# Part II, 2011

### Jump to course

Paper 1, Section II, H

comment(i) Let $X$ be an affine variety over an algebraically closed field. Define what it means for $X$ to be irreducible, and show that if $U$ is a non-empty open subset of an irreducible $X$, then $U$ is dense in $X$.

(ii) Show that $n \times n$ matrices with distinct eigenvalues form an affine variety, and are a Zariski open subvariety of affine space $\mathbb{A}^{n^{2}}$ over an algebraically closed field.

(iii) Let $\operatorname{char}_{A}(x)=\operatorname{det}(x I-A)$ be the characteristic polynomial of $A$. Show that the $n \times n$ matrices $A$ such that $\operatorname{char}_{A}(A)=0$ form a Zariski closed subvariety of $\mathbb{A}^{n^{2}}$. Hence conclude that this subvariety is all of $\mathbb{A}^{n^{2}}$.

Paper 2, Section II, H

comment(i) Let $k$ be an algebraically closed field, and let $I$ be an ideal in $k\left[x_{0}, \ldots, x_{n}\right]$. Define what it means for $I$ to be homogeneous.

Now let $Z \subseteq \mathbb{A}^{n+1}$ be a Zariski closed subvariety invariant under $k^{*}=k-\{0\}$; that is, if $z \in Z$ and $\lambda \in k^{*}$, then $\lambda z \in Z$. Show that $I(Z)$ is a homogeneous ideal.

(ii) Let $f \in k\left[x_{1}, \ldots, x_{n-1}\right]$, and let $\Gamma=\left\{(x, f(x)) \mid x \in \mathbb{A}^{n-1}\right\} \subseteq \mathbb{A}^{n}$ be the graph of $f$.

Let $\bar{\Gamma}$ be the closure of $\Gamma$ in $\mathbb{P}^{n}$.

Write, in terms of $f$, the homogeneous equations defining $\bar{\Gamma}$.

Assume that $k$ is an algebraically closed field of characteristic zero. Now suppose $n=3$ and $f(x, y)=y^{3}-x^{2} \in k[x, y]$. Find the singular points of the projective surface $\bar{\Gamma}$.

Paper 3, Section II, H

commentLet $X$ 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.]

Paper 4, Section II, H

commentLet $X$ be a smooth projective curve over an algebraically closed field $k$.

State the Riemann-Roch theorem, briefly defining all the terms that appear.

Now suppose $X$ has genus 1 , and let $P_{\infty} \in X$.

Compute $\mathcal{L}\left(n P_{\infty}\right)$ for $n \leqslant 6$. Show that $\phi_{3 P_{\infty}}$ defines an isomorphism of $X$ with a smooth plane curve in $\mathbb{P}^{2}$ which is defined by a polynomial of degree 3 .

Paper 1, Section II, H

commentAre the following statements true or false? Justify your answers.

(i) If $x$ and $y$ lie in the same path-component of $X$, then $\Pi_{1}(X, x) \cong \Pi_{1}(X, y)$.

(ii) If $x$ and $y$ are two points of the Klein bottle $K$, and $u$ and $v$ are two paths from $x$ to $y$, then $u$ and $v$ induce the same isomorphism from $\Pi_{1}(K, x)$ to $\Pi_{1}(K, y)$.

(iii) $\Pi_{1}(X \times Y,(x, y))$ is isomorphic to $\Pi_{1}(X, x) \times \Pi_{1}(Y, y)$ for any two spaces $X$ and $Y$.

(iv) If $X$ and $Y$ are connected polyhedra and $H_{1}(X) \cong H_{1}(Y)$, then $\Pi_{1}(X) \cong \Pi_{1}(Y)$.

Paper 2, Section II, H

commentExplain what is meant by a covering projection. State and prove the pathlifting property for covering projections, and indicate briefly how it generalizes to a lifting property for homotopies between paths. [You may assume the Lebesgue Covering Theorem.]

Let $X$ be a simply connected space, and let $G$ be a subgroup of the group of all homeomorphisms $X \rightarrow X$. Suppose that, for each $x \in X$, there exists an open neighbourhood $U$ of $x$ such that $U \cap g[U]=\emptyset$ for each $g \in G$ other than the identity. Show that the projection $p: X \rightarrow X / G$ is a covering projection, and deduce that $\Pi_{1}(X / G) \cong G$.

By regarding $S^{3}$ as the set of all quaternions of modulus 1 , or otherwise, show that there is a quotient space of $S^{3}$ whose fundamental group is a non-abelian group of order $8 .$

Paper 3, Section II, H

commentLet $K$ 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.

Paper 4, Section II, H

commentState the Mayer-Vietoris theorem, and use it to calculate, for each integer $q>1$, the homology group of the space $X_{q}$ obtained from the unit disc $B^{2} \subseteq \mathbb{C}$ by identifying pairs of points $\left(z_{1}, z_{2}\right)$ on its boundary whenever $z_{1}^{q}=z_{2}^{q}$. [You should construct an explicit triangulation of $X_{q}$.]

Show also how the theorem may be used to calculate the homology groups of the suspension $S K$ of a connected simplicial complex $K$ in terms of the homology groups of $K$, and of the wedge union $X \vee Y$ of two connected polyhedra. Hence show that, for any finite sequence $\left(G_{1}, G_{2}, \ldots, G_{n}\right)$ of finitely-generated abelian groups, there exists a polyhedron $X$ such that $H_{0}(X) \cong \mathbb{Z}, H_{i}(X) \cong G_{i}$ for $1 \leqslant i \leqslant n$ and $H_{i}(X)=0$ for $i>n$. [You may assume the structure theorem which asserts that any finitely-generated abelian group is isomorphic to a finite direct sum of (finite or infinite) cyclic groups.]

Paper 1, Section II, E

commentIn one dimension a particle of mass $m$ and momentum $\hbar k, k>0$, is scattered by a potential $V(x)$ where $V(x) \rightarrow 0$ as $|x| \rightarrow \infty$. Incoming and outgoing plane waves of positive $(+)$ and negative $(-)$ parity are given, respectively, by

$\begin{array}{ll} I_{+}(k, x)=e^{-i k|x|}, & I_{-}(k, x)=\operatorname{sgn}(x) e^{-i k|x|} \\ O_{+}(k, x)=e^{i k|x|}, & O_{-}(k, x)=-\operatorname{sgn}(x) e^{i k|x|} \end{array}$

The scattering solutions to the time-independent Schrödinger equation with positive and negative parity incoming waves are $\psi_{+}(x)$ and $\psi_{-}(x)$, respectively. State how the asymptotic behaviour of $\psi_{+}$and $\psi_{-}$can be expressed in terms of $I_{+}, I_{-}, O_{+}, O_{-}$and the S-matrix denoted by

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

In the case where $V(x)=V(-x)$ explain briefly why you expect $S_{+-}=S_{-+}=0$.

The potential $V(x)$ is given by

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

where $V_{0}$ is a constant. In this case, 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 $\left|S_{--}\right|^{2}=1$ and explain briefly the physical meaning of this result.

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

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

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

where, to leading order in $1 /\left(U_{0} a\right)$,

$\alpha=-\frac{\pi}{U_{0} a}, \quad \gamma=\left(\frac{\pi}{U_{0} a}\right)^{2}$

Explain briefy the physical meaning of this result, and why you expect that $\gamma>0$.

Paper 2, Section II, E

commentA beam of particles of mass $m$ and momentum $p=\hbar k$, incident along the $z$-axis, is scattered by a spherically symmetric potential $V(r)$, where $V(r)=0$ for large $r$. State the boundary conditions on the wavefunction as $r \rightarrow \infty$ and hence define the scattering amplitude $f(\theta)$, where $\theta$ is the scattering angle.

Given that, for large $r$,

$e^{i k r \cos \theta}=\frac{1}{2 i k r} \sum_{l=0}^{\infty}(2 l+1)\left(e^{i k r}-(-1)^{l} e^{-i k r}\right) P_{l}(\cos \theta)$

explain how the partial-wave expansion can be used to define the phase shifts $\delta_{l}(k)(l=$ $0,1,2, \ldots)$. Furthermore, given that $d \sigma / d \Omega=|f(\theta)|^{2}$, derive expressions for $f(\theta)$ and the total cross-section $\sigma$ in terms of the $\delta_{l}$.

In a particular case $V(r)$ is given by

$V(r)=\left\{\begin{array}{cl} \infty, & r<a \\ -V_{0}, & a<r<2 a \\ 0, & r>2 a \end{array}\right.$

where $V_{0}>0$. Show that the $\mathrm{S}$-wave phase shift $\delta_{0}$ satisfies

$\tan \left(\delta_{0}\right)=\frac{k \cos (2 k a)-\kappa \cot (\kappa a) \sin (2 k a)}{k \sin (2 k a)+\kappa \cot (\kappa a) \cos (2 k a)},$

where $\kappa^{2}=2 m V_{0} / \hbar^{2}+k^{2}$.

Derive an expression for the scattering length $a_{s}$ in terms of $\kappa$. Find the values of $\kappa$ for which $\left|a_{s}\right|$ diverges and briefly explain their physical significance.

Paper 3, Section II, E

commentAn 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.

Paper 4, Section II, E

commentA particle of charge $-e$ and mass $m$ moves in a magnetic field $\boldsymbol{B}(\boldsymbol{x}, t)$ and in an electric potential $\phi(\boldsymbol{x}, t)$. The time-dependent Schrödinger equation for the particle's wavefunction $\Psi(\boldsymbol{x}, t)$ is

$i \hbar\left(\frac{\partial}{\partial t}-\frac{i e}{\hbar} \phi\right) \Psi=-\frac{\hbar^{2}}{2 m}\left(\nabla+\frac{i e}{\hbar} \boldsymbol{A}\right)^{2} \Psi$

where $\boldsymbol{A}$ is the vector potential with $\boldsymbol{B}=\boldsymbol{\nabla} \wedge \boldsymbol{A}$. Show that this equation is invariant under the gauge transformations

$\begin{array}{ll} \boldsymbol{A}(\boldsymbol{x}, t) & \rightarrow \boldsymbol{A}(\boldsymbol{x}, t)+\boldsymbol{\nabla} f(\boldsymbol{x}, t) \\ \phi(\boldsymbol{x}, t) & \rightarrow \quad \phi(\boldsymbol{x}, t)-\frac{\partial}{\partial t} f(\boldsymbol{x}, t) \end{array}$

where $f$ is an arbitrary function, together with a suitable transformation for $\Psi$ which should be stated.

Assume now that $\partial \Psi / \partial z=0$, so that the particle motion is only in the $x$ and $y$ directions. Let $\boldsymbol{B}$ be the constant field $\boldsymbol{B}=(0,0, B)$ and let $\phi=0$. In the gauge where $\boldsymbol{A}=(-B y, 0,0)$ show that the stationary states are given by

$\Psi_{k}(\boldsymbol{x}, t)=\psi_{k}(\boldsymbol{x}) e^{-i E t / \hbar}$

with

$\psi_{k}(\boldsymbol{x})=e^{i k x} \chi_{k}(y)$

Show that $\chi_{k}(y)$ is the wavefunction for a simple one-dimensional harmonic oscillator centred at position $y_{0}=\hbar k / e B$. Deduce that the stationary states lie in infinitely degenerate levels (Landau levels) labelled by the integer $n \geqslant 0$, with energy

$E_{n}=(2 n+1) \frac{\hbar e B}{2 m}$

A uniform electric field $\mathcal{E}$ is applied in the $y$-direction so that $\phi=-\mathcal{E} y$. Show that the stationary states are given by $(*)$, where $\chi_{k}(y)$ is a harmonic oscillator wavefunction centred now at

$y_{0}=\frac{1}{e B}\left(\hbar k-m \frac{\mathcal{E}}{B}\right)$

Show also that the eigen-energies are given by

$E_{n, k}=(2 n+1) \frac{\hbar e B}{2 m}+e \mathcal{E} y_{0}+\frac{m \mathcal{E}^{2}}{2 B^{2}} .$

Why does this mean that the Landau energy levels are no longer degenerate in two dimensions?

Paper 1, Section II, J

comment(i) Let $X$ be a Markov chain with finitely many states. Define a stopping time and state the strong Markov property.

(ii) Let $X$ be a Markov chain with state-space $\{-1,0,1\}$ and Q-matrix

$Q=\left(\begin{array}{ccc} -(q+\lambda) & \lambda & q \\ 0 & 0 & 0 \\ q & \lambda & -(q+\lambda) \end{array}\right), \text { where } q, \lambda>0$

Consider the integral $\int_{0}^{t} X(s) \mathrm{d} s$, the signed difference between the times spent by the chain at states $+1$ and $-1$ by time $t$, and let

$\begin{aligned} Y &=\sup \left[\int_{0}^{t} X(s) \mathrm{d} s: t>0\right] \\ \psi_{\pm}(c) &=\mathbb{P}\left(Y>c \mid X_{0}=\pm 1\right), \quad c>0 \end{aligned}$

Derive the equation

$\psi_{-}(c)=\int_{0}^{\infty} q e^{-(\lambda+q) u_{+}} \psi_{+}(c+u) \mathrm{d} u$

(iii) Obtain another equation relating $\psi_{+}$to $\psi_{-}$.

(iv) Assuming that $\psi_{+}(c)=e^{-c A}, c>0$, where $A$ is a non-negative constant, calculate $A$.

(v) Give an intuitive explanation why the function $\psi_{+}$must have the exponential form $\psi_{+}(c)=e^{-c A}$ for some $A$.

Paper 2, Section II, J

comment(i) Explain briefly what is meant by saying that a continuous-time Markov chain $X(t)$ is a birth-and-death process with birth rates $\lambda_{i}>0, i \geqslant 0$, and death rates $\mu_{i}>0$, $i \geqslant 1$.

(ii) In the case where $X(t)$ is recurrent, find a sufficient condition on the birth and death parameters to ensure that

$\lim _{t \rightarrow \infty} \mathbb{P}(X(t)=j)=\pi_{j}>0, \quad j \geqslant 0$

and express $\pi_{j}$ in terms of these parameters. State the reversibility property of $X(t)$.

Jobs arrive according to a Poisson process of rate $\lambda>0$. They are processed individually, by a single server, the processing times being independent random variables, each with the exponential distribution of rate $\nu>0$. After processing, the job either leaves the system, with probability $p, 0<p<1$, or, with probability $1-p$, it splits into two separate jobs which are both sent to join the queue for processing again. Let $X(t)$ denote the number of jobs in the system at time $t$.

(iii) In the case $1+\lambda / \nu<2 p$, evaluate $\lim _{t \rightarrow \infty} \mathbb{P}(X(t)=j), j=0,1, \ldots$, and find the expected time that the processor is busy between two successive idle periods.

(iv) What happens if $1+\lambda / \nu \geqslant 2 p$ ?

Paper 3, Section II, J

comment(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.

Paper 4, Section II, J

commentAt an $\mathrm{M} / \mathrm{G} / 1$ queue, the arrival times form a Poisson process of rate $\lambda$ while service times $S_{1}, S_{2}, \ldots$ are independent of each other and of the arrival times and have a common distribution $G$ with mean $\mathbb{E} S_{1}<+\infty$.

(i) Show that the random variables $Q_{n}$ giving the number of customers left in the queue at departure times form a Markov chain.

(ii) Specify the transition probabilities of this chain as integrals in $\mathrm{d} G(t)$ involving parameter $\lambda$. [No proofs are needed.]

(iii) Assuming that $\rho=\lambda \mathbb{E} S_{1}<1$ and the chain $\left(Q_{n}\right)$ is positive recurrent, show that its stationary distribution $\left(\pi_{k}, k \geqslant 0\right)$ has the generating function given by

$\sum_{k \geqslant 0} \pi_{k} s^{k}=\frac{(1-\rho)(s-1) g(s)}{s-g(s)},|s| \leqslant 1$

for an appropriate function $g$, to be specified.

(iv) Deduce that, in equilibrium, $Q_{n}$ has the mean value

$\rho+\frac{\lambda^{2} \mathbb{E} S_{1}^{2}}{2(1-\rho)}$

Paper 1, Section II, A

commentA function $f(n)$, defined for positive integer $n$, has an asymptotic expansion for large $n$ of the following form:

$f(n) \sim \sum_{k=0}^{\infty} a_{k} \frac{1}{n^{2 k}}, \quad n \rightarrow \infty$

What precisely does this mean?

Show that the integral

$I(n)=\int_{0}^{2 \pi} \frac{\cos n t}{1+t^{2}} d t$

has an asymptotic expansion of the form $(*)$. [The Riemann-Lebesgue lemma may be used without proof.] Evaluate the coefficients $a_{0}, a_{1}$ and $a_{2}$.

Paper 3, Section II, A

commentLet

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

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 $n$.

Paper 4, Section II, A

commentDetermine the range of the integer $n$ for which the equation

$\frac{d^{2} y}{d z^{2}}=z^{n} y$

has an essential singularity at $z=\infty$.

Use the Liouville-Green method to find the leading asymptotic approximation to two independent solutions of

$\frac{d^{2} y}{d z^{2}}=z^{3} y$

for large $|z|$. Find the Stokes lines for these approximate solutions. For what range of $\arg z$ is the approximate solution which decays exponentially along the positive $z$-axis an asymptotic approximation to an exact solution with this exponential decay?

Paper 1, Section I, C

comment(i) A particle of mass $m$ and charge $q$, at position $\mathbf{x}$, moves in an electromagnetic field with scalar potential $\phi(\mathbf{x}, t)$ and vector potential $\mathbf{A}(\mathbf{x}, t)$. Verify that the Lagrangian

$L=\frac{1}{2} m \dot{\mathbf{x}}^{2}-q(\phi-\dot{\mathbf{x}} \cdot \mathbf{A})$

gives the correct equations of motion.

[Note that $\mathbf{E}=-\nabla \phi-\dot{\mathbf{A}}$ and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$.]

(ii) Consider the case of a constant uniform magnetic field, with $\mathbf{E}=\mathbf{0}$, given by $\phi=0$, $\mathbf{A}=(0, x B, 0)$, where $(x, y, z)$ are Cartesian coordinates and $B$ is a constant. Find the motion of the particle, and describe it carefully.

Paper 2, Section I, C

commentThree particles, each of mass $m$, move along a straight line. Their positions on the line containing the origin, $O$, are $x_{1}, x_{2}$ and $x_{3}$. They are subject to forces derived from the potential energy function

$V=\frac{1}{2} m \Omega^{2}\left[\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\left(x_{3}-x_{1}\right)^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right]$

Obtain Lagrange's equations for the system, and show that the frequency, $\omega$, of a normal mode satisfies

$f^{3}-9 f^{2}+24 f-16=0$

where $f=\left(\omega^{2} / \Omega^{2}\right)$. Find a complete set of normal modes for the system, and draw a diagram indicating the nature of the corresponding motions.

Paper 2, Section II, C

commentDerive Euler's equations governing the torque-free and force-free motion of a rigid body with principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$, where $I_{1}<I_{2}<I_{3}$. Identify two constants of the motion. Hence, or otherwise, find the equilibrium configurations such that the angular-momentum vector, as measured with respect to axes fixed in the body, remains constant. Discuss the stability of these configurations.

A spacecraft may be regarded as moving in a torque-free and force-free environment. Nevertheless, flexing of various parts of the frame can cause significant dissipation of energy. How does the angular-momentum vector ultimately align itself within the body?

Paper 3, Section I, $\mathbf{9 C}$

commentThe 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 $F(\mu) .$ ]

Paper 4, Section I, C

comment(i) A dynamical system is described by the Hamiltonian $H\left(q_{i}, p_{i}\right)$. Define the Poisson bracket $\{f, g\}$ of two functions $f\left(q_{i}, p_{i}, t\right), g\left(q_{i}, p_{i}, t\right)$. Assuming the Hamiltonian equations of motion, find an expression for $d f / d t$ in terms of the Poisson bracket.

(ii) A one-dimensional system has the Hamiltonian

$H=p^{2}+\frac{1}{q^{2}}$

Show that $u=p q-2 H t$ is a constant of the motion. Deduce the form of $(q(t), p(t))$ along a classical path, in terms of the constants $u$ and $H$.

Paper 4, Section II, C

commentGiven a Hamiltonian system with variables $\left(q_{i}, p_{i}\right), i=1, \ldots, n$, state the definition of a canonical transformation

$\left(q_{i}, p_{i}\right) \rightarrow\left(Q_{i}, P_{i}\right),$

where $\mathbf{Q}=\mathbf{Q}(\mathbf{q}, \mathbf{p}, t)$ and $\mathbf{P}=\mathbf{P}(\mathbf{q}, \mathbf{p}, t)$. Write down a matrix equation that is equivalent to the condition that the transformation is canonical.

Consider a harmonic oscillator of unit mass, with Hamiltonian

$H=\frac{1}{2}\left(p^{2}+\omega^{2} q^{2}\right) .$

Write down the Hamilton-Jacobi equation for Hamilton's principal function $S(q, E, t)$, and deduce the Hamilton-Jacobi equation

$\frac{1}{2}\left[\left(\frac{\partial W}{\partial q}\right)^{2}+\omega^{2} q^{2}\right]=E$

for Hamilton's characteristic function $W(q, E)$.

Solve (1) to obtain an integral expression for $W$, and deduce that, at energy $E$,

$S=\sqrt{2 E} \int d q \sqrt{\left(1-\frac{\omega^{2} q^{2}}{2 E}\right)}-E t$

Let $\alpha=E$, and define the angular coordinate

$\beta=\left(\frac{\partial S}{\partial E}\right)_{q, t}$

You may assume that (2) implies

$t+\beta=\left(\frac{1}{\omega}\right) \arcsin \left(\frac{\omega q}{\sqrt{2 E}}\right)$

Deduce that

$p=\frac{\partial S}{\partial q}=\frac{\partial W}{\partial q}=\sqrt{\left(2 E-\omega^{2} q^{2}\right)}$

from which

$p=\sqrt{2 E} \cos [\omega(t+\beta)] .$

Hence, or otherwise, show that the transformation from variables $(q, p)$ to $(\alpha, \beta)$ is canonical.

Paper 1, Section I, G

commentI think of an integer $n$ with $1 \leqslant n \leqslant 10^{6}$. Explain how to find $n$ using twenty questions (or less) of the form 'Is it true that $n \geqslant m$ ?' to which I answer yes or no.

I have watched a horse race with 15 horses. Is it possible to discover the order in which the horses finished by asking me twenty questions to which I answer yes or no?

Roughly how many questions of the yes/no type are required to discover the order in which $n$ horses finished if $n$ is large?

[You may assume that I answer honestly.]

Paper 1, Section II, $12 G$

commentDescribe the Rabin-Williams coding scheme. Show that any method for breaking it will enable us to factorise the product of two primes.

Explain how the Rabin-Williams scheme can be used for bit sharing (that is to say 'tossing coins by phone').

Paper 2, Section I, G

commentI happen to know that an apparently fair coin actually has probability $p$ of heads with $1>p>1 / 2$. I play a very long sequence of games of heads and tails in which my opponent pays me back twice my stake if the coin comes down heads and takes my stake if the coin comes down tails. I decide to bet a proportion $\alpha$ of my fortune at the end of the $n$th game in the $(n+1)$ st game. Determine, giving justification, the value $\alpha_{0}$ maximizing the expected logarithm of my fortune in the long term, assuming I use the same $\alpha_{0}$ at each game. Can it be actually disadvantageous for me to choose an $\alpha<\alpha_{0}$ (in the sense that I would be better off not playing)? Can it be actually disadvantageous for me to choose an $\alpha>\alpha_{0}$ ?

[Moral issues should be ignored.]

Paper 2, Section II, G

commentDefine a cyclic code. Show that there is a bijection between the cyclic codes of length $n$ and the factors of $X^{n}-1$ over the field $\mathbb{F}_{2}$ of order 2 .

What is meant by saying that $\alpha$ is a primitive $n$th root of unity in a finite field extension $K$ of $\mathbb{F}_{2}$ ? What is meant by saying that $C$ is a BCH code of length $n$ with defining set $\left\{\alpha, \alpha^{2}, \ldots, \alpha^{\delta-1}\right\}$ ? Show that such a code has minimum distance at least $\delta$.

Suppose that $K$ is a finite field extension of $\mathbb{F}_{2}$ in which $X^{7}-1$ factorises into linear factors. Show that if $\beta$ is a root of $X^{3}+X^{2}+1$ then $\beta$ is a primitive 7 th root of unity and $\beta^{2}$ is also a root of $X^{3}+X^{2}+1$. Quoting any further results that you need show that the $\mathrm{BCH}$ code of length 7 with defining set $\left\{\beta, \beta^{2}\right\}$ is the Hamming code.

[Results on the Vandermonde determinant may be used without proof provided they are quoted correctly.]

Paper 3, Section I, G

commentWhat 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 $n$.

Paper 4, Section I, G

commentDescribe a scheme for sending messages based on quantum theory which is not vulnerable to eavesdropping. You may ignore engineering problems.

Paper 1, Section I, E

commentLight of wavelength $\lambda_{e}$ emitted by a distant object is observed by us to have wavelength $\lambda_{0}$. The redshift $z$ of the object is defined by

$1+z=\frac{\lambda_{0}}{\lambda_{e}}$

Assuming that the object is at a fixed comoving distance from us in a homogeneous and isotropic universe with scale factor $a(t)$, show that

$1+z=\frac{a\left(t_{0}\right)}{a\left(t_{e}\right)}$

where $t_{e}$ is the time of emission and $t_{0}$ the time of observation (i.e. today).

[You may assume the non-relativistic Doppler shift formula $\Delta \lambda / \lambda=(v / c) \cos \theta$ for the shift $\Delta \lambda$ in the wavelength of light emitted by a nearby object travelling with velocity $v$ at angle $\theta$ to the line of sight.]

Given that the object radiates energy $L$ per unit time, explain why the rate at which energy passes through a sphere centred on the object and intersecting the Earth is $L /(1+z)^{2}$.

Paper 1, Section II, E

commentA homogeneous and isotropic universe, with scale factor $a$, curvature parameter $k$, energy density $\rho$ and pressure $P$, satisfies the Friedmann and energy conservation equations

$\begin{aligned} &H^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho \\ &\dot{\rho}+3 H\left(\rho+P / c^{2}\right)=0 \end{aligned}$

where $H=\dot{a} / a$, and the dot indicates a derivative with respect to cosmological time $t$.

(i) Derive the acceleration equation

$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)$

Given that the strong energy condition $\rho c^{2}+3 P \geqslant 0$ is satisfied, show that $(a H)^{2}$ is a decreasing function of $t$ in an expanding universe. Show also that the density parameter $\Omega=8 \pi G \rho /\left(3 H^{2}\right)$ satisfies

$\Omega-1=\frac{k c^{2}}{a^{2} H^{2}}$

Hence explain, briefly, the flatness problem of standard big bang cosmology.

(ii) A flat $(k=0)$ homogeneous and isotropic universe is filled with a radiation fluid $\left(w_{R}=1 / 3\right)$ and a dark energy fluid $\left(w_{\Lambda}=-1\right)$, each with an equation of state of the form $P_{i}=w_{i} \rho_{i} c^{2}$ and density parameters today equal to $\Omega_{R 0}$ and $\Omega_{\Lambda 0}$ respectively. Given that each fluid independently obeys the energy conservation equation, show that the total energy density $\left(\rho_{R}+\rho_{\Lambda}\right) c^{2}$ equals $\rho c^{2}$, where

$\rho(t)=\frac{3 H_{0}^{2}}{8 \pi G} \frac{\Omega_{R 0}}{a^{4}}\left(1+\frac{1-\Omega_{R 0}}{\Omega_{R 0}} a^{4}\right)$

with $H_{0}$ being the value of the Hubble parameter today. Hence solve the Friedmann equation to get

$a(t)=\alpha(\sinh \beta t)^{1 / 2}$

where $\alpha$ and $\beta$ should be expressed in terms $\Omega_{R 0}$ and $\Omega_{\Lambda 0}$. Show that this result agrees with the expected asymptotic solutions at both early $(t \rightarrow 0)$ and late $(t \rightarrow \infty)$ times.

[Hint: $\int d x / \sqrt{x^{2}+1}=\operatorname{arcsinh} x$.]

Paper 2, Section I, E

commentA spherically symmetric star in hydrostatic equilibrium has density $\rho(r)$ and pressure $P(r)$, which satisfy the pressure support equation,

$\frac{d P}{d r}=-\frac{G m \rho}{r^{2}}$

where $m(r)$ is the mass within a radius $r$. Show that this implies

$\frac{d}{d r}\left(\frac{r^{2}}{\rho} \frac{d P}{d r}\right)=-4 \pi G r^{2} \rho .$

Provide a justification for choosing the boundary conditions $d P / d r=0$ at the centre of the $\operatorname{star}(r=0)$ and $P=0$ at its outer radius $(r=R)$.

Use the pressure support equation $(*)$ to derive the virial theorem for a star,

$\langle P\rangle V=-\frac{1}{3} E_{\mathrm{grav}}$

where $\langle P\rangle$ is the average pressure, $V$ is the total volume of the star and $E_{\text {grav }}$ is its total gravitational potential energy.

Paper 3, Section I, E

commentFor 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.

Paper 3, Section II, E

commentAn 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.

Paper 4, Section I, 10E

commentThe equilibrium number density of fermions at temperature $T$ is

$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{\infty} \frac{p^{2} d p}{\exp [(\epsilon(p)-\mu) / k T]+1}$

where $g_{s}$ is the spin degeneracy and $\epsilon(p)=c \sqrt{p^{2}+m^{2} c^{2}}$. For a non-relativistic gas with $p c \ll m c^{2}$ and $k T \ll m c^{2}-\mu$, show that the number density becomes

$n=g_{s}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} \exp \left[\left(\mu-m c^{2}\right) / k T\right]$

[You may assume that $\int_{0}^{\infty} d x x^{2} e^{-x^{2} / \alpha}=(\sqrt{\pi} / 4) \alpha^{3 / 2}$ for $\alpha>0$.]

Before recombination, equilibrium is maintained between neutral hydrogen, free electrons, protons and photons through the interaction

$p+e^{-} \leftrightarrow H+\gamma$

Using the non-relativistic number density $(*)$, deduce Saha's equation relating the electron and hydrogen number densities,

$\frac{n_{e}^{2}}{n_{H}} \approx\left(\frac{2 \pi m_{e} k T}{h^{2}}\right)^{3 / 2} \exp (-I / k T)$

where $I=\left(m_{p}+m_{e}-m_{H}\right) c^{2}$ is the ionization energy of hydrogen. State clearly any assumptions you have made.

Paper 1, Section II, I

commentLet $X$ and $Y$ be manifolds and $f: X \rightarrow Y$ a smooth map. Define the notions critical point, critical value, regular value of $f$. Prove that if $y$ is a regular value of $f$, then $f^{-1}(y)$ (if non-empty) is a smooth manifold of $\operatorname{dimension} \operatorname{dim} X-\operatorname{dim} Y$.

[The Inverse Function Theorem may be assumed without proof if accurately stated.]

Let $M_{n}(\mathbb{R})$ be the set of all real $n \times n$ matrices and $\operatorname{SO}(n) \subset M_{n}(\mathbb{R})$ the group of all orthogonal matrices with determinant 1 . Show that $\mathrm{SO}(n)$ is a smooth manifold and find its dimension.

Show further that $\mathrm{SO}(n)$ is compact and that its tangent space at $A \in \operatorname{SO}(n)$ is given by all matrices $H$ such that $A H^{t}+H A^{t}=0$.

Paper 2, Section II, I

commentLet $\alpha: I \rightarrow \mathbb{R}^{3}$ be a smooth curve parametrized by arc-length, with $\alpha^{\prime \prime}(s) \neq 0$ for all $s \in I$. Define what is meant by the Frenet frame $t(s), n(s), b(s)$, the curvature and torsion of $\alpha$. State and prove the Frenet formulae.

By considering $\langle\alpha, t \times n\rangle$, or otherwise, show that, if for each $s \in I$ the vectors $\alpha(s)$, $t(s)$ and $n(s)$ are linearly dependent, then $\alpha(s)$ is a plane curve.

State and prove the isoperimetric inequality for $C^{1}$ regular plane curves.

[You may assume Wirtinger's inequality, provided you state it accurately.]

Paper 3, Section II, I

commentFor 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.

Paper 4, Section II, I

commentDefine what is meant by a geodesic. Let $S \subset \mathbb{R}^{3}$ be an oriented surface. Define the geodesic curvature $k_{g}$ of a smooth curve $\gamma: I \rightarrow S$ parametrized by arc-length.

Explain without detailed proofs what are the exponential map $\exp _{p}$ and the geodesic polar coordinates $(r, \theta)$ at $p \in S$. Determine the derivative $d\left(\exp _{p}\right)_{0}$. Prove that the coefficients of the first fundamental form of $S$ in the geodesic polar coordinates satisfy

$E=1, \quad F=0, \quad G(0, \theta)=0, \quad(\sqrt{G})_{r}(0, \theta)=1$

State the global Gauss-Bonnet formula for compact surfaces with boundary. [You should identify all terms in the formula.]

Suppose that $S$ is homeomorphic to a cylinder $S^{1} \times \mathbb{R}$ and has negative Gaussian curvature at each point. Prove that $S$ has at most one simple (i.e. without selfintersections) closed geodesic.

[Basic properties of geodesics may be assumed, if accurately stated.]

Paper 1, Section I, C

commentFind the fixed points of the dynamical system (with $\mu \neq 0$ )

$\begin{aligned} &\dot{x}=\mu^{2} x-x y \\ &\dot{y}=-y+x^{2} \end{aligned}$

and determine their type as a function of $\mu$.

Find the stable and unstable manifolds of the origin correct to order $4 .$

Paper 2, Section I, C

commentState the Poincaré-Bendixson theorem for two-dimensional dynamical systems.

A dynamical system can be written in polar coordinates $(r, \theta)$ as

$\begin{aligned} &\dot{r}=r-r^{3}(1+\alpha \cos \theta) \\ &\dot{\theta}=1-r^{2} \beta \cos \theta \end{aligned}$

where $\alpha$ and $\beta$ are constants with $0<\alpha<1$.

Show that trajectories enter the annulus $(1+\alpha)^{-1 / 2}<r<(1-\alpha)^{-1 / 2}$.

Show that if there is a fixed point $\left(r_{0}, \theta_{0}\right)$ inside the annulus then $r_{0}^{2}=(\beta-\alpha) / \beta$ and $\cos \theta_{0}=1 /(\beta-\alpha)$.

Use the Poincaré-Bendixson theorem to derive conditions on $\beta$ that guarantee the existence of a periodic orbit.

Paper 3, Section I, C

commentFor 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.]

Paper 3, Section II, C

Explain what is meant by a steady-state bifurcation of a fixed point $\mathbf{x}_{0}(\mu)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$