Part II, 2003, Paper 1

# Part II, 2003, Paper 1

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B1.23

commentDefine the differential cross section $\frac{d \sigma}{d \Omega}$. Show how it may be related to a scattering amplitude $f$ defined in terms of the behaviour of a wave function $\psi$ satisfying suitable boundary conditions as $r=|\mathbf{r}| \rightarrow \infty$.

For a particle scattering off a potential $V(r)$ show how $f(\theta)$, where $\theta$ is the scattering angle, may be expanded, for energy $E=\hbar^{2} k^{2} / 2 m$, as

$f(\theta)=\sum_{\ell=0}^{\infty} f_{\ell}(k) P_{\ell}(\cos \theta),$

and find $f_{\ell}(k)$ in terms of the phase shift $\delta_{\ell}(k)$. Obtain the optical theorem relating $\sigma_{\text {total }}$ and $f(0)$.

Suppose that

$e^{2 i \delta_{1}}=\frac{E-E_{0}-\frac{1}{2} i \Gamma}{E-E_{0}+\frac{1}{2} i \Gamma}$

Why for $E \approx E_{0}$ may $f_{1}(k)$ be dominant, and what is the expected behaviour of $\frac{d \sigma}{d \Omega}$ for $E \approx E_{0}$ ?

[For large $r$

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

Legendre polynomials satisfy

$\left.\int_{-1}^{1} P_{\ell}(t) P_{\ell^{\prime}}(t) d t=\frac{2}{2 \ell+1} \delta_{\ell \ell^{\prime}} \cdot\right]$

A1.10

comment(i) We work over the field of two elements. Define what is meant by a linear code of length $n$. What is meant by a generator matrix for a linear code?

Define what is meant by a parity check code of length $n$. Show that a code is linear if and only if it is a parity check code.

Give the original Hamming code in terms of parity checks and then find a generator matrix for it.

[You may use results from the theory of vector spaces provided that you quote them correctly.]

(ii) Suppose that $1 / 4>\delta>0$ and let $\alpha(n, n \delta)$ be the largest information rate of any binary error correcting code of length $n$ which can correct $[n \delta]$ errors.

Show that

$1-H(2 \delta) \leqslant \liminf _{n \rightarrow \infty} \alpha(n, n \delta) \leqslant 1-H(\delta)$

where

$H(\eta)=-\eta \log _{2} \eta-(1-\eta) \log _{2}(1-\eta)$

[You may assume any form of Stirling's theorem provided that you quote it correctly.]

B1.5

commentLet $G$ be a graph of order $n \geqslant 4$. Prove that if $G$ has $t_{2}(n)+1$ edges then it contains two triangles with a common edge. Here, $t_{2}(n)=\left\lfloor n^{2} / 4\right\rfloor$ is the Turán number.

Suppose instead that $G$ has exactly one triangle. Show that $G$ has at most $t_{2}(n-1)+2$ edges, and that this number can be attained.

A1.13

comment(i) Suppose $Y_{i}, 1 \leqslant i \leqslant n$, are independent binomial observations, with $Y_{i} \sim B i\left(t_{i}, \pi_{i}\right)$, $1 \leqslant i \leqslant n$, where $t_{1}, \ldots, t_{n}$ are known, and we wish to fit the model

$\omega: \log \frac{\pi_{i}}{1-\pi_{i}}=\mu+\beta^{T} x_{i} \quad \text { for each } i$

where $x_{1}, \ldots, x_{n}$ are given covariates, each of dimension $p$. Let $\hat{\mu}, \hat{\beta}$ be the maximum likelihood estimators of $\mu, \beta$. Derive equations for $\hat{\mu}, \hat{\beta}$ and state without proof the form of the approximate distribution of $\hat{\beta}$.

(ii) In 1975 , data were collected on the 3-year survival status of patients suffering from a type of cancer, yielding the following table

\begin{tabular}{ccrr} & & \multicolumn{2}{c}{ survive? } \ age in years & malignant & yes & no \ under 50 & no & 77 & 10 \ under 50 & yes & 51 & 13 \ $50-69$ & no & 51 & 11 \ $50-69$ & yes & 38 & 20 \ $70+$ & no & 7 & 3 \ $70+$ & yes & 6 & 3 \end{tabular}

Here the second column represents whether the initial tumour was not malignant or was malignant.

Let $Y_{i j}$ be the number surviving, for age group $i$ and malignancy status $j$, for $i=1,2,3$ and $j=1,2$, and let $t_{i j}$ be the corresponding total number. Thus $Y_{11}=77$, $t_{11}=87$. Assume $Y_{i j} \sim B i\left(t_{i j}, \pi_{i j}\right), 1 \leqslant i \leqslant 3,1 \leqslant j \leqslant 2$. The results from fitting the model

$\log \left(\pi_{i j} /\left(1-\pi_{i j}\right)\right)=\mu+\alpha_{i}+\beta_{j}$

with $\alpha_{1}=0, \beta_{1}=0$ give $\hat{\beta}_{2}=-0.7328(\mathrm{se}=0.2985)$, and deviance $=0.4941$. What do you conclude?

Why do we take $\alpha_{1}=0, \beta_{1}=0$ in the model?

What "residuals" should you compute, and to which distribution would you refer them?

B1.8

commentState the Implicit Function Theorem and outline how it produces submanifolds of Euclidean spaces.

Show that the unitary group $U(n) \subset G L(n, \mathbb{C})$ is a smooth manifold and find its dimension.

Identify the tangent space to $U(n)$ at the identity matrix as a subspace of the space of $n \times n$ complex matrices.

B1.17

commentConsider the one-dimensional map $f: \mathbb{R} \rightarrow \mathbb{R}$, where $f(x)=\mu x^{2}(1-x)$ with $\mu$ a real parameter. Find the range of values of $\mu$ for which the open interval $(0,1)$ is mapped into itself and contains at least one fixed point. Describe the bifurcation at $\mu=4$ and find the parameter value for which there is a period-doubling bifurcation. Determine whether the fixed point is an attractor at this bifurcation point.

A1.6

comment(i) State and prove Dulac's Criterion for the non-existence of periodic orbits in $\mathbb{R}^{2}$. Hence show (choosing a weighting factor of the form $x^{\alpha} y^{\beta}$ ) that there are no periodic orbits of the equations

$\dot{x}=x\left(2-6 x^{2}-5 y^{2}\right), \quad \dot{y}=y\left(-3+10 x^{2}+3 y^{2}\right)$

(ii) State the Poincaré-Bendixson Theorem. A model of a chemical reaction (the Brusselator) is defined by the second order system

$\dot{x}=a-x(1+b)+x^{2} y, \quad \dot{y}=b x-x^{2} y$

where $a, b$ are positive parameters. Show that there is a unique fixed point. Show that, for a suitable choice of $p>0$, trajectories enter the closed region bounded by $x=p, y=b / p$, $x+y=a+b / p$ and $y=0$. Deduce that when $b>1+a^{2}$, the system has a periodic orbit.

B1.21

commentA particle of charge $q$ and mass $m$ moves non-relativistically with 4 -velocity $u^{a}(t)$ along a trajectory $x^{a}(t)$. Its electromagnetic field is determined by the Liénard-Wiechert potential

$A^{a}\left(\mathbf{x}^{\prime}, t^{\prime}\right)=\frac{q}{4 \pi \epsilon_{0}} \frac{u^{a}(t)}{u_{b}(t)\left(x^{\prime}-x(t)\right)^{b}}$

where $t^{\prime}=t+\left|\mathbf{x}-\mathbf{x}^{\prime}\right|$ and $\mathbf{x}$ denotes the spatial part of the 4 -vector $x^{a}$.

Derive a formula for the Poynting vector at very large distances from the particle. Hence deduce Larmor's formula for the rate of loss of energy due to electromagnetic radiation by the particle.

A particle moves in the $(x, y)$ plane in a constant magnetic field $\mathbf{B}=(0,0, B)$. Initially it has kinetic energy $E_{0}$; derive a formula for the kinetic energy of this particle as a function of time.

B1.25

commentConsider a two-dimensional horizontal vortex sheet of strength $U$ at height $h$ above a horizontal rigid boundary at $y=0$, so that the inviscid fluid velocity is

$\mathbf{u}= \begin{cases}(U, 0) & 0<y<h \\ (0,0) & y>h\end{cases}$

Examine the temporal linear instabililty of the sheet and determine the relevant dispersion relationship.

For what wavelengths is the sheet unstable?

Evaluate the temporal growth rate and the wave propagation speed in the limit of both short and long waves. Comment briefly on the significance of your results.

A1.3

comment(i) Let $T: H_{1} \rightarrow H_{2}$ be a continuous linear map between two Hilbert spaces $H_{1}, H_{2}$. Define the adjoint $T^{*}$ of $T$. Explain what it means to say that $T$ is Hermitian or unitary.

Let $\phi: \mathbb{R} \rightarrow \mathbb{C}$ be a bounded continuous function. Show that the map

$T: L^{2}(\mathbb{R}) \rightarrow L^{2}(\mathbb{R})$

with $T f(x)=\phi(x) f(x+1)$ is a continuous linear map and find its adjoint. When is $T$ Hermitian? When is it unitary?

(ii) Let $C$ be a closed, non-empty, convex subset of a real Hilbert space $H$. Show that there exists a unique point $x_{o} \in C$ with minimal norm. Show that $x_{o}$ is characterised by the property

$\left\langle x_{o}-x, x_{o}\right\rangle \leqslant 0 \quad \text { for all } x \in C .$

Does this result still hold when $C$ is not closed or when $C$ is not convex? Justify your answers.

B1.7

commentWhat does it mean to say that a field is algebraically closed? Show that a field $M$ is algebraically closed if and only if, for any finite extension $L / K$ and every homomorphism $\sigma: K \hookrightarrow M$, there exists a homomorphism $L \hookrightarrow M$ whose restriction to $K$ is $\sigma$.

Let $K$ be a field of characteristic zero, and $M / K$ an algebraic extension such that every nonconstant polynomial over $K$ has at least one root in $M$. Prove that $M$ is algebraically closed.

A1.15 B1.24

comment(i) The worldline $x^{a}(\lambda)$ of a massive particle moving in a spacetime with metric $g_{a b}$ obeys the geodesic equation

$\frac{d^{2} x^{a}}{d \tau^{2}}+\Gamma^a_{bc} \frac{d x^{b}}{d \tau} \frac{d x^{c}}{d \tau}=0$

where $\tau$ is the particle's proper time and $\Gamma^a_{bc}$ are the Christoffel symbols; these are the equations of motion for the Lagrangian

$L_{1}=-m \sqrt{-g_{a b} \dot{x}^{a} \dot{x}^{b}}$

where $m$ is the particle's mass, and $\dot{x}^{a}=d x^{a} / d \lambda$. Why is the choice of worldline parameter $\lambda$ irrelevant? Among all possible worldlines passing through points $A$ and $B$, why is $x^{a}(\lambda)$ the one that extremizes the proper time elapsed between $A$ and $B$ ?

Explain how the equations of motion for a massive particle may be obtained from the alternative Lagrangian

$L_{2}=\frac{1}{2} g_{a b} \dot{x}^{a} \dot{x}^{b} .$

What can you conclude from the fact that $L_{2}$ has no explicit dependence on $\lambda$ ? How are the equations of motion for a massless particle obtained from $L_{2}$ ?

(ii) A photon moves in the Schwarzschild metric

$d s^{2}=\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)-\left(1-\frac{2 M}{r}\right) d t^{2}$

Given that the motion is confined to the plane $\theta=\pi / 2$, obtain the radial equation

$\left(\frac{d r}{d \lambda}\right)^{2}=E^{2}-\frac{h^{2}}{r^{2}}\left(1-\frac{2 M}{r}\right)$

where $E$ and $h$ are constants, the physical meaning of which should be stated.

Setting $u=1 / r$, obtain the equation

$\frac{d^{2} u}{d \phi^{2}}+u=3 M u^{2}$

Using the approximate solution

$u=\frac{1}{b} \sin \phi+\frac{M}{2 b^{2}}(3+\cos 2 \phi)+\ldots$

obtain the standard formula for the deflection of light passing far from a body of mass $M$ with impact parameter $b$. Reinstate factors of $G$ and $c$ to give your result in physical units.

A1.8

comment(i) State Brooks' Theorem, and prove it in the case of a 3 -connected graph.

(ii) Let $G$ be a bipartite graph, with vertex classes $X$ and $Y$, each of order $n$. If $G$ contains no cycle of length 4 show that

$e(G) \leqslant \frac{n}{2}(1+\sqrt{4 n-3})$

For which integers $n \leq 12$ are there examples where equality holds?

A1.4

comment(i) Let $p$ be a prime number. Show that a group $G$ of order $p^{n}(n \geqslant 2)$ has a nontrivial normal subgroup, that is, $G$ is not a simple group.

(ii) Let $p$ and $q$ be primes, $p>q$. Show that a group $G$ of order $p q$ has a normal Sylow $p$-subgroup. If $G$ has also a normal Sylow $q$-subgroup, show that $G$ is cyclic. Give a necessary and sufficient condition on $p$ and $q$ for the existence of a non-abelian group of order $p q$. Justify your answer.

B1.3

comment(i) Let $p$ be a prime number. Show that a group $G$ of order $p^{n}(n \geqslant 2)$ has a nontrivial normal subgroup, that is, $G$ is not a simple group.

(ii) Let $p$ and $q$ be primes, $p>q$. Show that a group $G$ of order $p q$ has a normal Sylow $p$-subgroup. If $G$ has also a normal Sylow $q$-subgroup, show that $G$ is cyclic. Give a necessary and sufficient condition on $p$ and $q$ for the existence of a non-abelian group of order $p q$. Justify your answer.

B1.10

commentLet $H$ be a Hilbert space and let $T \in \mathcal{B}(H)$.

(a) Define what it means for $T$ to be (i) invertible, and (ii) bounded below. Prove that $T$ is invertible if and only if both $T$ and $T^{*}$ are bounded below.

(b) Define what it means for $T$ to be normal. Prove that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for all $x \in H$. Deduce that, if $T$ is normal, then every point of Sp $T$ is an approximate eigenvalue of $T$.

(c) Let $S \in \mathcal{B}(H)$ be a self-adjoint operator, and let $\left(x_{n}\right)$ be a sequence in $H$ such that $\left\|x_{n}\right\|=1$ for all $n$ and $\left\|S x_{n}\right\| \rightarrow\|S\|$ as $n \rightarrow \infty$. Show, by direct calculation, that

$\left\|\left(S^{2}-\|S\|^{2}\right) x_{n}\right\|^{2} \rightarrow 0 \quad \text { as } n \rightarrow \infty$

and deduce that at least one of $\pm\|S\|$ is an approximate eigenvalue of $S$.

(d) Deduce that, with $S$ as in (c),

$r(S)=\|S\|=\sup \{|\langle S x, x\rangle|: x \in H,\|x\|=1\}$

B1.14

commentA binary Huffman code is used for encoding symbols $1, \ldots, m$ occurring with probabilities $p_{1} \geqslant \ldots \geqslant p_{m}>0$ where $\sum_{1 \leqslant j \leqslant m} p_{j}=1$. Let $s_{1}$ be the length of a shortest codeword and $s_{m}$ of a longest codeword. Determine the maximal and minimal values of $s_{1}$ and $s_{m}$, and find binary trees for which they are attained.

A $1 . 7 \quad$ B1.12

comment(i) State Zorn's Lemma. Use Zorn's Lemma to prove that every real vector space has a basis.

(ii) State the Bourbaki-Witt Theorem, and use it to prove Zorn's Lemma, making clear where in the argument you appeal to the Axiom of Choice.

Conversely, deduce the Bourbaki-Witt Theorem from Zorn's Lemma.

If $X$ is a non-empty poset in which every chain has an upper bound, must $X$ be chain-complete?

A1.1 B1.1

comment(i) Let $\left(X_{n}, Y_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk in $\mathbb{Z}^{2}$, starting from $(0,0)$, and set $T=\inf \left\{n \geqslant 0: \max \left\{\left|X_{n}\right|,\left|Y_{n}\right|\right\}=2\right\}$. Determine the quantities $\mathbb{E}(T)$ and $\mathbb{P}\left(X_{T}=2\right.$ and $\left.Y_{T}=0\right)$.

(ii) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a discrete-time Markov chain with state-space $I$ and transition matrix $P$. What does it mean to say that a state $i \in I$ is recurrent? Prove that $i$ is recurrent if and only if $\sum_{n=0}^{\infty} p_{i i}^{(n)}=\infty$, where $p_{i i}^{(n)}$ denotes the $(i, i)$ entry in $P^{n}$.

Show that the simple symmetric random walk in $\mathbb{Z}^{2}$ is recurrent.

B1.19

commentBy considering the integral

$\int_{C}\left(\frac{t}{1-t}\right)^{i} d t$

where $C$ is a large circle centred on the origin, show that

$B(1+i, 1-i)=\pi \operatorname{cosech} \pi$

where

$B(p, q)=\int_{0}^{1} t^{p-1}(1-t)^{q-1} d t, \operatorname{Re}(p)>0, \operatorname{Re}(q)>0$

By using $B(p, q)=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$, deduce that $\Gamma(i) \Gamma(-i)=\pi \operatorname{cosech} \pi$.

B1.9

commentLet $K=\mathbb{Q}(\alpha)$, where $\alpha=\sqrt[3]{10}$, and let $\mathcal{O}_{K}$ be the ring of algebraic integers of $K$. Show that the field polynomial of $r+s \alpha$, with $r$ and $s$ rational, is $(x-r)^{3}-10 s^{3}$.

Let $\beta=\frac{1}{3}\left(\alpha^{2}+\alpha+1\right)$. By verifying that $\beta=3 /(\alpha-1)$ and determining the field polynomial, or otherwise, show that $\beta$ is in $\mathcal{O}_{K}$.

By computing the traces of $\theta, \alpha \theta, \alpha^{2} \theta$, show that the elements of $\mathcal{O}_{K}$ have the form

$\theta=\frac{1}{3}\left(l+\frac{1}{10} m \alpha+\frac{1}{10} n \alpha^{2}\right)$

where $l, m, n$ are integers. By further computing the norm of $\frac{1}{10} \alpha(m+n \alpha)$, show that $\theta$ can be expressed as $\frac{1}{3}(u+v \alpha)+w \beta$ with $u, v, w$ integers. Deduce that $1, \alpha, \beta$ form an integral basis for $K$.

A1.9

comment(i) Let $p$ be an odd prime and $k$ a strictly positive integer. Prove that the multiplicative group of relatively prime residue classes modulo $p^{k}$ is cyclic.

[You may assume that the result is true for $k=1$.]

(ii) Let $n=p_{1} p_{2} \ldots p_{r}$, where $r \geqslant 2$ and $p_{1}, p_{2}, \ldots, p_{r}$ are distinct odd primes. Let $B$ denote the set of all integers which are relatively prime to $n$. We recall that $n$ is said to be an Euler pseudo-prime to the base $b \in B$ if

$b^{(n-1) / 2} \equiv\left(\frac{b}{n}\right) \bmod n .$

If $n$ is an Euler pseudo-prime to the base $b_{1} \in B$, but is not an Euler pseudo-prime to the base $b_{2} \in B$, prove that $n$ is not an Euler pseudo-prime to the base $b_{1} b_{2}$. Let $p$ denote any of the primes $p_{1}, p_{2}, \ldots, p_{r}$. Prove that there exists a $b \in B$ such that

$\left(\frac{b}{p}\right)=-1 \quad \text { and } \quad b \equiv 1 \bmod n / p$

and deduce that $n$ is not an Euler pseudo-prime to this base $b$. Hence prove that $n$ is not an Euler pseudo-prime to the base $b$ for at least half of all the relatively prime residue classes $b \bmod n$.

A1.20 B1.20

comment(i) The linear algebraic equations $A \mathbf{u}=\mathbf{b}$, where $A$ is symmetric and positive-definite, are solved with the Gauss-Seidel method. Prove that the iteration always converges.

(ii) The Poisson equation $\nabla^{2} u=f$ is given in the bounded, simply connected domain $\Omega \subseteq \mathbb{R}^{2}$, with zero Dirichlet boundary conditions on $\partial \Omega$. It is approximated by the fivepoint formula

$U_{m-1, n}+U_{m, n-1}+U_{m+1, n}+U_{m, n+1}-4 U_{m, n}=(\Delta x)^{2} f_{m, n}$

where $U_{m, n} \approx u(m \Delta x, n \Delta x), \quad f_{m, n}=f(m \Delta x, n \Delta x)$, and $(m \Delta x, n \Delta x)$ is in the interior of $\Omega$.

Assume for the sake of simplicity that the intersection of $\partial \Omega$ with the grid consists only of grid points, so that no special arrangements are required near the boundary. Prove that the method can be written in a vector notation, $A \mathbf{u}=\mathbf{b}$ with a negative-definite matrix $A$.

B1.18

comment(a) Define characteristic hypersurfaces and state a local existence and uniqueness theorem for a quasilinear partial differential equation with data on a non-characteristic hypersurface.

(b) Consider the initial value problem

$3 u_{x}+u_{y}=-y u, \quad u(x, 0)=f(x),$

for a function $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $C^{1}$ initial data $f$ given for $y=0$. Obtain a formula for the solution by the method of characteristics and deduce that a $C^{1}$ solution exists for all $(x, y) \in \mathbb{R}^{2}$.

Derive the following (well-posedness) property for solutions $u(x, y)$ and $v(x, y)$ corresponding to data $u(x, 0)=f(x)$ and $v(x, 0)=g(x)$ respectively:

$\sup _{x}|u(x, y)-v(x, y)| \leqslant \sup _{x}|f(x)-g(x)| \quad \text { for all } y .$

(c) Consider the initial value problem

$3 u_{x}+u_{y}=u^{2}, \quad u(x, 0)=f(x),$

for a function $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $C^{1}$ initial data $f$ given for $y=0$. Obtain a formula for the solution by the method of characteristics and hence show that if $f(x)<0$ for all $x$, then the solution exists for all $y>0$. Show also that if there exists $x_{0}$ with $f\left(x_{0}\right)>0$, then the solution does not exist for all $y>0$.

A1.2 B1.2

comment(i) Consider $N$ particles moving in 3 dimensions. The Cartesian coordinates of these particles are $x^{A}(t), A=1, \ldots, 3 N$. Now consider an invertible change of coordinates to coordinates $q^{a}\left(x^{A}, t\right), \quad a=1, \ldots, 3 N$, so that one may express $x^{A}$ as $x^{A}\left(q^{a}, t\right)$. Show that the velocity of the system in Cartesian coordinates $\dot{x}^{A}(t)$ is given by the following expression:

$\dot{x}^{A}\left(\dot{q}^{a}, q^{a}, t\right)=\sum_{b=1}^{3 N} \dot{q}^{b} \frac{\partial x^{A}}{\partial q^{b}}\left(q^{a}, t\right)+\frac{\partial x^{A}}{\partial t}\left(q^{a}, t\right)$

Furthermore, show that Lagrange's equations in the two coordinate systems are related via

$\frac{\partial L}{\partial q^{a}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}^{a}}\right)=\sum_{A=1}^{3 N} \frac{\partial x^{A}}{\partial q^{a}}\left(\frac{\partial L}{\partial x^{A}}-\frac{d}{d t} \frac{\partial L}{\partial \dot{x}^{A}}\right)$

(ii) Now consider the case where there are $p<3 N$ constraints applied, $f^{\ell}\left(x^{A}, t\right)=$ $0, \ell=1, \ldots, p$. By considering the $f^{\ell}, \ell=1, \ldots, p$, and a set of independent coordinates $q^{a}, a=1, \ldots, 3 N-p$, as a set of $3 N$ new coordinates, show that the Lagrange equations of the constrained system, i.e.

$\begin{gathered} \frac{\partial L}{\partial x^{A}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{x}^{A}}\right)+\sum_{\ell=1}^{p} \lambda^{\ell} \frac{\partial f^{\ell}}{\partial x^{A}}=0, \quad A=1, \ldots, 3 N \\ f^{\ell}=0, \quad \ell=1, \ldots, p \end{gathered}$

(where the $\lambda^{\ell}$ are Lagrange multipliers) imply Lagrange's equations for the unconstrained coordinates, i.e.

$\frac{\partial L}{\partial q^{a}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}^{a}}\right)=0, \quad a=1, \ldots, 3 N-p .$

A1.12 B1.15

comment(i) A public health official is seeking a rational policy of vaccination against a relatively mild ailment which causes absence from work. Surveys suggest that $60 \%$ of the population are already immune, but accurate tests to detect vulnerability in any individual are too costly for mass screening. A simple skin test has been developed, but is not completely reliable. A person who is immune to the ailment will have a negligible reaction to the skin test with probability $0.4$, a moderate reaction with probability $0.5$ and a strong reaction with probability 0.1. For a person who is vulnerable to the ailment the corresponding probabilities are $0.1,0.4$ and $0.5$. It is estimated that the money-equivalent of workhours lost from failing to vaccinate a vulnerable person is 20 , that the unnecessary cost of vaccinating an immune person is 8 , and that there is no cost associated with vaccinating a vulnerable person or failing to vaccinate an immune person. On the basis of the skin test, it must be decided whether to vaccinate or not. What is the Bayes decision rule that the health official should adopt?

(ii) A collection of $I$ students each sit $J$ exams. The ability of the $i$ th student is represented by $\theta_{i}$ and the performance of the $i$ th student on the $j$ th exam is measured by $X_{i j}$. Assume that, given $\boldsymbol{\theta}=\left(\theta_{1}, \ldots, \theta_{I}\right)$, an appropriate model is that the variables $\left\{X_{i j}, 1 \leqslant i \leqslant I, 1 \leqslant j \leqslant J\right\}$ are independent, and

$X_{i j} \sim N\left(\theta_{i}, \tau^{-1}\right),$

for a known positive constant $\tau$. It is reasonable to assume, a priori, that the $\theta_{i}$ are independent with

$\theta_{i} \sim N\left(\mu, \zeta^{-1}\right),$

where $\mu$ and $\zeta$ are population parameters, known from experience with previous cohorts of students.

Compute the posterior distribution of $\boldsymbol{\theta}$ given the observed exam marks vector $\mathbf{X}=\left\{X_{i j}, 1 \leqslant i \leqslant I, 1 \leqslant j \leqslant J\right\} .$

Suppose now that $\tau$ is also unknown, but assumed to have a $\operatorname{Gamma}\left(\alpha_{0}, \beta_{0}\right)$ distribution, for known $\alpha_{0}, \beta_{0}$. Compute the posterior distribution of $\tau$ given $\boldsymbol{\theta}$ and $\mathbf{X}$ Find, up to a normalisation constant, the form of the marginal density of $\boldsymbol{\theta}$ given $\mathbf{X}$.

B1.13

commentState and prove the first Borel-Cantelli Lemma.

Suppose that $\left(F_{n}\right)$ is a sequence of events in a common probability space such that $\mathbb{P}\left(F_{i} \cap F_{j}\right) \leqslant \mathbb{P}\left(F_{i}\right) \cdot \mathbb{P}\left(F_{j}\right)$ whenever $i \neq j$ and that $\sum_{n} \mathbb{P}\left(F_{n}\right)=\infty$.

Let $1_{F_{n}}$ be the indicator function of $F_{n}$ and let

$S_{n}=\sum_{k \leqslant n} 1_{F_{k}} ; \mu_{n}=\mathbb{E}\left(S_{n}\right)$

Use Chebyshev's inequality to show that

$\mathbb{P}\left(S_{n}<\frac{1}{2} \mu_{n}\right) \leqslant \mathbb{P}\left(\left|S_{n}-\mu_{n}\right|>\frac{1}{2} \mu_{n}\right) \leqslant \frac{4}{\mu_{n}}$

Deduce, using the first Borel-Cantelli Lemma, that $\mathbb{P}\left(F_{n}\right.$ infinitely often $)=1$.

A1.14

comment(i) An electron of mass $m$ and spin $\frac{1}{2}$ moves freely inside a cubical box of side $L$. Verify that the energy eigenstates of the system are $\phi_{l m n}(\mathbf{r}) \chi_{\pm}$where the spatial wavefunction is given by

$\phi_{l m n}(\mathbf{r})=\left(\frac{2}{L}\right)^{3 / 2} \sin \left(\frac{l \pi x}{L}\right) \sin \left(\frac{m \pi y}{L}\right) \sin \left(\frac{n \pi z}{L}\right)$

and

$\chi_{+}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \quad \chi_{-}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$

Give the corresponding energy eigenvalues.

A second electron is inserted into the box. Explain how the Pauli principle determines the structure of the wavefunctions associated with the lowest energy level and the first excited energy level. What are the values of the energy in these two levels and what are the corresponding degeneracies?

(ii) When the side of the box, $L$, is large, the number of eigenstates available to the electron with energy in the range $E \rightarrow E+d E$ is $\rho(E) d E$. Show that

$\rho(E)=\frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3} E}$

A large number, $N$, of electrons are inserted into the box. Explain how the ground state is constructed and define the Fermi energy, $E_{F}$. Show that in the ground state

$N=\frac{2}{3} \frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3}}\left(E_{F}\right)^{3 / 2}$

When a magnetic field $H$ in the $z$-direction is applied to the system, an electron with spin up acquires an additional energy $+\mu H$ and an electron with spin down an energy $-\mu H$, where $-\mu$ is the magnetic moment of the electron and $\mu>0$. Describe, for the case $E_{F}>\mu H$, the structure of the ground state of the system of $N$ electrons in the box and show that

$N=\frac{1}{3} \frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3}}\left(\left(E_{F}+\mu H\right)^{3 / 2}+\left(E_{F}-\mu H\right)^{3 / 2}\right) \text {. }$

Calculate the induced magnetic moment, $M$, of the ground state of the system and show that for a weak magnetic field the magnetic moment is given by

$M \approx \frac{3}{2} N \frac{\mu^{2} H}{E_{F}}$

B1.6

commentDefine the inner product $\langle\varphi, \psi\rangle$ of two class functions from the finite group $G$ into the complex numbers. Prove that characters of the irreducible representations of $G$ form an orthonormal basis for the space of class functions.

Consider the representation $\pi: S_{n} \rightarrow G L_{n}(\mathbb{C})$ of the symmetric group $S_{n}$ by permutation matrices. Show that $\pi$ splits as a direct sum $1 \oplus \rho$ where 1 denotes the trivial representation. Is the $(n-1)$-dimensional representation $\rho$ irreducible?

B1.11

commentProve that a holomorphic map from $\mathbb{P}^{1}$ to itself is either constant or a rational function. Prove that a holomorphic map of degree 1 from $\mathbb{P}^{1}$ to itself is a Möbius transformation.

Show that, for every finite set of distinct points $z_{1}, z_{2}, \ldots, z_{N}$ in $\mathbb{P}^{1}$ and any values $w_{1}, w_{2}, \ldots, w_{N} \in \mathbb{P}^{1}$, there is a holomorphic function $f: \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}$ with $f\left(z_{n}\right)=w_{n}$ for $n=1,2, \ldots, N$.

B1.22

commentA gas in equilibrium at temperature $T$ and pressure $P$ has quantum stationary states $i$ with energies $E_{i}(V)$ in volume $V$. What does it mean to say that a change in volume from $V$ to $V+d V$ is reversible?

Write down an expression for the probability that the gas is in state $i$. How is the entropy $S$ defined in terms of these probabilities? Write down an expression for the energy $E$ of the gas, and establish the relation

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

for reversible changes.

By considering the quantity $F=E-T S$, derive the Maxwell relation

$\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{\partial T}\right)_{V}$

A gas obeys the equation of state

$P V=R T+\frac{B(T)}{V}$

where $R$ is a constant and $B(T)$ is a function of $T$ only. The gas is expanded isothermally, at temperature $T$, from volume $V_{0}$ to volume $2 V_{0}$. Find the work $\Delta W$ done on the gas. Show that the heat $\Delta Q$ absorbed by the gas is given by

$\Delta Q=R T \log 2+\frac{T}{2 V_{0}} \frac{d B}{d T}$

A1.16

comment(i) Explain briefly how the relative motion of galaxies in a homogeneous and isotropic universe is described in terms of the scale factor $a(t)$ (where $t$ is time). In particular, show that the relative velocity $\mathbf{v}(t)$ of two galaxies is given in terms of their relative displacement $\mathbf{r}(t)$ by the formula $\mathbf{v}(t)=H(t) \mathbf{r}(t)$, where $H(t)$ is a function that you should determine in terms of $a(t)$. Given that $a(0)=0$, obtain a formula for the distance $R(t)$ to the cosmological horizon at time $t$. Given further that $a(t)=\left(t / t_{0}\right)^{\alpha}$, for $0<\alpha<1$ and constant $t_{0}$, compute $R(t)$. Hence show that $R(t) / a(t) \rightarrow 0$ as $t \rightarrow 0$.

(ii) A homogeneous and isotropic model universe has energy density $\rho(t) c^{2}$ and pressure $P(t)$, where $c$ is the speed of light. The evolution of its scale factor $a(t)$ is governed by the Friedmann equation

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

where the overdot indicates differentiation with respect to $t$. Use the "Fluid" equation

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

to obtain an equation for the acceleration $\ddot{a}(t)$. Assuming $\rho>0$ and $P \geqslant 0$, show that $\rho a^{3}$ cannot increase with time as long as $\dot{a}>0$, nor decrease if $\dot{a}<0$. Hence determine the late time behaviour of $a(t)$ for $k<0$. For $k>0$ show that an initially expanding universe must collapse to a "big crunch" at which $a \rightarrow 0$. How does $\dot{a}$ behave as $a \rightarrow 0$ ? Given that $P=0$, determine the form of $a(t)$ near the big crunch. Discuss the qualitative late time behaviour for $k=0$.

Cosmological models are often assumed to have an equation of state of the form $P=\sigma \rho c^{2}$ for constant $\sigma$. What physical principle requires $\sigma \leqslant 1$ ? Matter with $P=\rho c^{2}$ $(\sigma=1)$ is called "stiff matter" by cosmologists. Given that $k=0$, determine $a(t)$ for a universe that contains only stiff matter. In our Universe, why would you expect stiff matter to be negligible now even if it were significant in the early Universe?

A1.11 B1.16

comment(i) In the context of a single-period financial market with $d$ traded assets, what is an arbitrage? What is an equivalent martingale measure?

A simple single-period financial market contains two assets, $S^{0}$ (a bond), and $S^{1}$ (a share). The period can be good, bad, or indifferent, with probabilities $1 / 3$ each. At the beginning of the period, time 0 , both assets are worth 1 , i.e.

$S_{0}^{0}=1=S_{0}^{1}$

and at the end of the period, time 1 , the share is worth

$S_{1}^{1}= \begin{cases}a & \text { if the period was bad, } \\ b & \text { if the period was indifferent } \\ c & \text { if the period was good }\end{cases}$

where $a<b<c$. The bond is always worth 1 at the end of the period. Show that there is no arbitrage in this market if and only if $a<1<c$.

(ii) An agent with $C^{2}$ strictly increasing strictly concave utility $U$ has wealth $w_{0}$ at time 0 , and wishes to invest his wealth in shares and bonds so as to maximise his expected utility of wealth at time 1 . Explain how the solution to his optimisation problem generates an equivalent martingale measure.

Assume now that $a=3 / 4, b=1$, and $c=3 / 2$. Characterise all equivalent martingale measures for this problem. Characterise all equivalent martingale measures which arise as solutions of an agent's optimisation problem.

Calculate the largest and smallest possible prices for a European call option with strike 1 and expiry 1, as the pricing measure ranges over all equivalent martingale measures. Calculate the corresponding bounds when the pricing measure is restricted to the set arising from expected-utility-maximising agents' optimisation problems.

A1.17

comment(i) Define the character $\chi$ of a representation $D$ of a finite group $G$. Show that $<\chi \mid \chi>=1$ if and only if $D$ is irreducible, where

$<\chi \mid \chi>=\frac{1}{|G|} \sum_{g \in G} \chi(g) \chi\left(g^{-1}\right)$

If $|G|=8$ and $<\chi \mid \chi>=2$, what are the possible dimensions of the representation $D ?$

(ii) State and prove Schur's first and second lemmas.

A1.19

comment(i) Explain the concepts of: traction on an element of surface; the stress tensor; the strain tensor in an elastic medium. Derive a relationship between the two tensors for a linear isotropic elastic medium, stating clearly any assumption you need to make.

(ii) State what is meant by an $\mathrm{SH}$ wave in a homogeneous isotropic elastic medium. An SH wave in a medium with shear modulus $\mu$ and density $\rho$ is incident at angle $\theta$ on an interface with a medium with shear modulus $\mu^{\prime}$ and density $\rho^{\prime}$. Evaluate the form and amplitude of the reflected wave and transmitted wave. Comment on the case $c^{\prime} \sin \theta / c>1$, where $c^{2}=\mu / \rho$ and $\left(c^{\prime}\right)^{2}=\mu^{\prime} / \rho^{\prime}$.

A1.18

comment(i) A solute occupying a domain $V_{0}$ has concentration $C(\boldsymbol{x}, t)$ and is created at a rate $S(\boldsymbol{x}, t)$ per unit volume; $\boldsymbol{J}(\boldsymbol{x}, t)$ is the flux of solute per unit area; $\boldsymbol{x}, t$ are position and time. Derive the transport equation

$C_{t}+\nabla \cdot \boldsymbol{J}=S$

State Fick's Law of diffusion and hence write down the diffusion equation for $C(\boldsymbol{x}, t)$ for a case in which the solute flux occurs solely by diffusion, with diffusivity $D(\boldsymbol{x})$.

In a finite domain $0 \leqslant x \leqslant L, D, S$ and the steady-state distribution of $C$ depend only on $x ; C$ is equal to $C_{0}$ at $x=0$ and $C_{1} \neq C_{0}$ at $x=L$. Find $C(x)$ in the following two cases: (a) $D=D_{0}, S=0$, (b) $D=D_{1} x^{1 / 2}, S=0$,

where $D_{0}$ and $D_{1}$ are positive constants.

Show that there is no steady solution satisfying the boundary conditions if $D=$ $D_{1} x, S=0 .$

(ii) For the problem of Part (i), consider the case $D=D_{0}, S=k C$, where $D_{0}$ and $k$ are positive constants. Calculate the steady-state solution, $C=C_{s}(x)$, assuming that $\sqrt{k / D_{0}} \neq n \pi / L$ for any integer $n$.

Now let

$C(x, 0)=C_{0} \frac{\sin \alpha(L-x)}{\sin \alpha L}$

where $\alpha=\sqrt{k / D_{0}}$. Find the equations, boundary and initial conditions satisfied by $C^{\prime}(x, t)=C(x, t)-C_{s}(x)$. Solve the problem using separation of variables and show that

$C^{\prime}(x, t)=\sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi x}{L} \exp \left[\left(\alpha^{2}-\frac{n^{2} \pi^{2}}{L^{2}}\right) D_{0} t\right]$

for some constants $A_{n}$. Write down an integral expression for $A_{n}$, show that

$A_{1}=-\frac{2 \pi C_{1}}{\alpha^{2} L^{2}-\pi^{2}},$

and comment on the behaviour of the solution for large times in the two cases $\alpha L<\pi$ and $\alpha L>\pi$.

B1.26

commentConsider the equation

$\frac{\partial^{2} \phi}{\partial t^{2}}+\alpha^{2} \frac{\partial^{4} \phi}{\partial x^{4}}+\beta^{2} \phi=0$

with $\alpha$ and $\beta$ real constants. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Find the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and sketch the graphs of these functions.

By multiplying $(*)$ by $\partial \phi / \partial t$, obtain an energy equation in the form

$\frac{\partial E}{\partial t}+\frac{\partial F}{\partial x}=0$

where $E$ represents the energy density and $F$ the energy flux.

Now let $\phi(x, t)=A \cos (k x-\omega t)$, where $A$ is a real constant. Evaluate the average values of $E$ and $F$ over a period of the wave to show that

$\langle F\rangle=c_{g}\langle E\rangle$

Comment on the physical meaning of this result.