# Part II, 2004

### Jump to course

B2.10

commentFor each of the following curves $C$

(i) $C=\left\{(x, y) \in \mathbb{A}^{2} \mid x^{3}-x=y^{2}\right\} \quad$ (ii) $C=\left\{(x, y) \in \mathbb{A}^{2} \mid x^{2} y+x y^{2}=x^{4}+y^{4}\right\}$

compute the points at infinity of $\bar{C} \subset \mathbb{P}^{2}$ (i.e. describe $\bar{C} \backslash C$ ), and find the singular points of the projective curve $\bar{C}$.

At which points of $\bar{C}$ is the rational map $\bar{C} \rightarrow \mathbb{P}^{1}$, given by $(X: Y: Z) \mapsto(X: Y)$, not defined? Justify your answer.

B3.10

comment(i) Let $f: X \rightarrow Y$ be a morphism of smooth projective curves. Define the divisor $f^{*}(D)$ if $D$ is a divisor on $Y$, and state the "finiteness theorem".

(ii) Suppose $f: X \rightarrow \mathbb{P}^{1}$ is a morphism of degree 2 , that $X$ is smooth projective, and that $X \neq \mathbb{P}^{1}$. Let $P, Q \in X$ be distinct ramification points for $f$. Show that, as elements of $c l(X)$, we have $[P] \neq[Q]$, but $2[P]=2[Q]$.

B4.9

commentLet $F(X, Y, Z)$ be an irreducible homogeneous polynomial of degree $n$, and write $C=\left\{p \in \mathbb{P}^{2} \mid F(p)=0\right\}$ for the curve it defines in $\mathbb{P}^{2}$. Suppose $C$ is smooth. Show that the degree of its canonical class is $n(n-3)$.

Hence, or otherwise, show that a smooth curve of genus 2 does not embed in $\mathbb{P}^{2}$.

B2.8

commentLet $K$ and $L$ be finite simplicial complexes. Define the $n$-th chain group $C_{n}(K)$ and the boundary homomorphism $d_{n}: C_{n}(K) \rightarrow C_{n-1}(K)$. Prove that $d_{n-1} d_{n}=0$ and define the homology groups of $K$. Explain briefly how a simplicial map $f: K \rightarrow L$ induces a homomorphism $f_{\star}$ of homology groups.

Suppose now that $K$ consists of the proper faces of a 3-dimensional simplex. Calculate from first principles the homology groups of $K$. If a simplicial map $f: K \rightarrow K$ gives a homeomorphism of the underlying polyhedron $|K|$, is the induced homology map $f_{\star}$ necessarily the identity?

B3.7

commentA finite simplicial complex $K$ is the union of subcomplexes $L$ and $M$. Describe the Mayer-Vietoris exact sequence that relates the homology groups of $K$ to those of $L$, $M$ and $L \cap M$. Define all the homomorphisms in the sequence, proving that they are well-defined (a proof of exactness is not required).

A surface $X$ is constructed by identifying together (by means of a homeomorphism) the boundaries of two Möbius strips $Y$ and $Z$. Assuming relevant triangulations exist, determine the homology groups of $X$.

B4.5

commentWrite down the definition of a covering space and a covering map. State and prove the path lifting property for covering spaces and state, without proof, the homotopy lifting property.

Suppose that a group $G$ is a group of homeomorphisms of a space $X$. Prove that, under conditions to be stated, the quotient map $X \rightarrow X / G$ is a covering map and that $\pi_{1}(X / G)$ is isomorphic to $G$. Give two examples in which this last result can be used to determine the fundamental group of a space.

A2.10

comment(i) Define the minimum path and the maximum tension problems for a network with span intervals specified for each arc. State without proof the connection between the two problems, and describe the Max Tension Min Path algorithm of solving them.

(ii) Find the minimum path between nodes $\mathbf{S}$ and $\mathbf{S}^{\prime}$ in the network below. The span intervals are displayed alongside the arcs.

Part II 2004

A3.10

comment(i) Consider the problem

$\begin{aligned} \text { minimise } & f(x) \\ \text { subject to } & g(x)=b, x \in X, \end{aligned}$

where $f: \mathbb{R}^{n} \longrightarrow \mathbb{R}, g: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{m}, X \subseteq \mathbb{R}^{n}, x \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. State the Lagrange Sufficiency Theorem for problem $(*)$. What is meant by saying that this problem is strong Lagrangian? How is this related to the Lagrange Sufficiency Theorem? Define a supporting hyperplane and state a condition guaranteeing that problem $(*)$ is strong Lagrangian.

(ii) Define the terms flow, divergence, circulation, potential and differential for a network with nodes $N$ and $\operatorname{arcs} A$.

State the feasible differential problem for a network with span intervals $D(j)=$ $\left[d^{-}(j), d^{+}(j)\right], j \in A .$

State, without proof, the Feasible Differential Theorem.

[You must carefully define all quantities used in your statements.]

Show that the network below does not support a feasible differential.

Part II 2004

A4.11

comment(i) Consider an unrestricted geometric programming problem

$\min g(t), \quad t=\left(t_{1}, \ldots, t_{m}\right)>0,$

where $g(t)$ is given by

$g(t)=\sum_{i=1}^{n} c_{i} t_{1}^{a_{i 1}} \ldots t_{m}^{a_{i m}}$

with $n \geq m$ and positive coefficients $c_{1} \ldots, c_{n}$. State the dual problem of $(*)$ and show that if $\lambda^{*}=\left(\lambda_{1}^{*}, \ldots, \lambda_{n}^{*}\right)$ is a dual optimum then any positive solution to the system

$t_{1}^{a_{i 1}} \ldots t_{m}^{a_{i m}}=\frac{1}{c_{i}} \lambda_{i}^{*} v\left(\lambda^{*}\right), \quad i=1, \ldots, n,$

gives an optimum for primal problem $(*)$. Here $v(\lambda)$ is the dual objective function.

(ii) An amount of ore has to be moved from a pit in an open rectangular skip which is to be ordered from a supplier.

The skip cost is $£ 36$ per $1 \mathrm{~m}^{2}$ for the bottom and two side walls and $£ 18$ per $1 \mathrm{~m}^{2}$ for the front and the back walls. The cost of loading ore into the skip is $£ 3$ per $1 \mathrm{~m}^{3}$, the cost of lifting is $£ 2$ per $1 \mathrm{~m}^{3}$, and the cost of unloading is $£ 1$ per $1 \mathrm{~m}^{3}$. The cost of moving an empty skip is negligible.

Write down an unconstrained geometric programming problem for the optimal size (length, width, height) of skip minimizing the cost of moving $48 \mathrm{~m}^{3}$ of ore. By considering the dual problem, or otherwise, find the optimal cost and the optimal size of the skip.

B1.23

commentThe operator corresponding to a rotation through an angle $\theta$ about an axis $\mathbf{n}$, where $\mathbf{n}$ is a unit vector, is

$U(\mathbf{n}, \theta)=e^{i \theta \mathbf{n} \cdot \mathbf{J} / \hbar}$

If $U$ is unitary show that $\mathbf{J}$ must be hermitian. Let $\mathbf{V}=\left(V_{1}, V_{2}, V_{3}\right)$ be a vector operator such that

$U(\mathbf{n}, \delta \theta) \mathbf{V} U(\mathbf{n}, \delta \theta)^{-1}=\mathbf{V}+\delta \theta \mathbf{n} \times \mathbf{V} .$

Work out the commutators $\left[J_{i}, V_{j}\right]$. Calculate

$U(\hat{\mathbf{z}}, \theta) \mathbf{V U}(\hat{\mathbf{z}}, \theta)^{-1}$

for each component of $\mathbf{V}$.

If $|j m\rangle$ are standard angular momentum states determine $\left\langle j m^{\prime}|U(\hat{\mathbf{z}}, \theta)| j m\right\rangle$ for any $j, m, m^{\prime}$ and also determine $\left\langle\frac{1}{2} m^{\prime}|U(\hat{\mathbf{y}}, \theta)| \frac{1}{2} m\right\rangle$.

$\left[\right.$ Hint $\left.: J_{3}|j m\rangle=m \hbar|j m\rangle, J_{+}\left|\frac{1}{2}-\frac{1}{2}\right\rangle=\hbar\left|\frac{1}{2} \frac{1}{2}\right\rangle, J_{-}\left|\frac{1}{2} \frac{1}{2}\right\rangle=\hbar\left|\frac{1}{2}-\frac{1}{2}\right\rangle \cdot\right]$

B2.23

commentThe wave function for a single particle with a potential $V(r)$ has the asymptotic form for large $r$

$\psi(r, \theta) \sim e^{i k r \cos \theta}+f(\theta) \frac{e^{i k r}}{r} .$

How is $f(\theta)$ related to observable quantities? Show how $f(\theta)$ can be expressed in terms of phase shifts $\delta_{\ell}(k)$ for $\ell=0,1,2, \ldots$..

Assume that $V(r)=0$ for $r \geq a$, and let $R_{\ell}(r)$ denote the solution of the radial Schrödinger equation, regular at $r=0$, with energy $\hbar^{2} k^{2} / 2 m$ and angular momentum $\ell$. Let $N_{\ell}(k)=a R_{\ell}^{\prime}(a) / R_{\ell}(a)$. Show that

$\tan \delta_{\ell}(k)=\frac{N_{\ell}(k) j_{\ell}(k a)-k a j_{\ell}^{\prime}(k a)}{N_{\ell}(k) n_{\ell}(k a)-k a n_{\ell}^{\prime}(k a)} .$

Assuming that $N_{\ell}(k)$ is a smooth function for $k \approx 0$, determine the expected behaviour of $\delta_{\ell}(k)$ as $k \rightarrow 0$. Show that for $k \rightarrow 0$ then $f(\theta) \rightarrow c$, with $c$ a constant, and determine $c$ in terms of $N_{0}(0)$.

[For $V=0$ the two independent solutions of the radial Schrödinger equation are $j_{\ell}(k r)$ and $n_{\ell}(k r)$ with

$\begin{aligned} j_{\ell}(\rho) & \sim \frac{1}{\rho} \sin \left(\rho-\frac{1}{2} \ell \pi\right), n_{\ell}(\rho) \sim-\frac{1}{\rho} \cos \left(\rho-\frac{1}{2} \ell \pi\right) \text { as } \rho \rightarrow \\ j_{\ell}(\rho) & \propto \rho^{\ell}, n_{\ell}(\rho) \propto \rho^{-\ell-1} \text { as } \rho \rightarrow 0 \\ e^{i \rho \cos \theta} &=\sum_{\ell=0}^{\infty}(2 \ell+1) i^{\ell} j_{\ell}(\rho) P_{\ell}(\cos \theta), \\ j_{0}(\rho) &=\frac{\sin \rho}{\rho}, \quad n_{0}(\rho)=-\frac{\cos \rho}{\rho} \end{aligned}$

B3.23

commentFor a periodic potential $V(\mathbf{r})=V(\mathbf{r}+\ell)$, where $\ell$ is a lattice vector, show that we may write

$V(\mathbf{r})=\sum_{\mathbf{g}} a_{\mathbf{g}} e^{i \mathbf{g} \cdot \mathbf{r}}, \quad a_{\mathbf{g}}^{*}=a_{-\mathbf{g}}$

where the set of $g$ should be defined.

Show how to construct general wave functions satisfying $\psi(\mathbf{r}+\boldsymbol{\ell})=e^{i \mathbf{k} \cdot \boldsymbol{\ell}} \psi(\mathbf{r})$ in terms of free plane-wave wave-functions.

Show that the nearly free electron model gives an energy gap $2\left|a_{\mathbf{g}}\right|$ when $\mathbf{k}=\frac{1}{2} \mathbf{g}$.

Explain why, for a periodic potential, the allowed energies form bands $E_{n}(\mathbf{k})$ where $\mathbf{k}$ may be restricted to a single Brillouin zone. Show that $E_{n}(\mathbf{k})=E_{n}(\mathbf{k}+\mathbf{g})$ if $\mathbf{k}$ and $\mathbf{k}+\mathbf{g}$ belong to the Brillouin zone.

How are bands related to whether a material is a conductor or an insulator?

B4.24

commentDescribe briefly the variational approach to determining approximate energy eigenvalues for a Hamiltonian $H$.

Consider a Hamiltonian $H$ and two states $\left|\psi_{1}\right\rangle,\left|\psi_{2}\right\rangle$ such that

$\begin{array}{cl} \left\langle\psi_{1}|H| \psi_{1}\right\rangle=\left\langle\psi_{2}|H| \psi_{2}\right\rangle=\mathcal{E}, & \left\langle\psi_{2}|H| \psi_{1}\right\rangle=\left\langle\psi_{1}|H| \psi_{2}\right\rangle=\varepsilon \\ \left\langle\psi_{1} \mid \psi_{1}\right\rangle=\left\langle\psi_{2} \mid \psi_{2}\right\rangle=1, & \left\langle\psi_{2} \mid \psi_{1}\right\rangle=\left\langle\psi_{1} \mid \psi_{2}\right\rangle=s \end{array}$

Show that, by considering a linear combination $\alpha\left|\psi_{1}\right\rangle+\beta\left|\psi_{2}\right\rangle$, the variational method gives

$\frac{\mathcal{E}-\varepsilon}{1-s}, \quad \frac{\mathcal{E}+\varepsilon}{1+s}$

as approximate energy eigenvalues.

Consider the Hamiltonian for an electron in the presence of two protons at $\mathbf{0}$ and $\mathbf{R}$,

$H=\frac{\mathbf{p}^{2}}{2 m}+\frac{e^{2}}{4 \pi \epsilon_{0}}\left(\frac{1}{R}-\frac{1}{|\mathbf{r}|}-\frac{1}{|\mathbf{r}-\mathbf{R}|}\right), \quad R=|\mathbf{R}|$

Let $\psi_{0}(\mathbf{r})=e^{-r / a} /\left(\pi a^{3}\right)^{\frac{1}{2}}$ be the ground state hydrogen atom wave function which satisfies

$\left(\frac{\mathbf{p}^{2}}{2 m}-\frac{e^{2}}{4 \pi \epsilon_{0}|\mathbf{r}|}\right) \psi_{0}(\mathbf{r})=E_{0} \psi_{0}(\mathbf{r}) .$

It is given that

$\begin{aligned} &S=\int \mathrm{d}^{3} r \psi_{0}(\mathbf{r}) \psi_{0}(\mathbf{r}-\mathbf{R})=\left(1+\frac{R}{a}+\frac{R^{2}}{3 a^{2}}\right) e^{-R / a} \\ &U=\int \mathrm{d}^{3} r \frac{1}{|\mathbf{r}|} \psi_{0}(\mathbf{r}) \psi_{0}(\mathbf{r}-\mathbf{R})=\frac{1}{a}\left(1+\frac{R}{a}\right) e^{-R / a} \end{aligned}$

and, for large $R$, that

$\int \mathrm{d}^{3} r \frac{1}{|\mathbf{r}-\mathbf{R}|} \psi_{0}(\mathbf{r})^{2}-\frac{1}{R}=\mathrm{O}\left(e^{-2 R / a}\right)$

Consider the trial wave function $\alpha \psi_{0}(\mathbf{r})+\beta \psi_{0}(\mathbf{r}-\mathbf{R})$. Show that the variational estimate for the ground state energy for large $R$ is

$E(R)=E_{0}+\frac{e^{2}}{4 \pi \epsilon_{0} R}(S-R U)+\mathrm{O}\left(e^{-2 R / a}\right) .$

Explain why there is an attractive force between the two protons for large $R$.

B2.13

commentLet $M$ be a Poisson random measure of intensity $\lambda$ on the plane $\mathbb{R}^{2}$. Denote by $C(r)$ the circle $\left\{x \in \mathbb{R}^{2}:\|x\|<r\right\}$ of radius $r$ in $\mathbb{R}^{2}$ centred at the origin and let $R_{k}$ be the largest radius such that $C\left(R_{k}\right)$ contains precisely $k$ points of $M$. [Thus $C\left(R_{0}\right)$ is the largest circle about the origin containing no points of $M, C\left(R_{1}\right)$ is the largest circle about the origin containing a single point of $M$, and so on.] Calculate $\mathbb{E} R_{0}, \mathbb{E} R_{1}$ and $\mathbb{E} R_{2}$.

Now let $N$ be a Poisson random measure of intensity $\lambda$ on the line $\mathbb{R}^{1}$. Let $L_{k}$ be the length of the largest open interval that covers the origin and contains precisely $k$ points of $N$. [Thus $L_{0}$ gives the length of the largest interval containing the origin but no points of $N, L_{1}$ gives the length of the largest interval containing the origin and a single point of $N$, and so on.] Calculate $\mathbb{E} L_{0}, \mathbb{E} L_{1}$ and $\mathbb{E} L_{2}$.

B3.13

commentLet $\left(X_{t}, t \geq 0\right)$ be a renewal process with holding times $\left(S_{n}, n=1,2, \ldots\right)$ and $\left(Y_{t}, t \geq 0\right)$ be a renewal-reward process over $\left(X_{t}\right)$ with a sequence of rewards $\left(W_{n}, n=1,2, \ldots\right)$. Under assumptions on $\left(S_{n}\right)$ and $\left(W_{n}\right)$ which you should state clearly, prove that the ratios

$X_{t} / t \text { and } Y_{t} / t$

converge as $t \rightarrow \infty$. You should specify the form of convergence guaranteed by your assumptions. The law of large numbers, in the appropriate form, for sums $S_{1}+\ldots+S_{n}$ and $W_{1}+\ldots+W_{n}$ can be used without proof.

In a mountain resort, when you rent skiing equipment you are given two options. (1) You buy an insurance waiver that costs $C / 4$ where $C$ is the daily equipment rent. Under this option, the shop will immediately replace, at no cost to you, any piece of equipment you break during the day, no matter how many breaks you had. (2) If you don't buy the waiver, you'll pay $3 C$ in the case of any break.

To find out which option is better for me, I decided to set up two models of renewalreward process $\left(Y_{t}\right)$. In the first model, (Option 1), all of the holding times $S_{n}$ are equal to 6 . In the second model, given that there is no break on day $n$ (an event of probability $4 / 5)$, we have $S_{n}=6, W_{n}=C$, but given that there is a break on day $n$, we have that $S_{n}$ is uniformly distributed on $(0,6)$, and $W_{n}=4 C$. (In the second model, I would not continue skiing after a break, whereas in the first I would.)

Calculate in each of these models the limit

$\lim _{t \rightarrow \infty} Y_{t} / t$

representing the long-term average cost of a unit of my skiing time.

B4.12

commentConsider an $M / G / 1$ queue with $\rho=\lambda \mathbb{E} S<1$. Here $\lambda$ is the arrival rate and $\mathbb{E} S$ is the mean service time. Prove that in equilibrium, the customer's waiting time $W$ has the moment-generating function $M_{W}(t)=\mathbb{E} e^{t W}$ given by

$M_{W}(t)=\frac{(1-\rho) t}{t+\lambda\left(1-M_{S}(t)\right)}$

where $M_{S}(t)=\mathbb{E} e^{t S}$ is the moment-generating function of service time $S$.

[You may assume that in equilibrium, the $M / G / 1$ queue size $X$ at the time immediately after the customer's departure has the probability generating function

$\left.\mathbb{E} z^{X}=\frac{(1-\rho)(1-z) M_{S}(\lambda(z-1))}{M_{S}(\lambda(z-1))-z}, \quad 0 \leqslant z<1 .\right]$

Deduce that when the service times are exponential of rate $\mu$ then

$M_{W}(t)=1-\rho+\frac{\lambda(1-\rho)}{\mu-\lambda-t}, \quad-\infty<t<\mu-\lambda .$

Further, deduce that $W$ takes value 0 with probability $1-\rho$ and that

$\mathbb{P}(W>x \mid W>0)=e^{-(\mu-\lambda) x}, \quad x>0 .$

Sketch the graph of $\mathbb{P}(W>x)$ as a function of $x$.

Now consider the $M / G / 1$ queue in the heavy traffic approximation, when the service-time distribution is kept fixed and the arrival rate $\lambda \rightarrow 1 / \mathbb{E} S$, so that $\rho \rightarrow 1$. Assuming that the second moment $\mathbb{E} S^{2}<\infty$, check that the limiting distribution of the re-scaled waiting time $\tilde{W}_{\lambda}=(1-\lambda \mathbb{E} S) W$ is exponential, with rate $2 \mathbb{E} S / \mathbb{E} S^{2}$.

A1.10

comment(i) What is a linear code? What does it mean to say that a linear code has length $n$ and minimum weight $d$ ? When is a linear code perfect? Show that, if $n=2^{r}-1$, there exists a perfect linear code of length $n$ and minimum weight 3 .

(ii) Describe the construction of a Reed-Muller code. Establish its information rate and minimum weight.

A2.9

comment(i) Describe how a stream cypher operates. What is a one-time pad?

A one-time pad is used to send the message $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} y_{7}$ which is encoded as 0101011. By mistake, it is reused to send the message $y_{0} x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$ which is encoded as 0100010. Show that $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$ is one of two possible messages, and find the two possibilities.

(ii) Describe the RSA system associated with a public key $e$, a private key $d$ and the product $N$ of two large primes.

Give a simple example of how the system is vulnerable to a homomorphism attack. Explain how a signature system prevents such an attack. [You are not asked to give an explicit signature system.]

Explain how to factorise $N$ when $e, d$ and $N$ are known.

B1.5

commentState and prove Menger's theorem (vertex form).

Let $G$ be a graph of connectivity $\kappa(G) \geq k$ and let $S, T$ be disjoint subsets of $V(G)$ with $|S|,|T| \geq k$. Show that there exist $k$ vertex disjoint paths from $S$ to $T$.

The graph $H$ is said to be $k$-linked if, for every sequence $s_{1}, \ldots, s_{k}, t_{1}, \ldots, t_{k}$ of $2 k$ distinct vertices, there exist $s_{i}-t_{i}$ paths, $1 \leq i \leq k$, that are vertex disjoint. By removing an edge from $K_{2 k}$, or otherwise, show that, for $k \geqslant 2$, $H$ need not be $k$-linked even if $\kappa(H) \geq 2 k-2$.

Prove that if $|H|=n$ and $\delta(H) \geq \frac{1}{2}(n+3 k)-2$ then $H$ is $k$-linked.

B2.5

commentState and prove Sperner's lemma on antichains.

The family $\mathcal{A} \subset \mathcal{P}[n]$ is said to split $[n]$ if, for all distinct $i, j \in[n]$, there exists $A \in \mathcal{A}$ with $i \in A$ but $j \notin A$. Prove that if $\mathcal{A}$ splits $[n]$ then $n \leq\left(\begin{array}{c}a \\ \lfloor a / 2\rfloor\end{array}\right)$, where $a=|\mathcal{A}|$.

Show moreover that, if $\mathcal{A}$ splits $[n]$ and no element of $[n]$ is in more than $k<\lfloor a / 2\rfloor$ members of $\mathcal{A}$, then $n \leq\left(\begin{array}{l}a \\ k\end{array}\right)$.

B4.1

commentWrite an essay on Ramsey's theorem. You should include the finite and infinite versions, together with some discussion of bounds in the finite case, and give at least one application.

A1.13

comment(i) Assume that the $n$-dimensional vector $Y$ may be written as $Y=X \beta+\epsilon$, where $X$ is a given $n \times p$ matrix of $\operatorname{rank} p, \beta$ is an unknown vector, and

$\epsilon \sim N_{n}\left(0, \sigma^{2} I\right)$

Let $Q(\beta)=(Y-X \beta)^{T}(Y-X \beta)$. Find $\hat{\beta}$, the least-squares estimator of $\beta$, and state without proof the joint distribution of $\hat{\beta}$ and $Q(\hat{\beta})$.

(ii) Now suppose that we have observations $\left(Y_{i j}, 1 \leqslant i \leqslant I, 1 \leqslant j \leqslant J\right)$ and consider the model

$\Omega: Y_{i j}=\mu+\alpha_{i}+\beta_{j}+\epsilon_{i j},$

where $\left(\alpha_{i}\right),\left(\beta_{j}\right)$ are fixed parameters with $\Sigma \alpha_{i}=0, \Sigma \beta_{j}=0$, and $\left(\epsilon_{i j}\right)$ may be assumed independent normal variables, with $\epsilon_{i j} \sim N\left(0, \sigma^{2}\right)$, where $\sigma^{2}$ is unknown.

(a) Find $\left(\hat{\alpha}_{i}\right),\left(\hat{\beta}_{j}\right)$, the least-squares estimators of $\left(\alpha_{i}\right),\left(\beta_{j}\right)$.

(b) Find the least-squares estimators of $\left(\alpha_{i}\right)$ under the hypothesis $H_{0}: \beta_{j}=0$ for all $j$.

(c) Quoting any general theorems required, explain carefully how to test $H_{0}$, assuming $\Omega$ is true.

(d) What would be the effect of fitting the model $\Omega_{1}: Y_{i j}=\mu+\alpha_{i}+\beta_{j}+\gamma_{i j}+\epsilon_{i j}$, where now $\left(\alpha_{i}\right),\left(\beta_{j}\right),\left(\gamma_{i j}\right)$ are all fixed unknown parameters, and $\left(\epsilon_{i j}\right)$ has the distribution given above?

A2.12

comment(i) Suppose we have independent observations $Y_{1}, \ldots, Y_{n}$, and we assume that for $i=1, \ldots, n, Y_{i}$ is Poisson with mean $\mu_{i}$, and $\log \left(\mu_{i}\right)=\beta^{T} x_{i}$, where $x_{1}, \ldots, x_{n}$ are given covariate vectors each of dimension $p$, where $\beta$ is an unknown vector of dimension $p$, and $p<n$. Assuming that $\left\{x_{1}, \ldots, x_{n}\right\}$ span $\mathbb{R}^{p}$, find the equation for $\hat{\beta}$, the maximum likelihood estimator of $\beta$, and write down the large-sample distribution of $\hat{\beta}$.

(ii) A long-term agricultural experiment had 90 grassland plots, each $25 \mathrm{~m} \times 25 \mathrm{~m}$, differing in biomass, soil pH, and species richness (the count of species in the whole plot). While it was well-known that species richness declines with increasing biomass, it was not known how this relationship depends on soil pH, which for the given study has possible values "low", "medium" or "high", each taken 30 times. Explain the commands input, and interpret the resulting output in the (slightly edited) $R$ output below, in which "species" represents the species count.

(The first and last 2 lines of the data are reproduced here as an aid. You may assume that the factor pH has been correctly set up.)

A4.14

commentSuppose that $Y_{1}, \ldots, Y_{n}$ are independent observations, with $Y_{i}$ having probability density function of the following form

$f\left(y_{i} \mid \theta_{i}, \phi\right)=\exp \left[\frac{y_{i} \theta_{i}-b\left(\theta_{i}\right)}{\phi}+c\left(y_{i}, \phi\right)\right]$

where $\mathbb{E}\left(Y_{i}\right)=\mu_{i}$ and $g\left(\mu_{i}\right)=\beta^{T} x_{i}$. You should assume that $g()$ is a known function, and $\beta, \phi$ are unknown parameters, with $\phi>0$, and also $x_{1}, \ldots, x_{n}$ are given linearly independent covariate vectors. Show that

$\frac{\partial \ell}{\partial \beta}=\sum \frac{\left(y_{i}-\beta_{i}\right)}{g^{\prime}\left(\mu_{i}\right) V_{i}} x_{i}$

where $\ell$ is the log-likelihood and $V_{i}=\operatorname{var}\left(Y_{i}\right)=\phi b^{\prime \prime}\left(\theta_{i}\right)$.

Discuss carefully the (slightly edited) $\mathrm{R}$ output given below, and briefly suggest another possible method of analysis using the function $\mathrm{glm}$ ( ).

$>s<-\operatorname{scan}()$

1: $\begin{array}{llllll}33 & 63 & 157 & 38 & 108 & 159\end{array}$

7:

Read 6 items

$>r<-\operatorname{scan}()$

1: 327172565065248688773520

$7:$

Read 6 items

$>$ gender <- $\operatorname{scan}(, " \|)$

1: b b b g g g

$7:$

Read 6 items

$>$ age <- $\operatorname{scan}(, " \prime)$

1: 13&under 14-18 19&over

4: 13&under 14-18 19&over

7 :

Read 6 items

$>$ gender <- factor (gender) ; age <- factor (age)

$>\operatorname{summary}(\mathrm{glm}(\mathrm{s} / \mathrm{r} \sim$ gender $+$ age, binomial, weights $=\mathrm{r}))$

Coefficients:

Null deviance: $221.797542$ on 5 degrees of freedom

Residual deviance: $0.098749$ on 2 degrees of freedom

Number of Fisher Scoring iterations: 3

B1.8

commentWhat is a smooth vector bundle over a manifold $M$ ?

Assuming the existence of "bump functions", prove that every compact manifold embeds in some Euclidean space $\mathbb{R}^{n}$.

By choosing an inner product on $\mathbb{R}^{n}$, or otherwise, deduce that for any compact manifold $M$ there exists some vector bundle $\eta \rightarrow M$ such that the direct sum $T M \oplus \eta$ is isomorphic to a trivial vector bundle.

B2 7

commentFor each of the following assertions, either provide a proof or give and justify a counterexample.

[You may use, without proof, your knowledge of the de Rham cohomology of surfaces.]

(a) A smooth map $f: S^{2} \rightarrow T^{2}$ must have degree zero.

(b) An embedding $\varphi: S^{1} \rightarrow \Sigma_{g}$ extends to an embedding $\bar{\varphi}: D^{2} \rightarrow \Sigma_{g}$ if and only if the map

$\int_{\varphi\left(S^{1}\right)}: H^{1}\left(\Sigma_{g}\right) \rightarrow \mathbb{R}$

is the zero map.

(c) $\mathbb{R} \mathbb{P}^{1} \times \mathbb{R P}^{2}$ is orientable.

(d) The surface $\Sigma_{g}$ admits the structure of a Lie group if and only if $g=1$.

B4.4

commentDefine what it means for a manifold to be oriented, and define a volume form on an oriented manifold.

Prove carefully that, for a closed connected oriented manifold of dimension $n$, $H^{n}(M)=\mathbb{R}$.

[You may assume the existence of volume forms on an oriented manifold.]

If $M$ and $N$ are closed, connected, oriented manifolds of the same dimension, define the degree of a map $f: M \rightarrow N$.

If $f$ has degree $d>1$ and $y \in N$, can $f^{-1}(y)$ be

(i) infinite? (ii) a single point? (iii) empty?

Briefly justify your answers.

B2.4

comment(i) Define carefully what is meant by a Hopf bifurcation in a two-dimensional dynamical system. Write down the normal form for this bifurcation, correct to cubic order, and distinguish between bifurcations of supercritical and subcritical type. Describe, without detailed calculations, how a general two-dimensional system with a Hopf bifurcation at the origin can be reduced to normal form by a near-identity transformation.

(ii) A Takens-Bogdanov bifurcation of a fixed point of a two-dimensional system is characterised by a Jacobian with the canonical form

$A=\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)$

at the bifurcation point. Consider the system

$\begin{aligned} &\dot{x}=y+\alpha_{1} x^{2}+\beta_{1} x y+\gamma_{1} y^{2} \\ &\dot{y}=\alpha_{2} x^{2}+\beta_{2} x y+\gamma_{2} y^{2} \end{aligned}$

Show that a near-identity transformation of the form

$\begin{aligned} &\xi=x+a_{1} x^{2}+b_{1} x y+c_{1} y^{2} \\ &\eta=y+a_{2} x^{2}+b_{2} x y+c_{2} y^{2} \end{aligned}$

exists that reduces the system to the normal (canonical) form, correct up to quadratic terms,

$\dot{\xi}=\eta, \quad \dot{\eta}=\alpha_{2} \xi^{2}+\left(\beta_{2}+2 \alpha_{1}\right) \xi \eta .$

It is known that the general form of the equations near the bifurcation point can be written (setting $p=\alpha_{2}, q=\beta_{2}+2 \alpha_{1}$ )

$\dot{\xi}=\eta, \quad \dot{\eta}=\lambda \xi+\mu \eta+p \xi^{2}+q \xi \eta .$

Find all the fixed points of this system, and the values of $\lambda, \mu$ for which these fixed points have (a) steady state bifurcations and (b) Hopf bifurcations.

B3.4

comment(i) Describe the use of the stroboscopic method for obtaining approximate solutions to the second order equation

$\ddot{x}+x=\epsilon f(x, \dot{x}, t)$

when $|\epsilon| \ll 1$. In particular, by writing $x=R \cos (t+\phi), \dot{x}=-R \sin (t+\phi)$, obtain expressions in terms of $f$ for the rate of change of $R$ and $\phi$. Evaluate these expressions when $f=x^{2} \cos t$.

(ii) In planetary orbit theory a crude model of an orbit subject to perturbation from a distant body is given by the equation

$\frac{d^{2} u}{d \theta^{2}}+u=\lambda-\delta^{2} u^{-2}-2 \delta^{3} u^{-3} \cos \theta$

where $0<\delta \ll 1,\left(u^{-1}, \theta\right)$ are polar coordinates in the plane, and $\lambda$ is a positive constant.

(a) Show that when $\delta=0$ all bounded orbits are closed.

(b) Now suppose $\delta \neq 0$, and look for almost circular orbits with $u=\lambda+\delta w(\theta)+a \delta^{2}$, where $a$ is a constant. By writing $w=R(\theta) \cos (\theta+\phi(\theta))$, and by making a suitable choice of the constant $a$, use the stroboscopic method to find equations for $d w / d \theta$ and $d \phi / d \theta$. By writing $z=R \exp (i \phi)$ and considering $d z / d \theta$, or otherwise, determine $R(\theta)$ and $\phi(\theta)$ in the case $R(0)=R_{0}, \phi(0)=0$. Hence describe the orbits of the system.

B1.21

commentThe Maxwell field tensor is

$F^{a b}=\left(\begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & -B_{x} \\ E_{z} & -B_{y} & B_{x} & 0 \end{array}\right)$

and the 4-current density is $J^{a}=(\rho, \mathbf{j})$. Write down the 3-vector form of Maxwell's equations and the continuity equation, and obtain the equivalent 4-vector equations.

Consider a Lorentz transformation from a frame $\mathcal{F}$ to a frame $\mathcal{F}^{\prime}$ moving with relative (coordinate) velocity $v$ in the $x$-direction

$L_{b}^{a}=\left(\begin{array}{cccc} \gamma & \gamma v & 0 & 0 \\ \gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

where $\gamma=1 / \sqrt{1-v^{2}}$. Obtain the transformation laws for $\mathbf{E}$ and $\mathbf{B}$. Which quantities, quadratic in $\mathbf{E}$ and $\mathbf{B}$, are Lorentz scalars?

B2.21

commentA particle of rest mass $m$ and charge $q$ moves along a path $x^{a}(s)$, where $s$ is the particle's proper time. The equation of motion is

$m \ddot{x}^{a}=q F^{a b} \eta_{b c} \dot{x}^{c}$

where $\dot{x}^{a}=d x^{a} / d s$ etc., $F^{a b}$ is the Maxwell field tensor $\left(F^{01}=-E_{x}, F^{23}=-B_{x}\right.$, where $E_{x}$ and $B_{x}$ are the $x$-components of the electric and magnetic fields) and $\eta_{b c}$ is the Minkowski metric tensor. Show that $\dot{x}_{a} \ddot{x}^{a}=0$ and interpret both the equation of motion and this equation in the classical limit.

The electromagnetic field is given in cartesian coordinates by $\mathbf{E}=(0, E, 0)$ and $\mathbf{B}=(0,0, E)$, where $E$ is constant and uniform. The particle starts from rest at the origin. Show that the orbit is given by

$9 x^{2}=2 \alpha y^{3}, \quad z=0$

where $\alpha=q E / m$.

B4.21

commentUsing Lorentz gauge, $A_{, a}^{a}=0$, Maxwell's equations for a current distribution $J^{a}$ can be reduced to $\square A^{a}(x)=\mu_{0} J^{a}(x)$. The retarded solution is

$A^{a}(x)=\frac{\mu_{0}}{2 \pi} \int d^{4} y \theta\left(z^{0}\right) \delta\left(z_{c} z^{c}\right) J^{a}(y)$

where $z^{a}=x^{a}-y^{a}$. Explain, heuristically, the rôle of the $\delta$-function and Heaviside step function $\theta$ in this formula.

The current distribution is produced by a point particle of charge $q$ moving on a world line $r^{a}(s)$, where $s$ is the particle's proper time, so that

$J^{a}(y)=q \int d s V^{a}(s) \delta^{(4)}(y-r(s))$

where $V^{a}=\dot{r}^{a}(s)=d r^{a} / d s$. Show that

$A^{a}(x)=\frac{\mu_{0} q}{2 \pi} \int d s \theta\left(X^{0}\right) \delta\left(X_{c} X^{c}\right) V^{a}(s),$

where $X^{a}=x^{a}-r^{a}(s)$, and further that, setting $\alpha=X_{c} V^{c}$,

$A^{a}(x)=\frac{\mu_{0} q}{4 \pi}\left[\frac{V^{a}}{\alpha}\right]_{s=s^{*}}$

where $s^{*}$ should be defined. Verify that

$s^{*}, a=\left[\frac{X_{a}}{\alpha}\right]_{s=s^{*}} .$

Evaluating quantities at $s=s^{*}$ show that

$\left[\frac{V^{a}}{\alpha}\right]_{, b}=\frac{1}{\alpha^{2}}\left[-V^{a} V_{b}+S^{a} X_{b}\right]$

where $S^{a}=\dot{V}^{a}+V^{a}\left(1-X_{c} \dot{V}^{c}\right) / \alpha$. Hence verify that $A^{a}{ }_{, a}(x)=0$ and

$F_{a b}=\frac{\mu_{0} q}{4 \pi \alpha^{2}}\left(S_{a} X_{b}-S_{b} X_{a}\right) .$

Verify this formula for a stationary point charge at the origin.

[Hint: If $f(s)$ has simple zeros at $s_{i}, i=1,2, \ldots$ then

$\delta(f(s))=\sum_{i} \frac{\delta\left(s_{i}\right)}{\left|f^{\prime}\left(s_{i}\right)\right|}$

A $1 . 5 \quad$ B $1 . 4 \quad$

comment(i) Show that the work done in assembling a localised charge distribution $\rho(\mathbf{r})$ in a region $V$ with an associated potential $\phi(\mathbf{r})$ is

$W=\frac{1}{2} \int_{V} \rho(\mathbf{r}) \phi(\mathbf{r}) d \tau$

and that this can be written as an integral over all space

$W=\frac{1}{2} \epsilon_{0} \int|\mathbf{E}|^{2} d \tau$

where the electric field $\mathbf{E}=-\nabla \phi$.

(ii) What is the force per unit area on an infinite plane conducting sheet with a charge density $\sigma$ per unit area (a) if it is isolated in space and (b) if the electric field vanishes on one side of the sheet?

An infinite cylindrical capacitor consists of two concentric cylindrical conductors with radii $a, b(a<b)$, carrying charges $\pm q$ per unit length respectively. Calculate the capacitance per unit length and the energy per unit length. Next determine the total force on each conductor, and calculate the rate of change of energy of the inner and outer conductors if they are moved radially inwards and outwards respectively with speed $v$. What is the corresponding rate of change of the capacitance?

A2.5

comment(i) Write down the general solution of Poisson's equation. Derive from Maxwell's equations the Biot-Savart law for the magnetic field of a steady localised current distribution.

(ii) A plane rectangular loop with sides of length $a$ and $b$ lies in the plane $z=0$ and is centred on the origin. Show that when $r=|\mathbf{r}| \gg a, b$, the vector potential $\mathbf{A}(\mathbf{r})$ is given approximately by

$\mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \frac{\mathbf{m} \wedge \mathbf{r}}{r^{3}}$

where $\mathbf{m}=I a b \hat{\mathbf{z}}$ is the magnetic moment of the loop.

Hence show that the magnetic field $\mathbf{B}(\mathbf{r})$ at a great distance from an arbitrary small plane loop of area $A$, situated in the $x y$-plane near the origin and carrying a current $I$, is given by

$\mathbf{B}(\mathbf{r})=\frac{\mu_{0} I A}{4 \pi r^{5}}\left(3 x z, 3 y z, 2 r^{2}-3 x^{2}-3 y^{2}\right)$

A3.5 B3.3

comment(i) State Maxwell's equations and show that the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ can be expressed in terms of a scalar potential $\phi$ and a vector potential $\mathbf{A}$. Hence derive the inhomogeneous wave equations that are satisfied by $\phi$ and $\mathbf{A}$ respectively.

(ii) The plane $x=0$ separates a vacuum in the half-space $x<0$ from a perfectly conducting medium occupying the half-space $x>0$. Derive the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at $x=0$.

A plane electromagnetic wave with a magnetic field $\mathbf{B}=B(t, x, z) \hat{\mathbf{y}}$, travelling in the $x z$-plane at an angle $\theta$ to the $x$-direction, is incident on the interface at $x=0$. If the wave has frequency $\omega$ show that the total magnetic field is given by

$\mathbf{B}=B_{0} \cos \left(\frac{\omega x}{c} \cos \theta\right) \exp \left[i\left(\frac{\omega z}{c} \sin \theta-\omega t\right)\right] \hat{\mathbf{y}}$

where $B_{0}$ is a constant. Hence find the corresponding electric field $\mathbf{E}$, and obtain the surface charge density and the surface current at the interface.

A4.5

commentConsider a frame $S^{\prime}$ moving with velocity v relative to the laboratory frame $S$ where $|\mathbf{v}|^{2} \ll c^{2}$. The electric and magnetic fields in $S$ are $\mathbf{E}$ and $\mathbf{B}$, while those measured in $S^{\prime}$ are $\mathbf{E}^{\prime}$ and $\mathbf{B}^{\prime}$. Given that $\mathbf{B}^{\prime}=\mathbf{B}$, show that

$\oint_{\Gamma} \mathbf{E}^{\prime} \cdot d \mathbf{l}=\oint_{\Gamma}(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot d \mathbf{l},$

for any closed circuit $\Gamma$ and hence that $\mathbf{E}^{\prime}=\mathbf{E}+\mathbf{v} \wedge \mathbf{B}$.

Now consider a fluid with electrical conductivity $\sigma$ and moving with velocity $\mathbf{v}(\mathbf{r})$. Use Ohm's law in the moving frame to relate the current density $\mathbf{j}$ to the electric field $\mathbf{E}$ in the laboratory frame, and show that if $\mathbf{j}$ remains finite in the limit $\sigma \rightarrow \infty$ then

$\frac{\partial \mathbf{B}}{\partial t}=\nabla \wedge(\mathbf{v} \wedge \mathbf{B})$

The magnetic helicity $H$ in a volume $V$ is given by $\int_{V} \mathbf{A} \cdot \mathbf{B} d \tau$ where $\mathbf{A}$ is the vector potential. Show that if the normal components of $\mathbf{v}$ and $\mathbf{B}$ both vanish on the surface bounding $V$ then $d H / d t=0$.

B1.25

commentConsider a uniform stream of inviscid incompressible fluid incident onto a twodimensional body (such as a circular cylinder). Sketch the flow in the region close to the stagnation point, $S$, at the front of the body.

Let the fluid now have a small but non-zero viscosity. Using local co-ordinates $x$ along the boundary and $y$ normal to it, with the stagnation point as origin and $y>0$ in the fluid, explain why the local outer, inviscid flow is approximately of the form

$\mathbf{u}=(E x,-E y)$

for some positive constant $E$.

Use scaling arguments to find the thickness $\delta$ of the boundary layer on the body near $S$. Hence show that there is a solution of the boundary layer equations of the form

$u(x, y)=E x f^{\prime}(\eta)$

where $\eta$ is a suitable similarity variable and $f$ satisfies

$f^{\prime \prime \prime}+f f^{\prime \prime}-f^{\prime^{2}}=-1 .$

What are the appropriate boundary conditions for $(*)$ and why? Explain briefly how you would obtain a numerical solution to $(*)$ subject to the appropriate boundary conditions.

Explain why it is neither possible nor appropriate to perform a similar analysis near the rear stagnation point of the inviscid flow.

B2.25

commentAn incompressible fluid with density $\rho$ and viscosity $\mu$ is forced by a pressure difference $\Delta p$ through the narrow gap between two parallel circular cylinders of radius $a$ with axes $2 a+b$ apart. Explaining any approximations made, show that, provided $b \ll a$ and $\rho b^{3} \Delta p \ll \mu^{2} a$, the volume flux (per unit length of cylinder) is

$\frac{2 b^{5 / 2} \Delta p}{9 \pi a^{1 / 2} \mu}$

when the cylinders are stationary.

Show also that when the two cylinders rotate with angular velocities $\Omega$ and $-\Omega$ respectively, the change in the volume flux is

$\frac{4}{3} b a \Omega .$

For the case $\Delta p=0$, find and sketch the function $f(x)=u_{0}(x) /(a \Omega)$, where $u_{0}$ is the centreline velocity at position $x$ along the gap in the direction of flow. Comment on the values taken by $f$.

B3.24

commentUsing the Milne-Thompson circle theorem, or otherwise, write down the complex potential $w$ describing inviscid incompressible two-dimensional flow past a circular cylinder of radius $a$ centred on the origin, with circulation $\kappa$ and uniform velocity $(U, V)$ in the far field.

Hence, or otherwise, find an expression for the velocity field if the cylinder is replaced by a flat plate of length $4 a$, centred on the origin and aligned with the $x$-axis. Evaluate the velocity field on the two sides of the plate and confirm that the normal velocity is zero.

Explain the significance of the Kutta condition, and determine the value of the circulation that satisfies the Kutta condition when $U>0$.

With this value of the circulation, calculate the difference in pressure between the upper and lower sides of the plate at position $x(-2 a \leq x \leq 2 a)$. Comment briefly on the value of the pressure at the leading edge and the force that this would produce if the plate had a small non-zero thickness.

Determine the force on the plate, explaining carefully the direction in which it acts.

[The Blasius formula $F_{x}-i F_{y}=\frac{i \rho}{2} \oint_{C}\left(\frac{d w}{d z}\right)^{2} d z$, where $C$ is a closed contour lying just outside the body, may be used without proof.]

B4.26

commentWrite an essay on the Kelvin-Helmholtz instability of a vortex sheet. Your essay should include a detailed linearised analysis, a physical interpretation of the instability, and an informal discussion of nonlinear effects and of the effects of viscosity.

A2.13 B2.22

comment(i) The creation and annihilation operators for a harmonic oscillator of angular frequency $\omega$ satisfy the commutation relation $\left[a, a^{\dagger}\right]=1$. Write down an expression for the Hamiltonian $H$ in terms of $a$ and $a^{\dagger}$.

There exists a unique ground state $|0\rangle$ of $H$ such that $a|0\rangle=0$. Explain how the space of eigenstates $|n\rangle, n=0,1,2, \ldots$ of $H$ is formed, and deduce the eigenenergies for these states. Show that

$a|n\rangle=\sqrt{n}|n-1\rangle, \quad a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle$

(ii) Write down the number operator $N$ of the harmonic oscillator in terms of $a$ and $a^{\dagger}$. Show that

$N|n\rangle=n|n\rangle$

The operator $K_{r}$ is defined to be

$K_{r}=\frac{a^{\dagger r} a^{r}}{r !}, \quad r=0,1,2, \ldots$

Show that $K_{r}$ commutes with $N$. Show also that

$K_{r}|n\rangle= \begin{cases}\frac{n !}{(n-r) ! r !}|n\rangle & r \leq n \\ 0 & r>n\end{cases}$

By considering the action of $K_{r}$ on the state $|n\rangle$ show that

$\sum_{r=0}^{\infty}(-1)^{r} K_{r}=|0\rangle\langle 0|$

A3.13 B3.21

(i) A quantum mechanical system consists of two identical non-interacting particles with associated single-particle wave functions $\psi_{i}(x)$ and energies $E_{i}, i=1,2, \ldots$, where $E_{1}<E_{2}<\ldots$