# Part II, 2010

### Jump to course

Paper 1, Section II, G

comment(i) Let $X=\left\{(x, y) \in \mathbb{C}^{2} \mid x^{2}=y^{3}\right\}$. Show that $X$ is birational to $\mathbf{A}^{1}$, but not isomorphic to it.

(ii) Let $X$ be an affine variety. Define the dimension of $X$ in terms of the tangent spaces of $X$.

(iii) Let $f \in k\left[x_{1}, \ldots, x_{n}\right]$ be an irreducible polynomial, where $k$ is an algebraically closed field of arbitrary characteristic. Show that $\operatorname{dim} Z(f)=n-1$.

[You may assume the Nullstellensatz.]

Paper 2, Section II, G

commentLet $X=X_{n, m, r}$ be the set of $n \times m$ matrices of rank at most $r$ over a field $k$. Show that $X_{n, m, r}$ is naturally an affine subvariety of $\mathbf{A}^{n m}$ and that $X_{n, m, r}$ is a Zariski closed subvariety of $X_{n, m, r+1}$.

Show that if $r<\min (n, m)$, then 0 is a singular point of $X$.

Determine the dimension of $X_{5,2,1}$.

Paper 3, Section II, G

comment(i) Let $X$ be a curve, and $p \in X$ be a smooth point on $X$. Define what a local parameter at $p$ is.

Now let $f: X \rightarrow Y$ be a rational map to a quasi-projective variety $Y$. Show that if $Y$ is projective, $f$ extends to a morphism defined at $p$.

Give an example where this fails if $Y$ is not projective, and an example of a morphism $f: \mathbb{C}^{2} \backslash\{0\} \rightarrow \mathbf{P}^{1}$ which does not extend to $0 .$

(ii) Let $V=Z\left(X_{0}^{8}+X_{1}^{8}+X_{2}^{8}\right)$ and $W=Z\left(X_{0}^{4}+X_{1}^{4}+X_{2}^{4}\right)$ be curves in $\mathbf{P}^{2}$ over a field of characteristic not equal to 2 . Let $\phi: V \rightarrow W$ be the map $\left[X_{0}: X_{1}: X_{2}\right] \mapsto\left[X_{0}^{2}: X_{1}^{2}: X_{2}^{2}\right]$. Determine the degree of $\phi$, and the ramification $e_{p}$ for all $p \in V$.

Paper 4, Section II, G

commentLet $E \subseteq \mathbf{P}^{2}$ be the projective curve obtained from the affine curve $y^{2}=\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)\left(x-\lambda_{3}\right)$, where the $\lambda_{i}$ are distinct and $\lambda_{1} \lambda_{2} \lambda_{3} \neq 0$.

(i) Show there is a unique point at infinity, $P_{\infty}$.

(ii) Compute $\operatorname{div}(x), \operatorname{div}(y)$.

(iii) Show $\mathcal{L}\left(P_{\infty}\right)=k$.

(iv) Compute $l\left(n P_{\infty}\right)$ for all $n$.

[You may not use the Riemann-Roch theorem.]

Paper 1, Section II, H

commentState the path lifting and homotopy lifting lemmas for covering maps. Suppose that $X$ is path connected and locally path connected, that $p_{1}: Y_{1} \rightarrow X$ and $p_{2}: Y_{2} \rightarrow X$ are covering maps, and that $Y_{1}$ and $Y_{2}$ are simply connected. Using the lemmas you have stated, but without assuming the correspondence between covering spaces and subgroups of $\pi_{1}$, prove that $Y_{1}$ is homeomorphic to $Y_{2}$.

Paper 2, Section II, $\mathbf{2 1 H}$

commentLet $G$ be the finitely presented group $G=\left\langle a, b \mid a^{2} b^{3} a^{3} b^{2}=1\right\rangle$. Construct a path connected space $X$ with $\pi_{1}(X, x) \cong G$. Show that $X$ has a unique connected double cover $\pi: Y \rightarrow X$, and give a presentation for $\pi_{1}(Y, y)$.

Paper 3, Section II, H

commentSuppose $X$ is a finite simplicial complex and that $H_{*}(X)$ is a free abelian group for each value of $*$. Using the Mayer-Vietoris sequence or otherwise, compute $H_{*}\left(S^{1} \times X\right)$ in terms of $H_{*}(X)$. Use your result to compute $H_{*}\left(T^{n}\right)$.

[Note that $T^{n}=S^{1} \times \ldots \times S^{1}$, where there are $n$ factors in the product.]

Paper 4, Section II, $\mathbf{2 1 H}$

commentState the Snake Lemma. Explain how to define the boundary map which appears in it, and check that it is well-defined. Derive the Mayer-Vietoris sequence from the Snake Lemma.

Given a chain complex $C$, let $A \subset C$ be the span of all elements in $C$ with grading greater than or equal to $n$, and let $B \subset C$ be the span of all elements in $C$ with grading less than $n$. Give a short exact sequence of chain complexes relating $A, B$, and $C$. What is the boundary map in the corresponding long exact sequence?

Paper 1, Section II, B

commentGive an account of the variational principle for establishing an upper bound on the ground-state energy, $E_{0}$, of a particle moving in a potential $V(x)$ in one dimension.

Explain how an upper bound on the energy of the first excited state can be found in the case that $V(x)$ is a symmetric function.

A particle of mass $2 m=\hbar^{2}$ moves in the potential

$V(x)=-V_{0} e^{-x^{2}}, \quad V_{0}>0$

Use the trial wavefunction

$\psi(x)=e^{-\frac{1}{2} a x^{2}}$

where $a$ is a positive real parameter, to establish the upper bound $E_{0} \leqslant E(a)$ for the energy of the ground state, where

$E(a)=\frac{1}{2} a-V_{0} \frac{\sqrt{a}}{\sqrt{1+a}} .$

Show that, for $a>0, E(a)$ has one zero and find its position.

Show that the turning points of $E(a)$ are given by

$(1+a)^{3}=\frac{V_{0}^{2}}{a}$

and deduce that there is one turning point in $a>0$ for all $V_{0}>0$.

Sketch $E(a)$ for $a>0$ and hence deduce that $V(x)$ has at least one bound state for all $V_{0}>0$.

For $0<V_{0} \ll 1$ show that

$-V_{0}<E_{0} \leqslant \epsilon\left(V_{0}\right),$

where $\epsilon\left(V_{0}\right)=-\frac{1}{2} V_{0}^{2}+\mathrm{O}\left(V_{0}^{4}\right)$.

[You may use the result that $\int_{-\infty}^{\infty} e^{-b x^{2}} d x=\sqrt{\frac{\pi}{b}}$ for $b>0 .$ ]

Paper 2, Section II, B

commentA beam of particles of mass $m$ and momentum $p=\hbar k$ is incident along the $z$-axis. Write down the asymptotic form of the wave function which describes scattering under the influence of a spherically symmetric potential $V(r)$ and which defines the scattering amplitude $f(\theta)$.

Given that, for large $r$,

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

show how to derive the partial-wave expansion of the scattering amplitude in the form

$f(\theta)=\frac{1}{k} \sum_{l=0}^{\infty}(2 l+1) e^{i \delta_{l}} \sin \delta_{l} P_{l}(\cos \theta)$

Obtain an expression for the total cross-section, $\sigma$.

Let $V(r)$ have the form

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

where $V_{0}=\frac{\hbar^{2}}{2 m} \gamma^{2}$

Show that the $l=0$ phase-shift $\delta_{0}$ satisfies

$\frac{\tan \left(k a+\delta_{0}\right)}{k a}=\frac{\tan \kappa a}{\kappa a},$

where $\kappa^{2}=k^{2}+\gamma^{2}$.

Assume $\gamma$ to be large compared with $k$ so that $\kappa$ may be approximated by $\gamma$. Show, using graphical methods or otherwise, that there are values for $k$ for which $\delta_{0}(k)=n \pi$ for some integer $n$, which should not be calculated. Show that the smallest value, $k_{0}$, of $k$ for which this condition holds certainly satisfies $k_{0}<3 \pi / 2 a$.

Paper 3, Section II, B

commentState Bloch's theorem for a one dimensional lattice which is invariant under translations by $a$.

A simple model of a crystal consists of a one-dimensional linear array of identical sites with separation $a$. At the $n$th site the Hamiltonian, neglecting all other sites, is $H_{n}$ and an electron may occupy either of two states, $\phi_{n}(x)$ and $\chi_{n}(x)$, where

$H_{n} \phi_{n}(x)=E_{0} \phi_{n}(x), \quad H_{n} \chi_{n}(x)=E_{1} \chi_{n}(x),$

and $\phi_{n}$ and $\chi_{n}$ are orthonormal. How are $\phi_{n}(x)$ and $\chi_{n}(x)$ related to $\phi_{0}(x)$ and $\chi_{0}(x)$ ?

The full Hamiltonian is $H$ and is invariant under translations by $a$. Write trial wavefunctions $\psi(x)$ for the eigenstates of this model appropriate to a tight binding approximation if the electron has probability amplitudes $b_{n}$ and $c_{n}$ to be in the states $\phi_{n}$ and $\chi_{n}$ respectively.

Assume that the only non-zero matrix elements in this model are, for all $n$,

$\begin{aligned} &\left(\phi_{n}, H_{n} \phi_{n}\right)=E_{0}, \quad\left(\chi_{n}, H_{n} \chi_{n}\right)=E_{1} \\ &\left(\phi_{n}, V \phi_{n \pm 1}\right)=\left(\chi_{n}, V \chi_{n \pm 1}\right)=\left(\phi_{n}, V \chi_{n \pm 1}\right)=\left(\chi_{n}, V \phi_{n \pm 1}\right)=-A, \end{aligned}$

where $H=H_{n}+V$ and $A>0$. Show that the time-dependent Schrödinger equation governing the amplitudes becomes

$\begin{aligned} i \hbar \dot{b}_{n} &=E_{0} b_{n}-A\left(b_{n+1}+b_{n-1}+c_{n+1}+c_{n-1}\right) \\ i \hbar \dot{c}_{n} &=E_{1} c_{n}-A\left(c_{n+1}+c_{n-1}+b_{n+1}+b_{n-1}\right) \end{aligned}$

By examining solutions of the form

$\left(\begin{array}{l} b_{n} \\ c_{n} \end{array}\right)=\left(\begin{array}{l} B \\ C \end{array}\right) e^{i(k n a-E t / \hbar)}$

show that the allowed energies of the electron are two bands given by

$E=\frac{1}{2}\left(E_{0}+E_{1}-4 A \cos k a\right) \pm \frac{1}{2} \sqrt{\left(E_{0}-E_{1}\right)^{2}+16 A^{2} \cos ^{2} k a}$

Define the Brillouin zone for this system and find the energies at the top and bottom of both bands. Hence, show that the energy gap between the bands is

$\Delta E=-4 A+\sqrt{\left(E_{1}-E_{0}\right)^{2}+16 A^{2}}$

Show that the wavefunctions $\psi(x)$ satisfy Bloch's theorem.

Describe briefly what are the crucial differences between insulators, conductors and semiconductors.

Paper 4, Section II, B

commentThe scattering amplitude for electrons of momentum $\hbar \mathbf{k}$ incident on an atom located at the origin is $f(\hat{\mathbf{r}})$ where $\hat{\mathbf{r}}=\mathbf{r} / r$. Explain why, if the atom is displaced by a position vector a, the asymptotic form of the scattering wave function becomes

$\psi_{\mathbf{k}}(\mathbf{r}) \sim e^{i \mathbf{k} \cdot \mathbf{r}}+e^{i \mathbf{k} \cdot \mathbf{a}} \frac{e^{i k r^{\prime}}}{r^{\prime}} f\left(\hat{\mathbf{r}}^{\prime}\right) \sim e^{i \mathbf{k} \cdot \mathbf{r}}+e^{i\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \cdot \mathbf{a}} \frac{e^{i k r}}{r} f(\hat{\mathbf{r}}),$

where $\mathbf{r}^{\prime}=\mathbf{r}-\mathbf{a}, r^{\prime}=\left|\mathbf{r}^{\prime}\right|, \hat{\mathbf{r}}^{\prime}=\mathbf{r}^{\prime} / r^{\prime}$ and $k=|\mathbf{k}|, \mathbf{k}^{\prime}=k \hat{\mathbf{r}}$. For electrons incident on $N$ atoms in a regular Bravais crystal lattice show that the differential cross-section for scattering in the direction $\hat{\mathbf{r}}$ is

$\frac{d \sigma}{d \Omega}=N|f(\hat{\mathbf{r}})|^{2} \Delta\left(\mathbf{k}-\mathbf{k}^{\prime}\right) .$

Derive an explicit form for $\Delta(\mathbf{Q})$ and show that it is strongly peaked when $\mathbf{Q} \approx \mathbf{b}$ for $\mathbf{b}$ a reciprocal lattice vector.

State the Born approximation for $f(\hat{\mathbf{r}})$ when the scattering is due to a potential $V(\mathbf{r})$. Calculate the Born approximation for the case $V(\mathbf{r})=-a \delta(\mathbf{r}) .$

Electrons with de Broglie wavelength $\lambda$ are incident on a target composed of many randomly oriented small crystals. They are found to be scattered strongly through an angle of $60^{\circ}$. What is the likely distance between planes of atoms in the crystal responsible for the scattering?

Paper 1, Section II, I

comment(a) Define what it means to say that $\pi$ is an equilibrium distribution for a Markov chain on a countable state space with Q-matrix $Q=\left(q_{i j}\right)$, and give an equation which is satisfied by any equilibrium distribution. Comment on the possible non-uniqueness of equilibrium distributions.

(b) State a theorem on convergence to an equilibrium distribution for a continuoustime Markov chain.

A continuous-time Markov chain $\left(X_{t}, t \geqslant 0\right)$ has three states $1,2,3$ and the Qmatrix $Q=\left(q_{i j}\right)$ is of the form

$Q=\left(\begin{array}{ccc} -\lambda_{1} & \lambda_{1} / 2 & \lambda_{1} / 2 \\ \lambda_{2} / 2 & -\lambda_{2} & \lambda_{2} / 2 \\ \lambda_{3} / 2 & \lambda_{3} / 2 & -\lambda_{3} \end{array}\right)$

where the rates $\lambda_{1}, \lambda_{2}, \lambda_{3} \in[0, \infty)$ are not all zero.

[Note that some of the $\lambda_{i}$ may be zero, and those cases may need special treatment.]

(c) Find the equilibrium distributions of the Markov chain in question. Specify the cases of uniqueness and non-uniqueness.

(d) Find the limit of the transition matrix $P(t)=\exp (t Q)$ when $t \rightarrow \infty$.

(e) Describe the jump chain $\left(Y_{n}\right)$ and its equilibrium distributions. If $\widehat{P}$is the jump probability matrix, find the limit of $\widehat{P}^{n}$ as $n \rightarrow \infty$.

Paper 2, Section II, I

comment(a) Let $S_{k}$ be the sum of $k$ independent exponential random variables of rate $k \mu$. Compute the moment generating function $\phi_{S_{k}}(\theta)=\mathbb{E} e^{\theta S_{k}}$ of $S_{k}$. Show that, as $k \rightarrow \infty$, functions $\phi_{S_{k}}(\theta)$ converge to a limit. Describe the random variable $S$ for which the limiting function $\lim _{k \rightarrow \infty} \phi_{S_{k}}(\theta)$ coincides with $\mathbb{E} e^{\theta S}$.

(b) Define the $M / G / 1$ queue with infinite capacity (sometimes written $M / G / 1 / \infty$ ). Introduce the embedded discrete-time Markov chain $\left(X_{n}\right)$ and write down the recursive relation between $X_{n}$ and $X_{n-1}$.

Consider, for each fixed $k$ and for $0<\lambda<\mu$, an $\mathrm{M} / \mathrm{G} / 1 / \infty$ queue with arrival rate $\lambda$ and with service times distributed as $S_{k}$. Assume that the queue is empty at time 0 . Let $T_{k}$ be the earliest time at which a customer departs leaving the queue empty. Let $A$ be the first arrival time and $B_{k}=T_{k}-A$ the length of the busy period.

(c) Prove that the moment generating functions $\phi_{B_{k}}(\theta)=\mathbb{E} e^{\theta B_{k}}$ and $\phi_{S_{k}}(\theta)$ are related by the equation

$\phi_{B_{k}}(\theta)=\phi_{S_{k}}\left(\theta-\lambda\left(1-\phi_{B_{k}}(\theta)\right)\right)$

(d) Prove that the moment generating functions $\phi_{T_{k}}(\theta)=\mathbb{E} e^{\theta T_{k}}$ and $\phi_{S_{k}}(\theta)$ are related by the equation

$\frac{\lambda-\theta}{\lambda} \phi_{T_{k}}(\theta)=\phi_{S_{k}}\left((\lambda-\theta)\left(\phi_{T_{k}}(\theta)-1\right)\right)$

(e) Assume that, for all $\theta<\lambda$,

$\lim _{k \rightarrow \infty} \phi_{B_{k}}(\theta)=\mathbb{E} e^{\theta B}, \quad \lim _{k \rightarrow \infty} \phi_{T_{k}}(\theta)=\mathbb{E} e^{\theta T},$

for some random variables $B$ and $T$. Calculate $\mathbb{E} B$ and $\mathbb{E} T$. What service time distribution do these values correspond to?

Paper 3, Section II, I

commentCars looking for a parking space are directed to one of three unlimited parking lots A, B and C. First, immediately after the entrance, the road forks: one direction is to lot A, the other to B and C. Shortly afterwards, the latter forks again, between B and C. See the diagram below.

The policeman at the first road fork directs an entering car with probability $1 / 3$ to A and with probability $2 / 3$ to the second fork. The policeman at the second fork sends the passing cars to $\mathrm{B}$ or $\mathrm{C}$ alternately: cars $1,3,5, \ldots$ approaching the second fork go to $\mathrm{B}$ and cars $2,4,6, \ldots$ to $\mathrm{C}$.

Assuming that the total arrival process $(N(t))$ of cars is Poisson of rate $\lambda$, consider the processes $\left(X^{\mathrm{A}}(t)\right),\left(X^{\mathrm{B}}(t)\right)$ and $\left(X^{\mathrm{C}}(t)\right), t \geqslant 0$, where $X^{i}(t)$ is the number of cars directed to lot $i$ by time $t$, for $i=\mathrm{A}, \mathrm{B}, \mathrm{C}$. The times for a car to travel from the first to the second fork, or from a fork to the parking lot, are all negligible.

(a) Characterise each of the processes $\left(X^{\mathrm{A}}(t)\right),\left(X^{\mathrm{B}}(t)\right)$ and $\left(X^{\mathrm{C}}(t)\right)$, by specifying if it is (i) Poisson, (ii) renewal or (iii) delayed renewal. Correspondingly, specify the rate, the holding-time distribution and the distribution of the delay.

(b) In the case of a renewal process, determine the equilibrium delay distribution.

(c) Given $s, t>0$, write down explicit expressions for the probability $\mathbb{P}\left(X^{i}(s)=X^{i}(s+t)\right)$ that the interval $(s, t+s)$ is free of points in the corresponding process, $i=\mathrm{A}, \mathrm{B}, \mathrm{C}$.

Paper 4, Section II, I

comment(a) Let $\left(X_{t}\right)$ be an irreducible continuous-time Markov chain on a finite or countable state space. What does it mean to say that the chain is (i) transient, (ii) recurrent, (iii) positive recurrent, (iv) null recurrent? What is the relation between equilibrium distributions and properties (iii) and (iv)?

A population of microorganisms develops in continuous time; the size of the population is a Markov chain $\left(X_{t}\right)$ with states $0,1,2, \ldots$ Suppose $X_{t}=n$. It is known that after a short time $s$, the probability that $X_{t}$ increased by one is $\lambda(n+1) s+o(s)$ and (if $n \geqslant 1$ ) the probability that the population was exterminated between times $t$ and $t+s$ and never revived by time $t+s$ is $\mu s+o(s)$. Here $\lambda$ and $\mu$ are given positive constants. All other changes in the value of $X_{t}$ have a combined probability $o(s)$.

(b) Write down the Q-matrix of Markov chain $\left(X_{t}\right)$ and determine if $\left(X_{t}\right)$ is irreducible. Show that $\left(X_{t}\right)$ is non-explosive. Determine the jump chain.

(c) Now assume that

$\mu=\lambda$

Determine whether the chain is transient or recurrent, and in the latter case whether it is positive or null recurrent. Answer the same questions for the jump chain. Justify your answers.

Paper 1, Section II, C

commentFor $\lambda>0$ let

$I(\lambda)=\int_{0}^{b} f(x) \mathrm{e}^{-\lambda x} d x, \quad \text { with } \quad 0<b<\infty$

Assume that the function $f(x)$ is continuous on $0<x \leqslant b$, and that

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

as $x \rightarrow 0_{+}$, where $\alpha>-1$ and $\beta>0$.

(a) Explain briefly why in this case straightforward partial integrations in general cannot be applied for determining the asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$.

(b) Derive with proof an asymptotic expansion for $I(\lambda)$ as $\lambda \rightarrow \infty$.

(c) For the function

$B(s, t)=\int_{0}^{1} u^{s-1}(1-u)^{t-1} d u, \quad s, t>0$

obtain, using the substitution $u=e^{-x}$, the first two terms in an asymptotic expansion as $s \rightarrow \infty$. What happens as $t \rightarrow \infty$ ?

[Hint: The following formula may be useful

$\Gamma(y)=\int_{0}^{\infty} x^{y-1} \mathrm{e}^{-x} d t, \quad \text { for } \quad x>0$

Paper 3, Section II, C

commentConsider the ordinary differential equation

$y^{\prime \prime}=(|x|-E) y$

subject to the boundary conditions $y(\pm \infty)=0$. Write down the general form of the Liouville-Green solutions for this problem for $E>0$ and show that asymptotically the eigenvalues $E_{n}, n \in \mathbb{N}$ and $E_{n}<E_{n+1}$, behave as $E_{n}=\mathrm{O}\left(n^{2 / 3}\right)$ for large $n$.

Paper 4, Section II, C

comment(a) Consider for $\lambda>0$ the Laplace type integral

$I(\lambda)=\int_{a}^{b} f(t) \mathrm{e}^{-\lambda \phi(t)} d t$

for some finite $a, b \in \mathbb{R}$ and smooth, real-valued functions $f(t), \phi(t)$. Assume that the function $\phi(t)$ has a single minimum at $t=c$ with $a<c<b$. Give an account of Laplace's method for finding the leading order asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$ and briefly discuss the difference if instead $c=a$ or $c=b$, i.e. when the minimum is attained at the boundary.

(b) Determine the leading order asymptotic behaviour of

$I(\lambda)=\int_{-2}^{1} \cos t \mathrm{e}^{-\lambda t^{2}} d t$

as $\lambda \rightarrow \infty$

(c) Determine also the leading order asymptotic behaviour when cos $t$ is replaced by $\sin t$ in $(*)$.

Paper 1, Section I, D

commentA system with coordinates $q_{i}, i=1, \ldots, n$, has the Lagrangian $L\left(q_{i}, \dot{q}_{i}\right)$. Define the energy $E$.

Consider a charged particle, of mass $m$ and charge $e$, moving with velocity $\mathbf{v}$ in the presence of a magnetic field $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. The usual vector equation of motion can be derived from the Lagrangian

$L=\frac{1}{2} m \mathbf{v}^{2}+e \mathbf{v} \cdot \mathbf{A}$

where $\mathbf{A}$ is the vector potential.

The particle moves in the presence of a field such that

$\mathbf{A}=(0, r g(z), 0), \quad g(z)>0$

referred to cylindrical polar coordinates $(r, \phi, z)$. Obtain two constants of the motion, and write down the Lagrangian equations of motion obtained by variation of $r, \phi$ and $z$.

Show that, if the particle is projected from the point $\left(r_{0}, \phi_{0}, z_{0}\right)$ with velocity $\left(0,-2(e / m) r_{0} g\left(z_{0}\right), 0\right)$, it will describe a circular orbit provided that $g^{\prime}\left(z_{0}\right)=0$.

Paper 2, Section I, D

commentGiven the form

$T=\frac{1}{2} T_{i j} \dot{q}_{i} \dot{q}_{j}, \quad V=\frac{1}{2} V_{i j} q_{i} q_{j}$

for the kinetic energy $T$ and potential energy $V$ of a mechanical system, deduce Lagrange's equations of motion.

A light elastic string of length $4 b$, fixed at both ends, has three particles, each of mass $m$, attached at distances $b, 2 b, 3 b$ from one end. Gravity can be neglected. The particles vibrate with small oscillations transversely to the string, the tension $S$ in the string providing the restoring force. Take the displacements of the particles, $q_{i}, i=1,2,3$, to be the generalized coordinates. Take units such that $m=1, S / b=1$ and show that

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

Find the normal-mode frequencies for this system.

Paper 2, Section II, D

commentAn axially-symmetric top of mass $m$ is free to rotate about a fixed point $O$ on its axis. The principal moments of inertia about $O$ are $A, A, C$, and the centre of gravity $G$ is at a distance $\ell$ from $O$. Define Euler angles $\theta, \phi$ and $\psi$ which specify the orientation of the top, where $\theta$ is the inclination of $O G$ to the upward vertical. Show that there are three conserved quantities for the motion, and give their physical meaning.

Initially, the top is spinning with angular velocity $n$ about $O G$, with $G$ vertically above $O$, before being disturbed slightly. Show that, in the subsequent motion, $\theta$ will remain close to zero provided $C^{2} n^{2}>4 m g \ell A$, but that if $C^{2} n^{2}<4 m g \ell A$, then $\theta$ will attain a maximum value given by

$\cos \theta \simeq\left(C^{2} n^{2} / 2 m g \ell A\right)-1$

Paper 3, Section I, D

commentEuler's equations for the angular velocity $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of a rigid body, viewed in the body frame, are

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

and cyclic permutations, where the principal moments of inertia are assumed to obey $I_{1}<I_{2}<I_{3}$.

Write down two quadratic first integrals of the motion.

There is a family of solutions $\boldsymbol{\omega}(t)$, unique up to time-translations $t \rightarrow\left(t-t_{0}\right)$, which obey the boundary conditions $\boldsymbol{\omega} \rightarrow(0, \Omega, 0)$ as $t \rightarrow-\infty$ and $\boldsymbol{\omega} \rightarrow(0,-\Omega, 0)$ as $t \rightarrow \infty$, for a given positive constant $\Omega$. Show that, for such a solution, one has

$\mathbf{L}^{2}=2 E I_{2},$

where $\mathbf{L}$ is the angular momentum and $E$ is the kinetic energy.

By eliminating $\omega_{1}$ and $\omega_{3}$ in favour of $\omega_{2}$, or otherwise, show that, in this case, the second Euler equation reduces to

$\frac{d s}{d \tau}=1-s^{2}$

where $s=\omega_{2} / \Omega$ and $\tau=\Omega t\left[\left(I_{1}-I_{2}\right)\left(I_{2}-I_{3}\right) / I_{1} I_{3}\right]^{1 / 2}$. Find the general solution $s(\tau)$.

[You are not expected to calculate $\omega_{1}(t)$ or $\left.\omega_{3}(t) .\right]$

Paper 4, Section I, D

commentA system with one degree of freedom has Lagrangian $L(q, \dot{q})$. Define the canonical momentum $p$ and the energy $E$. Show that $E$ is constant along any classical path.

Consider a classical path $q_{c}(t)$ with the boundary-value data

$q_{c}(0)=q_{I}, \quad q_{c}(T)=q_{F}, \quad T>0$

Define the action $S_{c}\left(q_{I}, q_{F}, T\right)$ of the path. Show that the total derivative $d S_{c} / d T$ along the classical path obeys

$\frac{d S_{c}}{d T}=L$

Using Lagrange's equations, or otherwise, deduce that

$\frac{\partial S_{c}}{\partial q_{F}}=p_{F}, \quad \frac{\partial S_{c}}{\partial T}=-E,$

where $p_{F}$ is the final momentum.

Paper 4, Section II, D

commentA system is described by the Hamiltonian $H(q, p)$. Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t), g(q, p, t)$, and show from Hamilton's equations that

$\frac{d f}{d t}=\{f, H\}+\frac{\partial f}{\partial t}$

Consider the Hamiltonian

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

and define

$a=(p-i \omega q) /(2 \omega)^{1 / 2}, \quad a^{*}=(p+i \omega q) /(2 \omega)^{1 / 2},$

where $i=\sqrt{-1}$. Evaluate $\{a, a\}$ and $\left\{a, a^{*}\right\}$, and show that $\{a, H\}=-i \omega a$ and $\left\{a^{*}, H\right\}=i \omega a^{*}$. Show further that, when $f(q, p, t)$ is regarded as a function of the independent complex variables $a, a^{*}$ and of $t$, one has

$\frac{d f}{d t}=i \omega\left(a^{*} \frac{\partial f}{\partial a^{*}}-a \frac{\partial f}{\partial a}\right)+\frac{\partial f}{\partial t}$

Deduce that both $\log a^{*}-i \omega t$ and $\log a+i \omega t$ are constant during the motion.

Paper 1, Section I, H

commentExplain what is meant by saying that a binary code $\mathcal{C}$ is a decodable code with words $C_{j}$ of length $l_{j}$ for $1 \leqslant j \leqslant n$. Prove the MacMillan inequality which states that, for such a code,

$\sum_{j=1}^{n} 2^{-l_{j}} \leqslant 1$

Paper 1, Section II, H

commentState and prove Shannon's theorem for the capacity of a noisy memoryless binary symmetric channel, defining the terms you use.

[You may make use of any form of Stirling's formula and any standard theorems from probability, provided that you state them exactly.]

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

commentDescribe the standard Hamming code of length 7 , proving that it corrects a single error. Find its weight enumeration polynomial.

Paper 2, Section II, H

commentThe Van der Monde matrix $V\left(x_{0}, x_{1}, \ldots, x_{r-1}\right)$ is the $r \times r$ matrix with $(i, j)$ th entry $x_{i-1}^{j-1}$. Find an expression for $\operatorname{det} V\left(x_{0}, x_{1}, \ldots, x_{r-1}\right)$ as a product. Explain why this expression holds if we work modulo $p$ a prime.

Show that $\operatorname{det} V\left(x_{0}, x_{1}, \ldots, x_{r-1}\right) \equiv 0$ modulo $p$ if $r>p$, and that there exist $x_{0}, \ldots, x_{p-1}$ such that $\operatorname{det} V\left(x_{0}, x_{1}, \ldots, x_{p-1}\right) \not \equiv 0$. By using Wilson's theorem, or otherwise, find the possible values of $\operatorname{det} V\left(x_{0}, x_{1}, \ldots, x_{p-1}\right)$ modulo $p$.

The Dark Lord Y'Trinti has acquired the services of the dwarf Trigon who can engrave pairs of very large integers on very small rings. The Dark Lord wishes Trigon to engrave $n$ rings in such a way that anyone who acquires $r$ of the rings and knows the Prime Perilous $p$ can deduce the Integer $N$ of Power, but owning $r-1$ rings will give no information whatsoever. The integers $N$ and $p$ are very large and $p>N$. Advise the Dark Lord.

For reasons to be explained in the prequel, Trigon engraves an $(n+1)$ st ring with random integers. A band of heroes (who know the Prime Perilous and all the information contained in this question) set out to recover the rings. What, if anything, can they say, with very high probability, about the Integer of Power if they have $r$ rings (possibly including the fake)? What can they say if they have $r+1$ rings? What if they have $r+2$ rings?

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

commentWhat is a linear code? What is a parity check matrix for a linear code? What is the minimum distance $d(C)$ for a linear code $C ?$

If $C_{1}$ and $C_{2}$ are linear codes having a certain relation (which you should specify), define the bar product $C_{1} \mid C_{2}$. Show that

$d\left(C_{1} \mid C_{2}\right)=\min \left\{2 d\left(C_{1}\right), d\left(C_{2}\right)\right\}$

If $C_{1}$ has parity check matrix $P_{1}$ and $C_{2}$ has parity check matrix $P_{2}$, find a parity check matrix for $C_{1} \mid C_{2}$.

Paper 4, Section I, H

commentWhat is the discrete logarithm problem?

Describe the Diffie-Hellman key exchange system for two people. What is the connection with the discrete logarithm problem? Why might one use this scheme rather than just a public key system or a classical (pre-1960) coding system?

Extend the Diffie-Hellman system to $n$ people using $n(n-1)$ transmitted numbers.

Paper 1, Section I, D

commentWhat is meant by the expression 'Hubble time'?

For $a(t)$ the scale factor of the universe and assuming $a(0)=0$ and $a\left(t_{0}\right)=1$, where $t_{0}$ is the time now, obtain a formula for the size of the particle horizon $R_{0}$ of the universe.

Taking

$a(t)=\left(t / t_{0}\right)^{\alpha},$

show that $R_{0}$ is finite for certain values of $\alpha$. What might be the physically relevant values of $\alpha$ ? Show that the age of the universe is less than the Hubble time for these values of $\alpha$.

Paper 1, Section II, D

commentA star has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is the distance from the centre of the star. These quantities are related by the pressure support equation

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

where $P^{\prime}=d P / d r$ and $m(r)$ is the mass within radius $r$. Use this to derive the virial theorem

$E_{\text {grav }}=-3\langle P\rangle V,$

where $E_{\text {grav }}$ is the total gravitational potential energy and $\langle P\rangle$ the average pressure.

The total kinetic energy of a spherically symmetric star is related to $\langle P\rangle$ by

$E_{\mathrm{kin}}=\alpha\langle P\rangle V$

where $\alpha$ is a constant. Use the virial theorem to determine the condition on $\alpha$ for gravitational binding. By considering the relation between pressure and 'internal energy' $U$ for an ideal gas, determine $\alpha$ for the cases of a) an ideal gas of non-relativistic particles, b) an ideal gas of ultra-relativistic particles.

Why does your result imply a maximum mass for any star? Briefly explain what is meant by the Chandrasekhar limit.

A white dwarf is in orbit with a companion star. It slowly accretes matter from the other star until its mass exceeds the Chandrasekhar limit. Briefly explain its subsequent evolution.

Paper 2, Section I, D

commentThe number density $n=N / V$ for a photon gas in equilibrium is given by

$n=\frac{8 \pi}{c^{3}} \int_{0}^{\infty} \frac{\nu^{2}}{e^{h \nu / k T}-1} d \nu$

where $\nu$ is the photon frequency. By letting $x=h \nu / k T$, show that

$n=\alpha T^{3}$

where $\alpha$ is a constant which need not be evaluated.

The photon entropy density is given by

$s=\beta T^{3}$

where $\beta$ is a constant. By considering the entropy, explain why a photon gas cools as the universe expands.

Paper 3, Section I, D

commentConsider a homogenous and isotropic universe with mass density $\rho(t)$, pressure $P(t)$ and scale factor $a(t)$. As the universe expands its energy changes according to the relation $d E=-P d V$. Use this to derive the fluid equation

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

Use conservation of energy applied to a test particle at the boundary of a spherical fluid element to derive the Friedmann equation

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

where $k$ is a constant. State any assumption you have made. Briefly state the significance of $k$.

Paper 3, Section II, D

commentThe number density for particles in thermal equilibrium, neglecting quantum effects, is

$n=g_{s} \frac{4 \pi}{h^{3}} \int p^{2} d p \exp (-(E(p)-\mu) / k T)$

where $g_{s}$ is the number of degrees of freedom for the particle with energy $E(p)$ and $\mu$ is its chemical potential. Evaluate $n$ for a non-relativistic particle.

Thermal equilibrium between two species of non-relativistic particles is maintained by the reaction

$a+\alpha \leftrightarrow b+\beta$

where $\alpha$ and $\beta$ are massless particles. Evaluate the ratio of number densities $n_{a} / n_{b}$ given that their respective masses are $m_{a}$ and $m_{b}$ and chemical potentials are $\mu_{a}$ and $\mu_{b}$.

Explain how a reaction like the one above is relevant to the determination of the neutron to proton ratio in the early universe. Why does this ratio not fall rapidly to zero as the universe cools?

Explain briefly the process of primordial nucleosynthesis by which neutrons are converted into stable helium nuclei. Letting

$Y_{H e}=\rho_{H e} / \rho$

be the fraction of the universe's helium, compute $Y_{H e}$ as a function of the ratio $r=n_{n} / n_{p}$ at the time of nucleosynthesis.

Paper 4, Section I, D

commentThe linearised equation for the growth of density perturbations, $\delta_{\mathbf{k}}$, in an isotropic and homogenous universe is

$\ddot{\delta}_{\mathbf{k}}+2 \frac{\dot{a}}{a} \dot{\delta}_{\mathbf{k}}+\left(\frac{c_{s}^{2} \mathbf{k}^{2}}{a^{2}}-4 \pi G \rho\right) \delta_{\mathbf{k}}=0$

where $\rho$ is the density of matter, $c_{s}$ the sound speed, $c_{s}^{2}=d P / d \rho$, and $\mathbf{k}$ is the comoving wavevector and $a(t)$ is the scale factor of the universe.

What is the Jean's length? Discuss its significance for the growth of perturbations.

Consider a universe filled with pressure-free matter with $a(t)=\left(t / t_{0}\right)^{2 / 3}$. Compute the resulting equation for the growth of density perturbations. Show that your equation has growing and decaying modes and comment briefly on the significance of this fact.

Paper 1, Section II, H

comment(i) State the definition of smooth manifold with boundary and define the notion of boundary. Show that the boundary $\partial X$ is a manifold (without boundary) with $\operatorname{dim} \partial X=\operatorname{dim} X-1$.

(ii) Let $0<a<1$ and let $x_{1}, x_{2}, x_{3}, x_{4}$ denote Euclidean coordinates on $\mathbb{R}^{4}$. Show that the set

$X=\left\{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2} \leqslant a\right\} \cap\left\{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\right\} \cap\left\{x_{1}^{2}+2 x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=3 / 2\right\}$

is a manifold with boundary and compute its dimension. You may appeal to standard results concerning regular values of smooth functions.

(iii) Determine if the following statements are true or false, giving reasons:

a. If $X$ and $Y$ are manifolds, $f: X \rightarrow Y$ smooth and $Z \subset Y$ a submanifold of codimension $r$ such that $f$ is not transversal to $Z$, then $f^{-1}(Z)$ is not a submanifold of codimension $r$ in $X$.

b. If $X$ and $Y$ are manifolds and $f: X \rightarrow Y$ is smooth, then the set of regular values of $f$ is open in $Y$.

c. If $X$ and $Y$ are manifolds and $f: X \rightarrow Y$ is smooth then the set of critical points is of measure 0 in $X$.

Paper 2, Section II, H

comment(i) State and prove the isoperimetric inequality for plane curves. You may appeal to Wirtinger's inequality as long as you state it precisely.

(ii) State Fenchel's theorem for curves in space.

(iii) Let $\alpha: I \rightarrow \mathbb{R}^{2}$ be a closed regular plane curve bounding a region $K$. Suppose $K \supset\left[p_{1}, p_{1}+d_{1}\right] \times\left[p_{2}, p_{2}+d_{2}\right]$, for $d_{1}>0, d_{2}>0$, i.e. $K$ contains a rectangle of dimensions $d_{1}, d_{2}$. Let $k(s)$ denote the signed curvature of $\alpha$ with respect to the inward pointing normal, where $\alpha$ is parametrised anticlockwise. Show that there exists an $s_{0} \in I$ such that $k\left(s_{0}\right) \leqslant \sqrt{\pi /\left(d_{1} d_{2}\right)}$.

Paper 3, Section II, H

comment(i) State and prove the Theorema Egregium.

(ii) Define the notions principal curvatures, principal directions and umbilical point.

(iii) Let $S \subset \mathbb{R}^{3}$ be a connected compact regular surface (without boundary), and let $D \subset S$ be a dense subset of $S$ with the following property. For all $p \in D$, there exists an open neighbourhood $\mathcal{U}_{p}$ of $p$ in $S$ such that for all $\theta \in[0,2 \pi), \psi_{p, \theta}\left(\mathcal{U}_{p}\right)=\mathcal{U}_{p}$, where $\psi_{p, \theta}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ denotes rotation by $\theta$ around the line through $p$ perpendicular to $T_{p} S$. Show that $S$ is in fact a sphere.

Paper 4, Section II, H

comment(i) Let $S \subset \mathbb{R}^{3}$ be a regular surface. Define the notions exponential map, geodesic polar coordinates, geodesic circles.

(ii) State and prove Gauss' lemma.

(iii) Let $S$ be a regular surface. For fixed $r>0$, and points $p, q$ in $S$, let $S_{r}(p)$, $S_{r}(q)$ denote the geodesic circles around $p, q$, respectively, of radius $r$. Show the following statement: for each $p \in S$, there exists an $r=r(p)>0$ and a neighborhood $\mathcal{U}_{p}$ containing $p$ such that for all $q \in \mathcal{U}_{p}$, the sets $S_{r}(p)$ and $S_{r}(q)$ are smooth 1-dimensional manifolds which intersect transversally. What is the cardinality $\bmod 2$ of $S_{r}(p) \cap S_{r}(q)$ ?

Paper 1, Section I, D

commentConsider the 2-dimensional flow

$\dot{x}=-\mu x+y, \quad \dot{y}=\frac{x^{2}}{1+x^{2}}-\nu y$

where $x(t)$ and $y(t)$ are non-negative, the parameters $\mu$ and $\nu$ are strictly positive and $\mu \neq \nu$. Sketch the nullclines in the $x, y$ plane. Deduce that for $\mu<\mu_{c}$ (where $\mu_{c}$ is to be determined) there are three fixed points. Find them and determine their type.

Sketch the phase portrait for $\mu<\mu_{c}$ and identify, qualitatively on your sketch, the stable and unstable manifolds of the saddle point. What is the final outcome of this system?

Paper 2, Section I, D

commentConsider the 2-dimensional flow

$\dot{x}=\mu\left(\frac{1}{3} x^{3}-x\right)+y, \quad \dot{y}=-x$

where the parameter $\mu>0$. Using Lyapunov's approach, discuss the stability of the fixed point and its domain of attraction. Relevant definitions or theorems that you use should be stated carefully, but proofs are not required.

Paper 3, Section I, D

commentLet $I=[0,1)$. The sawtooth (Bernoulli shift) map $F: I \rightarrow I$ is defined by

$F(x)=2 x[\bmod 1]$

Describe the effect of $F$ using binary notation. Show that $F$ is continuous on $I$ except at $x=\frac{1}{2}$. Show also that $F$ has $N$-periodic points for all $N \geqslant 2$. Are they stable?

Explain why $F$ is chaotic, using Glendinning's definition.

Paper 3, Section II, D

commentDescribe informally the concepts of extended stable manifold theory. Illustrate your discussion by considering the 2-dimensional flow

$\dot{x}=\mu x+x y-x^{3}, \quad \dot{y}=-y+y^{2}-x^{2},$

where $\mu$ is a parameter with $|\mu| \ll 1$, in a neighbourhood of the origin. Determine the nature of the bifurcation.

Paper 4, Section I, D

commentConsider the 2-dimensional flow

$\dot{x}=y+\frac{1}{4} x\left(1-2 x^{2}-2 y^{2}\right), \quad \dot{y}=-x+\frac{1}{2} y\left(1-x^{2}-y^{2}\right)$

Use the Poincaré-Bendixson theorem, which should be stated carefully, to obtain a domain $\mathcal{D}$ in the $x y$-plane, within which there is at least one periodic orbit.

Paper 4, Section II, D

commentLet $I=[0,1]$ and consider continuous maps $F: I \rightarrow I$. Give an informal outline description of the two different bifurcations of fixed points of $F$ that can occur.

Illustrate your discussion by considering in detail the logistic map

$F(x)=\mu x(1-x),$

for $\mu \in(0,1+\sqrt{6}]$.

Describe qualitatively what happens for $\mu \in(1+\sqrt{6}, 4]$.

[You may assume without proof that

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

Paper 1, Section II, B

commentThe vector potential $A^{\mu}$ is determined by a current density distribution $j^{\mu}$ in the gauge $\partial_{\mu} A^{\mu}=0$ by

$\square A^{\mu}=-\mu_{0} j^{\mu}, \quad \square=-\frac{\partial^{2}}{\partial t^{2}}+\nabla^{2},$

in units where $c=1$.

Describe how to justify the result

$A^{\mu}(\mathbf{x}, t)=\frac{\mu_{0}}{4 \pi} \int d^{3} x^{\prime} \frac{j^{\mu}\left(\mathbf{x}^{\prime}, t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}, \quad t^{\prime}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right|$

A plane square loop of thin wire, edge lengths $l$, has its centre at the origin and lies in the $(x, y)$ plane. For $t<0$, no current is flowing in the loop, but at $t=0$ a constant current $I$ is turned on.

Find the vector potential at the point $(0,0, z)$ as a function of time due to a single edge of the loop.

What is the electric field due to the entire loop at $(0,0, z)$ as a function of time? Give a careful justification of your answer.

Paper 3, Section II, B

commentA particle of rest-mass $m$, electric charge $q$, is moving relativistically along the path $x^{\mu}(s)$ where $s$ parametrises the path.

Write down an action for which the extremum determines the particle's equation of motion in an electromagnetic field given by the potential $A^{\mu}(x)$.

Use your action to derive the particle's equation of motion in a form where $s$ is the proper time.

Suppose that the electric and magnetic fields are given by

$\begin{aligned} \mathbf{E} &=(0,0, E) \\ \mathbf{B} &=(0, B, 0) \end{aligned}$

where $E$ and $B$ are constants and $B>E>0$.

Find $x^{\mu}(s)$ given that the particle starts at rest at the origin when $s=0$.

Describe qualitatively the motion of the particle.

Paper 4, Section II, B

In a superconductor the number density of charge carriers of charge $q$