# Part II, 2008

### Jump to course

1.II.21F

comment(i) State the van Kampen theorem.

(ii) Calculate the fundamental group of the wedge $S^{2} \vee S^{1}$.

(iii) Let $X=\mathbb{R}^{3} \backslash A$ where $A$ is a circle. Calculate the fundamental group of $X$.

2.II.21F

commentProve the Borsuk-Ulam theorem in dimension 2: there is no map $f: S^{2} \rightarrow S^{1}$ such that $f(-x)=-f(x)$ for every $x \in S^{2}$. Deduce that $S^{2}$ is not homeomorphic to any subset of $\mathbb{R}^{2}$.

3.II.20F

commentLet $X$ be the quotient space obtained by identifying one pair of antipodal points on $S^{2}$. Using the Mayer-Vietoris exact sequence, calculate the homology groups and the Betti numbers of $X$.

4.II.21F

commentLet $X$ and $Y$ be topological spaces.

(i) Show that a map $f: X \rightarrow Y$ is a homotopy equivalence if there exist maps $g, h: Y \rightarrow X$ such that $f g \simeq 1_{Y}$ and $h f \simeq 1_{X}$. More generally, show that a map $f: X \rightarrow Y$ is a homotopy equivalence if there exist maps $g, h: Y \rightarrow X$ such that $f g$ and $h f$ are homotopy equivalences.

(ii) Suppose that $\tilde{X}$ and $\tilde{Y}$ are universal covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. Using path-lifting properties, show that if $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$.

1.II.33E

commentA beam of particles each of mass $m$ and energy $\hbar^{2} k^{2} /(2 m)$ scatters off an axisymmetric potential $V$. In the first Born approximation the scattering amplitude is

$f(\theta)=-\frac{m}{2 \pi \hbar^{2}} \int e^{-i\left(\mathbf{k}-\mathbf{k}_{0}\right) \cdot \mathbf{x}^{\prime}} V\left(\mathbf{x}^{\prime}\right) d^{3} x^{\prime}$

where $\mathbf{k}_{0}=(0,0, k)$ is the wave vector of the incident particles and $\mathbf{k}=(k \sin \theta, 0, k \cos \theta)$ is the wave vector of the outgoing particles at scattering angle $\theta$ (and $\phi=0$ ). Let $\mathbf{q}=\mathbf{k}-\mathbf{k}_{0}$ and $q=|\mathbf{q}|$. Show that when the scattering potential $V$ is spherically symmetric the expression $(*)$ simplifies to

$f(\theta)=-\frac{2 m}{\hbar^{2} q} \int_{0}^{\infty} r^{\prime} V\left(r^{\prime}\right) \sin \left(q r^{\prime}\right) d r^{\prime}$

and find the relation between $q$ and $\theta$.

Calculate this scattering amplitude for the potential $V(r)=V_{0} e^{-r}$ where $V_{0}$ is a constant, and show that at high energies the particles emerge predominantly in a narrow cone around the forward beam direction. Estimate the angular width of the cone.

2.II.33E

commentConsider a large, essentially two-dimensional, rectangular sample of conductor of area $A$, and containing $2 N$ electrons of charge $-e$. Suppose a magnetic field of strength $B$ is applied perpendicularly to the sample. Write down the Landau Hamiltonian for one of the electrons assuming that the electron interacts just with the magnetic field.

[You may ignore the interaction of the electron spin with the magnetic field.]

Find the allowed energy levels of the electron.

Find the total energy of the $2 N$ electrons at absolute zero temperature as a function of $B$, assuming that $B$ is in the range

$\frac{\pi \hbar N}{e A} \leqslant B \leqslant \frac{2 \pi \hbar N}{e A} .$

Comment on the values of the total energy when $B$ takes the values at the two ends of this range.

3.II.33E

commentConsider the body-centred cuboidal lattice $L$ with lattice points $\left(n_{1} a, n_{2} a, n_{3} b\right)$ and $\left(\left(n_{1}+\frac{1}{2}\right) a,\left(n_{2}+\frac{1}{2}\right) a,\left(n_{3}+\frac{1}{2}\right) b\right)$, where $a$ and $b$ are positive and $n_{1}, n_{2}$ and $n_{3}$ take all possible integer values. Find the reciprocal lattice $\widetilde{L}$and describe its geometrical form. Calculate the volumes of the unit cells of the lattices $L$ and $\widetilde{L}$.

Find the reciprocal lattice vector associated with the lattice planes parallel to the plane containing the points $(0,0, b),(0, a, b),\left(\frac{1}{2} a, \frac{1}{2} a, \frac{1}{2} b\right),(a, 0,0)$ and $(a, a, 0)$. Deduce the allowed Bragg scattering angles of X-rays off these planes, assuming that $b=\frac{4}{3} a$ and that the X-rays have wavelength $\lambda=\frac{1}{2} a$.

4.II.33E

commentExplain why the allowed energies of electrons in a three-dimensional crystal lie in energy bands. What quantum numbers can be used to classify the electron energy eigenstates?

Describe the effect on the energy level structure of adding a small density of impurity atoms randomly to the crystal.

1.II.26I

commentLet $\left(X_{t}, t \geqslant 0\right)$ be an irreducible continuous-time Markov chain with initial probability distribution $\pi$ and Q-matrix $Q$ (for short: a $(\pi, Q)$ CTMC), on a finite state space $I$.

(i) Define the terms reversible CTMC and detailed balance equations (DBEs) and explain, without proof, the relation between them.

(ii) Prove that any solution of the DBEs is an equilibrium distribution (ED) for $\left(X_{t}\right)$.

Let $\left(Y_{n}, n=0,1, \ldots\right)$ be an irreducible discrete-time Markov chain with initial probability distribution $\widehat{\pi}$and transition probability matrix $\widehat{P}$(for short: a $(\widehat{\pi}, \widehat{P})$ DTMC), on the state space $I$.

(iii) Repeat the two definitions from (i) in the context of the DTMC $\left(Y_{n}\right)$. State also in this context the relation between them, and prove a statement analogous to (ii).

(iv) What does it mean to say that $\left(Y_{n}\right)$ is the jump chain for $\left(X_{t}\right)$ ? State and prove a relation between the ED $\pi$ for the $\operatorname{CTMC}\left(X_{t}\right)$ and the ED $\widehat{\pi}$for its jump chain $\left(Y_{n}\right)$.

(v) Prove that $\left(X_{t}\right)$ is reversible (in equilibrium) if and only if its jump chain $\left(Y_{n}\right)$ is reversible (in equilibrium).

(vi) Consider now a continuous time random walk on a graph. More precisely, consider a CTMC $\left(X_{t}\right)$ on an undirected graph, where some pairs of states $i, j \in I$ are joined by one or more non-oriented 'links' $e_{i j}(1), \ldots, e_{i j}\left(m_{i j}\right)$. Here $m_{i j}=m_{j i}$ is the number of links between $i$ and $j$. Assume that the jump rate $q_{i j}$ is proportional to $m_{i j}$. Can the chain $\left(X_{t}\right)$ be reversible? Identify the corresponding jump chain $\left(Y_{n}\right)$ (which determines a discrete-time random walk on the graph) and comment on its reversibility.

2.II.26I

commentConsider a continuous-time Markov chain $\left(X_{t}\right)$ given by the diagram below.

We will assume that the rates $\alpha, \beta, \lambda$ and $\mu$ are all positive.

(a) Is the chain $\left(X_{t}\right)$ irreducible?

(b) Write down the standard equations for the hitting probabilities

$h_{\mathrm{C} i}=\mathbb{P}_{\mathrm{C} i}(\text { hit W0) }, \quad i \geqslant 0,$

and

$h_{\mathrm{W} i}=\mathbb{P}_{\mathrm{W} i}(\text { hit W0) }, \quad i \geqslant 1 .$

Explain how to identify the probabilities $h_{\mathrm{C} i}$ and $h_{\mathrm{W} i}$ among the solutions to these equations.

[You should state the theorem you use but its proof is not required.]

(c) Set $h^{(i)}=\left(\begin{array}{c}h_{\mathrm{C} i} \\ h_{\mathrm{W} i}\end{array}\right)$ and find a matrix $A$ such that

$h^{(i)}=A h^{(i-1)}, \quad i=1,2, \ldots$

The recursion matrix $A$ has a 'standard' eigenvalue and a 'standard' eigenvector that do not depend on the transition rates: what are they and why are they always present?

(d) Calculate the second eigenvalue $\vartheta$ of the matrix $A$, and the corresponding eigenvector, in the form $\left(\begin{array}{l}b \\ 1\end{array}\right)$, where $b>0$.

(e) Suppose the second eigenvalue $\vartheta$ is $\geqslant 1$. What can you say about $h_{\mathrm{C} i}$ and $h_{\mathrm{W} i}$ ? Is the chain $\left(X_{t}\right)$ transient or recurrent? Justify your answer.

(f) Now assume the opposite: the second eigenvalue $\vartheta$ is $<1$. Check that in this case $b<1$. Is the chain transient or recurrent under this condition?

(g) Finally, specify, by means of inequalities between the parameters $\alpha, \beta, \lambda$ and $\mu$, when the chain $\left(X_{t}\right)$ is recurrent and when it is transient.

3.II.25I

commentLet $\left(X_{t}\right)$ be an irreducible continuous-time Markov chain with countably many states. What does it mean to say the chain is (i) positive recurrent, (ii) null recurrent? Consider the chain $\left(X_{t}\right)$ with the arrow diagram below.

In this question we analyse the existence of equilibrium probabilities $\pi_{i \mathrm{C}}$ and $\pi_{i \mathrm{~W}}$ of the chain $\left(X_{t}\right)$ being in state $i \mathrm{C}$ or $i \mathrm{~W}, i=0,1, \ldots$, and the impact of this fact on positive and null recurrence of the chain.

(a) Write down the invariance equations $\pi Q=0$ and check that they have the form

$\begin{aligned} \pi_{0 C} &=\frac{\beta}{\lambda+\alpha} \pi_{0 \mathrm{~W}}, \\ \left(\pi_{1 \mathrm{C}}, \pi_{1 \mathrm{~W}}\right) &=\frac{\beta \pi_{0 \mathrm{~W}}}{\lambda+\alpha}\left(\frac{\lambda(\mu+\beta)}{\mu(\lambda+\alpha)}, \frac{\lambda}{\mu}\right) \\ \left(\pi_{(i+1) \mathrm{C}}, \pi_{(i+1) \mathrm{W}}\right) &=\left(\pi_{i \mathrm{C}}, \pi_{i \mathrm{~W}}\right) B, \quad i=1,2, \ldots, \end{aligned}$

where $B$ is a $2 \times 2$ recursion matrix:

$B=\left(\begin{array}{cc} \frac{\lambda \mu-\beta \alpha}{\mu(\lambda+\alpha)} & -\frac{\alpha}{\mu} \\ \frac{\beta(\beta+\mu)}{\mu(\lambda+\alpha)} & \frac{\beta+\mu}{\mu} \end{array}\right)$

(b) Verify that the row vector $\left(\pi_{1 \mathrm{C}}, \pi_{1 \mathrm{~W}}\right)$ is an eigenvector of $B$ with the eigenvalue $\theta$ where

$\theta=\frac{\lambda(\mu+\beta)}{\mu(\lambda+\alpha)}$

Hence, specify the form of equilibrium probabilities $\pi_{i \mathrm{C}}$ and $\pi_{i \mathrm{~W}}$ and conclude that the chain $\left(X_{t}\right)$ is positive recurrent if and only if $\mu \alpha>\lambda \beta$.

4.II.26I

commentOn a hot summer night, opening my window brings some relief. This attracts hordes of mosquitoes who manage to negotiate a dense window net. But, luckily, I have a mosquito trapping device in my room.

Assume the mosquitoes arrive in a Poisson process at rate $\lambda$; afterwards they wander around for independent and identically distributed random times with a finite mean $\mathbb{E} S$, where $S$ denotes the random wandering time of a mosquito, and finally are trapped by the device.

(a) Identify a mathematical model, which was introduced in the course, for the number of mosquitoes present in the room at times $t \geqslant 0$.

(b) Calculate the distribution of $Q(t)$ in terms of $\lambda$ and the tail probabilities $\mathbb{P}(S>x)$ of the wandering time $S$, where $Q(t)$ is the number of mosquitoes in the room at time $t>0$ (assuming that at the initial time, $Q(0)=0$ ).

(c) Write down the distribution for $Q^{\mathrm{E}}$, the number of mosquitoes in the room in equilibrium, in terms of $\lambda$ and $\mathbb{E} S$.

(d) Instead of waiting for the number of mosquitoes to reach equilibrium, I close the window at time $t>0$. For $v \geqslant 0$ let $X(t+v)$ be the number of mosquitoes left at time $t+v$, i.e. $v$ time units after closing the window. Calculate the distribution of $X(t+v)$.

(e) Let $V(t)$ be the time needed to trap all mosquitoes in the room after closing the window at time $t>0$. By considering the event $\{X(t+v) \geqslant 1\}$, or otherwise, compute $\mathbb{P}[V(t)>v]$.

(f) Now suppose that the time $t$ at which I shut the window is very large, so that I can assume that the number of mosquitoes in the room has the distribution of $Q^{E}$. Let $V^{E}$ be the further time needed to trap all mosquitoes in the room. Show that

$\mathbb{P}\left[V^{E}>v\right]=1-\exp \left(-\lambda \mathbb{E}\left[(S-v)_{+}\right]\right),$

where $x_{+} \equiv \max (x, 0)$.

1.II $. 30 \mathrm{~A}$

commentObtain an expression for the $n$th term of an asymptotic expansion, valid as $\lambda \rightarrow \infty$, for the integral

$I(\lambda)=\int_{0}^{1} t^{2 \alpha} e^{-\lambda\left(t^{2}+t^{3}\right)} d t \quad(\alpha>-1 / 2) .$

Estimate the value of $n$ for the term of least magnitude.

Obtain the first two terms of an asymptotic expansion, valid as $\lambda \rightarrow \infty$, for the integral

$J(\lambda)=\int_{0}^{1} t^{2 \alpha} e^{-\lambda\left(t^{2}-t^{3}\right)} d t \quad(-1 / 2<\alpha<0)$

[Hint:

$\left.\Gamma(z)=\int_{0}^{\infty} t^{z-1} e^{-t} d t .\right]$

[Stirling's formula may be quoted.]

3.II $. 30 \mathrm{~A}$

commentDescribe how the leading-order approximation may be found by the method of stationary phase of

$I(\lambda)=\int_{a}^{b} f(t) \exp (i \lambda g(t)) d t$

for $\lambda \gg 1$, where $\lambda, f$ and $g$ are real. You should consider the cases for which: (a) $g^{\prime}(t)$ has one simple zero at $t=t_{0}$, where $a<t_{0}<b$; (b) $g^{\prime}(t)$ has more than one simple zero in the region $a<t<b$; and (c) $g^{\prime}(t)$ has only a simple zero at $t=b$.

What is the order of magnitude of $I(\lambda)$ if $g^{\prime}(t)$ is non zero for $a \leqslant t \leqslant b$ ?

Use the method of stationary phase to find the leading-order approximation for $\lambda \gg 1$ to

$J(\lambda)=\int_{0}^{1} \sin \left(\lambda\left(t^{3}-t\right)\right) d t$

[Hint:

$\left.\int_{-\infty}^{\infty} \exp \left(i u^{2}\right) d u=\sqrt{\pi} e^{i \pi / 4} .\right]$

4.II $. 31 \mathrm{~A}$

commentThe Bessel equation of order $n$ is

$z^{2} y^{\prime \prime}+z y^{\prime}+\left(z^{2}-n^{2}\right) y=0 .$

Here, $n$ is taken to be an integer, with $n \geqslant 0$. The transformation $w(z)=z^{\frac{1}{2}} y(z)$ converts (1) to the form

$w^{\prime \prime}+q(z) w=0$

where

$q(z)=1-\frac{\left(n^{2}-\frac{1}{4}\right)}{z^{2}}$

Find two linearly independent solutions of the form

$w=e^{s z} \sum_{k=0}^{\infty} c_{k} z^{\rho-k}$

where $c_{k}$ are constants, with $c_{0} \neq 0$, and $s$ and $\rho$ are to be determined. Find recurrence relationships for the $c_{k}$.

Find the first two terms of two linearly independent Liouville-Green solutions of (2) for $w(z)$ valid in a neighbourhood of $z=\infty$. Relate these solutions to those of the form (3).

1.I.9A

commentThe action for a system with generalized coordinates $q_{i}(t)$ for a time interval $\left[t_{1}, t_{2}\right]$ is given by

$S=\int_{t_{1}}^{t_{2}} L\left(q_{i}, \dot{q}_{i}, t\right) d t$

where $L$ is the Lagrangian. The end point values $q_{i}\left(t_{1}\right)$ and $q_{i}\left(t_{2}\right)$ are fixed.

Derive Lagrange's equations from the principle of least action by considering the variation of $S$ for all possible paths.

Define the momentum $p_{i}$ conjugate to $q_{i}$. Derive a condition for $p_{i}$ to be a constant of the motion.

A symmetric top moves under the action of a potential $V(\theta)$. The Lagrangian is given by

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

where the generalized coordinates are the Euler angles $(\theta, \phi, \psi)$ and the principal moments of inertia are $I_{1}$ and $I_{3}$.

Show that $\omega_{3}=\dot{\psi}+\dot{\phi} \cos \theta$ is a constant of the motion and give expressions for two others. Show further that it is possible for the top to move with both $\theta$ and $\dot{\phi}$ constant provided these satisfy the condition

$I_{1} \dot{\phi}^{2} \sin \theta \cos \theta-I_{3} \omega_{3} \dot{\phi} \sin \theta=\frac{d V}{d \theta}$

2.I.9A

commentA system of $N$ particles $i=1,2,3, \ldots, N$, with mass $m_{i}$, moves around a circle of radius $a$. The angle between the radius to particle $i$ and a fixed reference radius is $\theta_{i}$. The interaction potential for the system is

$V=\frac{1}{2} k \sum_{j=1}^{N}\left(\theta_{j+1}-\theta_{j}\right)^{2}$

where $k$ is a constant and $\theta_{N+1}=\theta_{1}+2 \pi$.

The Lagrangian for the system is

$L=\frac{1}{2} a^{2} \sum_{j=1}^{N} m_{j} \dot{\theta}_{j}^{2}-V$

Write down the equation of motion for particle $i$ and show that the system is in equilibrium when the particles are equally spaced around the circle.

Show further that the system always has a normal mode of oscillation with zero frequency. What is the form of the motion associated with this?

Find all the frequencies and modes of oscillation when $N=2, m_{1}=k m / a^{2}$ and $m_{2}=2 \mathrm{~km} / \mathrm{a}^{2}$, where $m$ is a constant.

2.II.15B

commentA particle of mass $m$, charge $e$ and position vector $\mathbf{r}=\left(x_{1}, x_{2}, x_{3}\right) \equiv \mathbf{q}$ moves in a magnetic field whose vector potential is A. Its Hamiltonian is given by

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

Write down Hamilton's equations and use them to derive the equations of motion for the charged particle.

Define the Poisson bracket $[F, G]$ for general $F(\mathbf{p}, \mathbf{q})$ and $G(\mathbf{p}, \mathbf{q})$. Show that for motion governed by the above Hamiltonian

$\left[m \dot{x}_{i}, x_{j}\right]=-\delta_{i j}, \quad \text { and } \quad\left[m \dot{x}_{i}, m \dot{x}_{j}\right]=\frac{e}{c}\left(\frac{\partial A_{j}}{\partial x_{i}}-\frac{\partial A_{i}}{\partial x_{j}}\right)$

Consider the vector potential to be given by $\mathbf{A}=(0,0, F(r))$, where $r=\sqrt{x_{1}^{2}+x_{2}^{2}}$. Use Hamilton's equations to show that $p_{3}$ is constant and that circular motion at radius $r$ with angular frequency $\Omega$ is possible provided that

$\Omega^{2}=-\left(p_{3}-\frac{e F}{c}\right) \frac{e}{m^{2} c r} \frac{d F}{d r}$

3.I.9E

commentWriting $\mathbf{x}=\left(p_{1}, p_{2}, p_{3}, \ldots, p_{n}, q_{1}, q_{2}, q_{3}, \ldots, q_{n}\right)$, Hamilton's equations may be written in the form

$\dot{\mathbf{x}}=\mathbf{J} \frac{\partial H}{\partial \mathbf{x}} \text {, }$

where the $2 n \times 2 n$ matrix

$\mathbf{J}=\left(\begin{array}{rr} 0 & -I \\ I & 0 \end{array}\right)$

and $I$ and 0 denote the $n \times n$ unit and zero matrices respectively.

Explain what is meant by the statement that the transformation $\mathbf{x} \rightarrow \mathbf{y}$,

$\left(p_{1}, p_{2}, p_{3}, \ldots, p_{n}, q_{1}, q_{2}, q_{3}, \ldots, q_{n}\right) \rightarrow\left(P_{1}, P_{2}, P_{3}, \ldots, P_{n}, Q_{1}, Q_{2}, Q_{3}, \ldots, Q_{n}\right)$

is canonical, and show that the condition for this is that

$\mathbf{J}=\mathcal{J} \mathbf{J} \mathcal{J}^{T}$

where $\mathcal{J}$ is the Jacobian matrix with elements

$\mathcal{J}_{i j}=\frac{\partial y_{i}}{\partial x_{j}}$

Use this condition to show that for a system with $n=1$ the transformation given by

$P=p+2 q, \quad Q=\frac{1}{2} q-\frac{1}{4} p$

is canonical.

4.I.9B

comment(a) Show that the principal moments of inertia for an infinitesimally thin uniform rectangular sheet of mass $M$ with sides of length $a$ and $b$ (with $b<a$ ) about its centre of mass are $I_{1}=M b^{2} / 12, I_{2}=M a^{2} / 12$ and $I_{3}=M\left(a^{2}+b^{2}\right) / 12$.

(b) Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of the sheet 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

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

A possible solution of these equations is such that the sheet rotates with $\omega_{1}=\omega_{3}=0$, and $\omega_{2}=\Omega=$ constant.

By linearizing, find the equations governing small motions in the neighbourhood of this solution that have $\left(\omega_{1}, \omega_{3}\right) \neq 0$. Use these to show that there are solutions corresponding to instability such that $\omega_{1}$ and $\omega_{3}$ are both proportional to exp $(\beta \Omega t)$, with $\beta=\sqrt{\left(a^{2}-b^{2}\right) /\left(a^{2}+b^{2}\right)} .$

4.II.15B

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

Show from Hamilton's equations that when $H$ does not depend on time explicitly, for any function $F=F(\mathbf{p}, \mathbf{q})$,

$\frac{d F}{d t}=[F, H],$

where $[F, H]$ denotes the Poisson bracket.

For a system of $N$ interacting vortices

$H(\mathbf{p}, \mathbf{q})=-\frac{\kappa}{4} \sum_{\substack{i=1 \\ N}}^{N} \sum_{\substack{j=1 \\ j \neq i}}^{N} \ln \left[\left(p_{i}-p_{j}\right)^{2}+\left(q_{i}-q_{j}\right)^{2}\right]$

where $\kappa$ is a constant. Show that the quantity defined by

$F=\sum_{j=1}^{N}\left(q_{j}^{2}+p_{j}^{2}\right)$

is a constant of the motion.

(b) The action for a Hamiltonian system with one degree of freedom with $H=H(p, q)$ for which the motion is periodic is

$I=\frac{1}{2 \pi} \oint p(H, q) d q .$

Show without assuming any specific form for $H$ that the period of the motion $T$ is given by

$\frac{2 \pi}{T}=\frac{d H}{d I}$

Suppose now that the system has a parameter that is allowed to vary slowly with time. Explain briefly what is meant by the statement that the action is an adiabatic invariant. Suppose that when this parameter is fixed, $H=0$ when $I=0$. Deduce that, if $T$ decreases on an orbit with any $I$ when the parameter is slowly varied, then $H$ increases.

1.I.4G

commentDefine the entropy $H(X)$ of a random variable $X$ that takes no more than $N$ different values. What are the maximum and the minimum values for the entropy for a fixed value of $N$ ? Explain when the maximum and minimum are attained. You should prove any inequalities that you use.

1.II.12G

commentState Shannon's Noisy Coding Theorem for a binary symmetric channel.

Define the mutual information of two discrete random variables $X$ and $Y$. Prove that the mutual information is symmetric and non-negative. Define also the information capacity of a channel.

A channel transmits numbers chosen from the alphabet $\mathcal{A}=\{0,1,2\}$ and has transition matrix

$\left(\begin{array}{ccc} 1-2 \beta & \beta & \beta \\ \beta & 1-2 \beta & \beta \\ \beta & \beta & 1-2 \beta \end{array}\right)$

for a number $\beta$ with $0 \leqslant \beta \leqslant \frac{1}{3}$. Calculate the information capacity of the channel.

2.I.4G

commentDescribe briefly the Shannon-Fano and Huffman binary codes for a finite alphabet. Find examples of such codes for the alphabet $\mathcal{A}=\{a, b, c, d\}$ when the four letters are taken with probabilities $0.4,0.3,0.2$ and $0.1$ respectively.

2.II.12G

commentDescribe the Rabin cipher with modulus $N$, explaining how it can be deciphered by the intended recipient and why it is difficult for an interceptor to decipher it.

The Bursars' Committee decides to communicate using Rabin ciphers to maintain confidentiality. The secretary of the committee encrypts a message, thought of as a positive integer $m$, using the Rabin cipher with modulus $N$ (with $0<m<N$ ) and publishes both the encrypted message and the modulus. A foolish bursar deciphers this message to read it but then encrypts it again using a Rabin cipher with a different modulus $N^{\prime}$ (with $\left.m<N^{\prime}\right)$ and publishes the newly encrypted message and $N^{\prime}$. The president of CUSU, who happens to be a talented mathematician, knows that this has happened. Explain how the president can work out what the original message was using the two different encrypted versions.

Can the president of CUSU also decipher other messages sent out by the Bursars' Committee?

3.I.4G

commentDefine the Hamming code $h: \mathbb{F}_{2}^{4} \rightarrow \mathbb{F}_{2}^{7}$ and prove that the minimum distance between two distinct code words is 3. Explain how the Hamming code allows one error to be corrected.

A new code $c: \mathbb{F}_{2}^{4} \rightarrow \mathbb{F}_{2}^{8}$ is obtained by using the Hamming code for the first 7 bits and taking the last bit as a check digit on the previous 7 . Find the minimum distance between two distinct code words for this code. How many errors can this code detect? How many errors can it correct?

4.I.4G

commentWhat is a binary cyclic code of length $N$ ? What is the generator polynomial for such a cyclic code? Prove that the generator polynomial is a factor of $X^{N}-1$ over the field $\mathbb{F}_{2}$.

Find all the binary cyclic codes of length 5 .

$3 . \mathrm{II} . 15 \mathrm{E} \quad$

commentSmall density perturbations $\delta_{\mathbf{k}}(t)$ in pressureless matter inside the cosmological horizon obey the following Fourier evolution equation

$\ddot{\delta}_{\mathbf{k}}+2 \frac{\dot{a}}{a} \dot{\delta}_{\mathbf{k}}-4 \pi G \bar{\rho}_{\mathrm{c}} \delta_{\mathbf{k}}=0$

where $\bar{\rho}_{\mathrm{c}}$ is the average background density of the pressureless gravitating matter and $\mathbf{k}$ is the comoving wavevector.

(i) Seek power law solutions $\delta_{\mathbf{k}} \propto t^{\beta}(\beta$ constant) during the matter-dominated epoch $\left(t_{\mathrm{eq}}<t<t_{0}\right)$ to find the approximate solution

$\delta_{\mathbf{k}}(t)=A(\mathbf{k})\left(\frac{t}{t_{\mathrm{eq}}}\right)^{2 / 3}+B(\mathbf{k})\left(\frac{t}{t_{\mathrm{eq}}}\right)^{-1}, \quad t \gg t_{\mathrm{eq}}$

where $A, B$ are functions of $\mathbf{k}$ only and $t_{\mathrm{eq}}$ is the time of equal matter-radiation.

By considering the behaviour of the scalefactor $a$ and the relative density $\bar{\rho}_{\mathrm{c}} / \bar{\rho}_{\text {total }}$, show that early in the radiation era $\left(t \ll t_{\mathrm{eq}}\right)$ there is effectively no significant perturbation growth in $\delta_{\mathbf{k}}$ on sub-horizon scales.

(ii) For a given wavenumber $k=|\mathbf{k}|$, show that the time $t_{\mathrm{H}}$ at which this mode crosses inside the horizon, i.e., $c t_{\mathrm{H}} \approx 2 \pi a\left(t_{\mathrm{H}}\right) / k$, is given by

$\frac{t_{\mathrm{H}}}{t_{0}} \approx \begin{cases}\left(\frac{k_{0}}{k}\right)^{3}, & t_{\mathrm{H}} \gg t_{\mathrm{eq}} \\ \left(1+z_{\mathrm{eq}}\right)^{-1 / 2}\left(\frac{k_{0}}{k}\right)^{2}, & t_{\mathrm{H}} \ll t_{\mathrm{eq}}\end{cases}$

where $k_{0} \equiv 2 \pi /\left(c t_{0}\right)$, and the equal matter-radiation redshift is given by $1+z_{\mathrm{eq}}=$ $\left(t_{0} / t_{\mathrm{eq}}\right)^{2 / 3}$.

Assume that primordial perturbations from inflation are scale-invariant with a constant amplitude as they cross the Hubble radius given by $\left\langle\left|\delta_{\mathbf{k}}\left(t_{\mathrm{H}}\right)\right|^{2}\right\rangle \approx V^{-1} A / k^{3}$, where $A$ is a constant and $V$ is a large volume. Use the results of (i) to project these perturbations forward to $t_{0}$, and show that the power spectrum for perturbations today will be given approximately by

$P(k) \equiv V\left\langle\left|\delta_{\mathbf{k}}\left(t_{0}\right)\right|^{2}\right\rangle \approx \frac{A}{k_{0}^{4}} \times \begin{cases}k, & k<k_{\mathrm{eq}} \\ k_{\mathrm{eq}}\left(\frac{k_{\mathrm{eq}}}{k}\right)^{3}, & k>k_{\mathrm{eq}}\end{cases}$

$3 . \mathrm{I} . 10 \mathrm{E} \quad$

commentThe energy density $\epsilon$ and pressure $P$ of photons in the early universe is given by

$\epsilon=\frac{4 \sigma}{c} T^{4}, \quad P=\frac{1}{3} \epsilon,$

where $\sigma$ is the Stefan-Boltzmann constant. By using the first law of thermodynamics $d E=T d S-P d V+\mu d N$, deduce that the entropy differential $d S$ can be expressed in the form

$d S=\frac{16 \sigma}{3 c} d\left(T^{3} V\right)$

With the third law, show that the entropy density is given by $s=(16 \sigma / 3 c) T^{3}$.

While particle interaction rates $\Gamma$ remain much greater than the Hubble parameter $H$, justify why entropy will be conserved during the expansion of the universe. Hence, in the early universe (radiation domination) show that the temperature $T \propto a^{-1}$ where $a(t)$ is the scale factor of the universe, and show that the Hubble parameter $H \propto T^{2}$.

1.I.10E

commentThe number density of particles of mass $m$ at equilibrium in the early universe is given by the integral

$n=\frac{4 \pi g_{\mathrm{s}}}{h^{3}} \int_{0}^{\infty} \frac{p^{2} d p}{\exp [(E(p)-\mu) / k T] \mp 1}, \quad \begin{cases}- & \text { bosons } \\ + & \text { fermions }\end{cases}$

where $E(p)=c \sqrt{p^{2}+m^{2} c^{2}}, \mu$ is the chemical potential, and $g_{\mathrm{s}}$ is the spin degeneracy. Assuming that the particles remain in equilibrium when they become non-relativistic $\left(k T, \mu \ll m c^{2}\right)$, show that the number density can be expressed as

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

[Hint: Recall that $\left.\int_{0}^{\infty} d x e^{-\sigma^{2} x^{2}}=\sqrt{\pi} /(2 \sigma), \quad(\sigma>0) .\right]$

At around $t=100$ seconds, deuterium $D$ forms through the nuclear fusion of nonrelativistic protons $p$ and neutrons $n$ via the interaction $p+n \leftrightarrow D$. In equilibrium, what is the relationship between the chemical potentials of the three species? Show that the ratio of their number densities can be expressed as

$\frac{n_{D}}{n_{n} n_{p}} \approx\left(\frac{\pi m_{p} k T}{h^{2}}\right)^{-3 / 2} e^{B_{D} / k T}$

where the deuterium binding energy is $B_{D}=\left(m_{n}+m_{p}-m_{D}\right) c^{2}$ and you may take $g_{D}=4$. Now consider the fractional densities $X_{a}=n_{a} / n_{B}$, where $n_{B}$ is the baryon density of the universe, to re-express the ratio above in the form $X_{D} /\left(X_{n} X_{p}\right)$, which incorporates the baryon-to-photon ratio $\eta$ of the universe.

[You may assume that the photon density is $n_{\gamma}=\left(16 \pi \zeta(3) /(h c)^{3}\right)(k T)^{3}$.]

Why does deuterium form only at temperatures much lower than that given by $k T \approx B_{D}$ ?

1.II.15E

comment(i) A homogeneous and isotropic universe has mass density $\rho(t)$ and scale factor $a(t)$. Show how the conservation of total energy (kinetic plus gravitational potential) when applied to a test particle on the edge of a spherical region in this universe can be used to obtain the Friedmann equation

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

where $k$ is a constant. State clearly any assumptions you have made.

(ii) Assume that the universe is flat $(k=0)$ and filled with two major components: pressure-free matter $\left(P_{\mathrm{M}}=0\right)$ and dark energy with equation of state $P_{\Lambda}=-\rho_{\Lambda} c^{2}$ where their mass densities today $\left(t=t_{0}\right)$ are given respectively by $\rho_{\mathrm{M} 0}$ and $\rho_{\Lambda 0}$. Assuming that each component independently satisfies the fluid conservation equation, $\dot{\rho}=-3 H\left(\rho+P / c^{2}\right)$, show that the total mass density can be expressed as

$\rho(t)=\frac{\rho_{\mathrm{M} 0}}{a^{3}}+\rho_{\Lambda 0}$

where we have set $a\left(t_{0}\right)=1$.

Hence, solve the Friedmann equation and show that the scale factor can be expressed in the form

$a(t)=\alpha(\sinh \beta t)^{2 / 3},$

where $\alpha$ and $\beta$ are constants which you should specify in terms of $\rho_{\mathrm{M} 0}, \rho_{\Lambda 0}$ and $t_{0}$.

[Hint: try the substitution $b=a^{3 / 2}$.]

Show that the scale factor $a(t)$ has the expected behaviour for a matter-dominated universe at early times $(t \rightarrow 0)$ and that the universe accelerates at late times $(t \rightarrow \infty)$.

2.I.10E

commentA spherically-symmetric star obeys the pressure-support equation

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

where $P(r)$ is the pressure at a distance $r$ from the centre, $\rho(r)$ is the density, and $m(r)$ is the mass within a sphere of radius $r$. Show that this implies

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

Propose and justify appropriate boundary conditions for the pressure $P(r)$ at the centre of the star $(r=0)$ and at its outer edge $r=R$.

Show that the function

$F(r)=P(r)+\frac{G m^{2}}{8 \pi r^{4}}$

is a decreasing function of $r$. Deduce that the central pressure $P_{\mathrm{c}} \equiv P(0)$ satisfies

$P_{\mathrm{c}}>\frac{G M^{2}}{8 \pi R^{4}},$

where $M \equiv m(R)$ is the mass of the star.

4.I.10E

commentThe Friedmann and Raychaudhuri equations are respectively

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3} \rho-\frac{k c^{2}}{a^{2}} \quad \text { and } \quad \frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+\frac{3 P}{c^{2}}\right) \text {, }$

where $\rho$ is the mass density, $P$ is the pressure, $k$ is the curvature and $\dot{a} \equiv d a / d t$ with $t$ the cosmic time. Using conformal time $\tau$ (defined by $d \tau=d t / a$ ) and the equation of state $P=w \rho c^{2}$, show that these can be rewritten as

$\frac{k c^{2}}{\mathcal{H}^{2}}=\Omega-1 \quad \text { and } \quad 2 \frac{d \mathcal{H}}{d \tau}=-(3 w+1)\left(\mathcal{H}^{2}+k c^{2}\right)$

where $\mathcal{H}=a^{-1} d a / d \tau$ and the relative density is $\Omega \equiv \rho / \rho_{\text {crit }}=8 \pi G \rho a^{2} /\left(3 \mathcal{H}^{2}\right)$.

Use these relations to derive the following evolution equation for $\Omega$

$\frac{d \Omega}{d \tau}=(3 w+1) \mathcal{H} \Omega(\Omega-1)$

For both $w=0$ and $w=-1$, plot the qualitative evolution of $\Omega$ as a function of $\tau$ in an expanding universe $\mathcal{H}>0$ (i.e. include curves initially with $\Omega>1$ and $\Omega<1$ ).

Hence, or otherwise, briefly describe the flatness problem of the standard cosmology and how it can be solved by inflation.

1.II.24H

commentLet $n \geqslant 1$ be an integer, and let $M(n)$ denote the set of $n \times n$ real-valued matrices. We make $M(n)$ into an $n^{2}$-dimensional smooth manifold via the obvious identification $M(n)=\mathbb{R}^{n^{2}}$.

(a) Let $G L(n)$ denote the subset

$G L(n)=\left\{A \in M(n): A^{-1} \text { exists }\right\}$

Show that $G L(n)$ is a submanifold of $M(n)$. What is $\operatorname{dim} G L(n)$ ?

(b) Now let $S L(n) \subset G L(n)$ denote the subset

$S L(n)=\{A \in G L(n): \operatorname{det} A=1\}$

Show that for $A \in G L(n)$,

$(d \text { det })_{A} B=\operatorname{tr}\left(A^{-1} B\right) \operatorname{det} A .$

Show that $S L(n)$ is a submanifold of $G L(n)$. What is the dimension of $S L(n) ?$

(c) Now consider the set $X=M(n) \backslash G L(n)$. For what values of $n \geqslant 1$ is $X$ a submanifold of $M(n)$ ?

2.II $. 24 \mathrm{H}$

comment(a) For a regular curve in $\mathbb{R}^{3}$, define curvature and torsion and state the Frenet formulas.

(b) State and prove the isoperimetric inequality for domains $\Omega \subset \mathbb{R}^{2}$ with compact closure and $C^{1}$ boundary $\partial \Omega$.

[You may assume Wirtinger's inequality.]

(c) Let $\gamma: I \rightarrow \mathbb{R}^{2}$ be a closed plane regular curve such that $\gamma$ is contained in a disc of radius $r$. Show that there exists $s \in I$ such that $|k(s)| \geqslant r^{-1}$, where $k(s)$ denotes the signed curvature. Show by explicit example that the assumption of closedness is necessary.

3.II.23H

commentLet $S \subset \mathbb{R}^{3}$ be a surface.

(a) Define the Gauss Map, principal curvatures $k_{i}$, Gaussian curvature $K$ and mean curvature $H$. State the Theorema Egregium.

(b) Define what is meant for $S$ to be minimal. Prove that if $S$ is minimal, then $K \leqslant 0$. Give an example of a minimal surface whose Gaussian curvature is not identically 0 , justifying your answer.

(c) Does there exist a compact minimal surface $S \subset \mathbb{R}^{3}$ ? Justify your answer.

4.II $. 24 \mathrm{H} \quad$

commentLet $S \subset \mathbb{R}^{3}$ be a surface.

(a) In the case where $S$ is compact, define the Euler characteristic $\chi$ and genus $g$ of $S$.

(b) Define the notion of geodesic curvature $k_{g}$ for regular curves $\gamma: I \rightarrow S$. When is $k_{g}=0$ ? State the Global Gauss-Bonnet Theorem (including boundary term).

(c) Let $S=\mathbb{S}^{2}$ (the standard 2-sphere), and suppose $\gamma \subset \mathbb{S}^{2}$ is a simple closed regular curve such that $\mathbb{S}^{2} \backslash \gamma$ is the union of two distinct connected components with equal areas. Can $\gamma$ have everywhere strictly positive or everywhere strictly negative geodesic curvature?

(d) Prove or disprove the following statement: if $S$ is connected with Gaussian curvature $K=1$ identically, then $S$ is a subset of a sphere of radius 1 .

1.I.7A

commentSketch the phase plane of the system

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

(i) for $k=0$ and (ii) for $k=1 / 10$. Include in your sketches any trajectories that are the separatrices of a saddle point. In case (ii) shade the domain of stability of the origin.

2.I.7A

commentExplain the difference between a stationary bifurcation and an oscillatory bifurcation for a fixed point $\mathbf{x}_{0}$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x} ; \mu)$ in $\mathbb{R}^{n}$ with a real parameter $\mu$.

The normal form of a Hopf bifurcation in polar coordinates is

$\begin{aligned} &\dot{r}=\mu r-a r^{3}+O\left(r^{5}\right) \\ &\dot{\theta}=\omega+c \mu-b r^{2}+O\left(r^{4}\right) \end{aligned}$

where $a, b, c$ and $\omega$ are constants, $a \neq 0$, and $\omega>0$. Sketch the phase plane near the bifurcation for each of the cases (i) $\mu<0<a$, (ii) $0<\mu, a$, (iii) $\mu, a<0$ and (iv) $a<0<\mu$.

Let $R$ be the radius and $T$ the period of the limit cycle when one exists. Sketch how $R$ varies with $\mu$ for the case when the limit cycle is subcritical. Find the leading-order approximation to $d T / d \mu$ for $|\mu| \ll 1$.

3.I.7A

commentState the normal-form equations for (i) a saddle-node bifurcation, (ii) a transcritical bifurcation and (iii) a pitchfork bifurcation, for a one-dimensional map $x_{n+1}=F\left(x_{n} ; \mu\right)$.

Consider a period-doubling bifurcation of the form

$x_{n+1}=-x_{n}+\alpha+\beta x_{n}+\gamma x_{n}^{2}+\delta x_{n}^{3}+O\left(x_{n}^{4}\right),$

where $x_{n}=O\left(\mu^{1 / 2}\right), \alpha, \beta=O(\mu)$, and $\gamma, \delta=O(1)$ as $\mu \rightarrow 0$. Show that

$X_{n+2}=X_{n}+\hat{\mu} X_{n}-A X_{n}^{3}+O\left(X_{n}^{4}\right),$

where $X_{n}=x_{n}-\frac{1}{2} \alpha$, and the parameters $\hat{\mu}$ and $A$ are to be identified in terms of $\alpha, \beta$, $\gamma$ and $\delta$. Deduce the condition for the bifurcation to be supercritical.

3.II.14A

commentDefine the Poincaré index of a simple closed curve, not necessarily a trajectory, and the Poincaré index of an isolated fixed point $\mathbf{x}_{0}$ for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$. State the Poincaré index of a periodic orbit.

Consider the system

$\begin{aligned} &\dot{x}=y+a x-b x^{3} \\ &\dot{y}=x^{3}-x \end{aligned}$

where $a$ and $b$ are constants and $a \neq 0$.

(a) Find and classify the fixed points, and state their Poincaré indices.

(b) By considering a suitable function $H(x, y)$, show that any periodic orbit $\Gamma$ satisfies

$\oint_{\Gamma}\left(x-x^{3}\right)\left(a x-b x^{3}\right) d t=0$

where $x(t)$ is evaluated along the orbit.

(c) Deduce that if $b / a<1$ then the second-order differential equation

$\ddot{x}-\left(a-3 b x^{2}\right) \dot{x}+x-x^{3}=0$

has no periodic solutions.

4.I.7A

commentLet $F: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. State when $F$ is chaotic according to Glendinning's definition.

Prove that if $F$ has a 3 -cycle then $F^{2}$ has a horseshoe.

[You may assume the Intermediate Value Theorem.]

4.II.14A

commentExplain the difference between a hyperbolic and a nonhyperbolic fixed point $\mathbf{x}_{0}$ for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$.

Consider the system in $\mathbb{R}^{2}$, where $\mu$ is a real parameter,

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

Show that the fixed point $(\mu, 0)$ has a bifurcation when $\mu=1$, while the fixed points $(0, \pm 1)$ have a bifurcation when $\mu=-1$.

[The fixed point at $(0,-1)$ should not be considered further.]

Analyse each of the bifurcations at $(\mu, 0)$ and $(0,1)$ in turn as follows. Make a change of variable of the form $\mathbf{X}=\mathbf{x}-\mathbf{x}_{0}(\mu), \nu=\mu-\mu_{0}$. Identify the (non-extended) stable and centre linear subspaces at the bifurcation in terms of $X$ and $Y$. By finding the leading-order approximation to the extended centre manifold, construct the evolution equation on the extended centre manifold, and determine the type of bifurcation. Sketch the local bifurcation diagram, showing which fixed points are stable.

[Hint: the leading-order approximation to the extended centre manifold of the bifurcation at $(0,1)$ is $Y=a X$ for some coefficient a.]

Show that there is another fixed point in $y>0$ for $\mu<1$, and that this fixed point connects the two bifurcations.

$3 . \mathrm{II} . 35 \mathrm{D} \quad$

The retarded scalar potential $\varphi(t, \mathbf{x})$ produced by a charge distribution $\rho(t, \mathbf{x})$ is given by

$\varphi(t, \mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \int_{\Omega} d^{3} x^{\prime} \frac{\rho\left(t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right|, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}$

where $\Omega$ denotes all 3 -space. Describe briefly and qualitatively the physics underlying this formula.

Write the integrand in the formula above as a 1-dimensional integral over a new time coordinate $\tau$. Next consider a special source, a point charge $q$ moving along a trajectory $\mathbf{x}=\mathbf{x}_{0}(t)$