Part II, 2004, Paper 1

# Part II, 2004, Paper 1

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

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.

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.

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?

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.

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?

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.

A1.3

comment(i) Let $H$ be a Hilbert space, and let $M$ be a non-zero closed vector subspace of $H$. For $x \in H$, show that there is a unique closest point $P_{M}(x)$ to $x$ in $M$.

(ii) (a) Let $x \in H$. Show that $x-P_{M}(x) \in M^{\perp}$. Show also that if $y \in M$ and $x-y \in M^{\perp}$ then $y=P_{M}(x)$.

(b) Deduce that $H=M \bigoplus M^{\perp}$.

(c) Show that the map $P_{M}$ from $H$ to $M$ is a continuous linear map, with $\left\|P_{M}\right\|=1$.

(d) Show that $P_{M}$ is the projection onto $M$ along $M^{\perp}$.

Now suppose that $A$ is a subspace of $H$ that is not necessarily closed. Explain why $A^{\perp}=\{0\}$ implies that $A$ is dense in $H .$

Give an example of a subspace of $l^{2}$ that is dense in $l^{2}$ but is not equal to $l^{2}$.

B1.7

commentLet $L / K$ be a finite extension of fields. Define the trace $\operatorname{Tr}_{L / K}(x)$ and norm $N_{L / K}(x)$ of an element $x \in L$.

Assume now that the extension $L / K$ is Galois, with cyclic Galois group of prime order $p$, generated by $\sigma$.

i) Show that $\operatorname{Tr}_{L / K}(x)=\sum_{n=0}^{p-1} \sigma^{n}(x)$.

ii) Show that $\{\sigma(x)-x \mid x \in L\}$ is a $K$-vector subspace of $L$ of dimension $p-1$. Deduce that if $y \in L$, then $\operatorname{Tr}_{L / K}(y)=0$ if and only if $y=\sigma(x)-x$ for some $x \in L$. [You may assume without proof that $\operatorname{Tr}_{L / K}$ is surjective for any finite separable extension $L / K$.]

iii) Suppose that $L$ has characteristic $p$. Deduce from (i) that every element of $K$ can be written as $\sigma(x)-x$ for some $x \in L$. Show also that if $\sigma(x)=x+1$, then $x^{p}-x$ belongs to $K$. Deduce that $L$ is the splitting field over $K$ of $X^{p}-X-a$ for some $a \in K$.

A1.15 B1.24

comment(i) What is an affine parameter $\lambda$ of a timelike or null geodesic? Prove that for a timelike geodesic one may take $\lambda$ to be proper time $\tau$. The metric

$d s^{2}=-d t^{2}+a^{2}(t) d \mathbf{x}^{2}$

with $\dot{a}(t)>0$ represents an expanding universe. Calculate the Christoffel symbols.

(ii) Obtain the law of spatial momentum conservation for a particle of rest mass $m$ in the form

$m a^{2} \frac{d \mathbf{x}}{d \tau}=\mathbf{p}=\text { constant }$

Assuming that the energy $E=m d t / d \tau$, derive an expression for $E$ in terms of $m, \mathbf{p}$ and $a(t)$ and show that the energy is not conserved but rather that it decreases with time. In particular, show that if the particle is moving extremely relativistically then the energy decreases as $a^{-1}(t)$, and if it is moving non-relativistically then the kinetic energy, $E-m$, decreases as $a^{-2}(t)$.

Show that the frequency $\omega_{e}$ of a photon emitted at time $t_{e}$ will be observed at time $t_{o}$ to have frequency

$\omega_{o}=\omega_{e} \frac{a\left(t_{e}\right)}{a\left(t_{o}\right)}$

A1.8

comment(i) Let $G$ be a connected graph of order $n \geq 3$ such that for any two vertices $x$ and $y$,

$d(x)+d(y) \geq k$

Show that if $k<n$ then $G$ has a path of length $k$, and if $k=n$ then $G$ is Hamiltonian.

(ii) State and prove Hall's theorem.

[If you use any form of Menger's theorem, you must state it clearly.]

Let $G$ be a graph with directed edges. For $S \subset V(G)$, let

$\Gamma_{+}(S)=\{y \in V(G): x y \in E(G) \text { for some } x \in S\}$

Find a necessary and sufficient condition, in terms of the sizes of the sets $\Gamma_{+}(S)$, for the existence of a set $F \subset E(G)$ such that at every vertex there is exactly one incoming edge and exactly one outgoing edge belonging to $F$.

A1.4 B1.3

comment(i) Let $R$ be a commutative ring. Define the terms prime ideal and maximal ideal, and show that if an ideal $M$ in $R$ is maximal then $M$ is also prime.

(ii) Let $P$ be a non-trivial prime ideal in the commutative ring $R$ ('non-trivial' meaning that $P \neq\{0\}$ and $P \neq R$ ). If $P$ has finite index as a subgroup of $R$, show that $P$ is also maximal. Give an example to show that this may fail, if the assumption of finite index is omitted. Finally, show that if $R$ is a principal ideal domain, then every non-trivial prime ideal in $R$ is maximal.

B1.10

commentSuppose that $\left(e_{n}\right)$ and $\left(f_{m}\right)$ are orthonormal bases of a Hilbert space $H$ and that $T \in L(H)$.

(a) Show that $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}=\sum_{m=1}^{\infty}\left\|T^{*}\left(f_{m}\right)\right\|^{2}$.

(b) Show that $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}=\sum_{m=1}^{\infty}\left\|T\left(f_{m}\right)\right\|^{2}$.

$T \in L(H)$ is a Hilbert-Schmidt operator if $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}<\infty$ for some (and hence every) orthonormal basis $\left(e_{n}\right)$.

(c) Show that the set HS of Hilbert-Schmidt operators forms a linear subspace of $L(H)$, and that $\langle T, S\rangle=\sum_{n=1}^{\infty}\left\langle T\left(e_{n}\right), S\left(e_{n}\right)\right\rangle$ is an inner product on $H S$; show that this inner product does not depend on the choice of the orthonormal basis $\left(e_{n}\right)$.

(d) Let $\|T\|_{H S}$ be the corresponding norm. Show that $\|T\| \leqslant\|T\|_{H S}$, and show that a Hilbert-Schmidt operator is compact.

B1.14

commentState the formula for the capacity of a memoryless channel.

(a) Consider a memoryless channel where there are two input symbols, $A$ and $B$, and three output symbols, $A, B$, *. Suppose each input symbol is left intact with probability $1 / 2$, and transformed into a $*$ with probability $1 / 2$. Write down the channel matrix, and calculate the capacity.

(b) Now suppose the output is further processed by someone who cannot distinguish $A$ and $*$, so that the matrix becomes

$\left(\begin{array}{cc} 1 & 0 \\ 1 / 2 & 1 / 2 \end{array}\right)$

Calculate the new capacity.

A $1 . 7 \quad$ B1.12

comment(i) State and prove the Knaster-Tarski Fixed-Point Theorem.

(ii) A subset $S$ of a poset $X$ is called an up-set if whenever $x, y \in X$ satisfy $x \in S$ and $x \leqslant y$ then also $y \in S$. Show that the set of up-sets of $X$ (ordered by inclusion) is a complete poset.

Let $X$ and $Y$ be totally ordered sets, such that $X$ is isomorphic to an up-set in $Y$ and $Y$ is isomorphic to the complement of an up-set in $X$. Prove that $X$ is isomorphic to $Y$. Indicate clearly where in your argument you have made use of the fact that $X$ and $Y$ are total orders, rather than just partial orders.

[Recall that posets $X$ and $Y$ are called isomorphic if there exists a bijection $f$ from $X$ to $Y$ such that, for any $x, y \in X$, we have $f(x) \leqslant f(y)$ if and only if $x \leqslant y$.]

A1.1 B1.1

comment(i) Give the definitions of a recurrent and a null recurrent irreducible Markov chain.

Let $\left(X_{n}\right)$ be a recurrent Markov chain with state space $I$ and irreducible transition matrix $P=\left(p_{i j}\right)$. Prove that the vectors $\gamma^{k}=\left(\gamma_{j}^{k}, j \in I\right), k \in I$, with entries $\gamma_{k}^{k}=1$ and

$\gamma_{i}^{k}=\mathbb{E}_{k}(\# \text { of visits to } i \text { before returning to } k), \quad i \neq k,$

are $P$-invariant:

$\gamma_{j}^{k}=\sum_{i \in I} \gamma_{i}^{k} p_{i j}$

(ii) Let $\left(W_{n}\right)$ be the birth and death process on $\mathbb{Z}_{+}=\{0,1,2, \ldots\}$ with the following transition probabilities:

$\begin{aligned} &p_{i, i+1}=p_{i, i-1}=\frac{1}{2}, i \geq 1 \\ &p_{01}=1 \end{aligned}$

By relating $\left(W_{n}\right)$ to the symmetric simple random walk $\left(Y_{n}\right)$ on $\mathbb{Z}$, or otherwise, prove that $\left(W_{n}\right)$ is a recurrent Markov chain. By considering invariant measures, or otherwise, prove that $\left(W_{n}\right)$ is null recurrent.

Calculate the vectors $\gamma^{k}=\left(\gamma_{i}^{k}, i \in \mathbb{Z}_{+}\right)$for the chain $\left(W_{n}\right), k \in \mathbb{Z}_{+}$.

Finally, let $W_{0}=0$ and let $N$ be the number of visits to 1 before returning to 0 . Show that $\mathbb{P}_{0}(N=n)=(1 / 2)^{n}, n \geq 1$.

[You may use properties of the random walk $\left(Y_{n}\right)$ or general facts about Markov chains without proof but should clearly state them.]

B1.19

commentState the convolution theorem for Laplace transforms.

The temperature $T(x, t)$ in a semi-infinite rod satisfies the heat equation

$\frac{\partial^{2} T}{\partial x^{2}}=\frac{1}{k} \frac{\partial T}{\partial t}, \quad x \geq 0, t \geq 0$

and the conditions $T(x, 0)=0$ for $x \geq 0, T(0, t)=f(t)$ for $t \geq 0$ and $T(x, t) \rightarrow 0$ as $x \rightarrow \infty$. Show that

$T(x, t)=\int_{0}^{t} f(\tau) G(x, t-\tau) d \tau$

where

$G(x, t)=\sqrt{\frac{x^{2}}{4 \pi k t^{3}}} e^{-x^{2} / 4 k t}$

A1.6 B1.17

comment(i) State Liapunov's First Theorem and La Salle's Invariance Principle. Use these results to show that the system

$\ddot{x}+k \dot{x}+\sin x=0, \quad k>0$

has an asymptotically stable fixed point at the origin.

(ii) Define the basin of attraction of an invariant set of a dynamical system.

Consider the equations

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

(a) Find the fixed points of the system and determine their type.

(b) Show that the basin of attraction of the origin includes the union over $\alpha$ of the regions

$x^{2}+\alpha^{2} y^{2}<\frac{4 \alpha^{2}\left(1+\alpha^{2}\right)(\beta-1)}{\beta^{2}\left(1+\alpha^{2}\right)^{2}-4 \alpha^{2}} .$

Sketch these regions for $\alpha^{2}=1,1 / 2,2$ in the case $\beta=3$.

B1.9

commentLet $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $X^{3}-4 X+1$. Prove that $K$ has degree 3 over $\mathbb{Q}$, and admits three distinct embeddings in $\mathbb{R}$. Find the discriminant of $K$ and determine the ring of integers $\mathcal{O}$ of $K$. Factorise $2 \mathcal{O}$ and $3 \mathcal{O}$ into a product of prime ideals.

Using Minkowski's bound, show that $K$ has class number 1 provided all prime ideals in $\mathcal{O}$ dividing 2 and 3 are principal. Hence prove that $K$ has class number $1 .$

[You may assume that the discriminant of $X^{3}+a X+b$ is $-4 a^{3}-27 b^{2}$.]

A1.9

comment(i) State the law of quadratic reciprocity. For $p \neq 5$ an odd prime, evaluate the Legendre symbol

$\left(\frac{5}{p}\right)$

(ii) (a) Let $p_{1}, \ldots, p_{m}$ and $q_{1}, \ldots, q_{n}$ be distinct odd primes. Show that there exists an integer $x$ that is a quadratic residue modulo each of $p_{1}, \ldots, p_{m}$ and a quadratic non-residue modulo each of $q_{1}, \ldots, q_{n}$.

(b) Let $p$ be an odd prime. Show that

$\sum_{a=1}^{p-1}\left(\frac{a}{p}\right)=0$

(c) Let $p$ be an odd prime. Using (b) or otherwise, evaluate

$\sum_{a=1}^{p-2}\left(\frac{a}{p}\right)\left(\frac{a+1}{p}\right)$

$\left[\right.$ Hint for $(c)$ : Use the equality $\left(\frac{x^{2} y}{p}\right)=\left(\frac{y}{p}\right)$, valid when $p$ does not divide $\left.x .\right]$

A1.20 B1.20

comment(i) Define the Backward Difference Formula (BDF) method for ordinary differential equations and derive its two-step version.

(ii) Prove that the interval $(-\infty, 0)$ belongs to the linear stability domain of the twostep BDF method.

B1.18

comment(a) State and prove the Mean Value Theorem for harmonic functions.

(b) Let $u \geqslant 0$ be a harmonic function on an open set $\Omega \subset \mathbb{R}^{n}$. Let $B(x, a)=\{y \in$ $\left.\mathbb{R}^{n}:|x-y|<a\right\}$. For any $x \in \Omega$ and for any $r>0$ such that $B(x, 4 r) \subset \Omega$, show that

$\sup _{\{y \in B(x, r)\}} u(y) \leqslant 3^{n} \inf _{\{y \in B(x, r)\}} u(y) .$

A1.2 B1.2

comment(i) In Hamiltonian mechanics the action is written

$S=\int d t\left(p^{a} \dot{q}^{a}-H\left(q^{a}, p^{a}, t\right)\right)$

Starting from Maupertius' principle $\delta S=0$, derive Hamilton's equations

$\dot{q}^{a}=\frac{\partial H}{\partial p^{a}}, \quad \dot{p}^{a}=-\frac{\partial H}{\partial q^{a}} .$

Show that $H$ is a constant of the motion if $\partial H / \partial t=0$. When is $p^{a}$ a constant of the motion?

(ii) Consider the action $S$ given in Part (i), evaluated on a classical path, as a function of the final coordinates $q_{f}^{a}$ and final time $t_{f}$, with the initial coordinates and the initial time held fixed. Show that $S\left(q_{f}^{a}, t_{f}\right)$ obeys

$\frac{\partial S}{\partial q_{f}^{a}}=p_{f}^{a}, \quad \frac{\partial S}{\partial t_{f}}=-H\left(q_{f}^{a}, p_{f}^{a}, t_{f}\right)$

Now consider a simple harmonic oscillator with $H=\frac{1}{2}\left(p^{2}+q^{2}\right)$. Setting the initial time and the initial coordinate to zero, find the classical solution for $p$ and $q$ with final coordinate $q=q_{f}$ at time $t=t_{f}$. Hence calculate $S\left(t_{f}, q_{f}\right)$, and explicitly verify (2) in this case.

A1.12 B1.15

comment(i) What does it mean to say that a family $\{f(\cdot \mid \theta): \theta \in \Theta\}$ of densities is an exponential family?

Consider the family of densities on $(0, \infty)$ parametrised by the positive parameters $a, b$ and defined by

$f(x \mid a, b)=\frac{a \exp \left(-(a-b x)^{2} / 2 x\right)}{\sqrt{2 \pi x^{3}}} \quad(x>0)$

Prove that this family is an exponential family, and identify the natural parameters and the reference measure.

(ii) Let $\left(X_{1}, \ldots, X_{n}\right)$ be a sample drawn from the above distribution. Find the maximum-likelihood estimators of the parameters $(a, b)$. Find the Fisher information matrix of the family (in terms of the natural parameters). Briefly explain the significance of the Fisher information matrix in relation to unbiased estimation. Compute the mean of $X_{1}$ and of $X_{1}^{-1}$.

B1.13

commentLet $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{A}=\left(A_{i}: i=1,2, \ldots\right)$ be a sequence of events.

(a) What is meant by saying that $\mathcal{A}$ is a family of independent events?

(b) Define the events $\left\{A_{n}\right.$ infinitely often $\}$ and $\left\{A_{n}\right.$ eventually $\}$. State and prove the two Borel-Cantelli lemmas for $\mathcal{A}$.

(c) Let $X_{1}, X_{2}, \ldots$ be the outcomes of a sequence of independent flips of a fair coin,

$\mathbb{P}\left(X_{i}=0\right)=\mathbb{P}\left(X_{i}=1\right)=\frac{1}{2} \quad \text { for } i \geqslant 1$

Let $L_{n}$ be the length of the run beginning at the $n^{\text {th }}$flip. For example, if the first fourteen outcomes are 01110010000110 , then $L_{1}=1, L_{2}=3, L_{3}=2$, etc.

Show that

$\mathbb{P}\left(\limsup _{n \rightarrow \infty} \frac{L_{n}}{\log _{2} n}>1\right)=0$

and furthermore that

$\mathbb{P}\left(\limsup _{n \rightarrow \infty} \frac{L_{n}}{\log _{2} n}=1\right)=1$

A1.14

comment(i) Each particle in a system of $N$ identical fermions has a set of energy levels $E_{i}$ with degeneracy $g_{i}$, where $i=1,2, \ldots$. Derive the expression

$\bar{N}_{i}=\frac{g_{i}}{e^{\beta\left(E_{i}-\mu\right)}+1},$

for the mean number of particles $\bar{N}_{i}$ with energy $E_{i}$. Explain the physical significance of the parameters $\beta$ and $\mu$.

(ii) The spatial eigenfunctions of energy for an electron of mass $m$ moving in two dimensions and confined to a square box of side $L$ are

$\psi_{n_{1} n_{2}}(\mathbf{x})=\frac{2}{L} \sin \left(\frac{n_{1} \pi x}{L}\right) \sin \left(\frac{n_{2} \pi y}{L}\right)$

where $n_{i}=1,2, \ldots(i=1,2)$. Calculate the associated energies.

Hence show that when $L$ is large the number of states in energy range $E \rightarrow E+d E$ is

$\frac{m L^{2}}{2 \pi \hbar^{2}} d E$

How is this formula modified when electron spin is taken into account?

The box is filled with $N$ electrons in equilibrium at temperature $T$. Show that the chemical potential $\mu$ is given by

$\mu=\frac{1}{\beta} \log \left(e^{\beta \pi \hbar^{2} \rho / m}-1\right)$

where $\rho$ is the number of particles per unit area in the box.

What is the value of $\mu$ in the limit $T \rightarrow 0$ ?

Calculate the total energy of the lowest state of the system of particles as a function of $N$ and $L$.

B1.6

comment(a) Show that every irreducible complex representation of an abelian group is onedimensional.

(b) Show, by example, that the analogue of (a) fails for real representations.

(c) Let the cyclic group of order $n$ act on $\mathbb{C}^{n}$ by cyclic permutation of the standard basis vectors. Decompose this representation explicitly into irreducibles.

B1.11

commentLet $\tau$ be a fixed complex number with positive imaginary part. For $z \in \mathbb{C}$, define

$v(z)=\sum_{n=-\infty}^{\infty} \exp \left(\pi i \tau n^{2}+2 \pi i n\left(z+\frac{1}{2}\right)\right) .$

Prove the identities

$v(z+1)=v(z), \quad v(-z)=v(z), \quad v(z+\tau)=-\exp (-\pi i \tau-2 \pi i z) \cdot v(z)$

and deduce that $v(\tau / 2)=0$. Show further that $\tau / 2$ is the only zero of $v$ in the parallelogram $P$ with vertices $-1 / 2,1 / 2,1 / 2+\tau,-1 / 2+\tau$.

[You may assume that $v$ is holomorphic on $\mathbb{C}$.]

Now let $\left\{a_{1}, \ldots, a_{k}\right\}$ and $\left\{b_{1}, \ldots, b_{k}\right\}$ be two sets of complex numbers and

$f(z)=\prod_{j=1}^{k} \frac{v\left(z-a_{j}\right)}{v\left(z-b_{j}\right)}$

Prove that $f$ is a doubly-periodic meromorphic function, with periods 1 and $\tau$, if and only if $\sum_{j=1}^{k}\left(a_{j}-b_{j}\right)$ is an integer.

B1.22

commentDefine the notions of entropy $S$ and thermodynamic temperature $T$ for a gas of particles in a variable volume $V$. Derive the fundamental relation

$d E=T d S-P d V$

The free energy of the gas is defined as $F=E-T S$. Why is it convenient to regard $F$ as a function of $T$ and $V$ ? By considering $F$, or otherwise, show that

$\left.\frac{\partial S}{\partial V}\right|_{T}=\left.\frac{\partial P}{\partial T}\right|_{V}$

Deduce that the entropy of an ideal gas, whose equation of state is $P V=N T$ (using energy units), has the form

$S=N \log \left(\frac{V}{N}\right)+N c(T)$

where $c(T)$ is independent of $N$ and $V$.

Show that if the gas is in contact with a heat bath at temperature $T$, then the probability of finding the gas in a particular quantum microstate of energy $E_{r}$ is

$P_{r}=e^{\left(F-E_{r}\right) / T}$

A1.16

comment(i) Consider a homogeneous and isotropic universe with mass density $\rho(t)$, pressure $P(t)$ and scale factor $a(t)$. As the universe expands its energy $E$ decreases according to the thermodynamic relation $d E=-P d V$ where $V$ is the volume. Deduce the fluid conservation law

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

Apply the conservation of total energy (kinetic plus gravitational potential) to a test particle on the edge of a spherical region in this universe to obtain the Friedmann equation

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

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

(ii) Our universe is believed to be 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 $P_{\mathrm{Q}}=-\rho_{\mathrm{Q}} c^{2}$ where the mass densities today $\left(t=t_{0}\right)$ are given respectively by $\rho_{\mathrm{M} 0}$ and $\rho_{\mathrm{Q} 0}$. Assume that each component independently satisfies the fluid conservation equation to show that the total mass density can be expressed as

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

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

Now consider the substitution $b=a^{3 / 2}$ in the Friedmann equation to show that the solution for the scale factor can be written in the form

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

where $\alpha$ and $\beta$ are constants. Setting $a\left(t_{0}\right)=1$, specify $\alpha$ and $\beta$ in terms of $\rho_{\mathrm{M} 0}, \rho_{\mathrm{Q} 0}$ and $t_{0}$. Show that the scale factor $a(t)$ has the expected behaviour for an Einstein-de Sitter universe at early times $(t \rightarrow 0)$ and that the universe accelerates at late times $(t \rightarrow \infty)$.

[Hint: Recall that $\int d x / \sqrt{x^{2}+1}=\sinh ^{-1} x$.]

A1.11 B1.16

comment(i) What does it mean to say that $U$ is a utility function? What is a utility function with constant absolute risk aversion (CARA)?

Let $S_{t} \equiv\left(S_{t}^{1}, \ldots, S_{t}^{d}\right)^{T}$ denote the prices at time $t=0,1$ of $d$ risky assets, and suppose that there is also a riskless zeroth asset, whose price at time 0 is 1 , and whose price at time 1 is $1+r$. Suppose that $S_{1}$ has a multivariate Gaussian distribution, with mean $\mu_{1}$ and non-singular covariance $V$. An agent chooses at time 0 a portfolio $\theta=\left(\theta^{1}, \ldots, \theta^{d}\right)^{T}$ of holdings of the $d$ risky assets, at total cost $\theta \cdot S_{0}$, and at time 1 realises his gain $X=\theta \cdot\left(S_{1}-(1+r) S_{0}\right)$. Given that he wishes the mean of $X$ to be equal to $m$, find the smallest value that the variance $v$ of $X$ can be. What is the portfolio that achieves this smallest variance? Hence sketch the region in the $(v, m)$ plane of pairs $(v, m)$ that can be achieved by some choice of $\theta$, and indicate the mean-variance efficient frontier.

(ii) Suppose that the agent has a CARA utility with coefficient $\gamma$ of absolute risk aversion. What portfolio will he choose in order to maximise $E U(X)$ ? What then is the mean of $X$ ?

Regulation requires that the agent's choice of portfolio $\theta$ has to satisfy the valueat-risk (VaR) constraint

$m \geqslant-L+a \sqrt{v},$

where $L>0$ and $a>0$ are determined by the regulatory authority. Show that this constraint has no effect on the agent's decision if $\kappa \equiv \sqrt{\mu \cdot V^{-1} \mu} \geqslant a$. If $\kappa<a$, will this constraint necessarily affect the agent's choice of portfolio?

A1.19

comment(i) State and prove Maschke's theorem for finite-dimensional representations of finite groups.

(ii) $S_{3}$ is the group of bijections on $\{1,2,3\}$. Find the irreducible representations of $S_{3}$, state their dimensions and give their character table.

Let $T_{2}$ be the set of objects $T_{2}=\left\{a_{i_{1} i_{2}}: i_{1}, i_{2}=1,2,3\right\}$. The operation of the permutation group $S_{3}$ on $T_{2}$ is defined by the operation of the elements of $S_{3}$ separately on each index $i_{1}$ and $i_{2}$. For example,

$P_{12}: a_{13} \rightarrow a_{23}, \quad P_{231}: a_{23} \rightarrow a_{31}, \quad P_{13}: a_{33} \rightarrow a_{11}$

By considering a representative operator from each conjugacy class of $S_{3}$, find the table of group characters for the representation $\mathcal{T}_{2}$ of $S_{3}$ acting on $T_{2}$. Hence, deduce the irreducible representations into which $\mathcal{T}_{2}$ decomposes.

A1.17

comment(i) What is the polarisation $\mathbf{P}$ and slowness $\mathbf{s}$ of the time-harmonic plane elastic wave $\mathbf{u}=\mathbf{A} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)] ?$

Use the equation of motion for an isotropic homogenous elastic medium,

$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \wedge(\boldsymbol{\nabla} \wedge \mathbf{u})$

to show that $\mathbf{s} \cdot \mathbf{s}$ takes one of two values and obtain the corresponding conditions on $\mathbf{P}$. If $\mathbf{s}$ is complex show that $\operatorname{Re}(\mathbf{s}) \cdot \operatorname{Im}(\mathbf{s})=0$.

(ii) A homogeneous elastic layer of uniform thickness $h, S$-wave speed $\beta_{1}$ and shear modulus $\mu_{1}$ has a stress-free surface $z=0$ and overlies a lower layer of infinite depth, $S$-wave speed $\beta_{2}\left(>\beta_{1}\right)$ and shear modulus $\mu_{2}$. Show that the horizontal phase speed $c$ of trapped Love waves satisfies $\beta_{1}<c<\beta_{2}$. Show further that

$\tan \left[\left(\frac{c^{2}}{\beta_{1}^{2}}-1\right)^{1 / 2} k h\right]=\frac{\mu_{2}}{\mu_{1}}\left(\frac{1-c^{2} / \beta_{2}^{2}}{c^{2} / \beta_{1}^{2}-1}\right)^{1 / 2}$

where $k$ is the horizontal wavenumber.

Assuming that (1) can be solved to give $c(k)$, explain how to obtain the propagation speed of a pulse of Love waves with wavenumber $k$.

A1.18

comment(i) In an experiment, a finite amount $M$ of marker gas of diffusivity $D$ is released at time $t=0$ into an infinite tube in the neighbourhood of the origin $x=0$. Starting from the one-dimensional diffusion equation for the concentration $C(x, t)$ of marker gas,

$C_{t}=D C_{x x}$

use dimensional analysis to show that

$C=\frac{M}{(D t)^{1 / 2}} f(\xi)$

for some dimensionless function $f$ of the similarity variable $\xi=x /(D t)^{1 / 2}$.

Write down the equation and boundary conditions satisfied by $f(\xi)$.

(ii) Consider the experiment of Part (i). Find $f(\xi)$ and sketch your answer in the form of a plot of $C$ against $x$ at a few different times $t$.

Calculate $C(x, t)$ for a second experiment in which the concentration of marker gas at $x=0$ is instead raised to the value $C_{0}$ at $t=0$ and maintained at that value thereafter. Show that the total amount of marker gas released in this case becomes greater than $M$ after a time

$t=\frac{\pi}{16 D}\left(\frac{M}{C_{0}}\right)^{2} .$

Show further that, at much larger times than this, the concentration in the first experiment still remains greater than that in the second experiment for positions $x$ with $|x|>$ ${ }_{4} C_{0} D t / M$.

[Hint: $\operatorname{erfc}(z) \equiv \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-u^{2}} d u \sim \frac{1}{\sqrt{\pi} z} e^{-z^{2}}$ as $\left.z \rightarrow \infty .\right]$

B1.26

commentA physical system permits one-dimensional wave propagation in the $x$-direction according to the equation

$\frac{\partial^{2} \psi}{\partial t^{2}}-\alpha^{2} \frac{\partial^{6} \psi}{\partial x^{6}}=0$

where $\alpha$ is a real positive constant. Derive the corresponding dispersion relation and sketch graphs of frequency, phase velocity and group velocity as functions of the wave number. Is it the shortest or the longest waves that are at the front of a dispersing wave train arising from a localised initial disturbance? Do the wave crests move faster or slower than a packet of waves?

Find the solution of the above equation for the initial disturbance given by

$\psi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k, \quad \frac{\partial \psi}{\partial t}(x, 0)=0$

where $A(k)$ is real and $A(-k)=A(k)$.

Use the method of stationary phase to obtain a leading-order approximation to this solution for large $t$ when $V=x / t$ is held fixed.

[Note that

$\int_{-\infty}^{\infty} e^{\pm i u^{2}} d u=\pi^{\frac{1}{2}} e^{\pm i \pi / 4}$