Part II, 2003, Paper 4

# Part II, 2003, Paper 4

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B4.9

commentLet $X$ be a smooth curve of genus 0 over an algebraically closed field $k$. Show that $k(X)=k\left(\mathbb{P}^{1}\right) .$

Now let $C$ be a plane projective curve defined by an irreducible homogeneous cubic polynomial.

(a) Show that if $C$ is smooth then $C$ is not isomorphic to $\mathbb{P}^{1}$. Standard results on the canonical class may be assumed without proof, provided these are clearly stated.

(b) Show that if $C$ has a singularity then there exists a non-constant morphism from $\mathbb{P}^{1}$ to $C$.

B4.5

commentState the Mayer-Vietoris theorem. You should give the definition of all the homomorphisms involved.

Compute the homology groups of the union of the 2 -sphere with the line segment from the North pole to the South pole.

A4.11

commentDefine the optimal distribution problem. State what it means for a circuit $P$ to be flow-augmenting, and what it means for $P$ to be unbalanced. State the optimality theorem for flows. Describe the simplex-on-a-graph algorithm, giving a brief justification of its stopping rules.

Consider the problem of finding, in the network shown below, a minimum-cost flow from $s$ to $t$ of value 2 . Here the circled numbers are the upper arc capacities, the lower arc capacities all being zero, and the uncircled numbers are costs. Apply the simplex-on-agraph algorithm to solve this problem, taking as initial flow the superposition of a unit flow along the path $s,(s, a), a,(a, t), t$ and a unit flow along the path $s,(s, a), a,(a, b), b,(b, t), t$.

Part II 2003

B4.24

commentAtoms of mass $m$ in an infinite one-dimensional periodic array, with interatomic spacing $a$, have perturbed positions $x_{n}=n a+y_{n}$, for integer $n$. The potential between neighbouring atoms is

$\frac{1}{2} \lambda\left(x_{n+1}-x_{n}-a\right)^{2}$

for positive constant $\lambda$. Write down the Lagrangian for the variables $y_{n}$. Find the frequency $\omega(k)$ of a normal mode of wavenumber $k$. Define the Brillouin zone and explain why $k$ may be restricted to lie within it.

Assume now that the array is periodically-identified, so that there are effectively only $N$ atoms in the array and the atomic displacements $y_{n}$ satisfy the periodic boundary conditions $y_{n+N}=y_{n}$. Determine the allowed values of $k$ within the Brillouin zone. Show, for allowed wavenumbers $k$ and $k^{\prime}$, that

$\sum_{n=0}^{N-1} e^{i n\left(k-k^{\prime}\right) a}=N \delta_{k, k^{\prime}}$

By writing $y_{n}$ as

$y_{n}=\frac{1}{\sqrt{N}} \sum_{k} q_{k} e^{i n k a}$

where the sum is over allowed values of $k$, find the Lagrangian for the variables $q_{k}$, and hence the Hamiltonian $H$ as a function of $q_{k}$ and the conjugate momenta $p_{k}$. Show that the Hamiltonian operator $\hat{H}$ of the quantum theory can be written in the form

$\hat{H}=E_{0}+\sum_{k} \hbar \omega(k) a_{k}^{\dagger} a_{k}$

where $E_{0}$ is a constant and $a_{k}, a_{k}^{\dagger}$ are harmonic oscillator annihilation and creation operators. What is the physical interpretation of $a_{k}$ and $a_{k}^{\dagger}$ ? How does this show that phonons have quantized energies?

B4.12

commentExplain what is meant by a renewal process and by a renewal-reward process.

State and prove the law of large numbers for renewal-reward processes.

A component used in a manufacturing process has a maximum lifetime of 2 years and is equally likely to fail at any time during that period. If the component fails whilst in use, it is replaced immediately by a similar component, at a cost of $£ 1000$. The factory owner may alternatively replace the component before failure, at a time of his choosing, at a cost of $£ 200$. What should the factory owner do?

B4.1

commentWrite an essay on the Kruskal-Katona theorem. As well as stating the theorem and giving a detailed sketch of a proof, you should describe some further results that may be derived from it.

A4.14

commentThe nave height $x$, and the nave length $y$ for 16 Gothic-style cathedrals and 9 Romanesque-style cathedrals, all in England, have been recorded, and the corresponding $R$ output (slightly edited) is given below.

You may assume that $x, y$ are in suitable units, and that "style" has been set up as a factor with levels 1,2 corresponding to Gothic, Romanesque respectively.

(a) Explain carefully, with suitable graph(s) if necessary, the results of this analysis.

(b) Using the general model $Y=X \beta+\epsilon$ (in the conventional notation) explain carefully the theory needed for (a).

[Standard theorems need not be proved.]

B4.4

commentDefine the 'pull-back' homomorphism of differential forms determined by the smooth map $f: M \rightarrow N$ and state its main properties.

If $\theta: W \rightarrow V$ is a diffeomorphism between open subsets of $\mathbb{R}^{m}$ with coordinates $x_{i}$ on $V$ and $y_{j}$ on $W$ and the $m$-form $\omega$ is equal to $f d x_{1} \wedge \ldots \wedge d x_{m}$ on $V$, state and prove the expression for $\theta^{*}(\omega)$ as a multiple of $d y_{1} \wedge \ldots \wedge d y_{m}$.

Define the integral of an $m$-form $\omega$ over an oriented $m$-manifold $M$ and prove that it is well-defined.

Show that the inclusion map $f: N \hookrightarrow M$, of an oriented $n$-submanifold $N$ (without boundary) into $M$, determines an element $\nu$ of $H_{n}(M) \cong \operatorname{Hom}\left(H^{n}(M), \mathbb{R}\right)$. If $M=N \times P$ and $f(x)=(x, p)$, for $x \in N$ and $p$ fixed in $P$, what is the relation between $\nu$ and $\pi^{*}\left(\left[\omega_{N}\right]\right)$, where $\left[\omega_{N}\right]$ is the fundamental cohomology class of $N$ and $\pi$ is the projection onto the first factor?

B4.17

commentLet $f: S^{1} \rightarrow S^{1}$ be an orientation-preserving invertible map of the circle onto itself, with a lift $F: \mathbb{R} \rightarrow \mathbb{R}$. Define the rotation numbers $\rho_{0}(F)$ and $\rho(f)$.

Suppose that $\rho_{0}(F)=p / q$, where $p$ and $q$ are coprime integers. Prove that the map $f$ has periodic points of least period $q$, and no periodic points with any least period not equal to $q$.

Now suppose that $\rho_{0}(F)$ is irrational. Explain the distinction between wandering and non-wandering points under $f$. Let $\Omega(x)$ be the set of limit points of the sequence $\left\{x, f(x), f^{2}(x), \ldots\right\}$. Prove

(a) that the set $\Omega(x)=\Omega$ is independent of $x$ and is the smallest closed, non-empty, $f$-invariant subset of $S^{1}$;

(b) that $\Omega$ is the set of non-wandering points of $S^{1}$;

(c) that $\Omega$ is either the whole of $S^{1}$ or a Cantor set in $S^{1}$.

A4.6

commentExplain what is meant by a steady-state bifurcation of a fixed point $\mathbf{x}_{0}(\mu)$ of an ODE $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$, in $\mathbb{R}^{n}$, where $\mu$ is a real parameter. Give examples for $n=1$ of equations exhibiting saddle-node, transcritical and pitchfork bifurcations.

Consider the system in $\mathbb{R}^{2}$, with $\mu>0$,

$\dot{x}=x\left(1-y-4 x^{2}\right), \quad \dot{y}=y\left(\mu-y-x^{2}\right) .$

Show that the fixed point $(0, \mu)$ has a bifurcation when $\mu=1$, while the fixed points $\left(\pm \frac{1}{2}, 0\right)$ have a bifurcation when $\mu=\frac{1}{4}$. By finding the first approximation to the extended centre manifold, construct the normal form at the bifurcation point in each case, and determine the respective bifurcation types. Deduce that for $\mu$ just greater than $\frac{1}{4}$, and for $\mu$ just less than 1 , there is a stable pair of "mixed-mode" solutions with $x^{2}>0, y>0$.

B4.21

commentDescribe the physical meaning of the various components of the stress-energy tensor $T^{a b}$ of the electromagnetic field.

Suppose that one is given an electric field $\mathbf{E}(\mathbf{x})$ and a magnetic field $\mathbf{B}(\mathbf{x})$. Show that the angular momentum about the origin of these fields is

$\mathbf{J}=\frac{1}{\mu_{0}} \int \mathbf{x} \times(\mathbf{E} \times \mathbf{B}) d^{3} \mathbf{x}$

where the integral is taken over all space.

A point electric charge $Q$ is at the origin, and has electric field

$\mathbf{E}=\frac{Q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}}{|\mathbf{x}|^{3}}$

A point magnetic monopole of strength $P$ is at $\mathbf{y}$ and has magnetic field

$\mathbf{B}=\frac{\mu_{0} P}{4 \pi} \frac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|^{3}}$

Find the component, along the axis between the electric charge and the magnetic monopole, of the angular momentum of the electromagnetic field about the origin.

[Hint: You may find it helpful to express both $\mathbf{E}$ and $\mathbf{B}$ as gradients of scalar potentials.]

A4.5

commentLet $\mathbf{E}(\mathbf{r})$ be the electric field due to a continuous static charge distribution $\rho(\mathbf{r})$ for which $|\mathbf{E}| \rightarrow 0$ as $|\mathbf{r}| \rightarrow \infty$. Starting from consideration of a finite system of point charges, deduce that the electrostatic energy of the charge distribution $\rho$ is

$W=\frac{1}{2} \varepsilon_{0} \int|\mathbf{E}|^{2} d \tau$

where the volume integral is taken over all space.

A sheet of perfectly conducting material in the form of a surface $S$, with unit normal $\mathbf{n}$, carries a surface charge density $\sigma$. Let $E_{\pm}=\mathbf{n} \cdot \mathbf{E}_{\pm}$denote the normal components of the electric field $\mathbf{E}$ on either side of $S$. Show that

$\frac{1}{\varepsilon_{0}} \sigma=E_{+}-E_{-} .$

Three concentric spherical shells of perfectly conducting material have radii $a, b, c$ with $a<b<c$. The innermost and outermost shells are held at zero electric potential. The other shell is held at potential $V$. Find the potentials $\phi_{1}(r)$ in $a<r<b$ and $\phi_{2}(r)$ in $b<r<c$. Compute the surface charge density $\sigma$ on the shell of radius $b$. Use the formula $(*)$ to compute the electrostatic energy of the system.

B4.26

commentShow that the complex potential in the complex $\zeta$ plane,

$w=(U-i V) \zeta+(U+i V) \frac{c^{2}}{\zeta}-\frac{i \kappa}{2 \pi} \log \zeta$

describes irrotational, inviscid flow past the rigid cylinder $|\zeta|=c$, placed in a uniform flow $(U, V)$ with circulation $\kappa$.

Show that the transformation

$z=\zeta+\frac{c^{2}}{\zeta}$

maps the circle $|\zeta|=c$ in the $\zeta$ plane onto the flat plate airfoil $-2 c<x<2 c, y=0$ in the $z$ plane $(z=x+i y)$. Hence, write down an expression for the complex potential, $w_{p}$, of uniform flow past the flat plate, with circulation $\kappa$. Indicate very briefly how the value of $\kappa$ might be chosen to yield a physical solution.

Calculate the turning moment, $M$, exerted on the flat plate by the flow.

(You are given that

$M=-\frac{1}{2} \rho \operatorname{Re}\left\{\oint\left[\frac{\left(\frac{d w}{d \zeta}\right)^{2}}{\frac{d z}{d \zeta}}\right] z(\zeta) d \zeta\right\} \text {, }$

where $\rho$ is the fluid density and the integral is to be completed around a contour enclosing the circle $|\zeta|=c$ ).

A4.15 B4.22

commentDiscuss the quantum mechanics of the one-dimensional harmonic oscillator using creation and annihilation operators, showing how the energy levels are calculated.

A quantum mechanical system consists of two interacting harmonic oscillators and has the Hamiltonian

$H=\frac{1}{2} \hat{p}_{1}^{2}+\frac{1}{2} \hat{x}_{1}^{2}+\frac{1}{2} \hat{p}_{2}^{2}+\frac{1}{2} \hat{x}_{2}^{2}+\lambda \hat{x}_{1} \hat{x}_{2}$

For $\lambda=0$, what are the degeneracies of the three lowest energy levels? For $\lambda \neq 0$ compute, to lowest non-trivial order in perturbation theory, the energies of the ground state and first excited state.

[Standard results for perturbation theory may be stated without proof.]

A4.3

comment(i) State the Monotone Convergence Theorem and explain briefly how to prove it.

(ii) For which real values of $\alpha$ is $x^{-\alpha} \log x \in L^{1}((1, \infty))$ ?

Let $p>0$. Using the Monotone Convergence Theorem and the identity

$\frac{1}{x^{p}(x-1)}=\sum_{n=0}^{\infty} \frac{1}{x^{p+n+1}}$

prove carefully that

$\int_{1}^{\infty} \frac{\log x}{x^{p}(x-1)} d x=\sum_{n=0}^{\infty} \frac{1}{(n+p)^{2}}$

B4.3

commentWrite an essay on finite fields and their Galois theory.

A4.17 B4.25

commentWhat are "inertial coordinates" and what is their physical significance? [A proof of the existence of inertial coordinates is not required.] Let $O$ be the origin of inertial coordinates and let $\left.R_{a b c d}\right|_{O}$ be the curvature tensor at $O$ (with all indices lowered). Show that $\left.R_{a b c d}\right|_{O}$ can be expressed entirely in terms of second partial derivatives of the metric $g_{a b}$, evaluated at $O$. Use this expression to deduce that (a) $R_{a b c d}=-R_{b a c d}$ (b) $R_{a b c d}=R_{c d a b}$ (c) $R_{a[b c d]}=0$.

Starting from the expression for $R_{b c d}^{a}$ in terms of the Christoffel symbols, show (again by using inertial coordinates) that

$R_{a b[c d ; e]}=0$

Obtain the contracted Bianchi identities and explain why the Einstein equations take the form

$R_{a b}-\frac{1}{2} R g_{a b}=8 \pi T_{a b}-\Lambda g_{a b},$

where $T_{a b}$ is the energy-momentum tensor of the matter and $\Lambda$ is an arbitrary constant.

A4.7

commentWrite an essay on the Theorema Egregium for surfaces in $\mathbb{R}^{3}$.

A4.9

commentWrite an essay on the vertex-colouring of graphs drawn on compact surfaces other than the sphere. You should include a proof of Heawood's bound, and an example of a surface for which this bound is not attained.

A4.4

commentWrite an essay on the theory of invariants. Your essay should discuss the theorem on the finite generation of the ring of invariants, the theorem on elementary symmetric functions, and some examples of calculation of rings of invariants.

B4.7

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

(a) Show that if $\|I-T\|<1$ then $T$ is invertible.

(b) Prove that if $T$ is invertible and if $S \in \mathcal{B}(H)$ satisfies $\|S-T\|<\left\|T^{-1}\right\|^{-1}$, then $S$ is invertible.

(c) Define what it means for $T$ to be compact. Prove that the set of compact operators on $H$ is a closed subset of $\mathcal{B}(H)$.

(d) Prove that $T$ is compact if and only if there is a sequence $\left(F_{n}\right)$ in $\mathcal{B}(H)$, where each operator $F_{n}$ has finite rank, such that $\left\|F_{n}-T\right\| \rightarrow 0$ as $n \rightarrow \infty$.

(e) Suppose that $T=A+K$, where $A$ is invertible and $K$ is compact. Prove that then, also, $T=B+F$, where $B$ is invertible and $F$ has finite rank.

B4.13

commentState and prove the Fano and generalized Fano inequalities.

A4.8 B4.10

commentWrite an essay on propositional logic. You should include all relevant definitions, and should cover the Completeness Theorem, as well as the Compactness Theorem and the Decidability Theorem.

[You may assume that the set of primitive propositions is countable. You do not need to give proofs of simple examples of syntactic implication, such as the fact that $p \Rightarrow p$ is a theorem or that $p \Rightarrow q$ and $q \Rightarrow r$ syntactically imply $p \Rightarrow r$.]

A4.1

commentConsider a pack of cards labelled $1, \ldots, 52$. We repeatedly take the top card and insert it uniformly at random in one of the 52 possible places, that is, either on the top or on the bottom or in one of the 50 places inside the pack. How long on average will it take for the bottom card to reach the top?

Let $p_{n}$ denote the probability that after $n$ iterations the cards are found in increasing order. Show that, irrespective of the initial ordering, $p_{n}$ converges as $n \rightarrow \infty$, and determine the limit $p$. You should give precise statements of any general results to which you appeal.

Show that, at least until the bottom card reaches the top, the ordering of cards inserted beneath it is uniformly random. Hence or otherwise show that, for all $n$,

$\left|p_{n}-p\right| \leqslant 52(1+\log 52) / n$

A4.21

commentLet $y(x, \lambda)$ denote the solution for $0 \leqslant x<\infty$ of

$\frac{d^{2} y}{d x^{2}}-\left(x+\lambda^{2}\right) y=0$

subject to the conditions that $y(0, \lambda)=a$ and $y(x, \lambda) \rightarrow 0$ as $x \rightarrow \infty$, where $a>0$; it may be assumed that $y(x, \lambda)>0$ for $x>0$. Write $y(x, \lambda)$ in the form

$y(x, \lambda)=\exp (z(x, \lambda))$

and consider an asymptotic expansion of the form

$z(x, \lambda) \sim \sum_{n=0}^{\infty} \lambda^{1-n} \phi_{n}(x),$

valid in the limit $\lambda \rightarrow \infty$ with $x=O(1)$. Find $\phi_{0}(x), \phi_{1}(x), \phi_{2}(x)$ and $\phi_{3}(x)$.

It is known that the solution $y(x, \lambda)$ is of the form

$y(x, \lambda)=c Y(X)$

where

$X=x+\lambda^{2}$

and the constant factor $c$ depends on $\lambda$. By letting $Y(X)=\exp (Z(X))$, show that the expression

$Z(X)=-\frac{2}{3} X^{3 / 2}-\frac{1}{4} \ln X$

satisfies the relevant differential equation with an error of $O\left(1 / X^{3 / 2}\right)$ as $X \rightarrow \infty$. Comment on the relationship between your answers for $z(x, \lambda)$ and $Z(X)$.

B4.19

commentBy setting $w(z)=\int_{\gamma} f(t) e^{-z t} d t$, where $\gamma$ and $f(t)$ are to be suitably chosen, explain how to find integral representations of the solutions of the equation

$z w^{\prime \prime}-k w=0$

where $k$ is a non-zero real constant and $z$ is complex. Discuss $\gamma$ in the particular case that $z$ is restricted to be real and positive and distinguish the different cases that arise according to the $\operatorname{sign}$ of $k$.

Show that in this particular case, by choosing $\gamma$ as a closed contour around the origin, it is possible to express a solution in the form

$w(z)=A \sum_{n=0}^{\infty} \frac{(z k)^{n+1}}{n !(n+1) !}$

where $A$ is a constant.

Show also that for $k>0$ there are solutions that satisfy

$w(z) \sim B z^{1 / 4} e^{-2 \sqrt{k z}} \quad \text { as } z \rightarrow \infty$

where $B$ is a constant.

A4.22

commentLet $\Phi^{+}(t), \Phi^{-}(t)$ denote the boundary values of functions which are analytic inside and outside a disc of radius $\frac{1}{2}$ centred at the origin. Let $C$ denote the boundary of this disc.

Suppose that $\Phi^{+}, \Phi^{-}$satisfy the jump condition

$\Phi^{+}(t)=\frac{t}{t^{2}-1} \Phi^{-}(t)+\frac{t^{3}-t^{2}+1}{t^{2}-t}, \quad t \in C .$

(a) Show that the associated index is 1 .

(b) Find the canonical solution of the homogeneous problem, i.e. the solution satisfying

$X(z) \sim z^{-1}, \quad z \rightarrow \infty$

(c) Find the general solution of the Riemann-Hilbert problem satisfying the above jump condition as well as

$\Phi(z)=O\left(z^{-1}\right), \quad z \rightarrow \infty$

(d) Use the above result to solve the linear singular integral problem

$\left(t^{2}+t-1\right) \phi(t)+\frac{t^{2}-t-1}{\pi i} \oint_{C} \frac{\phi(\tau)}{\tau-t} d \tau=\frac{2\left(t^{3}-t^{2}+1\right)(t+1)}{t}, \quad t \in C .$

B4.6

commentWrite an essay on the Dirichlet unit theorem with particular reference to quadratic fields.

A4.10

commentWrite an essay describing the factor base method for factorising a large odd positive integer $n$. Your essay should include a detailed explanation of how the continued fraction of $\sqrt{n}$ can be used to find a suitable factor base.

A4.23 B4.20

commentWrite an essay on the conjugate gradient method. Your essay should include:

(a) a statement of the method and a sketch of its derivation;

(b) discussion, without detailed proofs, but with precise statements of relevant theorems, of the conjugacy of the search directions;

(c) a description of the standard form of the algorithm;

(d) discussion of the connection of the method with Krylov subspaces.

B4.14

commentThe scalars $x_{t}, y_{t}, u_{t}$, are related by the equations

$x_{t}=x_{t-1}+u_{t-1}, \quad y_{t}=x_{t-1}+\eta_{t-1}, \quad t=1, \ldots, T,$

where $\left\{\eta_{t}\right\}$ is a sequence of uncorrelated random variables with means of 0 and variances of 1. Given that $\hat{x}_{0}$ is an unbiased estimate of $x_{0}$ of variance 1 , the control variable $u_{t}$ is to be chosen at time $t$ on the basis of the information $W_{t}$, where $W_{0}=\left(\hat{x}_{0}\right)$ and $W_{t}=\left(\hat{x}_{0}, u_{0}, \ldots, u_{t-1}, y_{1}, \ldots, y_{t}\right), t=1,2, \ldots, T-1$. Let $\hat{x}_{1}, \ldots, \hat{x}_{T}$ be the Kalman filter estimates of $x_{1}, \ldots, x_{T}$ computed from

$\hat{x}_{t}=\hat{x}_{t-1}+u_{t-1}+h_{t}\left(y_{t}-\hat{x}_{t-1}\right)$

by appropriate choices of $h_{1}, \ldots, h_{T}$. Show that the variance of $\hat{x}_{t}$ is $V_{t}=1 /(1+t)$.

Define $F\left(W_{T}\right)=E\left[x_{T}^{2} \mid W_{T}\right]$ and

$F\left(W_{t}\right)=\inf _{u_{t}, \ldots, u_{T-1}} E\left[\sum_{\tau=t}^{T-1} u_{\tau}^{2}+x_{T}^{2} \mid W_{t}\right], \quad t=0, \ldots, T-1$

Show that $F\left(W_{t}\right)=\hat{x}_{t}^{2} P_{t}+d_{t}$, where $P_{t}=1 /(T-t+1), d_{T}=1 /(1+T)$ and $d_{t-1}=V_{t-1} V_{t} P_{t}+d_{t} .$

How would the expression for $F\left(W_{0}\right)$ differ if $\hat{x}_{0}$ had a variance different from $1 ?$

B4.18

commentDiscuss the basic properties of the Fourier transform and how it is used in the study of partial differential equations.

The essay should include: definition and basic properties, inversion theorem, applications to establishing well-posedness of evolution partial differential equations with constant coefficients.

A4.2

commentThe action $S$ of a Hamiltonian system may be regarded as a function of the final coordinates $q^{a}, a=1, \ldots, N$, and the final time $t$ by setting

$S\left(q^{a}, t\right)=\int_{\left(q_{i}^{a}, t_{i}\right)}^{\left(q^{a}, t\right)} d t^{\prime}\left[p^{a}\left(t^{\prime}\right) \dot{q}^{a}\left(t^{\prime}\right)-H\left(p^{a}\left(t^{\prime}\right), q^{a}\left(t^{\prime}\right), t^{\prime}\right)\right]$

where the initial coordinates $q_{i}^{a}$ and time $t_{i}$ are held fixed, and $p^{a}\left(t^{\prime}\right), q^{a}\left(t^{\prime}\right)$ are the solutions to Hamilton's equations with Hamiltonian $H$, satisfying $q^{a}(t)=q^{a}, q^{a}\left(t_{i}\right)=q_{i}^{a}$.

(a) Show that under an infinitesimal change of the final coordinates $\delta q^{a}$ and time $\delta t$, the change in $S$ is

$\delta S=p_{a}(t) \delta q_{a}-H\left(p^{a}(t), q^{a}(t), t\right) \delta t$

(b) Hence derive the Hamilton-Jacobi equation

$\frac{\partial S}{\partial t}\left(q^{a}, t\right)+H\left(\frac{\partial S}{\partial q^{a}}\left(q^{a}, t\right), q^{a}, t\right)=0$

(c) If we can find a solution to $(*)$,

$S=S\left(q^{a}, t ; P^{a}\right),$

where $P^{a}$ are $N$ integration constants, then we can use $S$ as a generating function of type $I I$, where

$p^{a}=\frac{\partial S}{\partial q^{a}} \quad, \quad Q^{a}=-\frac{\partial S}{\partial P^{a}}$

Show that the Hamiltonian $K$ in the new coordinates $Q^{a}, P^{a}$ vanishes.

(d) Write down and solve the Hamilton-Jacobi equation for the one-dimensional simple harmonic oscillator, where $H=\frac{1}{2}\left(p^{2}+q^{2}\right)$. Show the solution takes the form $S(q, t ; E)=W(q, E)-E t$. Using this as a generating function $F_{I I}(q, t, P)$ show that the new coordinates $Q, P$ are constants of the motion and give their physical interpretation.

A4.13 B4.15

commentWrite an account, with appropriate examples, of inference in multiparameter exponential families. Your account should include a discussion of natural statistics and their properties and of various conditional tests on natural parameters.

B4.11

commentLet $f:[a, b] \rightarrow \mathbb{R}$ be integrable with respect to Lebesgue measure $\mu$ on $[a, b]$. Prove that, if

$\int_{J} f d \mu=0$

for every sub-interval $J$ of $[a, b]$, then $f=0$ almost everywhere on $[a, b]$.

Now define

$F(x)=\int_{a}^{x} f d \mu .$

Prove that $F$ is continuous on $[a, b]$. Show that, if $F$ is zero on $[a, b]$, then $f$ is zero almost everywhere on $[a, b]$.

Suppose now that $f$ is bounded and Lebesgue integrable on $[a, b]$. By applying the Dominated Convergence Theorem to

$F_{n}(x)=\frac{F\left(x+\frac{1}{n}\right)-F(x)}{\frac{1}{n}},$

or otherwise, show that, if $F$ is differentiable on $[a, b]$, then $F^{\prime}=f$ almost everywhere on $[a, b]$.

The functions $f_{n}:[a, b] \rightarrow \mathbb{R}$ have the properties:

(a) $f_{n}$ converges pointwise to a differentiable function $g$ on $[a, b]$,

(b) each $f_{n}$ has a continuous derivative $f_{n}^{\prime}$ with $\left|f_{n}^{\prime}(x)\right| \leqslant 1$ on $[a, b]$,

(c) $f_{n}^{\prime}$ converges pointwise to some function $h$ on $[a, b]$.

Deduce that

$h(x)=\lim _{n \rightarrow \infty}\left(\frac{d f_{n}(x)}{d x}\right)=\frac{d}{d x}\left(\lim _{n \rightarrow \infty} f_{n}(x)\right)=g^{\prime}(x)$

almost everywhere on $[a, b]$.

A4.16

commentDescribe the energy band structure available to electrons moving in crystalline materials. How can it be used to explain the properties of crystalline materials that are conductors, insulators and semiconductors?

Where does the Fermi energy lie in an intrinsic semiconductor? Describe the process of doping of semiconductors and explain the difference between $n$-type and $p$-type doping. What is the effect of the doping on the position of the Fermi energy in the two cases?

Why is there a potential difference across a junction of $n$-type and $p$-type semiconductors?

Derive the relation

$I=I_{0}\left(1-e^{-q V / k T}\right)$

between the current, $I$, and the voltage, $V$, across an $n p$ junction, where $I_{0}$ is the total minority current in the semiconductor and $q$ is the charge on the electron, $T$ is the temperature and $k$ is Boltzmann's constant. Your derivation should include an explanation of the terms majority current and minority current.

Why can the $n p$ junction act as a rectifier?

B4.2

commentAssume that the group $S L_{2}\left(\mathbb{F}_{3}\right)$ of $2 \times 2$ matrices of determinant 1 with entries from the field $\mathbb{F}_{3}$ has presentation

$\left\langle X, P, Q: X^{3}=P^{4}=1, P^{2}=Q^{2}, P Q P^{-1}=Q^{-1}, X P X^{-1}=Q, X Q X^{-1}=P Q\right\rangle$

Show that the subgroup generated by $P^{2}$ is central and that the quotient group can be identified with the alternating group $A_{4}$. Assuming further that $S L_{2}\left(\mathbb{F}_{3}\right)$ has seven conjugacy classes find the character table.

Is it true that every irreducible character is induced up from the character of a 1-dimensional representation of some subgroup?

[Hint: You may find it useful to note that $S L_{2}\left(\mathbb{F}_{3}\right)$ may be regarded as a subgroup of $S U_{2}$, providing a faithful 2-dimensional representation; the subgroup generated by $P$ and $Q$ is the quaternion group of order 8 , acting irreducibly.]

B4.8

comment(a) Define the degree $\operatorname{deg} f$ of a meromorphic function on the Riemann sphere $\mathbb{P}^{1}$. State the Riemann-Hurwitz theorem.

Let $f$ and $g$ be two rational functions on the sphere $\mathbb{P}^{1}$. Show that

$\operatorname{deg}(f+g) \leqslant \operatorname{deg} f+\operatorname{deg} g$

Deduce that

$|\operatorname{deg} f-\operatorname{deg} g| \leqslant \operatorname{deg}(f+g) \leqslant \operatorname{deg} f+\operatorname{deg} g .$

(b) Describe the topological type of the Riemann surface defined by the equation $w^{2}+2 w=z^{5}$ in $\mathbb{C}^{2}$. [You should analyse carefully the behaviour as $w$ and $z$ approach $\infty$.]

B4.23

commentA gas of non-interacting identical bosons in volume $V$, with one-particle energy levels $\epsilon_{r}, r=1,2, \ldots, \infty$, is in equilibrium at temperature $T$ and chemical potential $\mu$. Let $n_{r}$ be the number of particles in the $r$ th one-particle state. Write down an expression for the grand partition function $\mathcal{Z}$. Write down an expression for the probability of finding a given set of occupation numbers $n_{r}$ of the one-particle states. Hence determine the mean occupation numbers $\bar{n}_{r}$ (the Bose-Einstein distribution). Write down expressions, in terms of the mean occupation numbers, for the total energy $E$ and total number of particles $N$.

Write down an expression for the grand potential $\Omega$ in terms of $\mathcal{Z}$. Given that

$S=-\left(\frac{\partial \Omega}{\partial T}\right)_{V, \mu}$

show that $S$ can be written in the form

$S=k \sum_{r} f\left(\bar{n}_{r}\right)$

for some function $f$, which you should determine. Hence show that $d S=0$ for any change of the gas that leaves the mean occupation numbers unchanged. Consider a (quasi-static) change of $V$ with this property. Using the formula

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

and given that $\epsilon_{r} \propto V^{-\sigma}(\sigma>0)$ for each $r$, show that

$P=\sigma(E / V)$

What is the value of $\sigma$ for photons?

Let $\mu=0$, so that $E$ is a function only of $T$ and $V$. Why does the energy density $\varepsilon=E / V$ depend only on $T ?$ Using the Maxwell relation

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

and the first law of thermodynamics for reversible changes, show that

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

and hence that

$\varepsilon(T) \propto T^{\gamma}$

for some power $\gamma$ that you should determine. Show further that

$S \propto\left(T V^{\sigma}\right)^{\frac{1}{\sigma}} .$

Hence verify, given $\mu=0$, that $\bar{n}_{r}$ is left unchanged by a change of $V$ at constant $S$.

A4.18

commentLet $g(p)$ be the density of states of a particle in volume $V$ as a function of the magnitude $p$ of the particle's momentum. Explain why $g(p) \propto V p^{2} / h^{3}$, where $h$ is Planck's constant. Write down the Bose-Einstein and Fermi-Dirac distributions for the (average) number $\bar{n}(p)$ of particles of an ideal gas with momentum $p$. Hence write down integrals for the (average) total number $N$ of particles and the (average) total energy $E$ as functions of temperature $T$ and chemical potential $\mu$. Why do $N$ and $E$ also depend on the volume $V ?$

Electromagnetic radiation in thermal equilibrium can be regarded as a gas of photons. Why are photons "ultra-relativistic" and how is photon momentum $p$ related to the frequency $\nu$ of the radiation? Why does a photon gas have zero chemical potential? Use your formula for $\bar{n}(p)$ to express the energy density $\varepsilon_{\gamma}$ of electromagnetic radiation in the form

$\varepsilon_{\gamma}=\int_{0}^{\infty} \epsilon(\nu) d \nu$

where $\epsilon(\nu)$ is a function of $\nu$ that you should determine up to a dimensionless multiplicative constant. Show that $\epsilon(\nu)$ is independent of $h$ when $k T \gg h \nu$, where $k$ is Boltzmann's constant. Let $\nu_{\text {peak }}$ be the value of $\nu$ at the maximum of the function $\epsilon(\nu)$; how does $\nu_{\text {peak }}$ depend on $T$ ?

Let $n_{\gamma}$ be the photon number density at temperature $T$. Show that $n_{\gamma} \propto T^{q}$ for some power $q$, which you should determine. Why is $n_{\gamma}$ unchanged as the volume $V$ is increased quasi-statically? How does $T$ depend on $V$ under these circumstances? Applying your result to the Cosmic Microwave Background Radiation (CMBR), deduce how the temperature $T_{\gamma}$ of the CMBR depends on the scale factor $a$ of the Universe. At a time when $T_{\gamma} \sim 3000 K$, the Universe underwent a transition from an earlier time at which it was opaque to a later time at which it was transparent. Explain briefly the reason for this transition and its relevance to the CMBR.

An ideal gas of fermions $f$ of mass $m$ is in equilibrium at temperature $T$ and chemical potential $\mu_{f}$ with a gas of its own anti-particles $\bar{f}$ and photons $(\gamma)$. Assuming that chemical equilibrium is maintained by the reaction

$f+\bar{f} \leftrightarrow \gamma$

determine the chemical potential $\mu_{\bar{f}}$ of the antiparticles. Let $n_{f}$ and $n_{\bar{f}}$ be the number densities of $f$ and $\bar{f}$, respectively. What will their values be for $k T \ll m c^{2}$ if $\mu_{f}=0$ ? Given that $\mu_{f}>0$, but $\mu_{f} \ll k T$, show that

$n_{f} \approx n_{0}(T)\left[1+\frac{\mu_{f}}{k T} F\left(m c^{2} / k T\right)\right]$

where $n_{0}(T)$ is the fermion number density at zero chemical potential and $F$ is a positive function of the dimensionless ratio $m c^{2} / k T$. What is $F$ when $k T \ll m c^{2}$ ?

Given that $\mu_{f} \ll k T$, obtain an expression for the ratio $\left(n_{f}-n_{\bar{f}}\right) / n_{0}$ in terms of $\mu, T$ and the function $F$. Supposing that $f$ is either a proton or neutron, why should you expect the ratio $\left(n_{f}-n_{\bar{f}}\right) / n_{\gamma}$ to remain constant as the Universe expands?

A4.12 B4.16

commentA single-period market contains $d$ risky assets, $S^{1}, S^{2}, \ldots, S^{d}$, initially worth $\left(S_{0}^{1}, S_{0}^{2}, \ldots, S_{0}^{d}\right)$, and at time 1 worth random amounts $\left(S_{1}^{1}, S_{1}^{2}, \ldots, S_{1}^{d}\right)$ whose first two moments are given by

$\mu=E S_{1}, \quad V=\operatorname{cov}\left(S_{1}\right) \equiv E\left[\left(S_{1}-E S_{1}\right)\left(S_{1}-E S_{1}\right)^{T}\right]$

An agent with given initial wealth $w_{0}$ is considering how to invest in the available assets, and has asked for your advice. Develop the theory of the mean-variance efficient frontier far enough to exhibit explicitly the minimum-variance portfolio achieving a required mean return, assuming that $V$ is non-singular. How does your analysis change if a riskless asset $S^{0}$ is added to the market? Under what (sufficient) conditions would an agent maximising expected utility actually choose a portfolio on the mean-variance efficient frontier?

A4.20

commentDefine the Rossby number. Under what conditions will a fluid flow be at (a) high and (b) low values of the Rossby number? Briefly describe both an oceanographic and a meteorological example of each type of flow.

Explain the concept of quasi-geostrophy for a thin layer of homogeneous fluid in a rapidly rotating system. Write down the quasi-geostrophic approximation for the vorticity in terms of the pressure, the fluid density and the rate of rotation. Define the potential vorticity and state the associated conservation law.

A broad current flows directly eastwards ( $+x$ direction) with uniform velocity $U$ across a flat ocean basin of depth $H$. The current encounters a low, two-dimensional ridge of width $L$ and height $H h(x)(0<x<L)$, whose axis is aligned in the north-south $(y)$ direction. Neglecting any effects of stratification and assuming a constant vertical rate of rotation $\frac{1}{2} f$, such that the Rossby number is small, determine the effect of the ridge on the current. Show that the direction of the current after it leaves the ridge is dependent on the cross-sectional area of the ridge, but not on the explicit form of $h(x)$.

A4.19

commentA shallow layer of fluid of viscosity $\mu$, density $\rho$ and depth $h(x, t)$ lies on a rigid horizontal plane $y=0$ and is bounded by impermeable barriers at $x=-L$ and $x=L$ $(L \gg h)$. Gravity acts vertically and a wind above the layer causes a shear stress $\tau(x)$ to be exerted on the upper surface in the $+x$ direction. Surface tension is negligible compared to gravity.

(a) Assuming that the steady flow in the layer can be analysed using lubrication theory, show that the horizontal pressure gradient $p_{x}$ is given by $p_{x}=\rho g h_{x}$ and hence that

$h h_{x}=\frac{3}{2} \frac{\tau}{\rho g} .$

Show also that the fluid velocity at the surface $y=h$ is equal to $\tau h / 4 \mu$, and sketch the velocity profile for $0 \leqslant y \leqslant h$.

(b) In the case in which $\tau$ is a constant, $\tau_{0}$, and assuming that the difference between $h$ and its average value $h_{0}$ remains small compared with $h_{0}$, show that

$h \approx h_{0}\left(1+\frac{3 \tau_{0} x}{2 \rho g h_{0}^{2}}\right)$

provided that

$\frac{\tau_{0} L}{\rho g h_{0}^{2}} \ll 1$

(c) Surfactant at surface concentration $\Gamma(x)$ is added to the surface, so that now

$\tau=\tau_{0}-A \Gamma_{x},$

where $A$ is a positive constant. The surfactant is advected by the surface fluid velocity and also experiences a surface diffusion with diffusivity $D$. Write down the equation for conservation of surfactant, and hence show that

$\left(\tau_{0}-A \Gamma_{x}\right) h \Gamma=4 \mu D \Gamma_{x}$

From equations (1), (2) and (3) deduce that

$\frac{\Gamma}{\Gamma_{0}}=\exp \left[\frac{\rho g}{18 \mu D}\left(h^{3}-h_{0}^{3}\right)\right]$

where $\Gamma_{0}$ is a constant. Assuming once more that $h_{1} \equiv h-h_{0} \ll h_{0}$, and that $h=h_{0}$ at $x=0$, show further that

$h_{1} \approx \frac{3 \tau_{0} x}{2 \rho g h_{0}}\left[1+\frac{A \Gamma_{0} h_{0}}{4 \mu D}\right]^{-1}$

provided that

$\frac{\tau_{0} h_{0} L}{\mu D} \ll 1 \quad \text { as well as } \quad \frac{\tau_{0} L}{\rho g h_{0}^{2}} \ll 1$

B4.27

commentShow that the equations governing isotropic linear elasticity have plane-wave solutions, identifying them as $\mathrm{P}, \mathrm{SV}$ or $\mathrm{SH}$ waves.

A semi-infinite elastic medium in $y<0$ (where $y$ is the vertical coordinate) with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ is overlaid by a layer of thickness $h$ (in $0<y<h$ ) of a second elastic medium with density $\rho^{\prime}$ and Lamé moduli $\lambda^{\prime}$ and $\mu^{\prime}$. The top surface at $y=h$ is free, i.e. the surface tractions vanish there. The speed of S-waves is lower in the layer, i.e. $c_{S}^{\prime}{ }^{2}=\mu^{\prime} / \rho^{\prime}<\mu / \rho=c_{S}^{2}$. For a time-harmonic SH-wave with horizontal wavenumber $k$ and frequency $\omega$, which oscillates in the slow top layer and decays exponentially into the fast semi-infinite medium, derive the dispersion relation for the apparent wave speed $c(k)=\omega / k$,

$\tan \left(k h \sqrt{\frac{c^{2}}{c_{S}^{2}}-1}\right)=\frac{\mu \sqrt{1-\frac{c^{2}}{c_{S}^{2}}}}{\mu^{\prime} \sqrt{\frac{c^{2}}{c_{S}^{\prime}}-1}} .$

Show graphically that there is always one root, and at least one higher mode if $\sqrt{c_{S}^{2} / c_{S}^{\prime 2}-1}>\pi / k h$.