Part II, 2001, Paper 1

Applications of Quantum Mechanics

• B1.23

  Applications of Quantum Mechanics | Part II, 2001

  A 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

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

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Coding and Cryptography

• A1.10

  Coding and Cryptography | Part II, 2001

  (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.

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Combinatorics

• B1.5

  Combinatorics | Part II, 2001

  Let $\mathcal{A} \subset[n]^{(r)}$ 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?

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Computational Statistics and Statistical Modelling

• A1.13

  Computational Statistics and Statistical Modelling | Part II, 2001

  (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

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

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Differentiable Manifolds

• B1.8

  Differentiable Manifolds | Part II, 2001

  Define 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 $\mathbb{R}^{(n+1)^{2}}$.

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Dynamical Systems

• B1.17

  Dynamical Systems | Part II, 2001

  Define 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.

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Dynamics of Differential Equations

• A1.6

  Dynamics of Differential Equations | Part II, 2001

  (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 $\left(x_{0}, y_{0}, z_{0}\right) \in \mathbb{R}^{3}$.

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Electrodynamics

• B1.21

  Electrodynamics | Part II, 2001

  Explain 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.

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Fluid Dynamics II

• B1.25

  Fluid Dynamics II | Part II, 2001

  The 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,

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

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Galois Theory

• B1.7

  Galois Theory | Part II, 2001

  Prove 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 $p ?$

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General Relativity

• A1.15 B1.24

  General Relativity | Part II, 2001

  (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

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Graph Theory

• A1.8

  Graph Theory | Part II, 2001

  (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.

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Groups, Rings and Fields

• A1.4 B1.3

  Groups, Rings and Fields | Part II, 2001

  (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?

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Hilbert Spaces

• B1.10

  Hilbert Spaces | Part II, 2001

  State 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 $H$.]

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Information Theory

• B1.14

  Information Theory | Part II, 2001

  Let $p_{1}, \ldots, p_{n}$ 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

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

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Logic, Computation and Set Theory

• A $1 . 7 \quad$ B1.12

  Logic, Computation and Set Theory | Part II, 2001

  (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?

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Markov Chains

• A1.1 B1.1

  Markov Chains | Part II, 2001

  (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.]

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Methods of Mathematical Physics

• B1.19

  Methods of Mathematical Physics | Part II, 2001

  State 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

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

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Number Fields

• B1.9

  Number Fields | Part II, 2001

  Let $K=\mathbf{Q}(\alpha)$ 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 $f$.

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Number Theory

• A1.9

  Number Theory | Part II, 2001

  (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 .

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Numerical Analysis

• A1.20 B1.20

  Numerical Analysis | Part II, 2001

  (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.]

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Partial Differential Equations

• B1.18

  Partial Differential Equations | Part II, 2001

  (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:

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

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Principles of Dynamics

• A1.2 B1.2

  Principles of Dynamics | Part II, 2001

  (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.

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Principles of Statistics

• A1.12 B1.15

  Principles of Statistics | Part II, 2001

  (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?

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Probability and Measure

• B1.13

  Probability and Measure | Part II, 2001

  State 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 $\lim _{r \rightarrow 1^{+}}\|X\|_{r}=\|X\|_{1}$.

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Quantum Physics

• A1.14

  Quantum Physics | Part II, 2001

  (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

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Representation Theory

• B1.6

  Representation Theory | Part II, 2001

  Compute 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.]

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Riemann Surfaces

• B1.11

  Riemann Surfaces | Part II, 2001

  Recall 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.

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Statistical Physics

• B1.22

  Statistical Physics | Part II, 2001

  Write 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.

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Statistical Physics and Cosmology

• A1.16

  Statistical Physics and Cosmology | Part II, 2001

  (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

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Stochastic Financial Models

• A1.11 B1.16

  Stochastic Financial Models | Part II, 2001

  (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 $t>0$.

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Symmetries and Groups in Physics

• A1.17

  Symmetries and Groups in Physics | Part II, 2001

  (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, $\mathbf{L}^{2}$.

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Theoretical Geophysics

• A1.19

  Theoretical Geophysics | Part II, 2001

  (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 $\omega$.

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Transport Processes

• A1.18

  Transport Processes | Part II, 2001

  (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.

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Waves in Fluid and Solid Media

• B1.26

  Waves in Fluid and Solid Media | Part II, 2001

  Derive 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

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

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