Part II, 2001, Paper 1

# Part II, 2001, Paper 1

### Jump to course

B1.23

commentA steady beam of particles, having wavenumber $k$ and moving in the $z$ direction, scatters on a spherically-symmetric potential. Write down the asymptotic form of the wave function at large $r$.

The incoming wave is written as a partial-wave series

$\sum_{\ell=0}^{\infty} \chi_{\ell}(k r) P_{\ell}(\cos \theta)$

Show that for large $r$

$\chi_{\ell}(k r) \sim \frac{\ell+\frac{1}{2}}{i k r}\left(e^{i k r}-(-1)^{\ell} e^{-i k r}\right)$

and calculate $\chi_{0}(k r)$ and $\chi_{1}(k r)$ for all $r$.

Write down the second-order differential equation satisfied by the $\chi_{\ell}(k r)$. Construct a second linearly-independent solution for each $\ell$ that is singular at $r=0$ and, when it is suitably normalised, has large- $r$ behaviour

$\frac{\ell+\frac{1}{2}}{i k r}\left(e^{i k r}+(-1)^{\ell} e^{-i k r}\right)$

A1.10

comment(i) Explain briefly how and why a signature scheme is used. Describe the el Gamal scheme,

(ii) Define a cyclic code. Define the generator of a cyclic code and show that it exists. Prove a necessary and sufficient condition for a polynomial to be the generator of a cyclic code of length $n$.

What is the $\mathrm{BCH}$ code? Show that the $\mathrm{BCH}$ code associated with $\left\{\beta, \beta^{2}\right\}$, where $\beta$ is a root of $X^{3}+X+1$ in an appropriate field, is Hamming's original code.

B1.5

commentLet $\mathcal{A} \subset[n]^{(r)}$ where $r \leqslant n / 2$. Prove that, if $\mathcal{A}$ is 1-intersecting, then $|\mathcal{A}| \leqslant\left(\begin{array}{l}n-1 \\ r-1\end{array}\right)$. State an upper bound on $|\mathcal{A}|$ that is valid if $\mathcal{A}$ is $t$-intersecting and $n$ is large compared to $r$ and $t$.

Let $\mathcal{B} \subset \mathcal{P}([n])$ be maximal 1-intersecting; that is, $\mathcal{B}$ is 1-intersecting but if $\mathcal{B} \subset \mathcal{C} \subset \mathcal{P}([n])$ and $\mathcal{B} \neq \mathcal{C}$ then $\mathcal{C}$ is not 1-intersecting. Show that $|\mathcal{B}|=2^{n-1}$.

Let $\mathcal{B} \subset \mathcal{P}([n])$ be 2 -intersecting. Show that $|\mathcal{B}| \geqslant 2^{n-2}$ is possible. Can the inequality be strict?

A1.13

comment(i) Assume that the $n$-dimensional observation 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 $\widehat{\beta}$, the least-squares estimator of $\beta$, and show that

$Q(\widehat{\beta})=Y^{T}(I-H) Y$

where $H$ is a matrix that you should define.

(ii) Show that $\sum_{i} H_{i i}=p$. Show further for the special case of

$Y_{i}=\beta_{1}+\beta_{2} x_{i}+\beta_{3} z_{i}+\epsilon_{i}, \quad 1 \leqslant i \leqslant n$

where $\Sigma x_{i}=0, \Sigma z_{i}=0$, that

$H=\frac{1}{n} \mathbf{1 1}{ }^{T}+a x x^{T}+b\left(x z^{T}+z x^{T}\right)+c z z^{T} ;$

here, $\mathbf{1}$ is a vector of which every element is one, and $a, b, c$, are constants that you should derive.

Hence show that, if $\widehat{Y}=X \widehat{\beta}$ is the vector of fitted values, then

$\frac{1}{\sigma^{2}} \operatorname{var}\left(\widehat{Y}_{i}\right)=\frac{1}{n}+a x_{i}^{2}+2 b x_{i} z_{i}+c z_{i}^{2}, \quad 1 \leqslant i \leqslant n .$

B1.8

commentDefine an immersion and an embedding of one manifold in another. State a necessary and sufficient condition for an immersion to be an embedding and prove its necessity.

Assuming the existence of "bump functions" on Euclidean spaces, state and prove a version of Whitney's embedding theorem.

Deduce that $\mathbb{R}^{n}$ embeds in $\mathbb{R}^{(n+1)^{2}}$.

B1.17

commentDefine topological conjugacy and $C^{1}$-conjugacy.

Let $a, b$ be real numbers with $a>b>0$ and let $F_{a}, F_{b}$ be the maps of $(0, \infty)$ to itself given by $F_{a}(x)=a x, F_{b}(x)=b x$. For which pairs $a, b$ are $F_{a}$ and $F_{b}$ topologically conjugate? Would the answer be the same for $C^{1}$-conjugacy? Justify your statements.

A1.6

comment(i) Given a differential equation $\dot{x}=f(x)$ for $x \in \mathbb{R}^{n}$, explain what it means to say that the solution is given by a flow $\phi: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$. Define the orbit, $o(x)$, through a point $x$ and the $\omega$-limit set, $\omega(x)$, of $x$. Define also a homoclinic orbit to a fixed point $x_{0}$. Sketch a flow in $\mathbb{R}^{2}$ with a homoclinic orbit, and identify (without detailed justification) the $\omega$-limit sets $\omega(x)$ for each point $x$ in your diagram.

(ii) Consider the differential equations

$\dot{x}=z y, \quad \dot{y}=-z x, \quad \dot{z}=-z^{2} .$

Transform the equations to polar coordinates $(r, \theta)$ in the $(x, y)$ plane. Solve the equation for $z$ to find $z(t)$, and hence find $\theta(t)$. Hence, or otherwise, determine (with justification) the $\omega$-limit set for all points $\left(x_{0}, y_{0}, z_{0}\right) \in \mathbb{R}^{3}$.

B1.21

commentExplain the multipole expansion in electrostatics, and devise formulae for the total charge, dipole moments and quadrupole moments given by a static charge distribution $\rho(\mathbf{r})$.

A nucleus is modelled as a uniform distribution of charge inside the ellipsoid

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1$

The total charge of the nucleus is $Q$. What are the dipole moments and quadrupole moments of this distribution?

Describe qualitatively what happens if the nucleus starts to oscillate.

B1.25

commentThe energy equation for the motion of a viscous, incompressible fluid states that

$\frac{d}{d t} \int_{V(t)} \frac{1}{2} \rho u^{2} d V=\int_{S(t)} u_{i} \sigma_{i j} n_{j} d S-2 \mu \int_{V(t)} e_{i j} e_{i j} d V$

Interpret each term in this equation and explain the meaning of the symbols used.

For steady rectilinear flow in a (not necessarily circular) pipe having rigid stationary walls, deduce a relation between the viscous dissipation per unit length of the pipe, the pressure gradient $G$, and the volume flux $Q$.

Starting from the Navier-Stokes equations, calculate the velocity field for steady rectilinear flow in a circular pipe of radius $a$. Using the relationship derived above, or otherwise, find in terms of $G$ the viscous dissipation per unit length for this flow.

[In cylindrical polar coordinates,

$\left.\nabla^{2} w(r)=\frac{1}{r} \frac{d}{d r}\left(r \frac{d w}{d r}\right) .\right]$

B1.7

commentProve that the Galois group $G$ of the polynomial $X^{6}+3$ over $\mathbf{Q}$ is of order 6 . By explicitly describing the elements of $G$, show that they have orders 1,2 or 3 . Hence deduce that $G$ is isomorphic to $S_{3}$.

Why does it follow that $X^{6}+3$ is reducible over the finite field $\mathbf{F}_{p}$, for all primes $p ?$

A1.15 B1.24

comment(i) The metric of any two-dimensional curved space, rotationally symmetric about a point $P$, can by suitable choice of coordinates be written locally in the form

$d s^{2}=e^{2 \phi(r)}\left(d r^{2}+r^{2} d \theta^{2}\right),$

where $r=0$ at $P, r>0$ away from $P$, and $0 \leqslant \theta<2 \pi$. Labelling the coordinates as $\left(x^{1}, x^{2}\right)=(r, \theta)$, show that the Christoffel symbols $\Gamma_{12}^{1}, \Gamma_{11}^{2}$ and $\Gamma_{22}^{2}$ are each zero, and compute the non-zero Christoffel symbols $\Gamma_{11}^{1}, \Gamma_{22}^{1}$ and $\Gamma_{12}^{2}=\Gamma_{21}^{2}$.

The Ricci tensor $R_{a b}(a, b=1,2)$ is defined by

$R_{a b}=\Gamma_{a b, c}^{c}-\Gamma_{a c, b}^{c}+\Gamma_{c d}^{c} \Gamma_{a b}^{d}-\Gamma_{a c}^{d} \Gamma_{b d}^{c},$

where a comma denotes a partial derivative. Show that $R_{12}=0$ and that

$R_{11}=-\phi^{\prime \prime}-r^{-1} \phi^{\prime}, \quad R_{22}=r^{2} R_{11}$

(ii) Suppose further that, in a neighbourhood of $P$, the Ricci scalar $R$ takes the constant value $-2$. Find a second order differential equation, which you should denote by $(*)$, for $\phi(r)$.

This space of constant Ricci scalar can, by a suitable coordinate transformation $r \rightarrow \chi(r)$, leaving $\theta$ invariant, be written locally as

$d s^{2}=d \chi^{2}+\sinh ^{2} \chi d \theta^{2}$

By studying this coordinate transformation, or otherwise, find $\cosh \chi$ and $\sinh \chi$ in terms of $r$ (up to a constant of integration). Deduce that

$e^{\phi(r)}=\frac{2 A}{\left(1-A^{2} r^{2}\right)} \quad, \quad(0 \leqslant A r<1)$

where $\mathrm{A}$ is a positive constant and verify that your equation $(*)$ for $\phi$ holds.

[Note that

$\left.\int \frac{d \chi}{\sinh \chi}=\text { const. }+\frac{1}{2} \log (\cosh \chi-1)-\frac{1}{2} \log (\cosh \chi+1) .\right]$

Part II

A1.8

comment(i) Show that any graph $G$ with minimal degree $\delta$ contains a cycle of length at least $\delta+1$. Give examples to show that, for each possible value of $\delta$, there is a graph with minimal degree $\delta$ but no cycle of length greater than $\delta+1$.

(ii) Let $K_{N}$ be the complete graph with $N$ vertices labelled $v_{1}, v_{2}, \ldots, v_{N}$. Prove, from first principles, that there are $N^{N-2}$ different spanning trees in $K_{N}$. In how many of these spanning trees does the vertex $v_{1}$ have degree 1 ?

A spanning tree in $K_{N}$ is chosen at random, with each of the $N^{N-2}$ trees being equally likely. Show that the average number of vertices of degree 1 in the random tree is approximately $N / e$ when $N$ is large.

Find the average degree of vertices in the random tree.

A1.4 B1.3

comment(i) Define the notion of a Sylow $p$-subgroup of a finite group $G$, and state a theorem concerning the number of them and the relation between them.

(ii) Show that any group of order 30 has a non-trivial normal subgroup. Is it true that every group of order 30 is commutative?

B1.10

commentState and prove the Riesz representation theorem for bounded linear functionals on a Hilbert space $H$.

[You may assume, without proof, that $H=E \oplus E^{\perp}$, for every closed subspace $E$ of $H$.]

Prove that, for every $T \in \mathcal{B}(H)$, there is a unique $T^{*} \in \mathcal{B}(H)$ such that $\langle T x, y\rangle=\left\langle x, T^{*} y\right\rangle$ for every $x, y \in H$. Prove that $\left\|T^{*} T\right\|=\|T\|^{2}$ for every $T \in \mathcal{B}(H)$.

Define a normal operator $T \in \mathcal{B}(H)$. Prove that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for every $x \in H$. Deduce that every point in the spectrum of a normal operator $T$ is an approximate eigenvalue of $T$.

[You may assume, without proof, any general criterion for the invertibility of a bounded linear operator on $H$.]

B1.14

commentLet $p_{1}, \ldots, p_{n}$ be a probability distribution, with $p^{*}=\max _{i}\left[p_{i}\right]$. Prove that

$\begin{aligned} &(i)-\sum_{i} p_{i} \log p_{i} \geqslant-p^{*} \log p^{*}-\left(1-p^{*}\right) \log \left(1-p^{*}\right) \\ &(\text { ii })-\sum_{i} p_{i} \log p_{i} \geqslant \log \left(1 / p^{*}\right) ; \text { and } \\ &(\text { iii })-\sum_{i} p_{i} \log p_{i} \geqslant 2\left(1-p^{*}\right) \end{aligned}$

All logarithms are to base 2 .

[Hint: To prove (iii), it is convenient to use (i) for $p^{*} \geqslant \frac{1}{2}$ and (ii) for $p^{*} \leqslant \frac{1}{2}$.]

Random variables $X$ and $Y$ with values $x$ and $y$ from finite 'alphabets' $I$ and $J$ represent the input and output of a transmission channel, with the conditional probability $p(x \mid y)=\mathbb{P}(X=x \mid Y=y)$. Let $h(p(\cdot \mid y))$ denote the entropy of the conditional distribution $p(\cdot \mid y), y \in J$, and $h(X \mid Y)$ denote the conditional entropy of $X$ given $Y$. Define the ideal observer decoding rule as a map $f: J \rightarrow I$ such that $p(f(y) \mid y)=\max _{x \in I} p(x \mid y)$ for all $y \in J$. Show that under this rule the error probability

$\pi_{\mathrm{er}}(y)=\sum_{\substack{x \in I \\ x \neq f(y)}} p(x \mid y)$

satisfies $\pi_{\mathrm{er}}(y) \leqslant \frac{1}{2} h(p(\cdot \mid y))$, and the expected value satisfies

$\mathbb{E} \pi_{\mathrm{er}}(Y) \leqslant \frac{1}{2} h(X \mid Y)$

A $1 . 7 \quad$ B1.12

comment(i) What is the Halting Problem? What is an unsolvable problem?

(ii) Prove that the Halting Problem is unsolvable. Is it decidable whether or not a machine halts with input zero?

A1.1 B1.1

comment(i) Let $X=\left(X_{n}: 0 \leqslant n \leqslant N\right)$ be an irreducible Markov chain on the finite state space $S$ with transition matrix $P=\left(p_{i j}\right)$ and invariant distribution $\pi$. What does it mean to say that $X$ is reversible in equilibrium?

Show that $X$ is reversible in equilibrium if and only if $\pi_{i} p_{i j}=\pi_{j} p_{j i}$ for all $i, j \in S$.

(ii) A finite connected graph $G$ has vertex set $V$ and edge set $E$, and has neither loops nor multiple edges. A particle performs a random walk on $V$, moving at each step to a randomly chosen neighbour of the current position, each such neighbour being picked with equal probability, independently of all previous moves. Show that the unique invariant distribution is given by $\pi_{v}=d_{v} /(2|E|)$ where $d_{v}$ is the degree of vertex $v$.

A rook performs a random walk on a chessboard; at each step, it is equally likely to make any of the moves which are legal for a rook. What is the mean recurrence time of a corner square. (You should give a clear statement of any general theorem used.)

[A chessboard is an $8 \times 8$ square grid. A legal move is one of any length parallel to the axes.]

B1.19

commentState and prove the convolution theorem for Laplace transforms.

Use the convolution theorem to prove that the Beta function

$B(p, q)=\int_{0}^{1}(1-\tau)^{p-1} \tau^{q-1} d \tau$

may be written in terms of the Gamma function as

$B(p, q)=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$

B1.9

commentLet $K=\mathbf{Q}(\alpha)$ be a number field, where $\alpha \in \mathcal{O}_{K}$. Let $f$ be the (normalized) minimal polynomial of $\alpha$ over $Q$. Show that the discriminant $\operatorname{disc}(f)$ of $f$ is equal to $\left(\mathcal{O}_{K}: \mathbf{Z}[\alpha]\right)^{2} D_{K}$.

Show that $f(x)=x^{3}+5 x^{2}-19$ is irreducible over Q. Determine $\operatorname{disc}(f)$ and the ring of algebraic integers $\mathcal{O}_{K}$ of $K=\mathbf{Q}(\alpha)$, where $\alpha \in \mathbf{C}$ is a root of $f$.

A1.9

comment(i) Describe Euclid's algorithm.

Find, in the RSA algorithm, the deciphering key corresponding to the enciphering key 7,527 .

(ii) Explain what is meant by a primitive root modulo an odd prime $p$.

Show that, if $g$ is a primitive root modulo $p$, then all primitive roots modulo $p$ are given by $g^{m}$, where $1 \leqslant m<p$ and $(m, p-1)=1$.

Verify, by Euler's criterion, that 3 is a primitive root modulo 17 . Hence find all primitive roots modulo 17 .

A1.20 B1.20

comment(i) Let $A$ be a symmetric $n \times n$ matrix such that

$A_{k, k}>\sum_{\substack{l=1 \\ l \neq k}}^{n}\left|A_{k, l}\right| \quad 1 \leqslant k \leqslant n .$

Prove that it is positive definite.

(ii) Prove that both Jacobi and Gauss-Seidel methods for the solution of the linear system $A \mathrm{x}=\mathbf{b}$, where the matrix $A$ obeys the conditions of (i), converge.

[You may quote the Householder-John theorem without proof.]

B1.18

comment(a) Solve the equation

$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=u^{2}$

together with the boundary condition on the $x$-axis:

$u(x, 0)=f(x)$

where $f$ is a smooth function. You should discuss the domain on which the solution is smooth. For which functions $f$ can the solution be extended to give a smooth solution on the upper half plane $\{y>0\}$ ?

(b) Solve the equation

$x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=0$

together with the boundary condition on the unit circle:

$u(x, y)=x \quad \text { when } \quad x^{2}+y^{2}=1$

A1.2 B1.2

comment(i) Show that Newton's equations in Cartesian coordinates, for a system of $N$ particles at positions $\mathbf{x}_{i}(t), i=1,2 \ldots N$, in a potential $V(\mathbf{x}, t)$, imply Lagrange's equations in a generalised coordinate system

$q_{j}=q_{j}\left(\mathbf{x}_{i}, t\right) \quad, \quad j=1,2 \ldots 3 N$

that is,

$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)=\frac{\partial L}{\partial q_{j}} \quad, \quad j=1,2 \ldots 3 N$

where $L=T-V, T(q, \dot{q}, t)$ being the total kinetic energy and $V(q, t)$ the total potential energy.

(ii) Consider a light rod of length $L$, free to rotate in a vertical plane (the $x z$ plane), but with one end $P$ forced to move in the $x$-direction. The other end of the rod is attached to a heavy mass $M$ upon which gravity acts in the negative $z$ direction.

(a) Write down the Lagrangian for the system.

(b) Show that, if $P$ is stationary, the rod has two equilibrium positions, one stable and the other unstable.

(c) The end at $P$ is now forced to move with constant acceleration, $\ddot{x}=A$. Show that, once more, there is one stable equilibrium value of the angle the rod makes with the vertical, and find it.

A1.12 B1.15

comment(i) What are the main approaches by which prior distributions are specified in Bayesian inference?

Define the risk function of a decision rule $d$. Given a prior distribution, define what is meant by a Bayes decision rule and explain how this is obtained from the posterior distribution.

(ii) Dashing late into King's Cross, I discover that Harry must have already boarded the Hogwarts Express. I must therefore make my own way onto platform nine and threequarters. Unusually, there are two guards on duty, and I will ask one of them for directions. It is safe to assume that one guard is a Wizard, who will certainly be able to direct me, and the other a Muggle, who will certainly not. But which is which? Before choosing one of them to ask for directions to platform nine and three-quarters, I have just enough time to ask one of them "Are you a Wizard?", and on the basis of their answer I must make my choice of which guard to ask for directions. I know that a Wizard will answer this question truthfully, but that a Muggle will, with probability $\frac{1}{3}$, answer it untruthfully.

Failure to catch the Hogwarts Express results in a loss which I measure as 1000 galleons, there being no loss associated with catching up with Harry on the train.

Write down an exhaustive set of non-randomised decision rules for my problem and, by drawing the associated risk set, determine my minimax decision rule.

My prior probability is $\frac{2}{3}$ that the guard I ask "Are you a Wizard?" is indeed a Wizard. What is my Bayes decision rule?

B1.13

commentState and prove Hölder's Inequality.

[Jensen's inequality, and other standard results, may be assumed.]

Let $\left(X_{n}\right)$ be a sequence of random variables bounded in $L_{p}$ for some $p>1$. Prove that $\left(X_{n}\right)$ is uniformly integrable.

Suppose that $X \in L_{p}(\Omega, \mathcal{F}, \mathbb{P})$ for some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and some $p \in(1, \infty)$. Show that $X \in L_{r}(\Omega, \mathcal{F}, \mathbb{P})$ for all $1 \leqslant r<p$ and that $\|X\|_{r}$ is an increasing function of $r$ on $[1, p]$.

Show further that $\lim _{r \rightarrow 1^{+}}\|X\|_{r}=\|X\|_{1}$.

A1.14

comment(i) A spinless quantum mechanical particle of mass $m$ moving in two dimensions is confined to a square box with sides of length $L$. Write down the energy eigenfunctions for the particle and the associated energies.

Show that, for large $L$, the number of states in the energy range $E \rightarrow E+d E$ is $\rho(E) d E$, where

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

(ii) If, instead, the particle is an electron with magnetic moment $\mu$ moving in an external magnetic field, $H$, show that

$\begin{array}{rlr} \rho(E) & =\frac{m L^{2}}{2 \pi \hbar^{2}}, & -\mu H<E<\mu H \\ & =\frac{m L^{2}}{\pi \hbar^{2}}, & \mu H<E \end{array}$

Let there be $N$ electrons in the box. Explain briefly how to construct the ground state of the system. Let $E_{F}$ be the Fermi energy. Show that when $E_{F}>\mu H$,

$N=\frac{m L^{2}}{\pi \hbar^{2}} E_{F}$

Show also that the magnetic moment, $M$, of the system in the ground state is

$M=\frac{\mu^{2} m L^{2}}{\pi \hbar^{2}} H$

and that the ground state energy is

$\frac{1}{2} \frac{\pi \hbar^{2}}{m L^{2}} N^{2}-\frac{1}{2} M H$

Part II

B1.6

commentCompute the character table of $A_{5}$ (begin by listing the conjugacy classes and their orders).

[It is not enough to write down the result; you must justify your answer.]

B1.11

commentRecall that an automorphism of a Riemann surface is a bijective analytic map onto itself, and that the inverse map is then guaranteed to be analytic.

Let $\Delta$ denote the $\operatorname{disc}\{z \in \mathbb{C}|| z \mid<1\}$, and let $\Delta^{*}=\Delta-\{0\}$.

(a) Prove that an automorphism $\phi: \Delta \rightarrow \Delta$ with $\phi(0)=0$ is a Euclidian rotation.

[Hint: Apply the maximum modulus principle to the functions $\phi(z) / z$ and $\phi^{-1}(z) / z$.]

(b) Prove that a holomorphic map $\phi: \Delta^{*} \rightarrow \Delta$ extends to the entire disc, and use this to conclude that any automorphism of $\Delta^{*}$ is a Euclidean rotation.

[You may use the result stated in part (a).]

(c) Define an analytic map between Riemann surfaces. Show that a continuous map between Riemann surfaces, known to be analytic everywhere except perhaps at a single point $P$, is, in fact, analytic everywhere.

B1.22

commentWrite down the first law of thermodynamics in differential form for an infinitesimal reversible change in terms of the increments $d E, d S$ and $d V$, where $E, S$ and $V$ are to be defined. Briefly give an interpretation of each term and deduce that

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

Define the specific heat at constant volume $C_{V}$ and show that for an adiabatic change

$C_{V} d T+\left(\left(\frac{\partial E}{\partial V}\right)_{T}+P\right) d V=0$

Derive the Maxwell relation

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

where $T$ is temperature and hence show that

$\left(\frac{\partial E}{\partial V}\right)_{T}=-P+T\left(\frac{\partial P}{\partial T}\right)_{V}$

An imperfect gas of volume $V$ obeys the van der Waals equation of state

$\left(P+\frac{a}{V^{2}}\right)(V-b)=R T$

where $a$ and $b$ are non-negative constants. Show that

$\left(\frac{\partial C_{V}}{\partial V}\right)_{T}=0,$

and deduce that $C_{V}$ is a function of $T$ only. It can further be shown that in this case $C_{V}$ is independent of $T$. Hence show that

$T(V-b)^{R / C_{V}}$

is constant on adiabatic curves.

A1.16

comment(i) Introducing the concept of a co-moving distance co-ordinate, explain briefly how the velocity of a galaxy in an isotropic and homogeneous universe is determined by the scale factor $a(t)$. How is the scale factor related to the Hubble constant $H_{0}$ ?

A homogeneous and isotropic universe has an energy density $\rho(t) c^{2}$ and a pressure $P(t)$. Use the relation $d E=-P d V$ to derive the "fluid equation"

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

where the overdot indicates differentiation with respect to time, $t$. Given that $a(t)$ satisfies the "acceleration equation"

$\ddot{a}=-\frac{4 \pi G}{3} a\left(\rho+\frac{3 P}{c^{2}}\right)$

show that the quantity

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

is time-independent.

The pressure $P$ is related to $\rho$ by the "equation of state"

$P=\sigma \rho c^{2}, \quad|\sigma|<1 .$

Given that $a\left(t_{0}\right)=1$, find $a(t)$ for $k=0$, and hence show that $a(0)=0$.

(ii) What is meant by the expression "the Hubble time"?

Assuming that $a(0)=0$ and $a\left(t_{0}\right)=1$, where $t_{0}$ is the time now (of the current cosmological era), obtain a formula for the radius $R_{0}$ of the observable universe.

Given that

$a(t)=\left(\frac{t}{t_{0}}\right)^{\alpha}$

for constant $\alpha$, find the values of $\alpha$ for which $R_{0}$ is finite. Given that $R_{0}$ is finite, show that the age of the universe is less than the Hubble time. Explain briefly, and qualitatively, why this result is to be expected as long as

$\rho+3 \frac{P}{c^{2}}>0 .$

Part II

A1.11 B1.16

comment(i) The price of the stock in the binomial model at time $r, 1 \leqslant r \leqslant n$, is $S_{r}=S_{0} \prod_{j=1}^{r} Y_{j}$, where $Y_{1}, Y_{2}, \ldots, Y_{n}$ are independent, identically-distributed random variables with $\mathbb{P}\left(Y_{1}=u\right)=p=1-\mathbb{P}\left(Y_{1}=d\right)$ and the initial price $S_{0}$ is a constant. Denote the fixed interest rate on the bank account by $\rho$, where $u>1+\rho>d>0$, and let the discount factor $\alpha=1 /(1+\rho)$. Determine the unique value $p=\bar{p}$ for which the sequence $\left\{\alpha^{r} S_{r}, 0 \leqslant r \leqslant n\right\}$ is a martingale.

Explain briefly the significance of $\bar{p}$ for the pricing of contingent claims in the model.

(ii) Let $T_{a}$ denote the first time that a standard Brownian motion reaches the level $a>0$. Prove that for $t>0$,

$\mathbb{P}\left(T_{a} \leqslant t\right)=2[1-\Phi(a / \sqrt{t})],$

where $\Phi$ is the standard normal distribution function.

Suppose that $A_{t}$ and $B_{t}$ represent the prices at time $t$ of two different stocks with initial prices 1 and 2 , respectively; the prices evolve so that they may be represented as $A_{t}=e^{\sigma_{1} X_{t}+\mu t}$ and $B_{t}=2 e^{\sigma_{2} Y_{t}+\mu t}$, respectively, where $\left\{X_{t}\right\}_{t \geqslant 0}$ and $\left\{Y_{t}\right\}_{t \geqslant 0}$ are independent standard Brownian motions and $\sigma_{1}, \sigma_{2}$ and $\mu$ are constants. Let $T$ denote the first time, if ever, that the prices of the two stocks are the same. Determine $\mathbb{P}(T \leqslant t)$, for $t>0$.

A1.17

comment(i) Let $h: G \rightarrow G^{\prime}$ be a surjective homomorphism between two groups, $G$ and $G^{\prime}$. If $D^{\prime}: G^{\prime} \rightarrow G L\left(\mathbb{C}^{n}\right)$ is a representation of $G^{\prime}$, show that $D(g)=D^{\prime}(h(g))$ for $g \in G$ is a representation of $G$ and, if $D^{\prime}$ is irreducible, show that $D$ is also irreducible. Show further that $\widetilde{D}(\widetilde{g})=D^{\prime}(\widetilde{h}(\widetilde{g}))$ is a representation of $G / \operatorname{ker}(h)$, where $\tilde{h}(\widetilde{g})=h(g)$ for $g \in G$ and $\widetilde{g} \in G / \operatorname{ker}(h)$ (with $g \in \widetilde{g}$ ). Deduce that the characters $\chi, \widetilde{\chi}, \chi^{\prime}$ of $D, \widetilde{D}, D^{\prime}$, respectively, satisfy

$\chi(g)=\tilde{\chi}(\widetilde{g})=\chi^{\prime}(h(g))$

(ii) $D_{4}$ is the symmetry group of rotations and reflections of a square. If $c$ is a rotation by $\pi / 2$ about the centre of the square and $b$ is a reflection in one of its symmetry axes, then $D_{4}=\left\{e, c, c^{2}, c^{3}, b, b c, b c^{2}, b c^{3}\right\}$. Given that the conjugacy classes are $\{e\}\left\{c^{2}\right\},\left\{c, c^{3}\right\}$ $\left\{b, b c^{2}\right\}$ and $\left\{b c, b c^{3}\right\}$ derive the character table of $D_{4}$.

Let $H_{0}$ be the Hamiltonian of a particle moving in a central potential. The $S O(3)$ symmetry ensures that the energy eigenvalues of $H_{0}$ are the same for all the angular momentum eigenstates in a given irreducible $S O(3)$ representation. Therefore, the energy eigenvalues of $H_{0}$ are labelled $E_{n l}$ with $n \in \mathbb{N}$ and $l \in \mathbb{N}_{0}, l<n$. Assume now that in a crystal the symmetry is reduced to a $D_{4}$ symmetry by an additional term $H_{1}$ of the total Hamiltonian, $H=H_{0}+H_{1}$. Find how the $H_{0}$ eigenstates in the $S O(3)$ irreducible representation with $l=2$ (the D-wave orbital) decompose into irreducible representations of $H$. You may assume that the character, $g(\theta)$, of a group element of $S O(3)$, in a representation labelled by $l$ is given by

$\chi\left(g_{\theta}\right)=1+2 \cos \theta+2 \cos (2 \theta)+\ldots+2 \cos (l \theta)$

where $\theta$ is a rotation angle and $l(l+1)$ is the eigenvalue of the total angular momentum, $\mathbf{L}^{2}$.

A1.19

comment(i) From the surface of a flat Earth, an explosive source emits P-waves downward into a horizontal homogeneous elastic layer of uniform thickness $h$ and P-wave speed $\alpha_{1}$ overlying a lower layer of infinite depth and P-wave speed $\alpha_{2}$, where $\alpha_{2}>\alpha_{1}$. A line of seismometers on the surface records the travel time $t$ as a function of distance $x$ from the source for the various arrivals along different ray paths.

Sketch the ray paths associated with the direct, reflected and head waves arriving at a given position. Calculate the travel times $t(x)$ of the direct and reflected waves, and sketch the corresponding travel-time curves. Hence explain how to estimate $\alpha_{1}$ and $h$ from the recorded arrival times. Explain briefly why head waves are only observed beyond a minimum distance $x_{c}$ from the source and why they have a travel-time curve of the form $t=t_{c}+\left(x-x_{c}\right) / \alpha_{2}$ for $x>x_{c}$.

[You need not calculate $x_{c}$ or $t_{c}$.]

(ii) A plane $\mathrm{SH}$-wave in a homogeneous elastic solid has displacement proportional to $\exp [i(k x+m z-\omega t)]$. Express the slowness vector $\mathbf{s}$ in terms of the wavevector $\mathbf{k}=(k, 0, m)$ and $\omega$. Deduce an equation for $m$ in terms of $k, \omega$ and the S-wave speed $\beta$.

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 $\mu_{2}$. Find the vertical structure of Love waves with displacement proportional to $\exp [i(k x-\omega t)]$, and show that the horizontal phase speed $c$ obeys

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

By sketching both sides of the equation as a function of $c$ in $\beta_{1} \leqslant c \leqslant \beta_{2}$ show that at least one mode exists for every value of $\omega$.

A1.18

comment(i) The diffusion equation for a spherically-symmetric concentration field $C(r, t)$ is

$C_{t}=\frac{D}{r^{2}}\left(r^{2} C_{r}\right)_{r},$

where $r$ is the radial coordinate. Find and sketch the similarity solution to (1) which satisfies $C \rightarrow 0$ as $r \rightarrow \infty$ and $\int_{0}^{\infty} 4 \pi r^{2} C(r, t) d r=M=$ constant, assuming it to be of the form

$C=\frac{M}{(D t)^{a}} F(\eta), \quad \eta=\frac{r}{(D t)^{b}},$

where $a$ and $b$ are numbers to be found.

(ii) A two-dimensional piece of heat-conducting material occupies the region $a \leqslant r \leqslant$ $b,-\pi / 2 \leqslant \theta \leqslant \pi / 2$ (in plane polar coordinates). The surfaces $r=a, \theta=-\pi / 2, \theta=\pi / 2$ are maintained at a constant temperature $T_{1}$; at the surface $r=b$ the boundary condition on temperature $T(r, \theta)$ is

$T_{r}+\beta T=0,$

where $\beta>0$ is a constant. Show that the temperature, which satisfies the steady heat conduction equation

$T_{r r}+\frac{1}{r} T_{r}+\frac{1}{r^{2}} T_{\theta \theta}=0,$

is given by a Fourier series of the form

$\frac{T}{T_{1}}=K+\sum_{n=0}^{\infty} \cos \left(\alpha_{n} \theta\right)\left[A_{n}\left(\frac{r}{a}\right)^{2 n+1}+B_{n}\left(\frac{a}{r}\right)^{2 n+1}\right]$

where $K, \alpha_{n}, A_{n}, B_{n}$ are to be found.

In the limits $a / b \ll 1$ and $\beta b \ll 1$, show that

$\int_{-\pi / 2}^{\pi / 2} T_{r} r d \theta \approx-\pi \beta b T_{1}$

given that

$\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8} .$

Explain how, in these limits, you could have obtained this result much more simply.

B1.26

commentDerive Riemann's equations for finite amplitude, one-dimensional sound waves in a perfect gas with ratio of specific heats $\gamma$.

At time $t=0$ the gas is at rest and has uniform density $\rho_{0}$, pressure $p_{0}$ and sound speed $c_{0}$. A piston initially at $x=0$ starts moving backwards at time $t=0$ with displacement $x=-a \sin \omega t$, where $a$ and $\omega$ are positive constants. Explain briefly how to find the resulting disturbance using a graphical construction in the $x t$-plane, and show that prior to any shock forming $c=c_{0}+\frac{1}{2}(\gamma-1) u$.

For small amplitude $a$, show that the excess pressure $\Delta p=p-p_{0}$ and the excess sound speed $\Delta c=c-c_{0}$ are related by

$\frac{\Delta p}{p_{0}}=\frac{2 \gamma}{\gamma-1} \frac{\Delta c}{c_{0}}+\frac{\gamma(\gamma+1)}{(\gamma-1)^{2}}\left(\frac{\Delta c}{c_{0}}\right)^{2}+O\left(\left(\frac{\Delta c}{c_{0}}\right)^{3}\right)$

Deduce that the time-averaged pressure on the face of the piston exceeds $p_{0}$ by

$\frac{1}{8} \rho_{0} a^{2} \omega^{2}(\gamma+1)+O\left(a^{3}\right)$