Part II, 2011, Paper 3

Algebraic Geometry

• Paper 3, Section II, H

  Algebraic Geometry | Part II, 2011

  Let $X$ be a smooth projective curve over an algebraically closed field $k$ of characteristic 0 .

  (i) Let $D$ be a divisor on $X$.

  Define $\mathcal{L}(D)$, and show $\operatorname{dim} \mathcal{L}(D) \leqslant \operatorname{deg} D+1$.

  (ii) Define the space of rational differentials $\Omega_{k(X) / k}^{1}$.

  If $p$ is a point on $X$, and $t$ a local parameter at $p$, show that $\Omega_{k(X) / k}^{1}=k(X) d t$.

  Use that equality to give a definition of $v_{p}(\omega) \in \mathbb{Z}$, for $\omega \in \Omega_{k(X) / k}^{1}, p \in X$. [You need not show that your definition is independent of the choice of local parameter.]

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Algebraic Topology

• Paper 3, Section II, H

  Algebraic Topology | Part II, 2011

  Let $K$ and $L$ be (finite) simplicial complexes. Explain carefully what is meant by a simplicial approximation to a continuous map $f:|K| \rightarrow|L|$. Indicate briefly how the cartesian product $|K| \times|L|$ may be triangulated.

  Two simplicial maps $g, h: K \rightarrow L$ are said to be contiguous if, for each simplex $\sigma$ of $K$, there exists a simplex $\sigma *$ of $L$ such that both $g(\sigma)$ and $h(\sigma)$ are faces of $\sigma *$. Show that:

  (i) any two simplicial approximations to a given map $f:|K| \rightarrow|L|$ are contiguous;

  (ii) if $g$ and $h$ are contiguous, then they induce homotopic maps $|K| \rightarrow|L|$;

  (iii) if $f$ and $g$ are homotopic maps $|K| \rightarrow|L|$, then for some subdivision $K^{(n)}$ of $K$ there exists a sequence $\left(h_{1}, h_{2}, \ldots, h_{m}\right)$ of simplicial maps $K^{(n)} \rightarrow L$ such that $h_{1}$ is a simplicial approximation to $f, h_{m}$ is a simplicial approximation to $g$ and each pair $\left(h_{i}, h_{i+1}\right)$ is contiguous.

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Applications of Quantum Mechanics

• Paper 3, Section II, E

  Applications of Quantum Mechanics | Part II, 2011

  An electron of mass $m$ moves in a $D$-dimensional periodic potential that satisfies the periodicity condition

  $$V(\boldsymbol{r}+\boldsymbol{l})=V(\boldsymbol{r}) \quad \forall l \in \Lambda,$$

  where $\Lambda$ is a D-dimensional Bravais lattice. State Bloch's theorem for the energy eigenfunctions of the electron.

  For a one-dimensional potential $V(x)$ such that $V(x+a)=V(x)$, give a full account of how the "nearly free electron model" leads to a band structure for the energy levels.

  Explain briefly the idea of a Fermi surface and its rôle in explaining the existence of conductors and insulators.

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Applied Probability

• Paper 3, Section II, J

  Applied Probability | Part II, 2011

  (i) Define an inhomogeneous Poisson process with rate function $\lambda(u)$.

  (ii) Show that the number of arrivals in an inhomogeneous Poisson process during the interval $(0, t)$ has the Poisson distribution with mean

  $$\int_{0}^{t} \lambda(u) \mathrm{d} u$$

  (iii) Suppose that $\Lambda=\{\Lambda(t), t \geqslant 0\}$ is a non-negative real-valued random process. Conditional on $\Lambda$, let $N=\{N(t), t \geqslant 0\}$ be an inhomogeneous Poisson process with rate function $\Lambda(u)$. Such a process $N$ is called a doubly-stochastic Poisson process. Show that the variance of $N(t)$ cannot be less than its mean.

  (iv) Now consider the process $M(t)$ obtained by deleting every odd-numbered point in an ordinary Poisson process of rate $\lambda$. Check that

  $$\mathbb{E} M(t)=\frac{2 \lambda t+e^{-2 \lambda t}-1}{4}, \quad \operatorname{Var} M(t)=\frac{4 \lambda t-8 \lambda t e^{-2 \lambda t}-e^{-4 \lambda t}+1}{16}$$

  Deduce that $M(t)$ is not a doubly-stochastic Poisson process.

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Asymptotic Methods

• Paper 3, Section II, A

  Asymptotic Methods | Part II, 2011

  Let

  $$I_{0}=\int_{C_{0}} e^{x \phi(z)} d z \text {, }$$

  where $\phi(z)$ is a complex analytic function and $C_{0}$ is a steepest descent contour from a simple saddle point of $\phi(z)$ at $z_{0}$. Establish the following leading asymptotic approximation, for large real $x$ :

  $$I_{0} \sim i \sqrt{\frac{\pi}{2 \phi^{\prime \prime}\left(z_{0}\right) x}} e^{x \phi\left(z_{0}\right)}$$

  Let $n$ be a positive integer, and let

  $$I=\int_{C} e^{-t^{2}-2 n \ln t} d t$$

  where $C$ is a contour in the upper half $t$-plane connecting $t=-\infty$ to $t=\infty$, and $\ln t$ is real on the positive $t$-axis with a branch cut along the negative $t$-axis. Using the method of steepest descent, find the leading asymptotic approximation to $I$ for large $n$.

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Classical Dynamics

• Paper 3, Section I, $\mathbf{9 C}$

  Classical Dynamics | Part II, 2011

  The Lagrangian for a heavy symmetric top is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  State Noether's Theorem. Hence, or otherwise, find two conserved quantities linear in momenta, and a third conserved quantity quadratic in momenta.

  Writing $\mu=\cos \theta$, deduce that $\mu$ obeys an equation of the form

  $$\dot{\mu}^{2}=F(\mu)$$

  where $F(\mu)$ is cubic in $\mu$. [You need not determine the explicit form of $F(\mu) .$ ]

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Coding and Cryptography

• Paper 3, Section I, G

  Coding and Cryptography | Part II, 2011

  What is the rank of a binary linear code $C ?$ What is the weight enumeration polynomial $W_{C}$ of $C ?$

  Show that $W_{C}(1,1)=2^{r}$ where $r$ is the rank of $C$. Show that $W_{C}(s, t)=W_{C}(t, s)$ for all $s$ and $t$ if and only if $W_{C}(1,0)=1$.

  Find, with reasons, the weight enumeration polynomial of the repetition code of length $n$, and of the simple parity check code of length $n$.

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Cosmology

• Paper 3, Section I, E

  Cosmology | Part II, 2011

  For an ideal gas of fermions of mass $m$ in volume $V$, and at temperature $T$ and chemical potential $\mu$, the number density $n$ and kinetic energy $E$ are given by

  $$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{\infty} \bar{n}(p) p^{2} d p, \quad E=\frac{4 \pi g_{s}}{h^{3}} V \int_{0}^{\infty} \bar{n}(p) \epsilon(p) p^{2} d p$$

  where $g_{s}$ is the spin-degeneracy factor, $h$ is Planck's constant, $\epsilon(p)=c \sqrt{p^{2}+m^{2} c^{2}}$ is the single-particle energy as a function of the momentum $p$, and

  $$\bar{n}(p)=\left[\exp \left(\frac{\epsilon(p)-\mu}{k T}\right)+1\right]^{-1}$$

  where $k$ is Boltzmann's constant.

  (i) Sketch the function $\bar{n}(p)$ at zero temperature, explaining why $\bar{n}(p)=0$ for $p>p_{F}$ (the Fermi momentum). Find an expression for $n$ at zero temperature as a function of $p F$.

  Assuming that a typical fermion is ultra-relativistic $\left(p c \gg m c^{2}\right)$ even at zero temperature, obtain an estimate of the energy density $E / V$ as a function of $p_{F}$, and hence show that

  $$E \sim h c n^{4 / 3} V$$

  in the ultra-relativistic limit at zero temperature.

  (ii) A white dwarf star of radius $R$ has total mass $M=\frac{4 \pi}{3} m_{p} n_{p} R^{3}$, where $m_{p}$ is the proton mass and $n_{p}$ the average proton number density. On the assumption that the star's degenerate electrons are ultra-relativistic, so that $(*)$ applies with $n$ replaced by the average electron number density $n_{e}$, deduce the following estimate for the star's internal kinetic energy:

  $$E_{\mathrm{kin}} \sim h c\left(\frac{M}{m_{p}}\right)^{4 / 3} \frac{1}{R} .$$

  By comparing this with the total gravitational potential energy, briefly discuss the consequences for white dwarf stability.

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• Paper 3, Section II, E

  Cosmology | Part II, 2011

  An expanding universe with scale factor $a(t)$ is filled with (pressure-free) cold dark matter (CDM) of average mass density $\bar{\rho}(t)$. In the Zel'dovich approximation to gravitational clumping, the perturbed position $\mathbf{r}(\mathbf{q}, t)$ of a CDM particle with unperturbed comoving position $\mathbf{q}$ is given by

  $$\mathbf{r}(\mathbf{q}, t)=a(t)[\mathbf{q}+\boldsymbol{\psi}(\mathbf{q}, t)]$$

  where $\psi$ is the comoving displacement.

  (i) Explain why the conservation of CDM particles implies that

  $$\rho(\mathbf{r}, t) d^{3} r=a^{3} \bar{\rho}(t) d^{3} q,$$

  where $\rho(\mathbf{r}, t)$ is the CDM mass density. Use (1) to verify that $d^{3} q=a^{-3}\left[1-\nabla_{\mathbf{q}} \cdot \psi\right] d^{3} r$, and hence deduce that the fractional density perturbation is, to first order,

  $$\delta \equiv \frac{\rho-\bar{\rho}}{\bar{\rho}}=-\nabla_{\mathbf{q}} \cdot \psi \text {. }$$

  Use this result to integrate the Poisson equation $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ for the gravitational potential $\Phi$. Then use the particle equation of motion $\ddot{\mathbf{r}}=-\nabla \Phi$ to deduce a second-order differential equation for $\psi$, and hence that

  $$\ddot{\delta}+2\left(\frac{\dot{a}}{a}\right) \dot{\delta}-4 \pi G \bar{\rho} \delta=0 .$$

  [You may assume that $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ implies $\nabla \Phi=(4 \pi G / 3) \bar{\rho} \mathbf{r}$ and that the pressure-free acceleration equation is $\ddot{a}=-(4 \pi G / 3) \bar{\rho} a .]$

  (ii) A flat matter-dominated universe with background density $\bar{\rho}=\left(6 \pi G t^{2}\right)^{-1}$ has scale factor $a(t)=\left(t / t_{0}\right)^{2 / 3}$. The universe is filled with a pressure-free homogeneous (non-clumping) fluid of mass density $\rho_{H}(t)$, as well as cold dark matter of mass density $\rho_{C}(\mathbf{r}, t)$.

  Assuming that the Zel'dovich perturbation equation in this case is as in (2) but with $\bar{\rho}$ replaced by $\bar{\rho}_{C}$, i.e. that

  $$\ddot{\delta}+2\left(\frac{\dot{a}}{a}\right) \dot{\delta}-4 \pi G \bar{\rho}_{C} \delta=0,$$

  seek power-law solutions $\delta \propto t^{\alpha}$ to find growing and decaying modes with

  $$\alpha=\frac{1}{6}\left(-1 \pm \sqrt{25-24 \Omega_{H}}\right)$$

  where $\Omega_{H}=\rho_{H} / \bar{\rho}$.

  Given that matter domination starts $\left(t=t_{\text {eq }}\right)$ at a redshift $z \approx 10^{5}$, and given an initial perturbation $\delta\left(t_{\mathrm{eq}}\right) \approx 10^{-5}$, show that $\Omega_{H}=2 / 3$ yields a model that is not compatible with the large-scale structure observed today.

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Differential Geometry

• Paper 3, Section II, I

  Differential Geometry | Part II, 2011

  For an oriented surface $S$ in $\mathbb{R}^{3}$, define the Gauss map, the second fundamental form and the normal curvature in the direction $w \in T_{p} S$ at a point $p \in S$.

  Let $\tilde{k}_{1}, \ldots, \tilde{k}_{m}$ be normal curvatures at $p$ in the directions $v_{1}, \ldots, v_{m}$, such that the angle between $v_{i}$ and $v_{i+1}$ is $\pi / m$ for each $i=1, \ldots, m-1(m \geqslant 2)$. Show that

  $$\tilde{k}_{1}+\ldots+\tilde{k}_{m}=m H$$

  where $H$ is the mean curvature of $S$ at $p$.

  What is a minimal surface? Show that if $S$ is a minimal surface, then its Gauss $\operatorname{map} N$ at each point $p \in S$ satisfies

  $$\left\langle d N_{p}\left(w_{1}\right), d N_{p}\left(w_{2}\right)\right\rangle=\mu(p)\left\langle w_{1}, w_{2}\right\rangle, \quad \text { for all } w_{1}, w_{2} \in T_{p} S,$$

  where $\mu(p) \in \mathbb{R}$ depends only on $p$. Conversely, if the identity $(*)$ holds at each point in $S$, must $S$ be minimal? Justify your answer.

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Dynamical Systems

• Paper 3, Section I, C

  Dynamical Systems | Part II, 2011

  For the map $x_{n+1}=\lambda x_{n}\left(1-x_{n}^{2}\right)$, with $\lambda>0$, show the following:

  (i) If $\lambda<1$, then the origin is the only fixed point and is stable.

  (ii) If $\lambda>1$, then the origin is unstable. There are two further fixed points which are stable for $1<\lambda<2$ and unstable for $\lambda>2$.

  (iii) If $\lambda<3 \sqrt{3} / 2$, then $x_{n}$ has the same sign as the starting value $x_{0}$ if $\left|x_{0}\right|<1$.

  (iv) If $\lambda<3$, then $\left|x_{n+1}\right|<2 \sqrt{3} / 3$ when $\left|x_{n}\right|<2 \sqrt{3} / 3$. Deduce that iterates starting sufficiently close to the origin remain bounded, though they may change sign.

  [Hint: For (iii) and (iv) a graphical representation may be helpful.]

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• Paper 3, Section II, C

  Dynamical Systems | Part II, 2011

  Explain what is meant by a steady-state bifurcation of a fixed point $\mathbf{x}_{0}(\mu)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$ in $\mathbb{R}^{n}$, where $\mu$ is a real parameter.

  Consider the system in $x \geqslant 0, y \geqslant 0$, with $\mu>0$,

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

  (i) Show that both the fixed point $(0, \mu)$ and the fixed point $(1,0)$ have a steady-state bifurcation when $\mu=1$.

  (ii) By finding the first approximation to the extended centre manifold, construct the normal form near the bifurcation point $(1,0)$ when $\mu$ is close to unity, and show that there is a transcritical bifurcation there. Explain why the symmetries of the equations mean that the bifurcation at $(0,1)$ must be of pitchfork type.

  (iii) Show that two fixed points with $x, y>0$ exist in the range $1<\mu<5 / 4$. Show that the solution with $y<1 / 2$ is stable. Identify the bifurcation that occurs at $\mu=5 / 4$.

  (iv) Draw a sketch of the values of $y$ at the fixed points as functions of $\mu$, indicating the bifurcation points and the regions where each branch is stable. [Detailed calculations are not required.]

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Electrodynamics

• Paper 3, Section II, C

  Electrodynamics | Part II, 2011

  Explain how time-dependent distributions of electric charge $\rho(\mathbf{x}, t)$ and current $\mathbf{j}(\mathbf{x}, t)$ can be combined into a four-vector $j^{a}(x)$ that obeys $\partial_{a} j^{a}=0$.

  This current generates a four-vector potential $A^{a}(x)$. Explain how to find $A^{a}$ in the gauge $\partial_{a} A^{a}=0$.

  A small circular loop of wire of radius $r$ is centred at the origin. The unit vector normal to the plane of the loop is $\mathbf{n}$. A current $I_{o} \sin \omega t$ flows in the loop. Find the three-vector potential $\mathbf{A}(\mathbf{x}, t)$ to leading order in $r /|\mathbf{x}|$.

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Fluid Dynamics II

• Paper 3, Section II, B

  Fluid Dynamics II | Part II, 2011

  If $A_{i}\left(x_{j}\right)$ is harmonic, i.e. if $\nabla^{2} A_{i}=0$, show that

  $$u_{i}=A_{i}-x_{k} \frac{\partial A_{k}}{\partial x_{i}}, \quad \text { with } \quad p=-2 \mu \frac{\partial A_{n}}{\partial x_{n}},$$

  satisfies the incompressibility condition and the Stokes equation. Show that the stress tensor is

  $$\sigma_{i j}=2 \mu\left(\delta_{i j} \frac{\partial A_{n}}{\partial x_{n}}-x_{k} \frac{\partial^{2} A_{k}}{\partial x_{i} \partial x_{j}}\right)$$

  Consider the Stokes flow corresponding to

  $$A_{i}=V_{i}\left(1-\frac{a}{2 r}\right),$$

  where $V_{i}$ are the components of a constant vector $\mathbf{V}$. Show that on the sphere $r=a$ the normal component of velocity vanishes and the surface traction $\sigma_{i j} x_{j} / a$ is in the normal direction. Hence deduce that the drag force on the sphere is given by

  $$\mathbf{F}=4 \pi \mu a \mathbf{V}$$

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Further Complex Methods

• Paper 3, Section I, E

  Further Complex Methods | Part II, 2011

  Explain the meaning of $z_{j}$ in the Weierstrass canonical product formula

  $$f(z)=f(0) \exp \left[\frac{f^{\prime}(0)}{f(0)} z\right] \prod_{j=1}^{\infty}\left\{\left(1-\frac{z}{z_{j}}\right) e^{\frac{z}{z_{j}}}\right\}$$

  Show that

  $$\frac{\sin (\pi z)}{\pi z}=\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}}\right)$$

  Deduce that

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

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Galois Theory

• Paper 3, Section II, H

  Galois Theory | Part II, 2011

  Let $n \geqslant 1$ and $K=\mathbb{Q}\left(\boldsymbol{\mu}_{n}\right)$ be the cyclotomic field generated by the $n$th roots of unity. Let $a \in \mathbb{Q}$ with $a \neq 0$, and consider $F=K(\sqrt[n]{a})$.

  (i) State, without proof, the theorem which determines $\operatorname{Gal}(K / \mathbb{Q})$.

  (ii) Show that $F / \mathbb{Q}$ is a Galois extension and that $\operatorname{Gal}(F / \mathbb{Q})$ is soluble. [When using facts about general Galois extensions and their generators, you should state them clearly.]

  (iii) When $n=p$ is prime, list all possible degrees $[F: \mathbb{Q}]$, with justification.

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Geometry and Groups

• Paper 3, Section I, $3 G$

  Geometry and Groups | Part II, 2011

  Define a Kleinian group.

  Give an example of a Kleinian group that is a free group on two generators and explain why it has this property.

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Graph Theory

• Paper 3, Section II, $17 \mathrm{~F}$

  Graph Theory | Part II, 2011

  Define the Turán graph $T_{r}(n)$. State and prove Turán's theorem. Hence, or otherwise, find $\operatorname{ex}\left(K_{3} ; n\right)$.

  Let $G$ be a bipartite graph with $n$ vertices in each class. Let $k$ be an integer, $1 \leqslant k \leqslant n$, and assume $e(G)>(k-1) n$. Show that $G$ contains a set of $k$ independent edges.

  [Hint: Suppose $G$ contains a set $D$ of a independent edges but no set of a $+1$ independent edges. Let $U$ be the set of vertices of the edges in $D$ and let $F$ be the set of edges in $G$ with precisely one vertex in $U$; consider $|F| .]$

  Hence, or otherwise, show that if $H$ is a triangle-free tripartite graph with $n$ vertices in each class then $e(H) \leqslant 2 n^{2}$.

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Integrable Systems

• Paper 3, Section II, A

  Integrable Systems | Part II, 2011

  Let $U(\rho, \tau, \lambda)$ and $V(\rho, \tau, \lambda)$ be matrix-valued functions. Consider the following system of overdetermined linear partial differential equations:

  $$\frac{\partial}{\partial \rho} \psi=U \psi, \quad \frac{\partial}{\partial \tau} \psi=V \psi$$

  where $\psi$ is a column vector whose components depend on $(\rho, \tau, \lambda)$. Using the consistency condition of this system, derive the associated zero curvature representation (ZCR)

  $$\frac{\partial}{\partial \tau} U-\frac{\partial}{\partial \rho} V+[U, V]=0,$$

  where $[\cdot, \cdot]$ denotes the usual matrix commutator.

  (i) Let

  $$U=\frac{i}{2}\left(\begin{array}{cc} 2 \lambda & \partial_{\rho} \phi \\ \partial_{\rho} \phi & -2 \lambda \end{array}\right), \quad V=\frac{1}{4 i \lambda}\left(\begin{array}{cc} \cos \phi & -i \sin \phi \\ i \sin \phi & -\cos \phi \end{array}\right)$$

  Find a partial differential equation for $\phi=\phi(\rho, \tau)$ which is equivalent to the $\mathrm{ZCR}(*)$.

  (ii) Assuming that $U$ and $V$ in $(*)$ do not depend on $t:=\rho-\tau$, show that the trace of $(U-V)^{p}$ does not depend on $x:=\rho+\tau$, where $p$ is any positive integer. Use this fact to construct a first integral of the ordinary differential equation

  $$\phi^{\prime \prime}=\sin \phi, \quad \text { where } \quad \phi=\phi(x) .$$

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Linear Analysis

• Paper 3, Section II, G

  Linear Analysis | Part II, 2011

  Let $H$ be a complex Hilbert space with orthonormal basis $\left(e_{n}\right)_{n=-\infty}^{\infty}$ Let $T: H \rightarrow H$ be a bounded linear operator. What is meant by the spectrum $\sigma(T)$ of $T$ ?

  Define $T$ by setting $T\left(e_{n}\right)=e_{n-1}+e_{n+1}$ for $n \in \mathbb{Z}$. Show that $T$ has a unique extension to a bounded, self-adjoint linear operator on $H$. Determine the norm $\|T\|$. Exhibit, with proof, an element of $\sigma(T)$.

  Show that $T$ has no eigenvectors. Is $T$ compact?

  [General results from spectral theory may be used without proof. You may also use the fact that if a sequence $\left(x_{n}\right)$ satisfies a linear recurrence $\lambda x_{n}=x_{n-1}+x_{n+1}$ with $\lambda \in \mathbb{R}$, $|\lambda| \leqslant 2, \lambda \neq 0$, then it has the form $x_{n}=A \alpha^{n} \sin \left(\theta_{1} n+\theta_{2}\right)$ or $x_{n}=(A+n B) \alpha^{n}$, where $A, B, \alpha \in \mathbb{R}$ and $0 \leqslant \theta_{1}<\pi,\left|\theta_{2}\right| \leqslant \pi / 2$.]

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Logic and Set Theory

• Paper 3, Section II, H

  Logic and Set Theory | Part II, 2011

  State and prove the Upward Löwenheim-Skolem Theorem.

  [You may assume the Compactness Theorem, provided that you state it clearly.]

  A total ordering $(X,<)$ is called dense if for any $x<y$ there exists $z$ with $x<z<y$. Show that a dense total ordering (on more than one point) cannot be a well-ordering.

  For each of the following theories, either give axioms, in the language of posets, for the theory or prove carefully that the theory is not axiomatisable in the language of posets.

  (i) The theory of dense total orderings.

  (ii) The theory of countable dense total orderings.

  (iii) The theory of uncountable dense total orderings.

  (iv) The theory of well-orderings.

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Mathematical Biology

• Paper 3, Section I, B

  Mathematical Biology | Part II, 2011

  The dynamics of a directly transmitted microparasite can be modelled by the system

  $$\begin{aligned} \frac{d X}{d t} &=b N-\beta X Y-b X \\ \frac{d Y}{d t} &=\beta X Y-(b+r) Y \\ \frac{d Z}{d t} &=r Y-b Z \end{aligned}$$

  where $b, \beta$ and $r$ are positive constants and $X, Y$ and $Z$ are respectively the numbers of susceptible, infected and immune (i.e. infected by the parasite, but showing no further symptoms of infection) individuals in a population of size $N$, independent of $t$, where $N=X+Y+Z$.

  Consider the possible steady states of these equations. Show that there is a threshold population size $N_{c}$ such that if $N<N_{c}$ there is no steady state with the parasite maintained in the population. Show that in this case the number of infected and immune individuals decreases to zero for all possible initial conditions.

  Show that for $N>N_{c}$ there is a possible steady state with $X=X_{s}<N$ and $Y=Y_{s}>0$, and find expressions for $X_{s}$ and $Y_{s}$.

  By linearising the equations for $d X / d t$ and $d Y / d t$ about the steady state $X=X_{s}$ and $Y=Y_{s}$, derive a quadratic equation for the possible growth or decay rate in terms of $X_{s}$ and $Y_{s}$ and hence show that the steady state is stable.

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• Paper 3, Section II, B

  Mathematical Biology | Part II, 2011

  The number density of a population of amoebae is $n(\mathbf{x}, t)$. The amoebae exhibit chemotaxis and are attracted to high concentrations of a chemical which has concentration $a(\mathbf{x}, t)$. The equations governing $n$ and $a$ are

  $$\begin{aligned} &\frac{\partial n}{\partial t}=\alpha n\left(n_{0}^{2}-n^{2}\right)+\nabla^{2} n-\nabla \cdot(\chi(n) n \nabla a) \\ &\frac{\partial a}{\partial t}=\beta n-\gamma a+D \nabla^{2} a \end{aligned}$$

  where the constants $n_{0}, \alpha, \beta, \gamma$ and $D$ are all positive.

  (i) Give a biological interpretation of each term in these equations and discuss the sign of $\chi(n)$.

  (ii) Show that there is a non-trivial (i.e. $a \neq 0, n \neq 0$ ) steady-state solution for $n$ and $a$, independent of $\mathbf{x}$, and show further that it is stable to small disturbances that are also independent of $\mathbf{x}$.

  (iii) Consider small spatially varying disturbances to the steady state, with spatial structure such that $\nabla^{2} \psi=-k^{2} \psi$, where $\psi$ is any disturbance quantity. Show that if such disturbances also satisfy $\partial \psi / \partial t=p \psi$, where $p$ is a constant, then $p$ satisfies a quadratic equation, to be derived. By considering the conditions required for $p=0$ to be a possible solution of this quadratic equation, or otherwise, deduce that instability is possible if

  $$\beta \chi_{0} n_{0}>2 \alpha n_{0}^{2} D+\gamma+2\left(2 D \alpha n_{0}^{2} \gamma\right)^{1 / 2}$$

  where $\chi_{0}=\chi\left(n_{0}\right)$.

  (iv) Explain briefly how your conclusions might change if an additional geometric constraint implied that $k^{2}>k_{0}^{2}$, where $k_{0}$ is a given constant.

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Number Theory

• Paper 3, Section I, I

  Number Theory | Part II, 2011

  (i) State Lagrange's Theorem, and prove that, if $p$ is an odd prime,

  $$(p-1) ! \equiv-1 \quad \bmod p$$

  (ii) Still assuming $p$ is an odd prime, prove that

  $$3^{2} \cdot 5^{2} \cdots(p-2)^{2} \equiv(-1)^{\frac{p+1}{2}} \bmod p$$

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• Paper 3, Section II, I

  Number Theory | Part II, 2011

  Let $\zeta(s)$ be the Riemann zeta function, and put $s=\sigma+i t$ with $\sigma, t \in \mathbb{R}$.

  (i) If $\sigma>1$, prove that

  $$\zeta(s)=\prod_{p}\left(1-p^{-s}\right)^{-1}$$

  where the product is taken over all primes $p$.

  (ii) Assuming that, for $\sigma>1$, we have

  $$\zeta(s)=\sum_{n=1}^{\infty} n\left(n^{-s}-(n+1)^{-s}\right)$$

  prove that $\zeta(s)-\frac{1}{s-1}$ has an analytic continuation to the half plane $\sigma>0$.

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Numerical Analysis

• Paper 3, Section II, A

  Numerical Analysis | Part II, 2011

  (i) The difference equation

  $$u_{i}^{n+1}=u_{i}^{n}+\frac{3}{2} \mu\left(u_{i-1}^{n}-2 u_{i}^{n}+u_{i+1}^{n}\right)-\frac{1}{2} \mu\left(u_{i-1}^{n-1}-2 u_{i}^{n-1}+u_{i+1}^{n-1}\right)$$

  where $\mu=\Delta t /(\Delta x)^{2}$, is the basic equation used in the second-order AdamsBashforth method and can be employed to approximate a solution of the diffusion equation $u_{t}=u_{x x}$. Prove that, as $\Delta t \rightarrow 0$ with constant $\mu$, the local error of the method is $O(\Delta t)^{2}$.

  (ii) By applying the Fourier stability test, show that the above method is stable if and only if $\mu \leqslant 1 / 4$.

  (iii) Define the leapfrog scheme to approximate the diffusion equation and prove that it is unstable for every choice of $\mu>0$.

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Optimization and Control

• Paper 3, Section II, 28K

  Optimization and Control | Part II, 2011

  An observable scalar state variable evolves as $x_{t+1}=x_{t}+u_{t}, t=0,1, \ldots$ Let controls $u_{0}, u_{1}, \ldots$ be determined by a policy $\pi$ and define

  $$C_{s}\left(\pi, x_{0}\right)=\sum_{t=0}^{s-1}\left(x_{t}^{2}+2 x_{t} u_{t}+7 u_{t}^{2}\right) \quad \text { and } \quad C_{s}\left(x_{0}\right)=\inf _{\pi} C_{s}\left(\pi, x_{0}\right)$$

  Show that it is possible to express $C_{s}\left(x_{0}\right)$ in terms of $\Pi_{s}$, which satisfies the recurrence

  $$\Pi_{s}=\frac{6\left(1+\Pi_{s-1}\right)}{7+\Pi_{s-1}}, \quad s=1,2, \ldots$$

  with $\Pi_{0}=0$.

  Deduce that $C_{\infty}\left(x_{0}\right) \geqslant 2 x_{0}^{2} .\left[C_{\infty}\left(x_{0}\right)\right.$ is defined as $\left.\lim _{s \rightarrow \infty} C_{s}\left(x_{0}\right) .\right]$

  By considering the policy $\pi^{*}$ which takes $u_{t}=-(1 / 3)(2 / 3)^{t} x_{0}, t=0,1, \ldots$, show that $C_{\infty}\left(x_{0}\right)=2 x_{0}^{2}$.

  Give an alternative description of $\pi^{*}$ in closed-loop form.

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Partial Differential Equations

• Paper 3, Section II, A

  Partial Differential Equations | Part II, 2011

  (a) State the local existence theorem of a classical solution of the Cauchy problem

  $$\begin{aligned} &a\left(x_{1}, x_{2}, u\right) \frac{\partial u}{\partial x_{1}}+b\left(x_{1}, x_{2}, u\right) \frac{\partial u}{\partial x_{2}}=c\left(x_{1}, x_{2}, u\right) \\ &\left.u\right|_{\Gamma}=u_{0} \end{aligned}$$

  where $\Gamma$ is a smooth curve in $\mathbb{R}^{2}$.

  (b) Solve, by using the method of characteristics,

  $$\begin{aligned} &2 x_{1} \frac{\partial u}{\partial x_{1}}+4 x_{2} \frac{\partial u}{\partial x_{2}}=u^{2} \\ &u\left(x_{1}, 2\right)=h \end{aligned}$$

  where $h>0$ is a constant. What is the maximal domain of existence in which $u$ is a solution of the Cauchy problem?

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Principles of Quantum Mechanics

• Paper 3, Section II, D

  Principles of Quantum Mechanics | Part II, 2011

  The Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{x}, \sigma_{y}, \sigma_{z}\right)=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$, with

  $$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

  are used to represent angular momentum operators with respect to basis states $|\uparrow\rangle$ and $|\downarrow\rangle$ corresponding to spin up and spin down along the $z$-axis. They satisfy

  $$\sigma_{i} \sigma_{j}=\delta_{i j}+i \epsilon_{i j k} \sigma_{k}$$

  (i) How are $|\uparrow\rangle$ and $|\downarrow\rangle$ represented? How is the spin operator s related to $\sigma$ and $\hbar$ ? Check that the commutation relations between the spin operators are as desired. Check that $\mathbf{s}^{2}$ acting on a spin one-half state has the correct eigenvalue.

  What are the states obtained by applying $s_{x}, s_{y}$ to the eigenstates $|\uparrow\rangle$ and $|\downarrow\rangle$ of $s_{z}$ ?

  (ii) Let $V$ be the space of states for a spin one-half system. Consider a combination of three such systems with states belonging to $V^{(1)} \otimes V^{(2)} \otimes V^{(3)}$ and spin operators acting on each subsystem denoted by $s_{x}^{(i)}, s_{y}^{(i)}$ with $i=1,2,3$. Find the eigenvalues of the operators

  $$s_{x}^{(1)} s_{y}^{(2)} s_{y}^{(3)}, \quad s_{y}^{(1)} s_{x}^{(2)} s_{y}^{(3)}, \quad s_{y}^{(1)} s_{y}^{(2)} s_{x}^{(3)} \quad \text { and } \quad s_{x}^{(1)} s_{x}^{(2)} s_{x}^{(3)}$$

  of the state

  $$|\Psi\rangle=\frac{1}{\sqrt{2}}\left[|\uparrow\rangle_{1}|\uparrow\rangle_{2}|\uparrow\rangle_{3}-|\downarrow\rangle_{1}|\downarrow\rangle_{2}|\downarrow\rangle_{3}\right]$$

  (iii) Consider now whether these outcomes for measurements of particular combinations of the operators $s_{x}^{(i)}, s_{y}^{(i)}$ in the state $|\Psi\rangle$ could be reproduced by replacing the spin operators with classical variables $\tilde{s}_{x}^{(i)}, \tilde{s}_{y}^{(i)}$ which take values $\pm \hbar / 2$ according to some probabilities. Assume that these variables are identical to the quantum measurements of $s_{x}^{(1)} s_{y}^{(2)} s_{y}^{(3)}, s_{y}^{(1)} s_{x}^{(2)} s_{y}^{(3)}, s_{y}^{(1)} s_{y}^{(2)} s_{x}^{(3)}$ on $|\Psi\rangle$. Show that classically this implies a unique possibility for

  $$\tilde{s}_{x}^{(1)} \tilde{s}_{x}^{(2)} \tilde{s}_{x}^{(3)},$$

  and find its value.

  State briefly how this result could be used to experimentally test quantum mechanics.

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Principles of Statistics

• Paper 3, Section II, $27 \mathrm{~K}$

  Principles of Statistics | Part II, 2011

  Random variables $X_{1}, X_{2}, \ldots$ are independent and identically distributed from the exponential distribution $\mathcal{E}(\theta)$, with density function

  $$p_{X}(x \mid \theta)=\theta e^{-\theta x} \quad(x>0),$$

  when the parameter $\Theta$ takes value $\theta>0$. The following experiment is performed. First $X_{1}$ is observed. Thereafter, if $X_{1}=x_{1}, \ldots, X_{i}=x_{i}$ have been observed $(i \geqslant 1)$, a coin having probability $\alpha\left(x_{i}\right)$ of landing heads is tossed, where $\alpha: \mathbb{R} \rightarrow(0,1)$ is a known function and the coin toss is independent of the $X$ 's and previous tosses. If it lands heads, no further observations are made; if tails, $X_{i+1}$ is observed.

  Let $N$ be the total number of $X$ 's observed, and $\mathbf{X}:=\left(X_{1}, \ldots, X_{N}\right)$. Write down the likelihood function for $\Theta$ based on data $\mathbf{X}=\left(x_{1}, \ldots, x_{n}\right)$, and identify a minimal sufficient statistic. What does the likelihood principle have to say about inference from this experiment?

  Now consider the experiment that only records $Y:=X_{N}$. Show that the density function of $Y$ has the form

  $$p_{Y}(y \mid \theta)=\exp \{a(y)-k(\theta)-\theta y\}$$

  Assuming the function $a(\cdot)$ is twice differentiable and that both $p_{Y}(y \mid \theta)$ and $\partial p_{Y}(y \mid \theta) / \partial y$ vanish at 0 and $\infty$, show that $a^{\prime}(Y)$ is an unbiased estimator of $\Theta$, and find its variance.

  Stating clearly any general results you use, deduce that

  $$-k^{\prime \prime}(\theta) \mathbb{E}_{\theta}\left\{a^{\prime \prime}(Y)\right\} \geqslant 1 .$$

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Probability and Measure

• Paper 3, Section II, 25K

  Probability and Measure | Part II, 2011

  (i) State and prove Kolmogorov's zero-one law.

  (ii) Let $(E, \mathcal{E}, \mu)$ be a finite measure space and suppose that $\left(B_{n}\right)_{n \geqslant 1}$ is a sequence of events such that $B_{n+1} \subset B_{n}$ for all $n \geqslant 1$. Show carefully that $\mu\left(B_{n}\right) \rightarrow \mu(B)$, where $B=\cap_{n=1}^{\infty} B_{n}$.

  (iii) Let $\left(X_{i}\right)_{i \geqslant 1}$ be a sequence of independent and identically distributed random variables such that $\mathbb{E}\left(X_{1}^{2}\right)=\sigma^{2}<\infty$ and $\mathbb{E}\left(X_{1}\right)=0$. Let $K>0$ and consider the event $A_{n}$ defined by

  $$A_{n}=\left\{\frac{S_{n}}{\sqrt{n}} \geqslant K\right\}, \quad \text { where } \quad S_{n}=\sum_{i=1}^{n} X_{i}$$

  Prove that there exists $c>0$ such that for all $n$ large enough, $\mathbb{P}\left(A_{n}\right) \geqslant c$. Any result used in the proof must be stated clearly.

  (iv) Prove using the results above that $A_{n}$ occurs infinitely often, almost surely. Deduce that

  $$\limsup _{n \rightarrow \infty} \frac{S_{n}}{\sqrt{n}}=\infty$$

  almost surely.

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Representation Theory

• Paper 3, Section II, I

  Representation Theory | Part II, 2011

  Define the character $\operatorname{Ind}_{H}^{G} \psi$ of a finite group $G$ which is induced by a character $\psi$ of a subgroup $H$ of $G$.

  State and prove the Frobenius reciprocity formula for the characters $\psi$ of $H$ and $\chi$ of $G$.

  Now suppose that $H$ has index 2 in $G$. An irreducible character $\psi$ of $H$ and an irreducible character $\chi$ of $G$ are said to be 'related' if

  $$\left\langle\operatorname{Ind}_{H}^{G} \psi, \chi\right\rangle_{G}=\left\langle\psi, \operatorname{Res}_{H}^{G} \chi\right\rangle_{H}>0$$

  Show that each $\psi$ of degree $d$ is either 'monogamous' in the sense that it is related to one $\chi$ (of degree $2 d$ ), or 'bigamous' in the sense that it is related to precisely two distinct characters $\chi_{1}, \chi_{2}$ (of degree $\left.d\right)$. Show that each $\chi$ is related to one bigamous $\psi$, or to two monogamous characters $\psi_{1}, \psi_{2}$ (of the same degree).

  Write down the degrees of the complex irreducible characters of the alternating group $A_{5}$. Find the degrees of the irreducible characters of a group $G$ containing $A_{5}$ as a subgroup of index 2 , distinguishing two possible cases.

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Riemann Surfaces

• Paper 3, Section II, G

  Riemann Surfaces | Part II, 2011

  State the Classical Monodromy Theorem for analytic continuations in subdomains of the plane.

  Let $n, r$ be positive integers with $r>1$ and set $h(z)=z^{n}-1$. By removing $n$ semi-infinite rays from $\mathbb{C}$, find a subdomain $U \subset \mathbb{C}$ on which an analytic function $h^{1 / r}$ may be defined, justifying this assertion. Describe briefly a gluing procedure which will produce the Riemann surface $R$ for the complete analytic function $h^{1 / r}$.

  Let $Z$ denote the set of $n$th roots of unity and assume that the natural analytic covering map $\pi: R \rightarrow \mathbb{C} \backslash Z$ extends to an analytic map of Riemann surfaces $\tilde{\pi}: \tilde{R} \rightarrow \mathbb{C}_{\infty}$, where $\tilde{R}$ is a compactification of $R$ and $\mathbb{C}_{\infty}$ denotes the extended complex plane. Show that $\tilde{\pi}$ has precisely $n$ branch points if and only if $r$ divides $n$.

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Statistical Modelling

• Paper 3, Section I, J

  Statistical Modelling | Part II, 2011

  Define a generalised linear model for a sample $Y_{1}, \ldots, Y_{n}$ of independent random variables. Define further the concept of the link function. Define the binomial regression model with logistic and probit link functions. Which of these is the canonical link function?

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Statistical Physics

• Paper 3, Section II, D

  Statistical Physics | Part II, 2011

  A gas of non-interacting particles has energy-momentum relationship $E=A(\hbar k)^{\alpha}$ for some constants $A$ and $\alpha$. Determine the density of states $g(E) d E$ in a threedimensional volume $V$.

  Explain why the chemical potential $\mu$ satisfies $\mu<0$ for the Bose-Einstein distribution.

  Show that an ideal quantum Bose gas with the energy-momentum relationship above has

  $$p V=\frac{\alpha E}{3} .$$

  If the particles are bosons at fixed temperature $T$ and chemical potential $\mu$, write down an expression for the number of particles that do not occupy the ground state. Use this to determine the values of $\alpha$ for which there exists a Bose-Einstein condensate at sufficiently low temperatures.

  Discuss whether a gas of photons can undergo Bose-Einstein condensation.

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Stochastic Financial Models

• Paper 3, Section II, J

  Stochastic Financial Models | Part II, 2011

  First, what is a Brownian motion?

  (i) The price $S_{t}$ of an asset evolving in continuous time is represented as

  $$S_{t}=S_{0} \exp \left(\sigma W_{t}+\mu t\right)$$

  where $W$ is a standard Brownian motion, and $\sigma$ and $\mu$ are constants. If riskless investment in a bank account returns a continuously-compounded rate of interest $r$, derive a formula for the time-0 price of a European call option on the asset $S$ with strike $K$ and expiry $T$. You may use any general results, but should state them clearly.

  (ii) In the same financial market, consider now a derivative which pays

  $$Y=\left\{\exp \left(T^{-1} \int_{0}^{T} \log \left(S_{u}\right) d u\right)-K\right\}^{+}$$

  at time $T$. Find the time-0 price for this derivative. Show that it is less than the price of the European call option which you derived in (i).

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Topics in Analysis

• Paper 3, Section I, F

  Topics in Analysis | Part II, 2011

  Let $\Gamma=\{z \in \mathbb{C}: z \neq 1,|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=1\}$.

  (i) Prove that, for any $\zeta \in \mathbb{C}$ with $|\operatorname{Re}(\zeta)|+|\operatorname{Im}(\zeta)|>1$ and any $\epsilon>0$, there exists a complex polynomial $p$ such that

  $$\sup _{z \in \Gamma}\left|p(z)-(z-\zeta)^{-1}\right|<\epsilon$$

  (ii) Does there exist a sequence of polynomials $p_{n}$ such that $p_{n}(z) \rightarrow(z-1)^{-1}$ for every $z \in \Gamma ?$ Justify your answer.

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• Paper 3, Section II, F

  Topics in Analysis | Part II, 2011

  Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous and let $n$ be a positive integer. For $g:[0,1] \rightarrow \mathbb{R}$ a continuous function, write $\|f-g\|_{L^{\infty}}=\sup _{x \in[0,1]}|f(x)-g(x)|$.

  (i) Let $p$ be a polynomial of degree at most $n$ with the property that there are $(n+2)$ distinct points $x_{1}, x_{2}, \ldots, x_{n+2} \in[0,1]$ with $x_{1}<x_{2}<\ldots<x_{n+2}$ such that

  $$f\left(x_{j}\right)-p\left(x_{j}\right)=(-1)^{j}\|f-p\|_{L^{\infty}}$$

  for each $j=1,2, \ldots, n+2$. Prove that

  $$\|f-p\|_{L^{\infty}} \leqslant\|f-q\|_{L^{\infty}}$$

  for every polynomial $q$ of degree at most $n$.

  (ii) Prove that there exists a polynomial $p$ of degree at most $n$ such that

  $$\|f-p\|_{L^{\infty}} \leqslant\|f-q\|_{L^{\infty}}$$

  for every polynomial $q$ of degree at most $n$.

  [If you deduce this from a more general result about abstract normed spaces, you must prove that result.]

  (iii) Let $Y=\left\{y_{1}, y_{2}, \ldots, y_{n+2}\right\}$ be any set of $(n+2)$ distinct points in $[0,1]$.

  (a) For $j=1,2, \ldots, n+2$, let

  $$r_{j}(x)=\prod_{k=1, k \neq j}^{n+2} \frac{x-y_{k}}{y_{j}-y_{k}}$$

  $t(x)=\sum_{j=1}^{n+2} f\left(y_{j}\right) r_{j}(x)$ and $r(x)=\sum_{j=1}^{n+2}(-1)^{j} r_{j}(x)$. Explain why there is a unique number $\lambda \in \mathbb{R}$ such that the degree of the polynomial $t-\lambda r$ is at $\operatorname{most} n$.

  (b) Let $\|f-g\|_{L^{\infty}(Y)}=\sup _{x \in Y}|f(x)-g(x)|$. Deduce from part (a) that there exists a polynomial $p$ of degree at most $n$ such that

  $$\|f-p\|_{L^{\infty}(Y)} \leqslant\|f-q\|_{L^{\infty}(Y)}$$

  for every polynomial $q$ of degree at most $n$.

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Waves

• Paper 3, Section II, 38B

  Waves | Part II, 2011

  The dispersion relation in a stationary medium is given by $\omega=\Omega_{0}(\mathbf{k})$, where $\Omega_{0}$ is a known function. Show that, in the frame of reference where the medium has a uniform velocity $-\mathbf{U}$, the dispersion relation is given by $\omega=\Omega_{0}(\mathbf{k})-\mathbf{U} \cdot \mathbf{k}$.

  An aircraft flies in a straight line with constant speed $M c_{0}$ through air with sound speed $c_{0}$. If $M>1$ show that, in the reference frame of the aircraft, the steady waves lie behind it on a cone of semi-angle $\sin ^{-1}(1 / M)$. Show further that the unsteady waves are confined to the interior of the cone.

  A small insect swims with constant velocity $\mathbf{U}=(U, 0)$ over the surface of a pool of water. The resultant capillary waves have dispersion relation $\omega^{2}=T|\mathbf{k}|^{3} / \rho$ on stationary water, where $T$ and $\rho$ are constants. Show that, in the reference frame of the insect, steady waves have group velocity

  $$\mathbf{c}_{g}=U\left(\frac{3}{2} \cos ^{2} \beta-1, \frac{3}{2} \cos \beta \sin \beta\right),$$

  where $\mathbf{k} \propto(\cos \beta, \sin \beta)$. Deduce that the steady wavefield extends in all directions around the insect.

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