Part II, 2003, Paper 3

# Part II, 2003, Paper 3

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B3.10

comment(a) Let $X \subseteq \mathbb{A}^{n}$ be an affine algebraic variety. Define the tangent space $T_{p} X$ for $p \in X$. Show that the set

$\left\{p \in X \mid \operatorname{dim} T_{p} X \geqslant d\right\}$

is closed, for every $d \geqslant 0$.

(b) Let $C$ be an irreducible projective curve, $p \in C$, and $f: C \backslash\{p\} \rightarrow \mathbb{P}^{n}$ a rational map. Show, carefully quoting any theorems that you use, that if $C$ is smooth at $p$ then $f$ extends to a regular map at $p$.

B3.7

commentDefine a covering map. Prove that any covering map induces an injective homomorphisms of fundamental groups.

Show that there is a non-trivial covering map of the real projective plane. Explain how to use this to find the fundamental group of the real projective plane.

A3.10

comment(i) Consider the problem

$\begin{aligned} \operatorname{minimize} & f(x) \\ \text { subject to } & h(x)=b, \quad x \in X, \end{aligned}$

where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}, h: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}, X \subseteq \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. State and prove the Lagrangian sufficiency theorem.

In each of the following cases, where $n=2, m=1$ and $X=\{(x, y): x, y \geqslant 0\}$, determine whether the Lagrangian sufficiency theorem can be applied to solve the problem:

(ii) Consider the problem in $\mathbb{R}^{n}$

$\begin{aligned} \operatorname{minimize} & \frac{1}{2} x^{T} Q x+c^{T} x \\ \text { subject to } & A x=b \end{aligned}$

where $Q$ is a positive-definite symmetric $n \times n$ matrix, $A$ is an $m \times n$ matrix, $c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. Explain how to reduce this problem to the solution of simultaneous linear equations.

Consider now the problem

$\begin{aligned} \operatorname{minimize} & \frac{1}{2} x^{T} Q x+c^{T} x \\ \text { subject to } & A x \geqslant b \end{aligned}$

Describe the active set method for its solution.

Consider the problem

$\begin{aligned} \operatorname{minimize} &(x-a)^{2}+(y-b)^{2}+x y \\ \text { subject to } & 0 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant 1 \end{aligned}$

where $a, b \in \mathbb{R}$. Draw a diagram partitioning the $(a, b)$-plane into regions according to which constraints are active at the minimum.

$\begin{aligned} & \text { (a) } \quad f(x, y)=-x, \quad h(x, y)=x^{2}+y^{2}, \quad b=1 \text {; } \\ & \text { (b) } \quad f(x, y)=e^{-x y}, \quad h(x)=x, \quad b=0 \text {. } \end{aligned}$

B3.23

commentConsider the two Hamiltonians

$\begin{aligned} &H_{1}=\frac{\mathbf{p}^{2}}{2 m}+V(|\mathbf{r}|), \\ &H_{2}=\frac{\mathbf{p}^{2}}{2 m}+\sum_{n_{i} \in \mathbb{Z}} V\left(\left|\mathbf{r}-n_{1} \mathbf{a}_{1}-n_{2} \mathbf{a}_{2}-n_{3} \mathbf{a}_{3}\right|\right), \end{aligned}$

where $\mathbf{a}_{i}$ are three linearly independent vectors. For each of the Hamiltonians $H=H_{1}$ and $H=H_{2}$, what are the symmetries of $H$ and what unitary operators $U$ are there such that $U H U^{-1}=H$ ?

For $\mathrm{H}_{2}$ derive Bloch's theorem. Suppose that $H_{1}$ has energy eigenfunction $\psi_{0}(\mathbf{r})$ with energy $E_{0}$ where $\psi_{0}(\mathbf{r}) \sim N e^{-K r}$ for large $r=|\mathbf{r}|$. Assume that $K\left|\mathbf{a}_{i}\right| \gg 1$ for each $i$. In a suitable approximation derive the energy eigenvalues for $H_{2}$ when $E \approx E_{0}$. Verify that the energy eigenfunctions and energy eigenvalues satisfy Bloch's theorem. What happens if $K\left|\mathbf{a}_{i}\right| \rightarrow \infty$ ?

B3.13

commentState the product theorem for Poisson random measures.

Consider a system of $n$ queues, each with infinitely many servers, in which, for $i=1, \ldots, n-1$, customers leaving the $i$ th queue immediately arrive at the $(i+1)$ th queue. Arrivals to the first queue form a Poisson process of rate $\lambda$. Service times at the $i$ th queue are all independent with distribution $F$, and independent of service times at other queues, for all $i$. Assume that initially the system is empty and write $V_{i}(t)$ for the number of customers at queue $i$ at time $t \geqslant 0$. Show that $V_{1}(t), \ldots, V_{n}(t)$ are independent Poisson random variables.

In the case $F(t)=1-e^{-\mu t}$ show that

$\mathbb{E}\left(V_{i}(t)\right)=\frac{\lambda}{\mu} \mathbb{P}\left(N_{t} \geqslant i\right), \quad t \geqslant 0, \quad i=1, \ldots, n,$

where $\left(N_{t}\right)_{t \geqslant 0}$ is a Poisson process of rate $\mu$.

Suppose now that arrivals to the first queue stop at time $T$. Determine the mean number of customers at the $i$ th queue at each time $t \geqslant T$.

B3.17

commentLet $f: I \rightarrow I$ be a continuous one-dimensional map of the interval $I \subset \mathbb{R}$. Explain what is meant by saying (a) that the map $f$ is topologically transitive, and (b) that the map $f$ has a horseshoe.

Consider the tent map defined on the interval $[0,1]$ by

$f(x)= \begin{cases}\mu x & 0 \leqslant x<\frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}$

for $1<\mu \leqslant 2$. Show that if $\mu>\sqrt{2}$ then this map is topologically transitive, and also that $f^{2}$ has a horseshoe.

A3.6 B3.4

comment(i) Define the Poincaré index of a curve $\mathcal{C}$ for a vector field $\mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathbb{R}^{2}$. Explain why the index is uniquely given by the sum of the indices for small curves around each fixed point within $\mathcal{C}$. Write down the indices for a saddle point and for a focus (spiral) or node, and show that the index of a periodic solution of $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has index unity.

A particular system has a periodic orbit containing five fixed points, and two further periodic orbits. Sketch the possible arrangements of these orbits, assuming there are no degeneracies.

(ii) A dynamical system in $\mathbb{R}^{2}$ depending on a parameter $\mu$ has a homoclinic orbit when $\mu=\mu_{0}$. Explain how to determine the stability of this orbit, and sketch the different behaviours for $\mu<\mu_{0}$ and $\mu>\mu_{0}$ in the case that the orbit is stable.

Now consider the system

$\dot{x}=y, \quad \dot{y}=x-x^{2}+y(\alpha+\beta x)$

where $\alpha, \beta$ are constants. Show that the origin is a saddle point, and that if there is an orbit homoclinic to the origin then $\alpha, \beta$ are related by

$\oint y^{2}(\alpha+\beta x) d t=0$

where the integral is taken round the orbit. Evaluate this integral for small $\alpha, \beta$ by approximating $y$ by its form when $\alpha=\beta=0$. Hence give conditions on (small) $\alpha, \beta$ that lead to a stable homoclinic orbit at the origin. [Note that $y d t=d x$.]

A3.5 B3.3

comment(i) Given the electric field (in cartesian components)

$\mathbf{E}(\mathbf{r}, t)=\left(0, x / t^{2}, 0\right)$

use the Maxwell equation

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

to find $\mathbf{B}$ subject to the boundary condition that $|\mathbf{B}| \rightarrow 0$ as $t \rightarrow \infty$.

Let $S$ be the planar rectangular surface in the $x y$-plane with corners at

$(0,0,0), \quad(L, 0,0), \quad(L, a, 0), \quad(0, a, 0)$

where $a$ is a constant and $L=L(t)$ is some function of time. The magnetic flux through $S$ is given by the surface integral

$\Phi=\int_{S} \mathbf{B} \cdot d \mathbf{S}$

Compute $\Phi$ as a function of $t$.

Let $\mathcal{C}$ be the closed rectangular curve that bounds the surface $S$, taken anticlockwise in the $x y$-plane, and let $\mathbf{v}$ be its velocity (which depends, in this case, on the segment of $\mathcal{C}$ being considered). Compute the line integral

$\oint_{\mathcal{C}}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}$

Hence verify that

$\oint_{\mathcal{C}}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}=-\frac{d \Phi}{d t}$

(ii) A surface $S$ is bounded by a time-dependent closed curve $\mathcal{C}(t)$ such that in time $\delta t$ it sweeps out a volume $\delta V$. By considering the volume integral

$\int_{\delta V} \nabla \cdot \mathbf{B} d \tau$

and using the divergence theorem, show that the Maxwell equation $\nabla \cdot \mathbf{B}=0$ implies that

$\frac{d \Phi}{d t}=\int_{S} \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S}-\oint_{\mathcal{C}}(\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}$

where $\Phi$ is the magnetic flux through $S$ as given in Part (i). Hence show, using (1) and Stokes' theorem, that (2) is a consequence of Maxwell's equations.

B3.24

commentA steady two-dimensional jet is generated in an infinite, incompressible fluid of density $\rho$ and kinematic viscosity $\nu$ by a point source of momentum with momentum flux in the $x$ direction $F$ per unit length located at the origin.

Using boundary layer theory, analyse the motion in the jet and show that the $x$-component of the velocity is given by

$u=U(x) f^{\prime}(\eta)$

where

$\eta=y / \delta(x), \quad \delta(x)=\left(\rho \nu^{2} x^{2} / F\right)^{1 / 3} \text { and } U(x)=\left(F^{2} / \rho^{2} \nu x\right)^{1 / 3}$

Show that $f$ satisfies the differential equation

$f^{\prime \prime \prime}+\frac{1}{3}\left(f f^{\prime \prime}+f^{\prime^{2}}\right)=0 .$

Write down the appropriate boundary conditions for this equation. [You need not solve the equation.]

A3.13 B3.21

comment(i) What are the commutation relations satisfied by the components of an angular momentum vector $\mathbf{J}$ ? State the possible eigenvalues of the component $J_{3}$ when $\mathbf{J}^{2}$ has eigenvalue $j(j+1) \hbar^{2}$.

Describe how the Pauli matrices

$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$

are used to construct the components of the angular momentum vector $\mathbf{S}$ for a spin $\frac{1}{2}$ system. Show that they obey the required commutation relations.

Show that $S_{1}, S_{2}$ and $S_{3}$ each have eigenvalues $\pm \frac{1}{2} \hbar$. Verify that $\mathbf{S}^{2}$ has eigenvalue $\frac{3}{4} \hbar^{2} .$

(ii) Let $\mathbf{J}$ and $|j m\rangle$ denote the standard operators and state vectors of angular momentum theory. Assume units where $\hbar=1$. Consider the operator

$U(\theta)=e^{-i \theta J_{2}}$

Show that

$\begin{aligned} &U(\theta) J_{1} U(\theta)^{-1}=\cos \theta J_{1}-\sin \theta J_{3} \\ &U(\theta) J_{3} U(\theta)^{-1}=\sin \theta J_{1}+\cos \theta J_{3} \end{aligned}$

Show that the state vectors $U\left(\frac{\pi}{2}\right)|j m\rangle$ are eigenvectors of $J_{1}$. Suppose that $J_{1}$ is measured for a system in the state $|j m\rangle$; show that the probability that the result is $m^{\prime}$ equals

$\left|\left\langle j m^{\prime}\left|e^{i \frac{\pi}{2} J_{2}}\right| j m\right\rangle\right|^{2}$

Consider the case $j=m=\frac{1}{2}$. Evaluate the probability that the measurement of $J_{1}$ will result in $m^{\prime}=-\frac{1}{2}$.

A3.3 B3.2

comment(i) Let $p$ be a point of the compact interval $I=[a, b] \subset \mathbb{R}$ and let $\delta_{p}: C(I) \rightarrow \mathbb{R}$ be defined by $\delta_{p}(f)=f(p)$. Show that

$\delta_{p}:\left(C(I),\|\cdot\|_{\infty}\right) \rightarrow \mathbb{R}$

is a continuous, linear map but that

$\delta_{p}:\left(C(I),\|\cdot\|_{1}\right) \rightarrow \mathbb{R}$

is not continuous.

(ii) Consider the space $C^{(n)}(I)$ of $n$-times continuously differentiable functions on the interval $I$. Write

$\|f\|_{\infty}^{(n)}=\sum_{k=0}^{n}\left\|f^{(k)}\right\|_{\infty} \quad \text { and } \quad\|f\|_{1}^{(n)}=\sum_{r=0}^{n}\left\|f^{(k)}\right\|_{1}$

for $f \in C^{(n)}(I)$. Show that $\left(C^{(n)}(I),\left\|^{\prime} \cdot\right\|_{\infty}^{(n)}\right)$ is a complete normed space. Is the space $\left(C^{(n)}(I),\|\cdot\|_{1}^{(n)}\right)$ also complete?

Let $f: I \rightarrow I$ be an $n$-times continuously differentiable map and define

$\mu_{f}: C^{(n)}(I) \rightarrow C^{(n)}(I) \quad \text { by } \quad g \mapsto g \circ f .$

Show that $\mu_{f}$ is a continuous linear map when $C^{(n)}(I)$ is equipped with the norm $\|\cdot\|_{\infty}^{(n)}$.

B3.6

commentLet $f$ be a separable polynomial of degree $n \geqslant 1$ over a field $K$. Explain what is meant by the Galois group $\operatorname{Gal}(f / K)$ of $f$ over $K$. Explain how $\operatorname{Gal}(f / K)$ can be identified with a subgroup of the symmetric group $S_{n}$. Show that as a permutation group, $\operatorname{Gal}(f / K)$ is transitive if and only if $f$ is irreducible over $K$.

Show that the Galois group of $f(X)=X^{5}+20 X^{2}-2$ over $\mathbb{Q}$ is $S_{5}$, stating clearly any general results you use.

Now let $K / \mathbb{Q}$ be a finite extension of prime degree $p>5$. By considering the degrees of the splitting fields of $f$ over $K$ and $\mathbb{Q}$, show that $\operatorname{Gal}(f / K)=S_{5}$ also.

A3.7

comment(i) Suppose that $C$ is a curve in the Euclidean $(\xi, \eta)$-plane and that $C$ is parameterized by its arc length $\sigma$. Suppose that $S$ in Euclidean $\mathbb{R}^{3}$ is the surface of revolution obtained by rotating $C$ about the $\xi$-axis. Take $\sigma, \theta$ as coordinates on $S$, where $\theta$ is the angle of rotation.

Show that the Riemannian metric on $S$ induced from the Euclidean metric on $\mathbb{R}^{3}$ is

$d s^{2}=d \sigma^{2}+\eta(\sigma)^{2} d \theta^{2}$

(ii) For the surface $S$ described in Part (i), let $e_{\sigma}=\partial / \partial \sigma$ and $e_{\theta}=\partial / \partial \theta$. Show that, along any geodesic $\gamma$ on $S$, the quantity $g\left(\dot{\gamma}, e_{\theta}\right)$ is constant. Here $g$ is the metric tensor on $S$.

[You may wish to compute $\left[X, e_{\theta}\right]=X e_{\theta}-e_{\theta} X$ for any vector field $X=A e_{\sigma}+B e_{\theta}$, where $A, B$ are functions of $\sigma, \theta$. Then use symmetry to compute $D_{\dot{\gamma}}\left(g\left(\dot{\gamma}, e_{\theta}\right)\right)$, which is the rate of change of $g\left(\dot{\gamma}, e_{\theta}\right)$ along $\gamma$.]

A3.4

comment(i) Let $K$ be the splitting field of the polynomial $f=X^{3}-2$ over the rationals. Find the Galois group $G$ of $K / \mathbb{Q}$ and describe its action on the roots of $f$.

(ii) Let $K$ be the splitting field of the polynomial $X^{4}+a X^{2}+b$ (where $a, b \in \mathbb{Q}$ ) over the rationals. Assuming that the polynomial is irreducible, prove that the Galois group $G$ of the extension $K / \mathbb{Q}$ is either $C_{4}$, or $C_{2} \times C_{2}$, or the dihedral group $D_{8}$.

B3.8

commentLet $\mathcal{H}$ be the space of all functions on the real line $\mathbb{R}$ of the form $p(x) e^{-x^{2} / 2}$, where $p$ is a polynomial with complex coefficients. Make $\mathcal{H}$ into an inner-product space, in the usual way, by defining the inner product to be

$\langle f, g\rangle=\int_{-\infty}^{\infty} f(t) \overline{g(t)} d t, \quad f, g \in \mathcal{H}$

You should assume, without proof, that this equation does define an inner product on $\mathcal{H}$. Define the norm by $\|f\|_{2}=\langle f, f\rangle^{1 / 2}$ for $f \in \mathcal{H}$. Now define a sequence of functions $\left(F_{n}\right)_{n \geqslant 0}$ on $\mathbb{R}$ by

$F_{n}(x)=(-1)^{n} e^{x^{2} / 2} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$

Prove that $\left(F_{n}\right)$ is an orthogonal sequence in $\mathcal{H}$ and that it spans $\mathcal{H}$.

For every $f \in \mathcal{H}$ define the Fourier transform $\widehat{f}$ of $f$ by

$\widehat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) e^{-i t x} d x, \quad t \in \mathbb{R}$

Show that

(a) $\widehat{F}_{n}=(-i)^{n} F_{n}$ for $n=0,1,2, \ldots$;

(b) for all $f \in \mathcal{H}$ and $x \in \mathbb{R}$,

$\widehat{\widehat{f}}(x)=f(-x)$

(c) $\|\widehat{f}\|_{2}=\|f\|_{2}$ for all $f \in \mathcal{H}$.

A3.8 B3.11

comment(i) What does it mean for a function from $\mathbb{N}^{k}$ to $\mathbb{N}$ to be recursive? Write down a function that is not recursive. You should include a proof that your example is not recursive.

(ii) What does it mean for a subset of $\mathbb{N}^{k}$ to be recursive, and what does it mean for it to be recursively enumerable? Give, with proof, an example of a set that is recursively enumerable but not recursive. Prove that a set is recursive if and only if both it and its complement are recursively enumerable. If a set is recursively enumerable, must its complement be recursively enumerable?

[You may assume the existence of any universal recursive functions or universal register machine programs that you wish.]

A3.1 B3.1

comment(i) Consider the continuous-time Markov chain $\left(X_{t}\right)_{t \geqslant 0}$ with state-space $\{1,2,3,4\}$ and $Q$-matrix

$Q=\left(\begin{array}{rrrr} -2 & 0 & 0 & 2 \\ 1 & -3 & 2 & 0 \\ 0 & 2 & -2 & 0 \\ 1 & 5 & 2 & -8 \end{array}\right)$

Set

$Y_{t}= \begin{cases}X_{t} & \text { if } X_{t} \in\{1,2,3\} \\ 2 & \text { if } X_{t}=4\end{cases}$

and

$Z_{t}= \begin{cases}X_{t} & \text { if } X_{t} \in\{1,2,3\} \\ 1 & \text { if } X_{t}=4\end{cases}$

Determine which, if any, of the processes $\left(Y_{t}\right)_{t \geqslant 0}$ and $\left(Z_{t}\right)_{t \geqslant 0}$ are Markov chains.

(ii) Find an invariant distribution for the chain $\left(X_{t}\right)_{t \geqslant 0}$ given in Part (i). Suppose $X_{0}=1$. Find, for all $t \geqslant 0$, the probability that $X_{t}=1$.

A3.17

comment(i) Explain what is meant by the assertion: "the series $\sum_{0}^{\infty} b_{n} x^{n}$ is asymptotic to $f(x)$ as $x \rightarrow 0 "$.

Consider the integral

$I(\lambda)=\int_{0}^{A} e^{-\lambda x} g(x) d x$

where $A>0, \lambda$ is real and $g$ has the asymptotic expansion

$g(x) \sim a_{0} x^{\alpha}+a_{1} x^{\alpha+1}+a_{2} x^{\alpha+2}+\ldots$

as $x \rightarrow+0$, with $\alpha>-1$. State Watson's lemma describing the asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$, and determine an expression for the general term in the asymptotic series.

(ii) Let

$h(t)=\pi^{-1 / 2} \int_{0}^{\infty} \frac{e^{-x}}{x^{1 / 2}(1+2 x t)} d x$

for $t \geqslant 0$. Show that

$h(t) \sim \sum_{k=0}^{\infty}(-1)^{k} 1.3 . \cdots \cdot(2 k-1) t^{k}$

as $t \rightarrow+0$.

Suggest, for the case that $t$ is smaller than unity, the point at which this asymptotic series should be truncated so as to produce optimal numerical accuracy.

B3.19

commentLet

$f(\lambda)=\int_{\gamma} e^{\lambda\left(t-t^{3} / 3\right)} d t, \quad \lambda \text { real and positive }$

where $\gamma$ is a path beginning at $\infty e^{-2 i \pi / 3}$ and ending at $+\infty$ (on the real axis). Identify the saddle points and sketch the paths of constant phase through these points.

Hence show that $f(\lambda) \sim e^{2 \lambda / 3} \sqrt{\pi / \lambda}$ as $\lambda \rightarrow \infty$.

A3.18

comment(i) Write down a Lax pair for the equation

$i q_{t}+q_{x x}=0 .$

Discuss briefly, without giving mathematical details, how this pair can be used to solve the Cauchy problem on the infinite line for this equation. Discuss how this approach can be used to solve the analogous problem for the nonlinear Schrödinger equation.

(ii) Let $q(\zeta, \eta), \tilde{q}(\zeta, \eta)$ satisfy the equations

$\begin{aligned} &\tilde{q}_{\zeta}=q_{\zeta}+2 \lambda \sin \frac{\tilde{q}+q}{2} \\ &\tilde{q}_{\eta}=-q_{\eta}+\frac{2}{\lambda} \sin \frac{\tilde{q}-q}{2} \end{aligned}$

where $\lambda$ is a constant.

(a) Show that the above equations are compatible provided that $q, \tilde{q}$ both satisfy the Sine-Gordon equation

$q_{\zeta \eta}=\sin q$

(b) Use the above result together with the fact that

$\int \frac{d x}{\sin x}=\ln \left(\tan \frac{x}{2}\right)+\text { constant }$

to show that the one-soliton solution of the Sine-Gordon equation is given by

$\tan \frac{q}{4}=c \exp \left(\lambda \zeta+\frac{\eta}{\lambda}\right)$

where $c$ is a constant.

A3.9

comment(i) Let $x \geqslant 2$ be a real number and let $P(x)=\prod_{p \leqslant x}\left(1-\frac{1}{p}\right)^{-1}$, where the product is taken over all primes $p \leqslant x$. Prove that $P(x)>\log x$.

(ii) Define the continued fraction of any positive irrational real number $x$. Illustrate your definition by computing the continued fraction of $1+\sqrt{3}$.

Suppose that $a, b, c$ are positive integers with $b=a c$ and that $x$ has the periodic continued fraction $[b, a, b, a, \ldots]$. Prove that $x=\frac{1}{2}\left(b+\sqrt{b^{2}+4 c}\right)$.

A3.19 B3.20

comment(i) The diffusion equation

$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(a(x) \frac{\partial u}{\partial x}\right), \quad 0 \leqslant x \leqslant 1, \quad t \geqslant 0$

with the initial condition $u(x, 0)=\phi(x), 0 \leqslant x \leqslant 1$ and zero boundary conditions at $x=0$ and $x=1$, is solved by the finite-difference method

$\begin{gathered} u_{m}^{n+1}=u_{m}^{n}+\mu\left[a_{m-\frac{1}{2}} u_{m-1}^{n}-\left(a_{m-\frac{1}{2}}+a_{m+\frac{1}{2}}\right) u_{m}^{n}+a_{m+\frac{1}{2}} u_{m+1}^{n}\right] \\ m=1,2, \ldots, N \end{gathered}$

where $\mu=\Delta t /(\Delta x)^{2}, \quad \Delta x=\frac{1}{N+1}$ and $u_{m}^{n} \approx u(m \Delta x, n \Delta t), a_{\alpha}=a(\alpha \Delta x)$.

Assuming sufficient smoothness of the function $a$, and that $\mu$ remains constant as $\Delta x>0$ and $\Delta t>0$ become small, prove that the exact solution satisfies the numerical scheme with error $O\left((\Delta x)^{3}\right)$.

(ii) For the problem defined in Part (i), assume that there exist $0<a_{-}<a_{+}<\infty$ such that $a_{-} \leqslant a(x) \leqslant a_{+}, \quad 0 \leqslant x \leqslant 1$. Prove that the method is stable for $0<\mu \leqslant 1 /\left(2 a_{+}\right)$.

[Hint: You may use without proof the Gerschgorin theorem: All the eigenvalues of the matrix $A=\left(a_{k, l}\right)_{k, l=1, \ldots, M}$ are contained in $\bigcup_{k=1}^{m} \mathbb{S}_{k}$, where

$\left.\mathbb{S}_{k}=\left\{z \in \mathbb{C}:\left|z-a_{k, k}\right| \leqslant \sum_{\substack{l=1 \\ l \neq k}}^{m}\left|a_{k, l}\right|\right\}, \quad k=1,2, \ldots, m . \quad\right]$

B3.14

commentState Pontryagin's Maximum Principle (PMP).

In a given lake the tonnage of fish, $x$, obeys

$d x / d t=0.001(50-x) x-u, \quad 0<x \leqslant 50$

where $u$ is the rate at which fish are extracted. It is desired to maximize

$\int_{0}^{\infty} u(t) e^{-0.03 t} d t$

choosing $u(t)$ under the constraints $0 \leqslant u(t) \leqslant 1.4$, and $u(t)=0$ if $x(t)=0$. Assume the PMP with an appropriate Hamiltonian $H(x, u, t, \lambda)$. Now define $G(x, u, t, \eta)=$ $e^{0.03 t} H(x, u, t, \lambda)$ and $\eta(t)=e^{0.03 t} \lambda(t)$. Show that there exists $\eta(t), 0 \leqslant t$ such that on the optimal trajectory $u$ maximizes

$G(x, u, t, \eta)=\eta[0.001(50-x) x-u]+u$

and

$d \eta / d t=0.002(x-10) \eta$

Suppose that $x(0)=20$ and that under an optimal policy it is not optimal to extract all the fish. Argue that $\eta(0) \geqslant 1$ is impossible and describe qualitatively what must happen under the optimal policy.

B3.18

commentConsider the initial value problem

$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0$

to be solved for $u: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$, subject to the initial conditions

$u(0, x)=f(x), \quad \frac{\partial u}{\partial t}(0, x)=0$

for $f$ in the Schwarz space $\mathcal{S}\left(\mathbb{R}^{n}\right)$. Use the Fourier transform in $x$ to obtain a representation for the solution in the form

$u(t, x)=\int e^{i x \cdot \xi} A(t, \xi) \widehat{f}(\xi) d^{n} \xi$

where $A$ should be determined explicitly. Explain carefully why your formula gives a smooth solution to (1) and why it satisfies the initial conditions (2), referring to the required properties of the Fourier transform as necessary.

Next consider the case $n=1$. Find a tempered distribution $T$ (depending on $t, x$ ) such that (3) can be written

$u=<T, \widehat{f}>$

and (using the definition of Fourier transform of tempered distributions) show that the formula reduces to

$u(t, x)=\frac{1}{2}[f(x-t)+f(x+t)]$

State and prove the Duhamel principle relating to the solution of the $n$-dimensional inhomogeneous wave equation

$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=h$

to be solved for $u: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$, subject to the initial conditions

$u(0, x)=0, \quad \frac{\partial u}{\partial t}(0, x)=0$

for $h$ a $C^{\infty}$ function. State clearly assumptions used on the solvability of the homogeneous problem.

[Hint: it may be useful to consider the Fourier transform of the tempered distribution defined by the function $\xi \mapsto e^{i \xi \cdot a}$.]

A3.2

comment(i) An axisymmetric bowling ball of mass $M$ has the shape of a sphere of radius $a$. However, it is biased so that the centre of mass is located a distance $a / 2$ away from the centre, along the symmetry axis.

The three principal moments of inertia about the centre of mass are $(A, A, C)$. The ball starts out in a stable equilibrium at rest on a perfectly frictionless flat surface with the symmetry axis vertical. The symmetry axis is then tilted through $\theta_{0}$, the ball is spun about this axis with an angular velocity $n$, and the ball is released.

Explain why the centre of mass of the ball moves only in the vertical direction during the subsequent motion. Write down the Lagrangian for the ball in terms of the usual Euler angles $\theta, \phi$ and $\psi$.

(ii) Show that there are three independent constants of the motion. Eliminate two of the angles from the Lagrangian and find the effective Lagrangian for the coordinate $\theta$.

Find the maximum and minimum values of $\theta$ in the motion of the ball when the quantity $\frac{C^{2} n^{2}}{A M g a}$ is (a) very small and (b) very large.

A3.12 B3.15

comment(i) Let $X_{1}, \ldots, X_{n}$ be independent, identically distributed random variables, with the exponential density $f(x ; \theta)=\theta e^{-\theta x}, x>0$.

Obtain the maximum likelihood estimator $\hat{\theta}$ of $\theta$. What is the asymptotic distribution of $\sqrt{n}(\hat{\theta}-\theta)$ ?

What is the minimum variance unbiased estimator of $\theta ?$ Justify your answer carefully.

(ii) Explain briefly what is meant by the profile log-likelihood for a scalar parameter of interest $\gamma$, in the presence of a nuisance parameter $\xi$. Describe how you would test a null hypothesis of the form $H_{0}: \gamma=\gamma_{0}$ using the profile log-likelihood ratio statistic.

In a reliability study, lifetimes $T_{1}, \ldots, T_{n}$ are independent and exponentially distributed, with means of the form $E\left(T_{i}\right)=\exp \left(\beta+\xi z_{i}\right)$ where $\beta, \xi$ are unknown and $z_{1}, \ldots, z_{n}$ are known constants. Inference is required for the mean lifetime, $\exp \left(\beta+\xi z_{0}\right)$, for covariate value $z_{0}$.

Find, as explicitly as possible, the profile log-likelihood for $\gamma \equiv \beta+\xi z_{0}$, with nuisance parameter $\xi$.

Show that, under $H_{0}: \gamma=\gamma_{0}$, the profile $\log -$ likelihood ratio statistic has a distribution which does not depend on the value of $\xi$. How might the parametric bootstrap be used to obtain a test of $H_{0}$ of exact size $\alpha$ ?

[Hint: if $Y$ is exponentially distributed with mean 1 , then $\mu Y$ is exponentially distributed with mean $\mu$.]

B3.12

commentExplain what is meant by the characteristic function $\phi$ of a real-valued random variable and prove that $|\phi|^{2}$ is also a characteristic function of some random variable.

Let us say that a characteristic function $\phi$ is infinitely divisible when, for each $n \geqslant 1$, we can write $\phi=\left(\phi_{n}\right)^{n}$ for some characteristic function $\phi_{n}$. Prove that, in this case, the limit

$\psi(t)=\lim _{n \rightarrow \infty}\left|\phi_{2 n}(t)\right|^{2}$

exists for all real $t$ and is continuous at $t=0$.

Using Lévy's continuity theorem for characteristic functions, which you should state carefully, deduce that $\psi$ is a characteristic function. Hence show that, if $\phi$ is infinitely divisible, then $\phi(t)$ cannot vanish for any real $t$.

B3.5

commentIf $\rho_{1}: G_{1} \rightarrow G L\left(V_{1}\right)$ and $\rho_{2}: G_{2} \rightarrow G L\left(V_{2}\right)$ are representations of the finite groups $G_{1}$ and $G_{2}$ respectively, define the tensor product $\rho_{1} \otimes \rho_{2}$ as a representation of the group $G_{1} \times G_{2}$ and show that its character is given by

$\chi_{\rho_{1} \otimes \rho_{2}}\left(g_{1}, g_{2}\right)=\chi_{\rho_{1}}\left(g_{1}\right) \chi_{\rho_{2}}\left(g_{2}\right)$

Prove that

(a) if $\rho_{1}$ and $\rho_{2}$ are irreducible, then $\rho_{1} \otimes \rho_{2}$ is an irreducible representation of $G_{1} \times G_{2}$;

(b) each irreducible representation of $G_{1} \times G_{2}$ is equivalent to a representation $\rho_{1} \otimes \rho_{2}$ where each $\rho_{i}$ is irreducible $(i=1,2) .$

Is every representation of $G_{1} \times G_{2}$ the tensor product of a representation of $G_{1}$ and a representation of $G_{2}$ ?

B3.9

commentLet $L$ be the lattice $\mathbb{Z} \omega_{1}+\mathbb{Z} \omega_{2}$ for two non-zero complex numbers $\omega_{1}, \omega_{2}$ whose ratio is not real. Recall that the Weierstrass function $\wp$ is given by the series

$\wp(u)=\frac{1}{u^{2}}+\sum_{\omega \in L-\{0\}}\left(\frac{1}{(u-\omega)^{2}}-\frac{1}{\omega^{2}}\right)$

the function $\zeta$ is the (unique) odd anti-derivative of $-\wp$; and $\sigma$ is defined by the conditions

$\sigma^{\prime}(u)=\zeta(u) \sigma(u) \quad \text { and } \quad \sigma^{\prime}(0)=1$

(a) By writing a differential equation for $\sigma(-u)$, or otherwise, show that $\sigma$ is an odd function.

(b) Show that $\sigma\left(u+\omega_{i}\right)=-\sigma(u) \exp \left(a_{i}\left(u+b_{i}\right)\right)$ for some constants $a_{i}, b_{i}$. Use (a) to express $b_{i}$ in terms of $\omega_{i}$. [Do not attempt to express $a_{i}$ in terms of $\omega_{i}$.]

(c) Show that the function $f(u)=\sigma(2 u) / \sigma(u)^{4}$ is periodic with respect to the lattice $L$ and deduce that $f(u)=-\wp^{\prime}(u)$.

B3.22

commentA diatomic molecule, free to move in two space dimensions, has classical Hamiltonian

$H=\frac{1}{2 m}|\mathbf{p}|^{2}+\frac{1}{2 I} J^{2}$

where $\mathbf{p}=\left(p_{1}, p_{2}\right)$ is the particle's momentum and $J$ is its angular momentum. Write down the classical partition function $Z$ for an ideal gas of $N$ such molecules in thermal equilibrium at temperature $T$. Show that it can be written in the form

$Z=\left(z_{t} z_{r o t}\right)^{N}$

where $z_{t}$ and $z_{\text {rot }}$ are the one-molecule partition functions associated with the translational and rotational degrees of freedom, respectively. Compute $z_{t}$ and $z_{r o t}$ and hence show that the energy $E$ of the gas is given by

$E=\frac{3}{2} N k T$

where $k$ is Boltzmann's constant. How does this result illustrate the principle of equipartition of energy?

In an improved model of the two-dimensional gas of diatomic molecules, the angular momentum $J$ is quantized in integer multiples of $\hbar$ :

$J=j \hbar, \quad j=0, \pm 1, \pm 2, \ldots$

Write down an expression for $z_{\text {rot }}$ in this case. Given that $k T \ll\left(\hbar^{2} / 2 I\right)$, obtain an expression for the energy $E$ in the form

$E \approx A T+B e^{-\hbar^{2} / 2 I k T}$

where $A$ and $B$ are constants that should be computed. How is this result compatible with the principle of equipartition of energy? Find $C_{v}$, the specific heat at constant volume, for $k T \ll\left(\hbar^{2} / 2 I\right)$.

Why can the sum over $j$ in $z_{\text {rot }}$ be approximated by an integral when $k T \gg\left(\hbar^{2} / 2 I\right)$ ? Deduce that $E \approx \frac{3}{2} N k T$ in this limit.

A3.14

comment(i) The pressure $P(r)$ and mass density $\rho(r)$, at distance $r$ from the centre of a spherically-symmetric star, obey the pressure-support equation

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

where $m^{\prime}=4 \pi r^{2} \rho(r)$, and the prime indicates differentiation with respect to $r$. Let $V$ be the total volume of the star, and $\langle P\rangle$ its average pressure. Use the pressure-support equation to derive the "virial theorem"

$\langle P\rangle V=-\frac{1}{3} E_{g r a v}$

where $E_{g r a v}$ is the total gravitational potential energy [Hint: multiply by $4 \pi r^{3}$ ]. If a star is assumed to be a self-gravitating ball of a non-relativistic ideal gas then it can be shown that

$\langle P\rangle V=\frac{2}{3} E_{k i n}$

where $E_{k i n}$ is the total kinetic energy. Use this result to show that the total energy $U=E_{\text {grav }}+E_{k i n}$ is negative. When nuclear reactions have converted the hydrogen in a star's core to helium the core contracts until the helium is converted to heavier elements, thereby increasing the total energy $U$ of the star. Explain briefly why this converts the star into a "Red Giant". (ii) Write down the first law of thermodynamics for the change in energy $E$ of a system at temperature $T$, pressure $P$ and chemical potential $\mu$ as a result of small changes in the entropy $S$, volume $V$ and particle number $N$. Use this to show that

$P=-\left(\frac{\partial E}{\partial V}\right)_{N, S}$

The microcanonical ensemble is the set of all accessible microstates of a system at fixed $E, V, N$. Define the canonical and grand-canonical ensembles. Why are the properties of a macroscopic system independent of the choice of thermodynamic ensemble?

The Gibbs "grand potential" $\mathcal{G}(T, V, \mu)$ can be defined as

$\mathcal{G}=E-T S-\mu N$

Use the first law to find expressions for $S, P, N$ as partial derivatives of $\mathcal{G}$. A system with variable particle number $n$ has non-degenerate energy eigenstates labeled by $r^{(n)}$, for each $n$, with energy eigenvalues $E_{r}^{(n)}$. If the system is in equilibrium at temperature $T$ and chemical potential $\mu$ then the probability $p\left(r^{(n)}\right)$ that it will be found in a particular $n$-particle state $r^{(n)}$ is given by the Gibbs probability distribution

$p\left(r^{(n)}\right)=\mathcal{Z}^{-1} e^{\left(\mu n-E_{r}^{(n)}\right) / k T}$

where $k$ is Boltzmann's constant. Deduce an expression for the normalization factor $\mathcal{Z}$ as a function of $\mu$ and $\beta=1 / k T$, and hence find expressions for the partial derivatives

$\frac{\partial \log \mathcal{Z}}{\partial \mu}, \quad \frac{\partial \log \mathcal{Z}}{\partial \beta}$

in terms of $N, E, \mu, \beta$.

Why does $\mathcal{Z}$ also depend on the volume $V$ ? Given that a change in $V$ at fixed $N, S$ leaves unchanged the Gibbs probability distribution, deduce that

$\left(\frac{\partial \log \mathcal{Z}}{\partial V}\right)_{\mu, \beta}=\beta P$

Use your results to show that

$\mathcal{G}=-k T \log \left(\mathcal{Z} / \mathcal{Z}_{0}\right)$

for some constant $\mathcal{Z}_{0}$.

A3.11 B3.16

comment(i) What does it mean to say that the process $\left(W_{t}\right)_{t \geqslant 0}$ is a Brownian motion? What does it mean to say that the process $\left(M_{t}\right)_{t \geqslant 0}$ is a martingale?

Suppose that $\left(W_{t}\right)_{t \geqslant 0}$ is a Brownian motion and the process $\left(X_{t}\right)_{t \geqslant 0}$ is given in terms of $W$ as

$X_{t}=x_{0}+\sigma W_{t}+\mu t$

for constants $\sigma, \mu$. For what values of $\theta$ is the process

$M_{t}=\exp \left(\theta X_{t}-\lambda t\right)$

a martingale? (Here, $\lambda$ is a positive constant.)

(ii) In a standard Black-Scholes model, the price at time $t$ of a share is represented as $S_{t}=\exp \left(X_{t}\right)$. You hold a perpetual American put option on this share, with strike $K$; you may exercise at any stopping time $\tau$, and upon exercise you receive $\max \left\{0, K-S_{\tau}\right\}$. Let $0<a<\log K$. Suppose you plan to use the exercise policy: 'Exercise as soon as the price falls to $e^{a}$ or lower.' Calculate what the option would be worth if you were to follow this policy. (Assume that the riskless rate of interest is constant and equal to $r>0$.) For what choice of $a$ is this value maximised?

A3.15

comment(i) Given that the character of an $S U(2)$ transformation in the $(2 l+1)$-dimensional irreducible representation $d_{l}$ is given by

$\chi_{l}(\theta)=\frac{\sin \left(l+\frac{1}{2}\right) \theta}{\sin \frac{\theta}{2}}$

show how the direct product representation $d_{l_{1}} \otimes d_{l_{2}}$ decomposes into irreducible $S U(2)$ representations.

(ii) Find the decomposition of the direct product representation $3 \otimes \overline{3}$ of $S U(3)$ into irreducible $S U(3)$ representations.

Mesons consist of one quark and one antiquark. The scalar Meson Octet consists of the following particles: $K^{\pm}\left(Y=\pm 1, I_{3}=\pm \frac{1}{2}\right), K^{0}\left(Y=1, I_{3}=-\frac{1}{2}\right), \bar{K}^{0}(Y=-1$, $\left.I_{3}=\frac{1}{2}\right), \pi^{\pm}\left(Y=0, I_{3}=\pm 1\right), \pi^{0}\left(Y=0, I_{3}=0\right)$ and $\eta\left(Y=0, I_{3}=0\right)^{2}$.

Use the direct product representation $3 \otimes \overline{3}$ of $S U(3)$ to identify the quark-type of the particles in the scalar Meson Octet. Deduce the quark-type of the $S U(3)$ singlet state $\eta^{\prime}$ contained in $3 \otimes \overline{3}$.

A3.16

comment(i) When a solid crystal grows into a supercooled infinite melt, latent heat must be removed from the interface by diffusion into the melt. Write down the equation and boundary conditions satisfied by the temperature $\theta(\boldsymbol{x}, t)$ in the melt, where $\boldsymbol{x}$ is position and $t$ time, in terms of the following material properties: solid density $\rho_{s}$, specific heat capacity $C_{p}$, coefficient of latent heat per unit mass $L$, thermal conductivity $k$, melting temperature $\theta_{m}$. You may assume that the densities of the melt and the solid are the same and that temperature in the melt far from the interface is $\theta_{m}-\Delta \theta$, where $\Delta \theta$ is a positive constant.

A spherical crystal of radius $a(t)$ grows into such a melt with $a(0)=0$. Use dimensional analysis to show that $a(t)$ is proportional to $t^{1 / 2}$.

(ii) Show that the above problem should have a similarity solution of the form

$\theta=\theta_{m}-\Delta \theta(1-F(\xi))$

where $\xi=r(\kappa t)^{-1 / 2}, r$ is the radial coordinate in spherical polars and $\kappa=k / \rho_{s} C_{p}$ is the thermal diffusivity. Recalling that, for spherically symmetric $\theta, \nabla^{2} \theta=\frac{1}{r^{2}}\left(r^{2} \theta_{r}\right)_{r}$, write down the equation and boundary conditions to be satisfied by $F(\xi)$. Hence show that the radius of the crystal is given by $a(t)=\lambda(\kappa t)^{1 / 2}$, where $\lambda$ satisfies the equation

$\int_{\lambda}^{\infty} \frac{e^{-\frac{1}{4} u^{2}}}{u^{2}} d u=\frac{2}{S \lambda^{3}} e^{-\frac{1}{4} \lambda^{2}}$

and $S=L / C_{p} \Delta \theta$.

Integrate the left hand side of this equation by parts, to give

$\frac{\sqrt{\pi}}{2} \lambda e^{\frac{1}{4} \lambda^{2}} \operatorname{erfc}\left(\frac{1}{2} \lambda\right)=1-\frac{2}{S \lambda^{2}} .$

Hence show that a solution with $\lambda$ small must have $\lambda \approx(2 / S)^{\frac{1}{2}}$, which is self-consistent if $S$ is large.

B3.25

commentDerive the wave equation governing the velocity potential for linearised sound in a perfect gas. How is the pressure disturbance related to the velocity potential? Write down the spherically symmetric solution to the wave equation with time dependence $e^{i \omega t}$, which is regular at the origin.

A high pressure gas is contained, at density $\rho_{0}$, within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Ignore the low pressure gas outside. Let the metal shell have radius $a$, mass $m$ per unit surface area, and elastic stiffness which tries to restore the radius to its equilibrium value $a_{0}$ with a force $-\kappa\left(a-a_{0}\right)$ per unit surface area. Show that the frequency of these vibrations is given by

$\omega^{2}\left(m+\frac{\rho_{0} a_{0}}{\theta \cot \theta-1}\right)=\kappa \quad \text { where } \theta=\omega a_{0} / c_{0}$