# Part II, 2017, Paper 4

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Paper 4, Section II, I

comment(a) Let $X$ and $Y$ be non-singular projective curves over a field $k$ and let $\varphi: X \rightarrow Y$ be a non-constant morphism. Define the ramification degree $e_{P}$ of $\varphi$ at a point $P \in X$.

(b) Suppose char $k \neq 2$. Let $X=Z(f)$ be the plane cubic with $f=x_{0} x_{2}^{2}-x_{1}^{3}+x_{0}^{2} x_{1}$, and let $Y=\mathbb{P}^{1}$. Explain how the projection

$\left(x_{0}: x_{1}: x_{2}\right) \mapsto\left(x_{0}: x_{1}\right)$

defines a morphism $\varphi: X \rightarrow Y$. Determine the degree of $\varphi$ and the ramification degrees $e_{P}$ for all $P \in X$.

(c) Let $X$ be a non-singular projective curve and let $P \in X$. Show that there is a non-constant rational function on $X$ which is regular on $X \backslash\{P\}$.

Paper 4, Section II, I

commentRecall that $\mathbb{R} P^{n}$ is real projective $n$-space, the quotient of $S^{n}$ obtained by identifying antipodal points. Consider the standard embedding of $S^{n}$ as the unit sphere in $\mathbb{R}^{n+1}$.

(a) For $n$ odd, show that there exists a continuous map $f: S^{n} \rightarrow S^{n}$ such that $f(x)$ is orthogonal to $x$, for all $x \in S^{n}$.

(b) Exhibit a triangulation of $\mathbb{R} P^{n}$.

(c) Describe the map $H_{n}\left(S^{n}\right) \rightarrow H_{n}\left(S^{n}\right)$ induced by the antipodal map, justifying your answer.

(d) Show that, for $n$ even, there is no continuous map $f: S^{n} \rightarrow S^{n}$ such that $f(x)$ is orthogonal to $x$ for all $x \in S^{n}$.

Paper 4, Section II, $22 F$

commentConsider $\mathbb{R}^{n}$ with the Lebesgue measure. Denote by $\mathcal{F} f(\xi)=\int_{\mathbb{R}^{n}} e^{-2 i \pi x \cdot \xi} f(x) d x$ the Fourier transform of $f \in L^{1}\left(\mathbb{R}^{n}\right)$ and by $\hat{f}$ the Fourier-Plancherel transform of $f \in L^{2}\left(\mathbb{R}^{n}\right)$. Let $\chi_{R}(\xi):=\left(1-\frac{|\xi|}{R}\right) \chi_{|\xi| \leqslant R}$ for $R>0$ and define for $s \in \mathbb{R}_{+}$

$H^{s}\left(\mathbb{R}^{n}\right):=\left\{f \in L^{2}\left(\mathbb{R}^{n}\right) \mid\left(1+|\cdot|^{2}\right)^{s / 2} \hat{f}(\cdot) \in L^{2}\left(\mathbb{R}^{n}\right)\right\}$

(i) Prove that $H^{s}\left(\mathbb{R}^{n}\right)$ is a vector subspace of $L^{2}\left(\mathbb{R}^{n}\right)$, and is a Hilbert space for the inner product $\langle f, g\rangle:=\int_{\mathbb{R}^{n}}\left(1+|\xi|^{2}\right)^{s} \hat{f}(\xi) \overline{\hat{g}(\xi)} d \xi$, where $\bar{z}$ denotes the complex conjugate of $z \in \mathbb{C}$.

(ii) Construct a function $f \in H^{s}(\mathbb{R}), s \in(0,1 / 2)$, that is not almost everywhere equal to a continuous function.

(iii) For $f \in L^{1}\left(\mathbb{R}^{n}\right)$, prove that $F_{R}: x \mapsto \int_{\mathbb{R}^{n}} \mathcal{F} f(\xi) \chi_{R}(\xi) e^{2 i \pi x \cdot \xi} d \xi$ is a well-defined function and that $F_{R} \in L^{1}\left(\mathbb{R}^{n}\right)$ converges to $f$ in $L^{1}\left(\mathbb{R}^{n}\right)$ as $R \rightarrow+\infty$.

[Hint: Prove that $F_{R}=K_{R} * f$ where $K_{R}$ is an approximation of the unit as $R \rightarrow+\infty .]$

(iv) Deduce that if $f \in L^{1}\left(\mathbb{R}^{n}\right)$ and $\left(1+|\cdot|^{2}\right)^{s / 2} \mathcal{F} f(\cdot) \in L^{2}\left(\mathbb{R}^{n}\right)$ then $f \in H^{s}\left(\mathbb{R}^{n}\right)$.

[Hint: Prove that: (1) there is a sequence $R_{k} \rightarrow+\infty$ such that $K_{R_{k}} * f$ converges to $f$ almost everywhere; (2) $K_{R} * f$ is uniformly bounded in $L^{2}\left(\mathbb{R}^{n}\right)$ as $R \rightarrow+\infty$.]

Paper 4, Section II, C

comment(a) In one dimension, a particle of mass $m$ is scattered by a potential $V(x)$ where $V(x) \rightarrow 0$ as $|x| \rightarrow \infty$. For wavenumber $k>0$, the incoming $(\mathcal{I})$ and outgoing $(\mathcal{O})$ asymptotic plane wave states with positive $(+)$ and negative $(-)$ parity are given by

$\begin{array}{rr} \mathcal{I}_{+}(x)=e^{-i k|x|}, & \mathcal{I}_{-}(x)=\operatorname{sign}(x) e^{-i k|x|} \\ \mathcal{O}_{+}(x)=e^{+i k|x|}, & \mathcal{O}_{-}(x)=-\operatorname{sign}(x) e^{+i k|x|} \end{array}$

(i) Explain how this basis may be used to define the $S$-matrix,

$\mathcal{S}^{P}=\left(\begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array}\right)$

(ii) For what choice of potential would you expect $S_{+-}=S_{-+}=0$ ? Why?

(b) The potential $V(x)$ is given by

$V(x)=V_{0}[\delta(x-a)+\delta(x+a)]$

with $V_{0}$ a constant.

(i) Show that

$S_{--}(k)=e^{-2 i k a}\left[\frac{\left(2 k-i U_{0}\right) e^{i k a}+i U_{0} e^{-i k a}}{\left(2 k+i U_{0}\right) e^{-i k a}-i U_{0} e^{i k a}}\right]$

where $U_{0}=2 m V_{0} / \hbar^{2}$. Verify that $\left|S_{--}\right|^{2}=1$. Explain the physical meaning of this result.

(ii) For $V_{0}<0$, by considering the poles or zeros of $S_{--}(k)$, show that there exists one bound state of negative parity if $a U_{0}<-1$.

(iii) For $V_{0}>0$ and $a U_{0} \gg 1$, show that $S_{--}(k)$ has a pole at

$k a=\pi+\alpha-i \gamma$

where $\alpha$ and $\gamma$ are real and

$\alpha=-\frac{\pi}{a U_{0}}+O\left(\frac{1}{\left(a U_{0}\right)^{2}}\right) \quad \text { and } \quad \gamma=\left(\frac{\pi}{a U_{0}}\right)^{2}+O\left(\frac{1}{\left(a U_{0}\right)^{3}}\right)$

Explain the physical significance of this result.

Paper 4, Section II, $26 K$

comment(a) Give the definition of an $M / M / 1$ queue. Prove that if $\lambda$ is the arrival rate and $\mu$ the service rate and $\lambda<\mu$, then the length of the queue is a positive recurrent Markov chain. What is the equilibrium distribution?

If the queue is in equilibrium and a customer arrives at some time $t$, what is the distribution of the waiting time (time spent waiting in the queue plus service time)?

(b) We now modify the above queue: on completion of service a customer leaves with probability $\delta$, or goes to the back of the queue with probability $1-\delta$. Find the distribution of the total time a customer spends being served.

Hence show that equilibrium is possible if $\lambda<\delta \mu$ and find the stationary distribution.

Show that, in equilibrium, the departure process is Poisson.

[You may use relevant theorems provided you state them clearly.]

Paper 4, Section II, E

commentConsider solutions to the equation

$\frac{d^{2} y}{d x^{2}}=\left(\frac{1}{4}+\frac{\mu^{2}-\frac{1}{4}}{x^{2}}\right) y$

of the form

$y(x)=\exp \left[S_{0}(x)+S_{1}(x)+S_{2}(x)+\ldots\right]$

with the assumption that, for large positive $x$, the function $S_{j}(x)$ is small compared to $S_{j-1}(x)$ for all $j=1,2 \ldots$

Obtain equations for the $S_{j}(x), j=0,1,2 \ldots$, which are formally equivalent to ( $)$. Solve explicitly for $S_{0}$ and $S_{1}$. Show that it is consistent to assume that $S_{j}(x)=c_{j} x^{-(j-1)}$ for some constants $c_{j}$. Give a recursion relation for the $c_{j}$.

Deduce that there exist two linearly independent solutions to $(\star)$ with asymptotic expansions as $x \rightarrow+\infty$ of the form

$y_{\pm}(x) \sim e^{\pm x / 2}\left(1+\sum_{j=1}^{\infty} A_{j}^{\pm} x^{-j}\right)$

Determine a recursion relation for the $A_{j}^{\pm}$. Compute $A_{1}^{\pm}$and $A_{2}^{\pm}$.

Paper 4, Section I, $4 \mathrm{H}$

comment(a) Describe the process for converting a deterministic finite-state automaton $D$ into a regular expression $R$ defining the same language, $\mathcal{L}(D)=\mathcal{L}(R)$. [You need only outline the steps, without proof, but you should clearly define all terminology you introduce.]

(b) Consider the language $L$ over the alphabet $\{0,1\}$ defined via

$L:=\left\{w 01^{n} \mid w \in\{0,1\}^{*}, n \in \mathbb{K}\right\} \cup\{1\}^{*} .$

Show that $L$ satisfies the pumping lemma for regular languages but is not a regular language itself.

Paper 4, Section I, E

commentConsider the Poisson bracket structure on $\mathbb{R}^{3}$ given by

$\{x, y\}=z, \quad\{y, z\}=x, \quad\{z, x\}=y$

and show that $\left\{f, \rho^{2}\right\}=0$, where $\rho^{2}=x^{2}+y^{2}+z^{2}$ and $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ is any polynomial function on $\mathbb{R}^{3}$.

Let $H=\left(A x^{2}+B y^{2}+C z^{2}\right) / 2$, where $A, B, C$ are positive constants. Find the explicit form of Hamilton's equations

$\dot{\mathbf{r}}=\{\mathbf{r}, H\}, \quad \text { where } \quad \mathbf{r}=(x, y, z)$

Find a condition on $A, B, C$ such that the oscillation described by

$x=1+\alpha(t), \quad y=\beta(t), \quad z=\gamma(t)$

is linearly unstable, where $\alpha(t), \beta(t), \gamma(t)$ are small.

Paper 4, Section II, $14 \mathrm{E}$

commentExplain how geodesics of a Riemannian metric

$g=g_{a b}\left(x^{c}\right) d x^{a} d x^{b}$

arise from the kinetic Lagrangian

$\mathcal{L}=\frac{1}{2} g_{a b}\left(x^{c}\right) \dot{x}^{a} \dot{x}^{b}$

where $a, b=1, \ldots, n$.

Find geodesics of the metric on the upper half plane

$\Sigma=\left\{(x, y) \in \mathbb{R}^{2}, y>0\right\}$

with the metric

$g=\frac{d x^{2}+d y^{2}}{y^{2}}$

and sketch the geodesic containing the points $(2,3)$ and $(10,3)$.

[Hint: Consider $d y / d x .]$

Paper 4, Section I, G

commentDescribe the RSA system with public key $(N, e)$ and private key $d$.

Give a simple example of how the system is vulnerable to a homomorphism attack.

Describe the El-Gamal signature scheme and explain how this can defeat a homomorphism attack.

Paper 4, Section I, C

comment(a) By considering a spherically symmetric star in hydrostatic equilibrium derive the pressure support equation

$\frac{d P}{d r}=-\frac{G M(r) \rho}{r^{2}},$

where $r$ is the radial distance from the centre of the star, $M(r)$ is the stellar mass contained inside that radius, and $P(r)$ and $\rho(r)$ are the pressure and density at radius $r$ respectively.

(b) Propose, and briefly justify, boundary conditions for this differential equation, both at the centre of the star $r=0$, and at the stellar surface $r=R$.

Suppose that $P=K \rho^{2}$ for some $K>0$. Show that the density satisfies the linear differential equation

$\frac{1}{x^{2}} \frac{\partial}{\partial x}\left(x^{2} \frac{\partial \rho}{\partial x}\right)=-\rho$

where $x=\alpha r$, for some constant $\alpha$, is a rescaled radial coordinate. Find $\alpha$.

Paper 4, Section II, I

commentLet $S \subset \mathbb{R}^{3}$ be a surface and $p \in S$. Define the exponential map exp $p$ and compute its differential $\left.d \exp _{p}\right|_{0}$. Deduce that $\exp _{p}$ is a local diffeomorphism.

Give an example of a surface $S$ and a point $p \in S$ for which the exponential map $\exp _{p}$ fails to be defined globally on $T_{p} S$. Can this failure be remedied by extending the surface? In other words, for any such $S$, is there always a surface $S \subset \widehat{S} \subset \mathbb{R}^{3}$ such that the exponential map $\widehat{\exp }_{p}$ defined with respect to $\widehat{S}$is globally defined on $T_{p} S=T_{p} \widehat{S}$?

State the version of the Gauss-Bonnet theorem with boundary term for a surface $S \subset \mathbb{R}^{3}$ and a closed disc $D \subset S$ whose boundary $\partial D$ can be parametrized as a smooth closed curve in $S$.

Let $S \subset \mathbb{R}^{3}$ be a flat surface, i.e. $K=0$. Can there exist a closed disc $D \subset S$, whose boundary $\partial D$ can be parametrized as a smooth closed curve, and a surface $\tilde{S} \subset \mathbb{R}^{3}$ such that all of the following hold:

(i) $(S \backslash D) \cup \partial D \subset \tilde{S}$;

(ii) letting $\tilde{D}$ be $(\tilde{S} \backslash(S \backslash D)) \cup \partial D$, we have that $\tilde{D}$ is a closed disc in $\tilde{S}$ with boundary $\partial \tilde{D}=\partial D$

(iii) the Gaussian curvature $\tilde{K}$ of $\tilde{S}$ satisfies $\tilde{K} \geqslant 0$, and there exists a $p \in \tilde{S}$ such that $\tilde{K}(p)>0$ ?

Justify your answer.

Paper 4, Section II, A

commentConsider the one-dimensional map $F: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$x_{i+1}=F\left(x_{i} ; \mu\right)=x_{i}\left(a x_{i}^{2}+b x_{i}+\mu\right),$

where $a$ and $b$ are constants, $\mu$ is a parameter and $a \neq 0$.

(a) Find the fixed points of $F$ and determine the linear stability of $x=0$. Hence show that there are bifurcations at $\mu=1$, at $\mu=-1$ and, if $b \neq 0$, at $\mu=1+b^{2} /(4 a)$.

Sketch the bifurcation diagram for each of the cases:

$\text { (i) } a>b=0, \quad \text { (ii) } a, b>0 \text { and (iii) } a, b<0 \text {. }$

In each case show the locus and stability of the fixed points in the $(\mu, x)$-plane, and state the type of each bifurcation. [Assume that there are no further bifurcations in the region sketched.]

(b) For the case $F(x)=x\left(\mu-x^{2}\right)$ (i.e. $\left.a=-1, b=0\right)$, you may assume that

$F^{2}(x)=x+x\left(\mu-1-x^{2}\right)\left(\mu+1-x^{2}\right)\left(1-\mu x^{2}+x^{4}\right) .$

Show that there are at most three 2-cycles and determine when they exist. By considering $F^{\prime}\left(x_{i}\right) F^{\prime}\left(x_{i+1}\right)$, or otherwise, show further that one 2-cycle is always unstable when it exists and that the others are unstable when $\mu>\sqrt{5}$. Sketch the bifurcation diagram showing the locus and stability of the fixed points and 2-cycles. State briefly what you would expect to occur for $\mu>\sqrt{5}$.

Paper 4, Section II, D

commentA dielectric material has a real, frequency-independent relative permittivity $\epsilon_{r}$ with $\left|\epsilon_{r}-1\right| \ll 1$. In this case, the macroscopic polarization that develops when the dielectric is placed in an external electric field $\mathbf{E}_{\text {ext }}(t, \mathbf{x})$ is $\mathbf{P}(t, \mathbf{x}) \approx \epsilon_{0}\left(\epsilon_{r}-1\right) \mathbf{E}_{\text {ext }}(t, \mathbf{x})$. Explain briefly why the associated bound current density is

$\mathbf{J}_{\text {bound }}(t, \mathbf{x}) \approx \epsilon_{0}\left(\epsilon_{r}-1\right) \frac{\partial \mathbf{E}_{\text {ext }}(t, \mathbf{x})}{\partial t}$

[You should ignore any magnetic response of the dielectric.]

A sphere of such a dielectric, with radius $a$, is centred on $\mathbf{x}=0$. The sphere scatters an incident plane electromagnetic wave with electric field

$\mathbf{E}(t, \mathbf{x})=\mathbf{E}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}$

where $\omega=c|\mathbf{k}|$ and $\mathbf{E}_{0}$ is a constant vector. Working in the Lorenz gauge, show that at large distances $r=|\mathbf{x}|$, for which both $r \gg a$ and $k a^{2} / r \ll 2 \pi$, the magnetic vector potential $\mathbf{A}_{\text {scatt }}(t, \mathbf{x})$ of the scattered radiation is

$\mathbf{A}_{\mathrm{scatt}}(t, \mathbf{x}) \approx-i \omega \mathbf{E}_{0} \frac{e^{i(k r-\omega t)}}{r} \frac{\left(\epsilon_{r}-1\right)}{4 \pi c^{2}} \int_{\left|\mathbf{x}^{\prime}\right| \leqslant a} e^{i \mathbf{q} \cdot \mathbf{x}^{\prime}} d^{3} \mathbf{x}^{\prime}$

where $\mathbf{q}=\mathbf{k}-k \hat{\mathbf{x}}$ with $\hat{\mathbf{x}}=\mathbf{x} / r$.

In the far-field, where $k r \gg 1$, the electric and magnetic fields of the scattered radiation are given by

$\begin{aligned} &\mathbf{E}_{\text {scatt }}(t, \mathbf{x}) \approx-i \omega \hat{\mathbf{x}} \times\left[\hat{\mathbf{x}} \times \mathbf{A}_{\text {scatt }}(t, \mathbf{x})\right] \\ &\mathbf{B}_{\text {scatt }}(t, \mathbf{x}) \approx i k \hat{\mathbf{x}} \times \mathbf{A}_{\text {scatt }}(t, \mathbf{x}) \end{aligned}$

By calculating the Poynting vector of the scattered and incident radiation, show that the ratio of the time-averaged power scattered per unit solid angle to the time-averaged incident power per unit area (i.e. the differential cross-section) is

$\frac{d \sigma}{d \Omega}=\left(\epsilon_{r}-1\right)^{2} k^{4}\left(\frac{\sin (q a)-q a \cos (q a)}{q^{3}}\right)^{2}\left|\hat{\mathbf{x}} \times \hat{\mathbf{E}}_{0}\right|^{2}$

where $\hat{\mathbf{E}}_{0}=\mathbf{E}_{0} /\left|\mathbf{E}_{0}\right|$ and $q=|\mathbf{q}|$.

[You may assume that, in the Lorenz gauge, the retarded potential due to a localised current distribution is

$\mathbf{A}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d^{3} \mathbf{x}^{\prime},$

where the retarded time $\left.t_{\text {ret }}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right| / c .\right]$

Paper 4, Section II, B

commentA horizontal layer of inviscid fluid of density $\rho_{1}$ occupying $0<y<h$ flows with velocity $(U, 0)$ above a horizontal layer of inviscid fluid of density $\rho_{2}>\rho_{1}$ occupying $-h<y<0$ and flowing with velocity $(-U, 0)$, in Cartesian coordinates $(x, y)$. There are rigid boundaries at $y=\pm h$. The interface between the two layers is perturbed to position $y=\operatorname{Re}\left(A e^{i k x+\sigma t}\right)$.

Write down the full set of equations and boundary conditions governing this flow. Derive the linearised boundary conditions appropriate in the limit $A \rightarrow 0$. Solve the linearised equations to show that the perturbation to the interface grows exponentially in time if

$U^{2}>\frac{\rho_{2}^{2}-\rho_{1}^{2}}{\rho_{1} \rho_{2}} \frac{g}{4 k} \tanh k h .$

Sketch the right-hand side of this inequality as a function of $k$. Thereby deduce the minimum value of $U$ that makes the system unstable for all wavelengths.

Paper 4, Section I, $7 \mathbf{E} \quad$

commentConsider the differential equation

$z \frac{d^{2} y}{d z^{2}}-2 \frac{d y}{d z}+z y=0$

Laplace's method finds a solution of this differential equation by writing $y(z)$ in the form

$y(z)=\int_{C} e^{z t} f(t) d t$

where $C$ is a closed contour.

Determine $f(t)$. Hence find two linearly independent real solutions of $(\star)$ for $z$ real.

Paper 4, Section II, I

comment(a) State the Fundamental Theorem of Galois Theory.

(b) What does it mean for an extension $L$ of $\mathbb{Q}$ to be cyclotomic? Show that a cyclotomic extension $L$ of $\mathbb{Q}$ is a Galois extension and prove that its Galois group is Abelian.

(c) What is the Galois group $G$ of $\mathbb{Q}(\eta)$ over $\mathbb{Q}$, where $\eta$ is a primitive 7 th root of unity? Identify the intermediate subfields $M$, with $\mathbb{Q} \leqslant M \leqslant \mathbb{Q}(\eta)$, in terms of $\eta$, and identify subgroups of $G$ to which they correspond. Justify your answers.

Paper 4, Section II, D

comment(a) In the transverse traceless gauge, a plane gravitational wave propagating in the $z$ direction is described by a perturbation $h_{\alpha \beta}$ of the Minkowski metric $\eta_{\alpha \beta}=$ $\operatorname{diag}(-1,1,1,1)$ in Cartesian coordinates $x^{\alpha}=(t, x, y, z)$, where

$h_{\alpha \beta}=H_{\alpha \beta} e^{i k_{\mu} x^{\mu}}, \quad \text { where } \quad k^{\mu}=\omega(1,0,0,1)$

and $H_{\alpha \beta}$ is a constant matrix. Spacetime indices in this question are raised or lowered with the Minkowski metric.

The energy-momentum tensor of a gravitational wave is defined to be

$\tau_{\mu \nu}=\frac{1}{32 \pi}\left(\partial_{\mu} h^{\alpha \beta}\right)\left(\partial_{\nu} h_{\alpha \beta}\right)$

Show that $\partial^{\nu} \tau_{\mu \nu}=\frac{1}{2} \partial_{\mu} \tau_{\nu}^{\nu}$ and hence, or otherwise, show that energy and momentum are conserved.

(b) A point mass $m$ undergoes harmonic motion along the $z$-axis with frequency $\omega$ and amplitude $L$. Compute the energy flux emitted in gravitational radiation.

[Hint: The quadrupole formula for time-averaged energy flux radiated in gravitational waves is

\left\langle\frac{d E}{d t}\right\rangle=\frac{1}{5}\left\langle\dddot{Q}_{i j} \dddot{Q}_{i j}\right\rangle

where $Q_{i j}$ is the reduced quadrupole tensor.]

Paper 4, Section II, H

commentLet $G$ be a graph of maximum degree $\Delta$. Show the following:

(i) Every eigenvalue $\lambda$ of $G$ satisfies $|\lambda| \leqslant \Delta$.

(ii) If $G$ is regular then $\Delta$ is an eigenvalue.

(iii) If $G$ is regular and connected then the multiplicity of $\Delta$ as an eigenvalue is 1 .

(iv) If $G$ is regular and not connected then the multiplicity of $\Delta$ as an eigenvalue is greater than 1 .

Let $A$ be the adjacency matrix of the Petersen graph. Explain why $A^{2}+A-2 I=J$, where $I$ is the identity matrix and $J$ is the all-1 matrix. Find, with multiplicities, the eigenvalues of the Petersen graph.

Paper 4, Section II, F

commentLet $H$ be a complex Hilbert space with inner product $(\cdot, \cdot)$ and let $T: H \rightarrow H$ be a bounded linear map.

(i) Define the spectrum $\sigma(T)$, the point spectrum $\sigma_{p}(T)$, the continuous spectrum $\sigma_{c}(T)$, and the residual spectrum $\sigma_{r}(T)$.

(ii) Show that $T^{*} T$ is self-adjoint and that $\sigma\left(T^{*} T\right) \subset[0, \infty)$. Show that if $T$ is compact then so is $T^{*} T$.

(iii) Assume that $T$ is compact. Prove that $T$ has a singular value decomposition: for $N<\infty$ or $N=\infty$, there exist orthonormal systems $\left(u_{i}\right)_{i=1}^{N} \subset H$ and $\left(v_{i}\right)_{i=1}^{N} \subset H$ and $\left(\lambda_{i}\right)_{i=1}^{N} \subset[0, \infty)$ such that, for any $x \in H$,

$T x=\sum_{i=1}^{N} \lambda_{i}\left(u_{i}, x\right) v_{i}$

[You may use the spectral theorem for compact self-adjoint linear operators.]

Paper 4, Section II, H

commentProve that every set has a transitive closure. [If you apply the Axiom of Replacement to a function-class $F$, you must explain clearly why $F$ is indeed a function-class.]

State the Axiom of Foundation and the Principle of $\epsilon$-Induction, and show that they are equivalent (in the presence of the other axioms of $\mathrm{ZFC}$ ).

State the $\epsilon$-Recursion Theorem.

Sets $C_{\alpha}$ are defined for each ordinal $\alpha$ by recursion, as follows: $C_{0}=\emptyset, C_{\alpha+1}$ is the set of all countable subsets of $C_{\alpha}$, and $C_{\lambda}=\cup_{\alpha<\lambda} C_{\alpha}$ for $\lambda$ a non-zero limit. Does there exist an $\alpha$ with $C_{\alpha+1}=C_{\alpha}$ ? Justify your answer.

Paper 4, Section I, B

commentConsider an epidemic model with host demographics (natural births and deaths).

The system is given by

$\begin{aligned} &\frac{d S}{d t}=-\beta I S-\mu S+\mu N \\ &\frac{d I}{d t}=+\beta I S-\nu I-\mu I \end{aligned}$

where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, $N$ is the total population size and the parameters $\beta, \mu$ and $\nu$ are positive. The basic reproduction ratio is defined as $R_{0}=\beta N /(\mu+\nu) .$

Show that the system has an endemic equilibrium (where the disease is present) for $R_{0}>1$. Show that the endemic equilibrium is stable.

Interpret the meaning of the case $\nu \gg \mu$ and show that in this case the approximate period of (decaying) oscillation around the endemic equilibrium is given by

$T=\frac{2 \pi}{\sqrt{\mu \nu\left(R_{0}-1\right)}}$

Suppose now a vaccine is introduced which is given to some proportion of the population at birth, but not enough to eradicate the disease. What will be the effect on the period of (decaying) oscillations?

Paper 4, Section II, B

commentAn activator-inhibitor system is described by the equations

$\begin{aligned} &\frac{\partial u}{\partial t}=u(c+u-v)+\frac{\partial^{2} u}{\partial x^{2}} \\ &\frac{\partial v}{\partial t}=v(a u-b v)+d \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$

where $a, b, c, d>0$.

Find and sketch the range of $a, b$ for which the spatially homogeneous system has a stable stationary solution with $u>0$ and $v>0$.

Considering spatial perturbations of the form $\cos (k x)$ about the solution found above, find conditions for the system to be unstable. Sketch this region in the $(a, b)$-plane for fixed $d$ (for a value of $d$ such that the region is non-empty).

Show that $k_{c}$, the critical wavenumber at the onset of the instability, is given by

$k_{c}=\sqrt{\frac{2 a c}{d-a}}$

Paper 4, Section II, H

comment(a) Write down $\mathcal{O}_{K}$, when $K=\mathbb{Q}(\sqrt{\delta})$, and $\delta \equiv 2$ or $3(\bmod 4)$. [You need not prove your answer.]

Let $L=\mathbb{Q}(\sqrt{2}, \sqrt{\delta})$, where $\delta \equiv 3(\bmod 4)$ is a square-free integer. Find an integral basis of $\mathcal{O}_{L} \cdot$ [Hint: Begin by considering the relative traces $t r_{L / K}$, for $K$ a quadratic subfield of $L .]$

(b) Compute the ideal class group of $\mathbb{Q}(\sqrt{-14})$.

Paper 4, Section I, G

commentShow that, for $x \geqslant 2$ a real number,

$\prod_{\substack{p \leqslant x \\ p \text { is prime }}}\left(1-\frac{1}{p}\right)^{-1}>\log x$

Hence prove that

$\sum_{\substack{p \leqslant x, p \text { is prime }}} \frac{1}{p}>\log \log x+c,$

where $c$ is a constant you should make explicit.

Paper 4, Section II, 10G

comment(a) State Dirichlet's theorem on primes in arithmetic progression.

(b) Let $d$ be the discriminant of a binary quadratic form, and let $p$ be an odd prime. Show that $p$ is represented by some binary quadratic form of discriminant $d$ if and only if $x^{2} \equiv d(\bmod p)$ is soluble.

(c) Let $f(x, y)=x^{2}+15 y^{2}$ and $g(x, y)=3 x^{2}+5 y^{2}$. Show that $f$ and $g$ each represent infinitely many primes. Are there any primes represented by both $f$ and $g$ ?

Paper 4, Section II, A

comment(a) The diffusion equation

$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}, \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant t \leqslant T$

is approximated by the Crank-Nicolson scheme

$u_{m}^{n+1}-\frac{1}{2} \mu\left(u_{m-1}^{n+1}-2 u_{m}^{n+1}+u_{m+1}^{n+1}\right)=u_{m}^{n}+\frac{1}{2} \mu\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right)$

with $m=1, \ldots, M$. Here $\mu=k / h^{2}, k=\Delta t, h=\Delta x=\frac{1}{M+1}$, and $u_{m}^{n}$ is an approximation to $u(m h, n k)$. Assuming that $u(0, t)=u(1, t) \stackrel{M+1}{=} 0$, show that the above scheme can be written in the form

$B \mathbf{u}^{n+1}=C \mathbf{u}^{n}, \quad 0 \leqslant n \leqslant T / k-1$

where $\mathbf{u}^{n}=\left[u_{1}^{n}, \ldots, u_{M}^{n}\right]^{T}$ and the real matrices $B$ and $C$ should be found. Using matrix analysis, find the range of $\mu>0$ for which the scheme is stable.

[Hint: All Toeplitz symmetric tridiagonal (TST) matrices have the same set of orthogonal eigenvectors, and a TST matrix with the elements $a_{i, i}=a$ and $a_{i, i \pm 1}=b$ has the eigenvalues $\lambda_{k}=a+2 b \cos \frac{\pi k}{M+1}$. ]

(b) The wave equation

$\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}}, \quad x \in \mathbb{R}, \quad t \geqslant 0$

with given initial conditions for $u$ and $\partial u / \partial t$, is approximated by the scheme

$u_{m}^{n+1}-2 u_{m}^{n}+u_{m}^{n-1}=\mu\left(u_{m+1}^{n}-2 u_{m}^{n}+u_{m-1}^{n}\right),$

with the Courant number now $\mu=k^{2} / h^{2}$. Applying the Fourier technique, find the range of $\mu>0$ for which the method is stable.

Paper 4, Section II, $29 \mathrm{~K}$

commentA file of $X$ gigabytes (GB) is to be transmitted over a communications link. At each time $t$ the sender can choose a transmission rate $u(t)$ within the range $[0,1]$ GB per second. The charge for transmitting at rate $u(t)$ at time $t$ is $u(t) p(t)$. The function $p$ is fully known at time $t=0$. If it takes a total time $T$ to transmit the file then there is a delay cost of $\gamma T^{2}, \gamma>0$. Thus $u$ and $T$ are to be chosen to minimize

$\int_{0}^{T} u(t) p(t) d t+\gamma T^{2}$

where $u(t) \in[0,1], d x(t) / d t=-u(t), x(0)=X$ and $x(T)=0$. Using Pontryagin's maximum principle, or otherwise, show that a property of the optimal policy is that there exists $p^{*}$ such that $u(t)=1$ if $p(t)<p^{*}$ and $u(t)=0$ if $p(t)>p^{*}$.

Show that the optimal $p^{*}$ and $T$ are related by $p^{*}=p(T)+2 \gamma T$.

Suppose $p(t)=t+1 / t$ and $X=1$. Show that it is optimal to transmit at a constant rate $u(t)=1$ between times $T-1 \leqslant t \leqslant T$, where $T$ is the unique positive solution to the equation

$\frac{1}{(T-1) T}=2 \gamma T+1$

Paper 4, Section II, C

commentThe Hamiltonian for a quantum system in the Schrödinger picture is

$H_{0}+\lambda V(t),$

where $H_{0}$ is independent of time and the parameter $\lambda$ is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.

Let $|n\rangle$ and $|m\rangle$ be eigenstates of $H_{0}$ with distinct eigenvalues $E_{n}$ and $E_{m}$ respectively. Show that if the system was in the state $|n\rangle$ in the remote past, then the probability of measuring it to be in a different state $|m\rangle$ at a time $t$ is

$\frac{\lambda^{2}}{\hbar^{2}}\left|\int_{-\infty}^{t} d t^{\prime}\left\langle m\left|V\left(t^{\prime}\right)\right| n\right\rangle e^{i\left(E_{m}-E_{n}\right) t^{\prime} / \hbar}\right|^{2}+O\left(\lambda^{3}\right)$

Let the system be a simple harmonic oscillator with $H_{0}=\hbar \omega\left(a^{\dagger} a+\frac{1}{2}\right)$, where $\left[a, a^{\dagger}\right]=1$. Let $|0\rangle$ be the ground state which obeys $a|0\rangle=0$. Suppose

$V(t)=e^{-p|t|}\left(a+a^{\dagger}\right),$

with $p>0$. In the remote past the system was in the ground state. Find the probability, to lowest non-trivial order in $\lambda$, for the system to be in the first excited state in the far future.

Paper 4, Section II, $27 \mathrm{~K}$

commentFor the statistical model $\left\{\mathcal{N}_{d}(\theta, \Sigma), \theta \in \mathbb{R}^{d}\right\}$, where $\Sigma$ is a known, positive-definite $d \times d$ matrix, we want to estimate $\theta$ based on $n$ i.i.d. observations $X_{1}, \ldots, X_{n}$ with distribution $\mathcal{N}_{d}(\theta, \Sigma)$.

(a) Derive the maximum likelihood estimator $\hat{\theta}_{n}$ of $\theta$. What is the distribution of $\hat{\theta}_{n}$ ?

(b) For $\alpha \in(0,1)$, construct a confidence region $C_{n}^{\alpha}$ such that $\mathbf{P}_{\theta}\left(\theta \in C_{n}^{\alpha}\right)=1-\alpha$.

(c) For $\Sigma=I_{d}$, compute the maximum likelihood estimator of $\theta$ for the following parameter spaces:

(i) $\Theta=\left\{\theta:\|\theta\|_{2}=1\right\}$.

(ii) $\Theta=\left\{\theta: v^{\top} \theta=0\right\}$ for some unit vector $v \in \mathbb{R}^{d}$.

(d) For $\Sigma=I_{d}$, we want to test the null hypothesis $\Theta_{0}=\{0\}$ (i.e. $\left.\theta=0\right)$ against the composite alternative $\Theta_{1}=\mathbb{R}^{d} \backslash\{0\}$. Compute the likelihood ratio statistic $\Lambda\left(\Theta_{1}, \Theta_{0}\right)$ and give its distribution under the null hypothesis. Compare this result with the statement of Wilks' theorem.

Paper 4, Section II, J

comment(a) Suppose that $(E, \mathcal{E}, \mu)$ is a finite measure space and $\theta: E \rightarrow E$ is a measurable map. Prove that $\mu_{\theta}(A)=\mu\left(\theta^{-1}(A)\right)$ defines a measure on $(E, \mathcal{E})$.

(b) Suppose that $\mathcal{A}$ is a $\pi$-system which generates $\mathcal{E}$. Using Dynkin's lemma, prove that $\theta$ is measure-preserving if and only if $\mu_{\theta}(A)=\mu(A)$ for all $A \in \mathcal{A}$.

(c) State Birkhoff's ergodic theorem and the maximal ergodic lemma.

(d) Consider the case $(E, \mathcal{E}, \mu)=([0,1), \mathcal{B}([0,1)), \mu)$ where $\mu$ is Lebesgue measure on $[0,1)$. Let $\theta:[0,1) \rightarrow[0,1)$ be the following map. If $x=\sum_{n=1}^{\infty} 2^{-n} \omega_{n}$ is the binary expansion of $x$ (where we disallow infinite sequences of $1 \mathrm{~s}$ ), then $\theta(x)=$ $\sum_{n=1}^{\infty} 2^{-n}\left(\omega_{n-1} \mathbf{1}_{n \in E}+\omega_{n+1} \mathbf{1}_{n \in O}\right)$ where $E$ and $O$ are respectively the even and odd elements of $\mathbb{N}$.

(i) Prove that $\theta$ is measure-preserving. [You may assume that $\theta$ is measurable.]

(ii) Prove or disprove: $\theta$ is ergodic.

Paper 4, Section II, G

commentLet $G=\mathrm{SU}(2)$ and let $V_{n}$ be the vector space of complex homogeneous polynomials of degree $n$ in two variables.

(a) Prove that $V_{n}$ has the structure of an irreducible representation for $G$.

(b) State and prove the Clebsch-Gordan theorem.

(c) Quoting without proof any properties of symmetric and exterior powers which you need, decompose $\mathrm{S}^{2} V_{n}$ and $\Lambda^{2} V_{n}(n \geqslant 1)$ into irreducible $G$-spaces.

Paper 4, Section I, J

commentA Cambridge scientist is testing approaches to slow the spread of a species of moth in certain trees. Two groups of 30 trees were treated with different organic pesticides, and a third group of 30 trees was kept under control conditions. At the end of the summer the trees are classified according to the level of leaf damage, obtaining the following contingency table.

Which of the following Generalised Linear Model fitting commands is appropriate for these data? Why? Describe the model being fit.

Paper 4, Section II, J

commentThe dataset diesel records the number of diesel cars which go through a block of Hills Road in 6 disjoint periods of 30 minutes, between 8AM and 11AM. The measurements are repeated each day for 10 days. Answer the following questions based on the code below, which is shown with partial output.

(a) Can we reject the model fit. 1 at a $1 \%$ level? Justify your answer.

(b) What is the difference between the deviance of the models fit. 2 and fit.3?

(c) Which of fit. 2 and fit. 3 would you use to perform variable selection by backward stepwise selection? Why?

(d) How does the final plot differ from what you expect under the model in fit.2? Provide a possible explanation and suggest a better model.

$>$ head (diesel)

period num.cars day

$\begin{array}{llll}1 & 1 & 69 & 1\end{array}$

$\begin{array}{lllll}2 & 2 & 97 & 1\end{array}$

$\begin{array}{llll}3 & 3 & 103 & 1\end{array}$

$\begin{array}{llll}4 & 4 & 99 & 1\end{array}$

$\begin{array}{llll}5 & 5 & 67 & 1\end{array}$

$6 \quad 6 \quad 911$

$>$ fit. $1=$ glm(num.cars period, data=diesel, family=poisson)

summary (fit.1)

Deviance Residuals:

Min 1Q Median 3Q Max

$\begin{array}{lllll}-4.0188 & -1.4837 & -0.2117 & 1.6257 & 4.5965\end{array}$

Coefficients:

Estimate Std. Error $z$ value $\operatorname{Pr}(>|z|)$

(Intercept) $4.628535 \quad 0.029288158 .035<2 \mathrm{e}-16 * * *$

period $-0.006073 \quad 0.007551-0.804 \quad 0.421$

Signif. codes: 0 ? $* * *$ ? $0.001 ? * * ? 0.01$ ? $*$ ? $0.05$ ?.? $0.1$ ? ? 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: $262.36$ on 59 degrees of freedom

Residual deviance: $261.72$ on 58 degrees of freedom

AIC: $651.2$

$>$ diesel$period.factor = factor(diesel$period)

$>$ fit. $2=$ glm (num.cars period.factor, data=diesel, family=poisson)

$\operatorname{summary}$ (fit.2)

Coefficients:

Estimate Std. Error z value $\operatorname{Pr}(>|z|)$

Part II, $2017 \quad$ List of Questions

[TURN OVER

Paper 4, Section II, D

commentThe van der Waals equation of state is

$p=\frac{k T}{v-b}-\frac{a}{v^{2}}$

where $p$ is the pressure, $v=V / N$ is the volume divided by the number of particles, $T$ is the temperature, $k$ is Boltzmann's constant and $a, b$ are positive constants.

(i) Prove that the Gibbs free energy $G=E+p V-T S$ satisfies $G=\mu N$. Hence obtain an expression for $(\partial \mu / \partial p)_{T, N}$ and use it to explain the Maxwell construction for determining the pressure at which the gas and liquid phases can coexist at a given temperature.

(ii) Explain what is meant by the critical point and determine the values $p_{c}, v_{c}, T_{c}$ corresponding to this point.

(iii) By defining $\bar{p}=p / p_{c}, \bar{v}=v / v_{c}$ and $\bar{T}=T / T_{c}$, derive the law of corresponding states:

$\bar{p}=\frac{8 \bar{T}}{3 \bar{v}-1}-\frac{3}{\bar{v}^{2}} .$

(iv) To investigate the behaviour near the critical point, let $\bar{T}=1+t$ and $\bar{v}=1+\phi$, where $t$ and $\phi$ are small. Expand $\bar{p}$ to cubic order in $\phi$ and hence show that

$\left(\frac{\partial \bar{p}}{\partial \phi}\right)_{t}=-\frac{9}{2} \phi^{2}+\mathcal{O}\left(\phi^{3}\right)+t[-6+\mathcal{O}(\phi)] .$

At fixed small $t$, let $\phi_{l}(t)$ and $\phi_{g}(t)$ be the values of $\phi$ corresponding to the liquid and gas phases on the co-existence curve. By changing the integration variable from $p$ to $\phi$, use the Maxwell construction to show that $\phi_{l}(t)=-\phi_{g}(t)$. Deduce that, as the critical point is approached along the co-existence curve,

$\bar{v}_{\text {gas }}-\bar{v}_{\text {liquid }} \sim\left(T_{c}-T\right)^{1 / 2}$

Paper 4, Section II, J

comment(a) Describe the (Cox-Ross-Rubinstein) binomial model. When is the model arbitragefree? How is the equivalent martingale measure characterised in this case?

(b) What is the price and the hedging strategy for any given contingent claim $C$ in the binomial model?

(c) For any fixed $0<t<T$ and $K>0$, the payoff function of a forward-start-option is given by

$\left(\frac{S_{T}^{1}}{S_{t}^{1}}-K\right)^{+}$

Find a formula for the price of the forward-start-option in the binomial model.

Paper 4, Section I, F

commentIf $x \in(0,1]$, set

$x=\frac{1}{N(x)+T(x)}$

where $N(x)$ is an integer and $1>T(x) \geqslant 0$. Let $N(0)=T(0)=0$.

If $x$ is also irrational, write down the continued fraction expansion in terms of $N T^{j}(x)\left(\right.$ where $\left.N T^{0}(x)=N(x)\right)$.

Let $X$ be a random variable taking values in $[0,1]$ with probability density function

$f(x)=\frac{1}{(\log 2)(1+x)}$

Show that $T(X)$ has the same distribution as $X$.

Paper 4, Section II, 11F

comment(a) Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is continuous with $\gamma(0)=\gamma(1)$ and $\gamma(t) \neq 0$ for all $t \in[0,1]$. Show that if $\gamma(0)=|\gamma(0)| \exp \left(i \theta_{0}\right)$ (with $\theta_{0}$ real) we can define a continuous function $\theta:[0,1] \rightarrow \mathbb{R}$ such that $\theta(0)=\theta_{0}$ and $\gamma(t)=|\gamma(t)| \exp (i \theta(t))$. Hence define the winding number $w(\gamma)=w(0, \gamma)$ of $\gamma$ around 0 .

(b) Show that $w(\gamma)$ can take any integer value.

(c) If $\gamma_{1}$ and $\gamma_{2}$ satisfy the requirements of the definition, and $\left(\gamma_{1} \times \gamma_{2}\right)(t)=\gamma_{1}(t) \gamma_{2}(t)$, show that

$w\left(\gamma_{1} \times \gamma_{2}\right)=w\left(\gamma_{1}\right)+w\left(\gamma_{2}\right)$

(d) If $\gamma_{1}$ and $\gamma_{2}$ satisfy the requirements of the definition and $\left|\gamma_{1}(t)-\gamma_{2}(t)\right|<\left|\gamma_{1}(t)\right|$ for all $t \in[0,1]$, show that

$w\left(\gamma_{1}\right)=w\left(\gamma_{2}\right)$

(e) State and prove a theorem that says that winding number is unchanged under an appropriate homotopy.

Paper 4, Section II, 38B

Consider the Rossby-wave equation

$\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial x^{2}}-\ell^{2}\right) \varphi+\beta \frac{\partial \varphi}{\partial x}=0,$

where $\ell>0$ and $\beta>0$ are real constants. Find and sketch the dispersion relation for waves with wavenumber $k$ and frequency $\omega(k)$. Find and sketch the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and identify in which direction(s) the wave crests travel, and the corresponding direction(s) of the group velocity.

Write down the solution with initial value

$\varphi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$

where $A(k)$ is real and $A(-k)=A(k)$. Use the method of stationary phase to obtain leading-order approximations to $\varphi(x, t)$ for large $t$, with $x / t$ having the constant value $V$, for

(i) $0<V<\beta / 8 \ell^{2}$,

(ii) $-\beta / \ell^{2}<V \leqslant 0$,

where the solutions for the stationary points should be left in implicit form. [It is helpful to note that $\omega(-k)=-\omega(k)$.]

Briefly discuss the nature of the solution for $V>\beta / 8 \ell^{2}$ and $V<-\beta / \ell^{2}$