(\mu_{i}\right)\left(g^{\prime}\left(\mu_{i}\right)\right)^{2}}, \quad j, k=1, \ldots, p$$

  How do these expressions simplify when the canonical link is used?

  Explain briefly how these two expressions can be used to obtain the maximum likelihood estimate $\hat{\beta}$ for $\beta$.

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Statistical Physics

• 4.II.34D

  Statistical Physics | Part II, 2008

  Show that the Fermi momentum $p_{F}$ of a gas of $N$ non-interacting electrons in volume $V$ is

  $$p_{F}=\left(3 \pi^{2} \hbar^{3} \frac{N}{V}\right)^{1 / 3}$$

  Consider the electrons to be effectively massless, so that an electron of momentum $p$ has (relativistic) energy $c p$. Show that the mean energy per electron at zero temperature is $3 c p_{F} / 4$.

  When a constant external magnetic field of strength $B$ is applied to the electron gas, each electron gets an energy contribution $\pm \mu B$ depending on whether its spin is parallel or antiparallel to the field. Here $\mu$ is the magnitude of the magnetic moment of an electron. Calculate the total magnetic moment of the electron gas at zero temperature, assuming $\mu B$ is much less than $c p_{F}$.

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Stochastic Financial Models

• 4.II.28J

  Stochastic Financial Models | Part II, 2008

  (a) Consider a family $\left(X_{n}: n \geqslant 0\right)$ of independent, identically distributed, positive random variables and fix $z_{0}>0$. Define inductively

  $$z_{n+1}=z_{n} X_{n}, \quad n \geqslant 0$$

  Compute, for $n \in\{1, \ldots, N\}$, the conditional expectation $\mathbb{E}\left(z_{N} \mid z_{n}\right)$.

  (b) Fix $R \in[0,1)$. In the setting of part (a), compute also $\mathbb{E}\left(U\left(z_{N}\right) \mid z_{n}\right)$, where

  $$U(x)=x^{1-R} /(1-R), \quad x \geqslant 0 .$$

  (c) Let $U$ be as in part (b). An investor with wealth $w_{0}>0$ at time 0 wishes to invest it in such a way as to maximise $\mathbb{E}\left(U\left(w_{N}\right)\right)$ where $w_{N}$ is the wealth at the start of day $N$. Let $\alpha \in[0,1]$ be fixed. On day $n$, he chooses the proportion $\theta \in[\alpha, 1]$ of his wealth to invest in a single risky asset, so that his wealth at the start of day $n+1$ will be

  $$w_{n+1}=w_{n}\left\{\theta X_{n}+(1-\theta)(1+r)\right\}$$

  Here, $\left(X_{n}: n \geqslant 0\right)$ is as in part (a) and $r$ is the per-period riskless rate of interest. If $V_{n}(w)=\sup \mathbb{E}\left(U\left(w_{N}\right) \mid w_{n}=w\right)$ denotes the value function of this optimization problem, show that $V_{n}\left(w_{n}\right)=a_{n} U\left(w_{n}\right)$ and give a formula for $a_{n}$. Verify that, in the case $\alpha=1$, your answer is in agreement with part (b).

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Topics in Analysis

• 4.I.2F

  Topics in Analysis | Part II, 2008

  (a) State Runge's theorem on uniform approximation of analytic functions by polynomials.

  (b) Suppose $f$ is analytic on

  $$\Omega=\{z \in \mathbf{C}:|z|<1\} \backslash\{z \in \mathbf{C}: \operatorname{Im}(z)=0, \operatorname{Re}(z) \leqslant 0\} .$$

  Prove that there exists a sequence of polynomials which converges to $f$ uniformly on compact subsets of $\Omega$.

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Waves

• 4.II $. 38$ B

  Waves | Part II, 2008

  A layer of rock of shear modulus $\bar{\mu}$ and shear wave speed $\bar{c}_{s}$ occupies the region $0 \leqslant y \leqslant h$ with a free surface at $y=h$. A second rock having shear modulus $\mu$ and shear wave speed $c_{s}>\bar{c}_{s}$ occupies $y \leqslant 0$. Show that elastic $S H$ waves of wavenumber $k$ and phase speed $c$ can propagate in the layer with zero disturbance at $y=-\infty$ if $\bar{c}_{s}<c<c_{s}$ and $c$ satisfies the dispersion relation

  $$\tan \left[k h \sqrt{c^{2} / \bar{c}_{s}^{2}-1}\right]=\frac{\mu}{\bar{\mu}} \frac{\sqrt{1-c^{2} / c_{s}^{2}}}{\sqrt{c^{2} / \bar{c}_{s}^{2}-1}} .$$

  Show graphically, or otherwise, that this equation has at least one real solution for any value of $k h$, and determine the smallest value of $k h$ for which the equation has at least two real solutions.

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