# Part II, 2009

### Jump to course

Paper 1, Section II, G

commentDefine what is meant by a rational map from a projective variety $V \subset \mathbb{P}^{n}$ to $\mathbb{P}^{m}$. What is a regular point of a rational map?

Consider the rational map $\phi: \mathbb{P}^{2}-\rightarrow \mathbb{P}^{2}$ given by

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

Show that $\phi$ is not regular at the points $(1: 0: 0),(0: 1: 0),(0: 0: 1)$ and that it is regular elsewhere, and that it is a birational map from $\mathbb{P}^{2}$ to itself.

Let $V \subset \mathbb{P}^{2}$ be the plane curve given by the vanishing of the polynomial $X_{0}^{2} X_{1}^{3}+X_{1}^{2} X_{2}^{3}+X_{2}^{2} X_{0}^{3}$ over a field of characteristic zero. Show that $V$ is irreducible, and that $\phi$ determines a birational equivalence between $V$ and a nonsingular plane quartic.

Paper 2, Section II, G

commentLet $V$ be an irreducible variety over an algebraically closed field $k$. Define the tangent space of $V$ at a point $P$. Show that for any integer $r \geqslant 0$, the set $\left\{P \in V \mid \operatorname{dim} T_{V, P} \geqslant r\right\}$ is a closed subvariety of $V$.

Assume that $k$ has characteristic different from 2. Let $V=V(I) \subset \mathbb{P}^{4}$ be the variety given by the ideal $I=(F, G) \subset k\left[X_{0}, \ldots, X_{4}\right]$, where

$F=X_{1} X_{2}+X_{3} X_{4}, \quad G=X_{0} X_{1}+X_{3}^{2}+X_{4}^{2}$

Determine the singular subvariety of $V$, and compute $\operatorname{dim} T_{V, P}$ at each singular point $P$. [You may assume that $V$ is irreducible.]

Paper 3, Section II, G

commentLet $V$ be a smooth projective curve, and let $D$ be an effective divisor on $V$. Explain how $D$ defines a morphism $\phi_{D}$ from $V$ to some projective space. State the necessary and sufficient conditions for $\phi_{D}$ to be finite. State the necessary and sufficient conditions for $\phi_{D}$ to be an isomorphism onto its image.

Let $V$ have genus 2 , and let $K$ be an effective canonical divisor. Show that the morphism $\phi_{K}$ is a morphism of degree 2 from $V$ to $\mathbb{P}^{1}$.

By considering the divisor $K+P_{1}+P_{2}$ for points $P_{i}$ with $P_{1}+P_{2} \nsim K$, show that there exists a birational morphism from $V$ to a singular plane quartic.

[You may assume the Riemann-Roch Theorem.]

Paper 4, Section II, G

commentState the Riemann-Roch theorem for a smooth projective curve $V$, and use it to outline a proof of the Riemann-Hurwitz formula for a non-constant morphism between projective nonsingular curves in characteristic zero.

Let $V \subset \mathbb{P}^{2}$ be a smooth projective plane cubic over an algebraically closed field $k$ of characteristic zero, written in normal form $X_{0} X_{2}^{2}=F\left(X_{0}, X_{1}\right)$ for a homogeneous cubic polynomial $F$, and let $P_{0}=(0: 0: 1)$ be the point at infinity. Taking the group law on $V$ for which $P_{0}$ is the identity element, let $P \in V$ be a point of order 3 . Show that there exists a linear form $H \in k\left[X_{0}, X_{1}, X_{2}\right]$ such that $V \cap V(H)=\{P\}$.

Let $H_{1}, H_{2} \in k\left[X_{0}, X_{1}, X_{2}\right]$ be nonzero linear forms. Suppose the lines $\left\{H_{i}=0\right\}$ are distinct, do not meet at a point of $V$, and are nowhere tangent to $V$. Let $W \subset \mathbb{P}^{3}$ be given by the vanishing of the polynomials

$X_{0} X_{2}^{2}-F\left(X_{0}, X_{1}\right), \quad X_{3}^{2}-H_{1}\left(X_{0}, X_{1}, X_{2}\right) H_{2}\left(X_{0}, X_{1}, X_{2}\right)$

Show that $W$ has genus 4 . [You may assume without proof that $W$ is an irreducible smooth curve.]

Paper 1, Section II, G

commentLet $X$ be the space obtained by identifying two copies of the Möbius strip along their boundary. Use the Seifert-Van Kampen theorem to find a presentation of the fundamental group $\pi_{1}(X)$. Show that $\pi_{1}(X)$ is an infinite non-abelian group.

Paper 2, Section II, $21 G$

commentLet $p: X \rightarrow Y$ be a connected covering map. Define the notion of a deck transformation (also known as covering transformation) for $p$. What does it mean for $p$ to be a regular (normal) covering map?

If $p^{-1}(y)$ contains $n$ points for each $y \in Y$, we say $p$ is $n$-to-1. Show that $p$ is regular under either of the following hypotheses:

(1) $p$ is 2-to-1,

(2) $\pi_{1}(Y)$ is abelian.

Give an example of a 3 -to-1 cover of $S^{1} \vee S^{1}$ which is regular, and one which is not regular.

Paper 3, Section II, G

comment(i) Suppose that $(C, d)$ and $\left(C^{\prime}, d^{\prime}\right)$ are chain complexes, and $f, g: C \rightarrow C^{\prime}$ are chain maps. Define what it means for $f$ and $g$ to be chain homotopic.

Show that if $f$ and $g$ are chain homotopic, and $f_{*}, g_{*}: H_{*}(C) \rightarrow H_{*}\left(C^{\prime}\right)$ are the induced maps, then $f_{*}=g_{*}$.

(ii) Define the Euler characteristic of a finite chain complex.

Given that one of the sequences below is exact and the others are not, which is the exact one?

$\begin{aligned} &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{25} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \\ &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \\ &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{19} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{23} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \end{aligned}$

Justify your choice.

Paper 4, Section II, G

commentLet $X$ be the subset of $\mathbb{R}^{4}$ given by $X=A \cup B \cup C \subset \mathbb{R}^{4}$, where $A, B$ and $C$ are defined as follows:

$\begin{aligned} &A=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{R}^{4}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\right\} \\ &B=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{R}^{4}: x_{1}=x_{2}=0, x_{3}^{2}+x_{4}^{2} \leqslant 1\right\} \\ &C=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{R}^{4}: x_{3}=x_{4}=0, x_{1}^{2}+x_{2}^{2} \leqslant 1\right\} \end{aligned}$

Compute $H_{*}(X)$

Paper 1, Section II, $34 \mathrm{D}$

commentConsider the scaled one-dimensional Schrödinger equation with a potential $V(x)$ such that there is a complete set of real, normalized bound states $\psi_{n}(x), n=0,1,2, \ldots$, with discrete energies $E_{0}<E_{1}<E_{2}<\ldots$, satisfying

$-\frac{d^{2} \psi_{n}}{d x^{2}}+V(x) \psi_{n}=E_{n} \psi_{n}$

Show that the quantity

$\langle E\rangle=\int_{-\infty}^{\infty}\left(\left(\frac{d \psi}{d x}\right)^{2}+V(x) \psi^{2}\right) d x$

where $\psi(x)$ is a real, normalized trial function depending on one or more parameters $\alpha$, can be used to estimate $E_{0}$, and show that $\langle E\rangle \geqslant E_{0}$.

Let the potential be $V(x)=|x|$. Using a suitable one-parameter family of either Gaussian or piecewise polynomial trial functions, find a good estimate for $E_{0}$ in this case.

How could you obtain a good estimate for $E_{1}$ ? [ You should suggest suitable trial functions, but DO NOT carry out any further integration.]

Paper 2, Section II, D

commentA particle scatters quantum mechanically off a spherically symmetric potential $V(r)$. In the $l=0$ sector, and assuming $\hbar^{2} / 2 m=1$, the radial wavefunction $u(r)$ satisfies

$-\frac{d^{2} u}{d r^{2}}+V(r) u=k^{2} u$

and $u(0)=0$. The asymptotic behaviour of $u$, for large $r$, is

$u(r) \sim C\left(S(k) e^{i k r}-e^{-i k r}\right)$

where $C$ is a constant. Show that if $S(k)$ is analytically continued to complex $k$, then

$S(k) S(-k)=1 \quad \text { and } \quad S(k)^{*} S\left(k^{*}\right)=1$

Deduce that for real $k, S(k)=e^{2 i \delta_{0}(k)}$ for some real function $\delta_{0}(k)$, and that $\delta_{0}(k)=-\delta_{0}(-k) .$

For a certain potential,

$S(k)=\frac{(k+i \lambda)(k+3 i \lambda)}{(k-i \lambda)(k-3 i \lambda)}$

where $\lambda$ is a real, positive constant. Evaluate the scattering length $a$ and the total cross section $4 \pi a^{2}$.

Briefly explain the significance of the zeros of $S(k)$.

Paper 3, Section II, D

commentAn electron of charge $-e$ and mass $m$ is subject to a magnetic field of the form $\mathbf{B}=(0,0, B(y))$, where $B(y)$ is everywhere greater than some positive constant $B_{0}$. In a stationary state of energy $E$, the electron's wavefunction $\Psi$ satisfies

$-\frac{\hbar^{2}}{2 m}\left(\boldsymbol{\nabla}+\frac{i e}{\hbar} \mathbf{A}\right)^{2} \Psi+\frac{e \hbar}{2 m} \mathbf{B} \cdot \boldsymbol{\sigma} \Psi=E \Psi,$

where $\mathbf{A}$ is the vector potential and $\sigma_{1}, \sigma_{2}$ and $\sigma_{3}$ are the Pauli matrices.

Assume that the electron is in a spin down state and has no momentum along the $z$-axis. Show that with a suitable choice of gauge, and after separating variables, equation (*) can be reduced to

$-\frac{d^{2} \chi}{d y^{2}}+(k+a(y))^{2} \chi-b(y) \chi=\epsilon \chi,$

where $\chi$ depends only on $y, \epsilon$ is a rescaled energy, and $b(y)$ a rescaled magnetic field strength. What is the relationship between $a(y)$ and $b(y)$ ?

Show that $(* *)$ can be factorized in the form $M^{\dagger} M \chi=\epsilon \chi$ where

$M=\frac{d}{d y}+W(y)$

for some function $W(y)$, and deduce that $\epsilon$ is non-negative.

Show that zero energy states exist for all $k$ and are therefore infinitely degenerate.

Paper 4, Section II, D

commentWhat are meant by Bloch states and the Brillouin zone for a quantum mechanical particle moving in a one-dimensional periodic potential?

Derive an approximate value for the lowest-lying energy gap for the Schrödinger equation

$-\frac{d^{2} \psi}{d x^{2}}-V_{0}(\cos x+\cos 2 x) \psi=E \psi$

when $V_{0}$ is small and positive.

Estimate the width of this gap in the case that $V_{0}$ is large and positive.

Paper 1, Section II, J

comment(a) Let $\left(X_{t}, t \geqslant 0\right)$ be a continuous-time Markov chain on a countable state space I. Explain what is meant by a stopping time for the chain $\left(X_{t}, t \geqslant 0\right)$. State the strong Markov property. What does it mean to say that $X$ is irreducible?

(b) Let $\left(X_{t}, t \geqslant 0\right)$ be a Markov chain on $I=\{0,1, \ldots\}$ with $Q$-matrix given by $Q=\left(q_{i, j}\right)_{i, j \in I}$ such that:

(1) $q_{i, 0}>0$ for all $i \geqslant 1$, but $q_{0, j}=0$ for all $j \in I$, and

(2) $q_{i, i+1}>0$ for all $i \geqslant 1$, but $q_{i, j}=0$ if $j>i+1$.

Is $\left(X_{t}, t \geqslant 0\right)$ irreducible? Fix $M \geqslant 1$, and assume that $X_{0}=i$, where $1 \leqslant i \leqslant M$. Show that if $J_{1}=\inf \left\{t \geqslant 0: X_{t} \neq X_{0}\right\}$ is the first jump time, then there exists $\delta>0$ such that $\mathbb{P}_{i}\left(X_{J_{1}}=0\right) \geqslant \delta$, uniformly over $1 \leqslant i \leqslant M$. Let $T_{0}=0$ and define recursively for $m \geqslant 0$,

$T_{m+1}=\inf \left\{t \geqslant T_{m}: X_{t} \neq X_{T_{m}} \text { and } 1 \leqslant X_{t} \leqslant M\right\}$

Let $A_{m}$ be the event $A_{m}=\left\{T_{m}<\infty\right\}$. Show that $\mathbb{P}_{i}\left(A_{m}\right) \leqslant(1-\delta)^{m}$, for $1 \leqslant i \leqslant M$.

(c) Let $\left(X_{t}, t \geqslant 0\right)$ be the Markov chain from (b). Define two events $E$ and $F$ by

$E=\left\{X_{t}=0 \text { for all } t \text { large enough }\right\}, \quad F=\left\{\lim _{t \rightarrow \infty} X_{t}=+\infty\right\}$

Show that $\mathbb{P}_{i}(E \cup F)=1$ for all $i \in I$.

Paper 2, Section II, J

commentLet $X_{1}, X_{2}, \ldots$, be a sequence of independent, identically distributed positive random variables, with a common probability density function $f(x), x>0$. Call $X_{n}$ a record value if $X_{n}>\max \left\{X_{1}, \ldots, X_{n-1}\right\}$. Consider the sequence of record values

$V_{0}=0, V_{1}=X_{1}, \ldots, V_{n}=X_{i_{n}},$

where

$i_{n}=\min \left\{i \geqslant 1: X_{i}>V_{n-1}\right\}, n>1 .$

Define the record process $\left(R_{t}\right)_{t \geqslant 0}$ by $R_{0}=0$ and

$R_{t}=\max \left\{n \geqslant 1: V_{n}<t\right\}, \quad t>0$

(a) By induction on $n$, or otherwise, show that the joint probability density function of the random variables $V_{1}, \ldots, V_{n}$ is given by:

$f_{V_{1}, \ldots, V_{n}}\left(x_{1}, \ldots, x_{n}\right)=f\left(x_{1}\right) \frac{f\left(x_{2}\right)}{1-F\left(x_{1}\right)} \times \ldots \times \frac{f\left(x_{n}\right)}{1-F\left(x_{n-1}\right)},$

where $F(x)=\int_{0}^{x} f(y) \mathrm{d} y$ is the cumulative distribution function for $f(x)$.

(b) Prove that the random variable $R_{t}$ has a Poisson distribution with parameter $\Lambda(t)$ of the form

$\Lambda(t)=\int_{0}^{t} \lambda(s) \mathrm{d} s,$

and determine the 'instantaneous rate' $\lambda(s)$.

[Hint: You may use the formula

$\begin{aligned} &\mathbb{P}\left(R_{t}=k\right)=\mathbb{P}\left(V_{k} \leqslant t<V_{k+1}\right) \\ &=\int_{0}^{t} \ldots \int_{0}^{t} \mathbf{1}_{\left\{t_{1}<\ldots<t_{k}\right\}} f_{V_{1}, \ldots, V_{k}}\left(t_{1}, \ldots, t_{k}\right) \\ &\quad \times \mathbb{P}\left(V_{k+1}>t \mid V_{1}=t_{1}, \ldots, V_{k}=t_{k}\right) \prod_{j=1}^{k} \mathrm{~d} t_{j}, \end{aligned}$

for any $k \geqslant 1 .]$

Paper 3, Section II, J

comment(a) Define the Poisson process $\left(N_{t}, t \geqslant 0\right)$ with rate $\lambda>0$, in terms of its holding times. Show that for all times $t \geqslant 0, N_{t}$ has a Poisson distribution, with a parameter which you should specify.

(b) Let $X$ be a random variable with probability density function

$f(x)=\frac{1}{2} \lambda^{3} x^{2} e^{-\lambda x} \mathbf{1}_{\{x>0\}} .$

Prove that $X$ is distributed as the sum $Y_{1}+Y_{2}+Y_{3}$ of three independent exponential random variables of rate $\lambda$. Calculate the expectation, variance and moment generating function of $X$.

Consider a renewal process $\left(X_{t}, t \geqslant 0\right)$ with holding times having density $(*)$. Prove that the renewal function $m(t)=\mathbb{E}\left(X_{t}\right)$ has the form

$m(t)=\frac{\lambda t}{3}-\frac{1}{3} p_{1}(t)-\frac{2}{3} p_{2}(t)$

where $p_{1}(t)=\mathbb{P}\left(N_{t}=1 \bmod 3\right), p_{2}(t)=\mathbb{P}\left(N_{t}=2 \bmod 3\right)$ and $\left(N_{t}, t \geqslant 0\right)$ is the Poisson process of rate $\lambda$.

(c) Consider the delayed renewal process $\left(X_{t}^{\mathrm{D}}, t \geqslant 0\right)$ with holding times $S_{1}^{\mathrm{D}}, S_{2}, S_{3}, \ldots$ where $\left(S_{n}, n \geqslant 1\right)$, are the holding times of $\left(X_{t}, t \geqslant 0\right)$ from (b). Specify the distribution of $S_{1}^{\mathrm{D}}$ for which the delayed process becomes the renewal process in equilibrium.

[You may use theorems from the course provided that you state them clearly.]

Paper 4, Section II, J

commentA flea jumps on the vertices of a triangle $A B C$; its position is described by a continuous time Markov chain with a $Q$-matrix

$Q=\left(\begin{array}{ccc} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 1 & 0 & -1 \end{array}\right) \quad \begin{aligned} &A \\ &B \\ &C \end{aligned}$

(a) Draw a diagram representing the possible transitions of the flea together with the rates of each of these transitions. Find the eigenvalues of $Q$ and express the transition probabilities $p_{x y}(t), x, y=A, B, C$, in terms of these eigenvalues.

[Hint: $\operatorname{det}(Q-\mu \mathbf{I})=(-1-\mu)^{3}+1$. Specifying the equilibrium distribution may help.]

Hence specify the probabilities $\mathbb{P}\left(N_{t}=i \bmod 3\right)$ where $\left(N_{t}, t \geqslant 0\right)$ is a Poisson process of rate $1 .$

(b) A second flea jumps on the vertices of the triangle $A B C$ as a Markov chain with Q-matrix

$Q^{\prime}=\left(\begin{array}{ccc} -\rho & 0 & \rho \\ \rho & -\rho & 0 \\ 0 & \rho & -\rho \end{array}\right) \quad \begin{aligned} &A \\ &B \\ &C \end{aligned}$

where $\rho>0$ is a given real number. Let the position of the second flea at time $t$ be denoted by $Y_{t}$. We assume that $\left(Y_{t}, t \geqslant 0\right)$ is independent of $\left(X_{t}, t \geqslant 0\right)$. Let $p(t)=\mathbb{P}\left(X_{t}=Y_{t}\right)$. Show that $\lim _{t \rightarrow \infty} p(t)$ exists and is independent of the starting points of $X$ and $Y$. Compute this limit.

Paper 1, Section II, A

commentConsider the integral

$I(\lambda)=\int_{0}^{A} \mathrm{e}^{-\lambda t} f(t) d t, \quad A>0$

in the limit $\lambda \rightarrow \infty$, given that $f(t)$ has the asymptotic expansion

$f(t) \sim \sum_{n=0}^{\infty} a_{n} t^{n \beta}$

as $t \rightarrow 0_{+}$, where $\beta>0$. State Watson's lemma.

Now consider the integral

$J(\lambda)=\int_{a}^{b} \mathrm{e}^{\lambda \phi(t)} F(t) d t$

where $\lambda \gg 1$ and the real function $\phi(t)$ has a unique maximum in the interval $[a, b]$ at $c$, with $a<c<b$, such that

$\phi^{\prime}(c)=0, \phi^{\prime \prime}(c)<0$

By making a monotonic change of variable from $t$ to a suitable variable $\zeta$ (Laplace's method), or otherwise, deduce the existence of an asymptotic expansion for $J(\lambda)$ as $\lambda \rightarrow \infty$. Derive the leading term

$J(\lambda) \sim \mathrm{e}^{\lambda \phi(c)} F(c)\left(\frac{2 \pi}{\lambda\left|\phi^{\prime \prime}(c)\right|}\right)^{\frac{1}{2}}$

The gamma function is defined for $x>0$ by

$\Gamma(x+1)=\int_{0}^{\infty} \exp (x \log t-t) d t$

By means of the substitution $t=x s$, or otherwise, deduce Stirling's formula

$\Gamma(x+1) \sim x^{\left(x+\frac{1}{2}\right)} \mathrm{e}^{-x} \sqrt{2 \pi}\left(1+\frac{1}{12 x}+\cdots\right)$

as $x \rightarrow \infty$

Paper 3, Section II, A

commentConsider the contour-integral representation

$J_{0}(x)=\operatorname{Re} \frac{1}{i \pi} \int_{C} e^{i x \cosh t} d t$

of the Bessel function $J_{0}$ for real $x$, where $C$ is any contour from $-\infty-\frac{i \pi}{2}$ to $+\infty+\frac{i \pi}{2}$.

Writing $t=u+i v$, give in terms of the real quantities $u, v$ the equation of the steepest-descent contour from $-\infty-\frac{i \pi}{2}$ to $+\infty+\frac{i \pi}{2}$ which passes through $t=0$.

Deduce the leading term in the asymptotic expansion of $J_{0}(x)$, valid as $x \rightarrow \infty$

$J_{0}(x) \sim \sqrt{\frac{2}{\pi x}} \cos \left(x-\frac{\pi}{4}\right)$

Paper 4, Section II, A

commentThe differential equation

$f^{\prime \prime}=Q(x) f$

has a singular point at $x=\infty$. Assuming that $Q(x)>0$, write down the Liouville Green lowest approximations $f_{\pm}(x)$ for $x \rightarrow \infty$, with $f_{-}(x) \rightarrow 0$.

The Airy function $\operatorname{Ai}(x)$ satisfies $(*)$ with

$Q(x)=x$

and $\operatorname{Ai}(x) \rightarrow 0$ as $x \rightarrow \infty$. Writing

$\operatorname{Ai}(x)=w(x) f_{-}(x)$

show that $w(x)$ obeys

$x^{2} w^{\prime \prime}-\left(2 x^{5 / 2}+\frac{1}{2} x\right) w^{\prime}+\frac{5}{16} w=0$

Derive the expansion

$w \sim c\left(1-\frac{5}{48} x^{-3 / 2}\right) \quad \text { as } \quad x \rightarrow \infty$

where $c$ is a constant.

Paper 1, Section I, E

commentLagrange's equations for a system with generalized coordinates $q_{i}(t)$ are given by

$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0$

where $L$ is the Lagrangian. The Hamiltonian is given by

$H=\sum_{j} p_{j} \dot{q}_{j}-L,$

where the momentum conjugate to $q_{j}$ is

$p_{j}=\frac{\partial L}{\partial \dot{q}_{j}}$

Derive Hamilton's equations in the form

$\dot{q}_{i}=\frac{\partial H}{\partial p_{i}}, \quad \dot{p}_{i}=-\frac{\partial H}{\partial q_{i}}$

Explain what is meant by the statement that $q_{k}$ is an ignorable coordinate and give an associated constant of the motion in this case.

The Hamiltonian for a particle of mass $m$ moving on the surface of a sphere of radius $a$ under a potential $V(\theta)$ is given by

$H=\frac{1}{2 m a^{2}}\left(p_{\theta}^{2}+\frac{p_{\phi}^{2}}{\sin ^{2} \theta}\right)+V(\theta)$

where the generalized coordinates are the spherical polar angles $(\theta, \phi)$. Write down two constants of the motion and show that it is possible for the particle to move with constant $\theta$ provided that

$p_{\phi}^{2}=\left(\frac{m a^{2} \sin ^{3} \theta}{\cos \theta}\right) \frac{d V}{d \theta} .$

Paper 2, Section $\mathbf{I}$, E

commentA system of three particles of equal mass $m$ moves along the $x$ axis with $x_{i}$ denoting the $x$ coordinate of particle $i$. There is an equilibrium configuration for which $x_{1}=0$, $x_{2}=a$ and $x_{3}=2 a$.

Particles 1 and 2, and particles 2 and 3, are connected by springs with spring constant $\mu$ that provide restoring forces when the respective particle separations deviate from their equilibrium values. In addition, particle 1 is connected to the origin by a spring with spring constant $16 \mu / 3$. The Lagrangian for the system is

$L=\frac{m}{2}\left(\dot{x}_{1}^{2}+\dot{\eta}_{1}^{2}+\dot{\eta}_{2}^{2}\right)-\frac{\mu}{2}\left(\frac{16}{3} x_{1}^{2}+\left(\eta_{1}-x_{1}\right)^{2}+\left(\eta_{2}-\eta_{1}\right)^{2}\right)$

where the generalized coordinates are $x_{1}, \eta_{1}=x_{2}-a$ and $\eta_{2}=x_{3}-2 a$.

Write down the equations of motion. Show that the generalized coordinates can oscillate with a period $P=2 \pi / \omega$, where

$\omega^{2}=\frac{\mu}{3 m}$

and find the form of the corresponding normal mode in this case.

Paper 2, Section II, E

commentA symmetric top of unit mass moves under the action of gravity. The Lagrangian is given by

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

where the generalized coordinates are the Euler angles $(\theta, \phi, \psi)$, the principal moments of inertia are $I_{1}$ and $I_{3}$ and the distance from the centre of gravity of the top to the origin is $l$.

Show that $\omega_{3}=\dot{\psi}+\dot{\phi} \cos \theta$ and $p_{\phi}=I_{1} \dot{\phi} \sin ^{2} \theta+I_{3} \omega_{3} \cos \theta$ are constants of the motion. Show further that, when $p_{\phi}=I_{3} \omega_{3}$, with $\omega_{3}>0$, the equation of motion for $\theta$ is

$\frac{d^{2} \theta}{d t^{2}}=\frac{g l \sin \theta}{I_{1}}\left(1-\frac{I_{3}^{2} \omega_{3}^{2}}{4 I_{1} g l \cos ^{4}(\theta / 2)}\right)$

Find the possible equilibrium values of $\theta$ in the two cases:

(i) $I_{3}^{2} \omega_{3}^{2}>4 I_{1} g l$,

(ii) $I_{3}^{2} \omega_{3}^{2}<4 I_{1} g l$.

By considering linear perturbations in the neighbourhoods of the equilibria in each case, find which are unstable and give expressions for the periods of small oscillations about the stable equilibria.

Paper 3, Section I, E

comment(a) Show that the principal moments of inertia of a uniform circular cylinder of radius $a$, length $h$ and mass $M$ about its centre of mass are $I_{1}=I_{2}=M\left(a^{2} / 4+h^{2} / 12\right)$ and $I_{3}=M a^{2} / 2$, with the $x_{3}$ axis being directed along the length of the cylinder.

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

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

and

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

Show that, for the cylinder of part $(\mathrm{a}), \omega_{3}$ is constant. Show further that, when $\omega_{3} \neq 0$, the angular momentum vector precesses about the $x_{3}$ axis with angular velocity $\Omega$ given by

$\Omega=\left(\frac{3 a^{2}-h^{2}}{3 a^{2}+h^{2}}\right) \omega_{3}$

Paper 4, Section I, E

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

A second Hamiltonian system has the Hamiltonian $G=G(\mathbf{p}, \mathbf{q})$. Neither $H$ nor $G$ contains the time explicitly. Show that the condition for $H(\mathbf{p}, \mathbf{q})$ to be invariant under the evolution of the coordinates and momenta generated by the Hamiltonian $G(\mathbf{p}, \mathbf{q})$ is that the Poisson bracket $[H, G]$ vanishes. Deduce that $G$ is a constant of the motion for evolution under $H$.

Show that, when $G=\alpha \sum_{k=1}^{n} p_{k}$, where $\alpha$ is constant, the motion it generates is a translation of each $q_{k}$ by an amount $\alpha t$, while the corresponding $p_{k}$ remains fixed. What do you infer is conserved when $H$ is invariant under this transformation?

(b) When $n=3$ and $H$ is a function of $p_{1}^{2}+p_{2}^{2}+p_{3}^{2}$ and $q_{1}^{2}+q_{2}^{2}+q_{3}^{2}$ only, find $\left[H, L_{i}\right]$ when

$L_{i}=\epsilon_{i j k} q_{j} p_{k}$

Paper 4, Section II, E

commentThe Hamiltonian for a particle of mass $m$, charge $e$ and position vector $\mathbf{q}=(x, y, z)$, moving in an electromagnetic field, is given by

$H(\mathbf{p}, \mathbf{q}, t)=\frac{1}{2 m}\left(\mathbf{p}-\frac{e \mathbf{A}}{c}\right)^{2}$

where $\mathbf{A}(\mathbf{q}, t)$ is the vector potential. Write down Hamilton's equations and use them to derive the equations of motion for the charged particle.

Show that, when $\mathbf{A}=\left(-y B_{0}(z, t), 0,0\right)$, there are solutions for which $p_{x}=0$ and for which the particle motion is such that

$\frac{d^{2} y}{d t^{2}}=-\Omega^{2} y$

where $\Omega=e B_{0} /(m c)$. Show in addition that the Hamiltonian may be written as

$H=\frac{m}{2}\left(\frac{d z}{d t}\right)^{2}+E^{\prime}$

where

$E^{\prime}=\frac{m}{2}\left(\left(\frac{d y}{d t}\right)^{2}+\Omega^{2} y^{2}\right)$

Assuming that $B_{0}$ is constant, find the action

$I\left(E^{\prime}, B_{0}\right)=\frac{1}{2 \pi} \oint m\left(\frac{d y}{d t}\right) d y$

associated with the $y$ motion.

It is now supposed that $B_{0}$ varies on a time-scale much longer than $\Omega^{-1}$ and thus is slowly varying. Show by applying the theory of adiabatic invariance that the motion in the $z$ direction takes place under an effective potential and give an expression for it.

Paper 1, Section I, H

commentI am putting up my Christmas lights. If I plug in a set of bulbs and one is defective, none will light up. A badly written note left over from the previous year tells me that exactly one of my 10 bulbs is defective and that the probability that the $k$ th bulb is defective is $k / 55$.

(i) Find an explicit procedure for identifying the defective bulb in the least expected number of steps.

[You should explain your method but no proof is required.]

(ii) Is there a different procedure from the one you gave in (i) with the same expected number of steps? Either write down another procedure and explain briefly why it gives the same expected number or explain briefly why no such procedure exists.

(iii) Because I make such a fuss about each test, my wife wishes me to tell her the maximum number $N$ of trials that might be required. Will the procedure in (i) give the minimum $N$ ? Either write down another procedure and explain briefly why it gives a smaller $N$ or explain briefly why no such procedure exists.

Paper 1, Section II, H

comment(i) State and prove Gibbs' inequality.

(ii) A casino offers me the following game: I choose strictly positive numbers $a_{1}, \ldots, a_{n}$ with $\sum_{j=1}^{n} a_{j}=1$. I give the casino my entire fortune $f$ and roll an $n$-sided die. With probability $p_{j}$ the casino returns $u_{j}^{-1} a_{j} f$ for $j=1,2, \ldots, n$. If I intend to play the game many times (staking my entire fortune each time) explain carefully why I should choose $a_{1}, \ldots, a_{n}$ to maximise $\sum_{j=1}^{n} p_{j} \log \left(u_{j}^{-1} a_{j}\right)$.

[You should assume $n \geqslant 2$ and $u_{j}, p_{j}>0$ for each $j .$ ]

(iii) Determine the appropriate $a_{1}, \ldots, a_{n}$. Let $\sum_{i=1}^{n} u_{i}=U$. Show that, if $U<1$, then, in the long run with high probability, my fortune increases. Show that, if $U>1$, the casino can choose $u_{1}, \ldots, u_{n}$ in such a way that, in the long run with high probability, my fortune decreases. Is it true that, if $U>1$, any choice of $u_{1}, \ldots, u_{n}$ will ensure that, in the long run with high probability, my fortune decreases? Why?

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

commentKnowing that

$25 \equiv 2886^{2} \quad \bmod 3953$

and that 3953 is the product of two primes $p$ and $q$, find $p$ and $q$.

[You should explain your method in sufficient detail to show that it is reasonably general.]

Paper 2, Section II, H

commentDescribe the construction of the Reed-Miller code $R M(m, d)$. Establish its information rate and minimum weight.

Show that every codeword in $R M(d, d-1)$ has even weight. By considering $\mathbf{x} \wedge \mathbf{y}$ with $\mathbf{x} \in R M(m, r)$ and $\mathbf{y} \in R M(m, m-r-1)$, or otherwise, show that $R M(m, m-r-1) \subseteq R M(m, r)^{\perp}$. Show that, in fact, $R M(m, m-r-1)=R M(m, r)^{\perp} .$

Paper 3, Section I, $4 \mathrm{H}$

commentDefine a binary code of length 15 with information rate $11 / 15$ which will correct single errors. Show that it has the rate stated and give an explicit procedure for identifying the error. Show that the procedure works.

[Hint: You may wish to imitate the corresponding discussion for a code of length 7 .]

Paper 4, Section I, H

commentWhat is a general feedback register? What is a linear feedback register? Give an example of a general feedback register which is not a linear feedback register and prove that it has the stated property.

By giving proofs or counterexamples, establish which, if any, of the following statements are true and which, if any, are false.

(i) Given two linear feedback registers, there always exist non-zero initial fills for which the outputs are identical.

(ii) If two linear feedback registers have different lengths, there do not exist non-zero initial fills for which the outputs are identical.

(iii) If two linear feedback registers have different lengths, there exist non-zero initial fills for which the outputs are not identical.

(iv) There exist two linear feedback registers of different lengths and non-zero initial fills for which the outputs are identical.

Paper 1, Section I, D

commentPrior to a time $t \sim 100,000$ years, the Universe was filled with a gas of photons and non-relativistic free electrons and protons maintained in equilibrium by Thomson scattering. At around $t \sim 400,000$ years, the protons and electrons began combining to form neutral hydrogen,

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

[You may assume that the equilibrium number density of a non-relativistic species $\left(k T \ll m c^{2}\right)$ is given by

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

while the photon number density is

$n_{\gamma}=16 \pi \zeta(3)\left(\frac{k T}{h c}\right)^{3}, \quad(\zeta(3) \approx 1.20 \ldots)$

Deduce Saha's equation for the recombination process $(*)$ stating clearly your assumptions and the steps made in the calculation,

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

where $I$ is the ionization energy of hydrogen.

Consider now the fractional ionization $X_{e}=n_{e} / n_{B}$ where $n_{B}=n_{p}+n_{H}=\eta n_{\gamma}$ is the baryon number of the Universe and $\eta$ is the baryon to photon ratio. Find an expression for the ratio

$\left(1-X_{e}\right) / X_{e}^{2}$

in terms only of $k T$ and constants such as $\eta$ and $I$.

Suggest a reason why neutral hydrogen forms at a temperature $k T \approx 0.3 \mathrm{eV}$ which is much lower than the hydrogen ionization temperature $k T=I \approx 13 \mathrm{eV}$.

Paper 1, Section II, D

comment(i) In a homogeneous and isotropic universe, the scalefactor $a(t)$ obeys the Friedmann equation

$\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho,$

where $\rho(t)$ is the matter density which, together with the pressure $P(t)$, satisfies

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

Use these two equations to derive the Raychaudhuri equation,

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

(ii) Conformal time $\tau$ is defined by taking $d t / d \tau=a$, so that $\dot{a}=a^{\prime} / a \equiv \mathcal{H}$ where primes denote derivatives with respect to $\tau$. For matter obeying the equation of state $P=w \rho c^{2}$, show that the Friedmann and energy conservation equations imply

$\mathcal{H}^{2}+k c^{2}=\frac{8 \pi G}{3} \rho_{0} a^{-(1+3 w)}$

where $\rho_{0}=\rho\left(t_{0}\right)$ and we take $a\left(t_{0}\right)=1$ today. Use the Raychaudhuri equation to derive the expression

$\mathcal{H}^{\prime}+\frac{1}{2}(1+3 w)\left[\mathcal{H}^{2}+k c^{2}\right]=0$

For a $k c^{2}=1$ closed universe, by solving first for $\mathcal{H}$ (or otherwise), show that the scale factor satisfies

$a=\alpha(\sin \beta \tau)^{2 /(1+3 w)}$

where $\alpha, \beta$ are constants. [Hint: You may assume that $\int d x /\left(1+x^{2}\right)=-\cot ^{-1} x+$ const.]

For a closed universe dominated by pressure-free matter $(P=0)$, find the complete parametric solution

$a=\frac{1}{2} \alpha(1-\cos 2 \beta \tau), \quad t=\frac{\alpha}{4 \beta}(2 \beta \tau-\sin 2 \beta \tau)$

Paper 2, Section I, D

comment(a) The equilibrium distribution for the energy density of a massless neutrino takes the form

$\epsilon=\frac{4 \pi c}{h^{3}} \int_{0}^{\infty} \frac{p^{3} d p}{\exp (p c / k T)+1} .$

Show that this can be expressed in the form $\epsilon=\alpha T^{4}$, where the constant $\alpha$ need not be evaluated explicitly.

(b) In the early universe, the entropy density $s$ at a temperature $T$ is $s=$ $(8 \sigma / 3 c) \mathcal{N}_{S} T^{3}$ where $\mathcal{N}_{S}$ is the total effective spin degrees of freedom. Briefly explain why $\mathcal{N}_{S}=\mathcal{N}_{*}+\mathcal{N}_{S D}$, each term of which consists of two separate components as follows: the contribution from each massless species in equilibrium $\left(T_{i}=T\right)$ is

$\mathcal{N}_{*}=\sum_{\text {bosons }} g_{i}+\frac{7}{8} \sum_{\text {fermions }} g_{i}$

and a similar sum for massless species which have decoupled,

$\mathcal{N}_{S D}=\sum_{\text {bosons }} g_{i}\left(\frac{T_{i}}{T}\right)^{3}+\frac{7}{8} \sum_{\text {fermions }} g_{i}\left(\frac{T_{i}}{T}\right)^{3}$

where in each case $g_{i}$ is the degeneracy and $T_{i}$ is the temperature of the species $i$.

The three species of neutrinos and antineutrinos decouple from equilibrium at a temperature $T \approx 1 \mathrm{MeV}$, after which positrons and electrons annihilate at $T \approx 0.5 \mathrm{MeV}$, leaving photons in equilibrium with a small excess population of electrons. Using entropy considerations, explain why the ratio of the neutrino and photon temperatures today is given by

$\frac{T_{\nu}}{T_{\gamma}}=\left(\frac{4}{11}\right)^{1 / 3}$

Paper 3, Section I, D

(a) Write down an expression for the total gravitational potential energy $E_{\text {grav }}$ of a spherically symmetric star of outer radius $R$ in terms of its mass density $\rho(r)$ and the total mass $m(r)$ inside a radius $r$, satisfying the relation $d m / d r=4 \pi r^{2} \rho(r)$.

An isotropic mass distribution obeys the pressure-support equation,

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

where $P(r)$ is the pressure. Multiply this expression by $4 \pi r^{3}$ and integrate with respect to $r$ to derive the virial theorem relating the kinetic and gravitational energy of the star

$E_{\mathrm{kin}}=-\frac{1}{2} E_{\mathrm{grav}}$

where you may assume for a non-relativistic ideal gas that $E_{\mathrm{kin}}=\frac{3}{2}\langle P\rangle V$, with $\langle P\rangle$ the average pressure.

(b) Consider a white dwarf supported by electron Fermi degeneracy pressure $P \approx h^{2} n^{5 / 3} / m_{\mathrm{e}}$, where $m_{\mathrm{e}}$ is the electron mass and $n$ is the number density. Assume a uniform density $\rho(r)=m_{\mathrm{p}} n(r) \approx m_{\mathrm{p}}\langle n\rangle$, so the total mass of the star is given by $M=(4 \pi / 3)\langle n\rangle m_{\mathrm{p}} R^{3}$