Part II, 2001, Paper 4

Algebraic Curves

• B4.9

  Algebraic Curves | Part II, 2001

  Write an essay on curves of genus one (over an algebraically closed field $k$ of characteristic zero). Legendre's normal form should not be discussed.

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Algebraic Topology

• B4.5

  Algebraic Topology | Part II, 2001

  Write an essay on the definition of simplicial homology groups. The essay should include a discussion of orientations, of the action of a simplicial map and a proof of $\partial^{2}=0$.

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Algorithms and Networks

• A4.11

  Algorithms and Networks | Part II, 2001

  State the optimal distribution problem. Carefully describe the simplex-on-a-graph algorithm for solving optimal distribution problems when the flow in each arc in the network is constrained to lie in the interval $[0, \infty)$. Explain how the algorithm can be initialised if there is no obvious feasible solution with which to begin. Describe the adjustments that are needed for the algorithm to cope with more general capacity constraints $x(j) \in\left[c^{-}(j), c^{+}(j)\right]$ for each arc $j$ (where $c^{\pm}(j)$ may be finite or infinite).

  Part II

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Applications of Quantum Mechanics

• B4.24

  Applications of Quantum Mechanics | Part II, 2001

  Derive the Bloch form of the wave function $\psi(x)$ of an electron moving in a onedimensional crystal lattice.

  The potential in such an $N$-atom lattice is modelled by

  $$V(x)=\sum_{n}\left(-\frac{\hbar^{2} U}{2 m} \delta(x-n L)\right)$$

  Assuming that $\psi(x)$ is continuous across each lattice site, and applying periodic boundary conditions, derive an equation for the allowed electron energy levels. Show that for suitable values of $U L$ they have a band structure, and calculate the number of levels in each band when $U L>2$. Verify that when $U L \gg 1$ the levels are very close to those corresponding to a solitary atom.

  Describe briefly how the band structure in a real 3-dimensional crystal differs from that of this simple model.

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Applied Probability

• B4.12

  Applied Probability | Part II, 2001

  Define a renewal process and a renewal reward process.

  State and prove the strong law of large numbers for these processes.

  [You may assume the strong law of large numbers for independent, identically-distributed random variables.

  State and prove Little's formula.

  Customers arrive according to a Poisson process with rate $\nu$ at a single server, but a restricted waiting room causes those who arrive when $n$ customers are already present to be lost. Accepted customers have service times which are independent and identicallydistributed with mean $\alpha$ and independent of the arrival process. Let $P_{j}$ be the equilibrium probability that an arriving customer finds $j$ customers already present.

  Using Little's formula, or otherwise, determine a relationship between $P_{0}, P_{n}, \nu$ and

  Part II

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Combinatorics

• B4.1

  Combinatorics | Part II, 2001

  Write an essay on extremal graph theory. You should give proofs of at least two major theorems and you should also include a description of alternative proofs or further results.

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Computational Statistics and Statistical Modelling

• A4.14

  Computational Statistics and Statistical Modelling | Part II, 2001

  (i) Assume that independent observations $Y_{1}, \ldots, Y_{n}$ are such that

  $$Y_{i} \sim \operatorname{Binomial}\left(t_{i}, \pi_{i}\right), \log \frac{\pi_{i}}{1-\pi_{i}}=\beta^{T} x_{i} \quad \text { for } 1 \leqslant i \leqslant n$$

  where $x_{1}, \ldots, x_{n}$ are given covariates. Discuss carefully how to estimate $\beta$, and how to test that the model fits.

  (ii) Carmichael et al. (1989) collected data on the numbers of 5 -year old children with "dmft", i.e. with 5 or more decayed, missing or filled teeth, classified by social class, and by whether or not their tap water was fluoridated or non-fluoridated. The numbers of such children with dmft, and the total numbers, are given in the table below:

  \begin{tabular}{l|ll} Social Class & Fluoridated & Non-fluoridated \ \hline I & $12 / 117$ & $12 / 56$ \ II & $26 / 170$ & $48 / 146$ \ III & $11 / 52$ & $29 / 64$ \ Unclassified & $24 / 118$ & $49 / 104$ \end{tabular}

  A (slightly edited) version of the $R$ output is given below. Explain carefully what model is being fitted, whether it does actually fit, and what the parameter estimates and Std. Errors are telling you. (You may assume that the factors SClass (social class) and Fl (with/without) have been correctly set up.)

  

  $\begin{array}{lrrrr} & \text { Estimate } & \text { Std. } & \text { Error } & \text { z value } \\ \text { (Intercept) } & -2.2716 & 0.2396 & -9.480 \\ \text { SClassII } & 0.5099 & 0.2628 & 1.940 \\ \text { SClassIII } & 0.9857 & 0.3021 & 3.262 \\ \text { SClassUnc } & 1.0020 & 0.2684 & 3.734 \\ \text { Flwithout } & 1.0813 & 0.1694 & 6.383\end{array}$

  

  Here 'Yes' is the vector of numbers with dmft, taking values $12,12, \ldots, 24,49$, 'Total' is the vector of Total in each category, taking values $117,56, \ldots, 118,104$, and SClass, Fl are the factors corresponding to Social class and Fluoride status, defined in the obvious way.

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Differentiable Manifolds

• B4.4

  Differentiable Manifolds | Part II, 2001

  Describe the Mayer-Vietoris exact sequence for forms on a manifold $M$ and show how to derive from it the Mayer-Vietoris exact sequence for the de Rham cohomology.

  Calculate $H^{*}\left(\mathbb{R} \mathbb{P}^{n}\right)$.

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Dynamical Systems

• B4.17

  Dynamical Systems | Part II, 2001

  Define the rotation number $\rho(f)$ of an orientation-preserving circle map $f$ and the rotation number $\rho(F)$ of a lift $F$ of $f$. Prove that $\rho(f)$ and $\rho(F)$ are well-defined. Prove also that $\rho(F)$ is a continuous function of $F$.

  State without proof the main consequence of $\rho(f)$ being rational.

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Dynamics of Differential Equations

• A4.6

  Dynamics of Differential Equations | Part II, 2001

  Write a short essay about periodic orbits in flows in two dimensions. Your essay should include criteria for the existence and non-existence of periodic orbits, and should mention (with sketches) at least two bifurcations that create or destroy periodic orbits in flows as a parameter is altered (though a detailed analysis of any bifurcation is not required).

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Electrodynamics

• B4.21

  Electrodynamics | Part II, 2001

  The Liénard-Wiechert potential for a particle of charge $q$, assumed to be moving non-relativistically along the trajectory $y^{\mu}(\tau), \tau$ being the proper time along the trajectory,

  $$A^{\mu}(x, t)=\left.\frac{\mu_{0} q}{4 \pi} \frac{d y^{\mu} / d \tau}{(x-y(\tau))_{\nu} d y^{\nu} / d \tau}\right|_{\tau=\tau_{0}} .$$

  Explain how to calculate $\tau_{0}$ given $x^{\mu}=(x, t)$ and $y^{\mu}=\left(y, t^{\prime}\right)$.

  Derive Larmor's formula for the rate at which electromagnetic energy is radiated from a particle of charge $q$ undergoing an acceleration $a$.

  Suppose that one considers the classical non-relativistic hydrogen atom with an electron of mass $m$ and charge $-e$ orbiting a fixed proton of charge $+e$, in a circular orbit of radius $r_{0}$. What is the total energy of the electron? As the electron is accelerated towards the proton it will radiate, thereby losing energy and causing the orbit to decay. Derive a formula for the lifetime of the orbit.

  Part II

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Electromagnetism

• A4.5 $\quad$

  Electromagnetism | Part II, 2001

  Write down the form of Ohm's Law that applies to a conductor if at a point $\mathbf{r}$ it is moving with velocity $\mathbf{v}(\mathbf{r})$.

  Use two of Maxwell's equations to prove that

  $$\int_{C}(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot d \mathbf{r}=-\frac{d}{d t} \int_{S} \mathbf{B} \cdot d \mathbf{S}$$

  where $C(t)$ is a moving closed loop, $\mathbf{v}$ is the velocity at the point $\mathbf{r}$ on $C$, and $S$ is a surface spanning $C$. The time derivative on the right hand side accounts for changes in both $C$ and B. Explain briefly the physical importance of this result.

  Find and sketch the magnetic field $\mathbf{B}$ described in the vector potential

  $$\mathbf{A}(r, \theta, z)=\left(0, \frac{1}{2} b r z, 0\right)$$

  in cylindrical polar coordinates $(r, \theta, z)$, where $b>0$ is constant.

  A conducting circular loop of radius $a$ and resistance $R$ lies in the plane $z=h(t)$ with its centre on the $z$-axis.

  Find the magnitude and direction of the current induced in the loop as $h(t)$ changes with time, neglecting self-inductance.

  At time $t=0$ the loop is at rest at $z=0$. For time $t>0$ the loop moves with constant velocity $d h / d t=v>0$. Ignoring the inertia of the loop, use energy considerations to find the force $F(t)$ necessary to maintain this motion.

  [ In cylindrical polar coordinates

  $$\left.\operatorname{curl} \mathbf{A}=\left(\frac{1}{r} \frac{\partial A_{z}}{\partial \theta}-\frac{\partial A_{\theta}}{\partial z}, \frac{\partial A_{r}}{\partial z}-\frac{\partial A_{z}}{\partial r}, \frac{1}{r} \frac{\partial}{\partial r}\left(r A_{\theta}\right)-\frac{1}{r} \frac{\partial A_{r}}{\partial \theta}\right)\right]$$

  Part II

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Fluid Dynamics II

• B4.26

  Fluid Dynamics II | Part II, 2001

  Starting from the steady planar vorticity equation

  $$\mathbf{u} \cdot \nabla \omega=\nu \nabla^{2} \omega,$$

  outline briefly the derivation of the boundary layer equation

  $$u u_{x}+v u_{y}=U d U / d x+\nu u_{y y},$$

  explaining the significance of the symbols used.

  Viscous fluid occupies the region $y>0$ with rigid stationary walls along $y=0$ for $x>0$ and $x<0$. There is a line sink at the origin of strength $\pi Q, Q>0$, with $Q / \nu \gg 1$. Assuming that vorticity is confined to boundary layers along the rigid walls:

  (a) Find the flow outside the boundary layers.

  (b) Explain why the boundary layer thickness $\delta$ along the wall $x>0$ is proportional to $x$, and deduce that

  $$\delta=\left(\frac{\nu}{Q}\right)^{\frac{1}{2}} x$$

  (c) Show that the boundary layer equation admits a solution having stream function

  $$\psi=(\nu Q)^{1 / 2} f(\eta) \quad \text { with } \quad \eta=y / \delta$$

  Find the equation and boundary conditions satisfied by $f$.

  (d) Verify that a solution is

  $$f^{\prime}=\frac{6}{1+\cosh (\eta \sqrt{2}+c)}-1$$

  provided that $c$ has one of two values to be determined. Should the positive or negative value be chosen?

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Foundations of Quantum Mechanics

• A4.15 B4.22

  Foundations of Quantum Mechanics | Part II, 2001

  (i) The two states of a spin- $\frac{1}{2}$ particle corresponding to spin pointing along the $z$ axis are denoted by $|\uparrow\rangle$ and $|\downarrow\rangle$. Explain why the states

  $$|\uparrow, \theta\rangle=\cos \frac{\theta}{2}|\uparrow\rangle+\sin \frac{\theta}{2}|\downarrow\rangle, \quad \quad|\downarrow, \theta\rangle=-\sin \frac{\theta}{2}|\uparrow\rangle+\cos \frac{\theta}{2}|\downarrow\rangle$$

  correspond to the spins being aligned along a direction at an angle $\theta$ to the $z$ direction.

  The spin- 0 state of two spin- $\frac{1}{2}$ particles is

  $$|0\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle_{1}|\downarrow\rangle_{2}-|\downarrow\rangle_{1}|\uparrow\rangle_{2}\right)$$

  Show that this is independent of the direction chosen to define $|\uparrow\rangle_{1,2},|\downarrow\rangle_{1,2}$. If the spin of particle 1 along some direction is measured to be $\frac{1}{2} \hbar$ show that the spin of particle 2 along the same direction is determined, giving its value.

  [The Pauli matrices are given by

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

  (ii) Starting from the commutation relation for angular momentum in the form

  $$\left[J_{3}, J_{\pm}\right]=\pm \hbar J_{\pm}, \quad\left[J_{+}, J_{-}\right]=2 \hbar J_{3},$$

  obtain the possible values of $j, m$, where $m \hbar$ are the eigenvalues of $J_{3}$ and $j(j+1) \hbar^{2}$ are the eigenvalues of $\mathbf{J}^{2}=\frac{1}{2}\left(J_{+} J_{-}+J_{-} J_{+}\right)+J_{3}^{2}$. Show that the corresponding normalized eigenvectors, $|j, m\rangle$, satisfy

  $$J_{\pm}|j, m\rangle=\hbar((j \mp m)(j \pm m+1))^{1 / 2}|j, m \pm 1\rangle,$$

  and that

  $$\frac{1}{n !} J_{-}^{n}|j, j\rangle=\hbar^{n}\left(\frac{(2 j) !}{n !(2 j-n) !}\right)^{1 / 2}|j, j-n\rangle, \quad n \leq 2 j$$

  The state $|w\rangle$ is defined by

  $$|w\rangle=e^{w J_{-} / \hbar}|j, j\rangle$$

  for any complex $w$. By expanding the exponential show that $\langle w \mid w\rangle=\left(1+|w|^{2}\right)^{2 j}$. Verify that

  $$e^{-w J_{-} / \hbar} J_{3} e^{w J_{-} / \hbar}=J_{3}-w J_{-}$$

  and hence show that

  $$J_{3}|w\rangle=\hbar\left(j-w \frac{\partial}{\partial w}\right)|w\rangle$$

  If $H=\alpha J_{3}$ verify that $\left|e^{i \alpha t}\right\rangle e^{-i j \alpha t}$ is a solution of the time-dependent Schrödinger equation.

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Functional Analysis

• A4.3

  Functional Analysis | Part II, 2001

  Write an account of the classical sequence spaces: $\ell_{p}(1 \leqslant p \leqslant \infty)$ and $c_{0}$. You should define them, prove that they are Banach spaces, and discuss their properties, including their dual spaces. Show that $\ell_{\infty}$ is inseparable but that $c_{0}$ and $\ell_{p}$ for $1 \leqslant p<\infty$ are separable.

  Prove that, if $T: X \rightarrow Y$ is an isomorphism between two Banach spaces, then

  $$T^{*}: Y^{*} \rightarrow X^{*} ; \quad f \mapsto f \circ T$$

  is an isomorphism between their duals.

  Hence, or otherwise, show that no two of the spaces $c_{0}, \ell_{1}, \ell_{2}, \ell_{\infty}$ are isomorphic.

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Galois Theory

• B4.3

  Galois Theory | Part II, 2001

  Define the concept of separability and normality for algebraic field extensions. Suppose $K=k(\alpha)$ is a simple algebraic extension of $k$, and that $\operatorname{Aut}(K / k)$ denotes the group of $k$-automorphisms of $K$. Prove that $|\operatorname{Aut}(K / k)| \leqslant[K: k]$, with equality if and only if $K / k$ is normal and separable.

  [You may assume that the splitting field of a separable polynomial $f \in k[X]$ is normal and separable over $k$.]

  Suppose now that $G$ is a finite group of automorphisms of a field $F$, and $E=F^{G}$ is the fixed subfield. Prove:

  (i) $F / E$ is separable.

  (ii) $G=\operatorname{Aut}(F / E)$ and $[F: E]=|G|$.

  (iii) $F / E$ is normal.

  [The Primitive Element Theorem for finite separable extensions may be used without proof.]

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General Relativity

• A4.17 B4.25

  General Relativity | Part II, 2001

  Discuss how Einstein's theory of gravitation reduces to Newton's in the limit of weak fields. Your answer should include discussion of: (a) the field equations; (b) the motion of a point particle; (c) the motion of a pressureless fluid.

  [The metric in a weak gravitational field, with Newtonian potential $\phi$, may be taken as

  $$d s^{2}=d x^{2}+d y^{2}+d z^{2}-(1+2 \phi) d t^{2} .$$

  The Riemann tensor is

  $$\left.R_{b c d}^{a}=\Gamma_{b d, c}^{a}-\Gamma_{b c, d}^{a}+\Gamma_{c f}^{a} \Gamma_{b d}^{f}-\Gamma_{d f}^{a} \Gamma_{b c}^{f}\right]$$

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Graph Theory

• A4.9

  Graph Theory | Part II, 2001

  Write an essay on extremal graph theory. Your essay should include proofs of at least two major results and a discussion of variations on these results or their proofs.

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Groups, Rings and Fields

• A4.4

  Groups, Rings and Fields | Part II, 2001

  Show that the ring $\mathbf{Z}[\omega]$ is Euclidean, where $\omega=\exp (2 \pi i / 3)$.

  Show that a prime number $p \neq 3$ is reducible in $\mathbf{Z}[\omega]$ if and only if $p \equiv 1(\bmod 3)$.

  Which prime numbers $p$ can be written in the form $p=a^{2}+a b+b^{2}$ with $a, b \in \mathbf{Z}$ (and why)?

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Hilbert Spaces

• B4.7

  Hilbert Spaces | Part II, 2001

  Write an essay on the use of Hermite functions in the elementary theory of the Fourier transform on $L^{2}(\mathbb{R})$.

  [You should assume, without proof, any results that you need concerning the approximation of functions by Hermite functions.]

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Information Theory

• B4.13

  Information Theory | Part II, 2001

  State the Kraft inequality. Prove that it gives a necessary and sufficient condition for the existence of a prefix-free code with given codeword lengths.

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Logic, Computation and Set Theory

• A4.8 B4.10

  Logic, Computation and Set Theory | Part II, 2001

  What is a wellfounded relation, and how does wellfoundedness underpin wellfounded induction?

  A formula $\phi(x, y)$ with two free variables defines an $\in$-automorphism if for all $x$ there is a unique $y$, the function $f$, defined by $y=f(x)$ if and only if $\phi(x, y)$, is a permutation of the universe, and we have $(\forall x y)(x \in y \leftrightarrow f(x) \in f(y))$.

  Use wellfounded induction over $\in$ to prove that all formulæ defining $\in$-automorphisms are equivalent to $x=y$.

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Markov Chains

• A4.1

  Markov Chains | Part II, 2001

  Write an essay on the convergence to equilibrium of a discrete-time Markov chain on a countable state-space. You should include a discussion of the existence of invariant distributions, and of the limit theorem in the non-null recurrent case.

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Mathematical Methods

• A4.21

  Mathematical Methods | Part II, 2001

  The equation

  $$\mathbf{A x}=\lambda \mathbf{x}$$

  where $\mathbf{A}$ is a real square matrix and $\mathbf{x}$ a column vector, has a simple eigenvalue $\lambda=\mu$ with corresponding right-eigenvector $\mathbf{x}=\mathbf{X}$. Show how to find expressions for the perturbed eigenvalue and right-eigenvector solutions of

  $$\mathbf{A} \mathbf{x}+\epsilon \mathbf{b}(\mathbf{x})=\lambda \mathbf{x}, \quad|\epsilon| \ll 1$$

  to first order in $\epsilon$, where $\mathbf{b}$ is a vector function of $\mathbf{x}$. State clearly any assumptions you make.

  If $\mathbf{A}$ is $(n \times n)$ and has a complete set of right-eigenvectors $\mathbf{X}^{(j)}, j=1,2, \ldots n$, which span $\mathbb{R}^{n}$ and correspond to separate eigenvalues $\mu^{(j)}, j=1,2, \ldots n$, find an expression for the first-order perturbation to $\mathbf{X}^{(1)}$ in terms of the $\left\{\mathbf{X}^{(j)}\right\}$ and the corresponding lefteigenvectors of $\mathbf{A}$.

  Find the normalised eigenfunctions and eigenvalues of the equation

  $$\frac{d^{2} y}{d x^{2}}+\lambda y=0,0<x<1$$

  with $y(0)=y(1)=0$. Let these be the zeroth order approximations to the eigenfunctions of

  $$\frac{d^{2} y}{d x^{2}}+\lambda y+\epsilon b(y)=0,0<x<1$$

  with $y(0)=y(1)=0$ and where $b$ is a function of $y$. Show that the first-order perturbations of the eigenvalues are given by

  $$\lambda_{n}^{(1)}=-\epsilon \sqrt{2} \int_{0}^{1} \sin (n \pi x) \quad b(\sqrt{2} \sin n \pi x) d x$$

  Part II

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Methods of Mathematical Physics

• B4.19

  Methods of Mathematical Physics | Part II, 2001

  Show that $\int_{0}^{\pi} \mathrm{e}^{i x \cos t} d t$ satisfies the differential equation

  $$x y^{\prime \prime}+y^{\prime}+x y=0$$

  and find a second solution, in the form of an integral, for $x>0$.

  Show, by finding the asymptotic behaviour as $x \rightarrow+\infty$, that your two solutions are linearly independent.

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Number Fields

• B4.6

  Number Fields | Part II, 2001

  For a prime number $p>2$, set $\zeta=e^{2 \pi i / p}, K=\mathbf{Q}(\zeta)$ and $K^{+}=\mathbf{Q}\left(\zeta+\zeta^{-1}\right)$.

  (a) Show that the (normalized) minimal polynomial of $\zeta-1$ over $\mathbf{Q}$ is equal to

  $$f(x)=\frac{(x+1)^{p}-1}{x} .$$

  (b) Determine the degrees $[K: \mathbf{Q}]$ and $\left[K^{+}: \mathbf{Q}\right]$.

  (c) Show that

  $$\prod_{j=1}^{p-1}\left(1-\zeta^{j}\right)=p$$

  (d) Show that $\operatorname{disc}(f)=(-1)^{\frac{p-1}{2}} p^{p-2}$.

  (e) Show that $K$ contains $\mathbf{Q}\left(\sqrt{p^{*}}\right)$, where $p^{*}=(-1)^{\frac{p-1}{2}} p$.

  (f) If $j, k \in \mathbf{Z}$ are not divisible by $p$, show that $\frac{1-\zeta^{j}}{1-\zeta^{k}}$ lies in $\mathcal{O}_{K}^{*}$.

  (g) Show that the ideal $(p)=p \mathcal{O}_{K}$ is equal to $(1-\zeta)^{p-1}$.

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Number Theory

• A4.10

  Number Theory | Part II, 2001

  Attempt one of the following:

  (i) Discuss pseudoprimes and primality testing. Find all bases for which 57 is a Fermat pseudoprime. Determine whether 57 is also an Euler pseudoprime for these bases.

  (ii) Write a brief account of various methods for factoring large numbers. Use Fermat factorization to find the factors of 10033. Would Pollard's $p-1$ method also be practical in this instance?

  (iii) Show that $\sum 1 / p_{n}$ is divergent, where $p_{n}$ denotes the $n$-th prime.

  Write a brief account of basic properties of the Riemann zeta-function.

  State the prime number theorem. Show that it implies that for all sufficiently large positive integers $n$ there is a prime $p$ satisfying $n<p \leqslant 2 n$.

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Numerical Analysis

• A4.22 B4.20

  Numerical Analysis | Part II, 2001

  Write an essay on the computation of eigenvalues and eigenvectors of matrices.

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Optimization and Control

• B4.14

  Optimization and Control | Part II, 2001

  Consider the scalar system with plant equation $x_{t+1}=x_{t}+u_{t}, t=0,1, \ldots$ and cost

  $$C_{s}\left(x_{0}, u_{0}, u_{1}, \ldots\right)=\sum_{t=0}^{s}\left[u_{t}^{2}+\frac{4}{3} x_{t}^{2}\right]$$

  Show from first principles that $\min _{u_{0}, u_{1}, \ldots} C_{s}=V_{s} x_{0}^{2}$, where $V_{0}=4 / 3$ and for $s=0,1, \ldots$

  $$V_{s+1}=4 / 3+V_{s} /\left(1+V_{s}\right)$$

  Show that $V_{s} \rightarrow 2$ as $s \rightarrow \infty$.

  Prove that $C_{\infty}$ is minimized by the stationary control, $u_{t}=-2 x_{t} / 3$ for all $t$.

  Consider the stationary policy $\pi_{0}$ that has $u_{t}=-x_{t}$ for all $t$. What is the value of $C_{\infty}$ under this policy?

  Consider the following algorithm, in which steps 1 and 2 are repeated as many times as desired.

  1. For a given stationary policy $\pi_{n}$, for which $u_{t}=k_{n} x_{t}$ for all $t$, determine the value of $C_{\infty}$ under this policy as $V^{\pi_{n}} x_{0}^{2}$ by solving for $V^{\pi_{n}}$ in

  $$V^{\pi_{n}}=k_{n}^{2}+4 / 3+\left(1+k_{n}\right)^{2} V^{\pi_{n}}$$

  2. Now find $k_{n+1}$ as the minimizer of

  $$k_{n+1}^{2}+4 / 3+\left(1+k_{n+1}\right)^{2} V^{\pi_{n}}$$

  and define $\pi_{n+1}$ as the policy for which $u_{t}=k_{n+1} x_{t}$ for all $t$.

  Explain why $\pi_{n+1}$ is guaranteed to be a better policy than $\pi_{n}$.

  Let $\pi_{0}$ be the stationary policy with $u_{t}=-x_{t}$. Determine $\pi_{1}$ and verify that it minimizes $C_{\infty}$ to within $0.2 \%$ of its optimum.

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Partial Differential Equations

• B4.18

  Partial Differential Equations | Part II, 2001

  Write an essay on one of the following two topics:

  (a) The notion of well-posedness for initial and boundary value problems for differential equations. Your answer should include a definition and give examples and state precise theorems for some specific problems.

  (b) The concepts of distribution and tempered distribution and their use in the study of partial differential equations.

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Principles of Dynamics

• A4.2

  Principles of Dynamics | Part II, 2001

  (i) Consider a particle of charge $q$ and mass $m$, moving in a stationary magnetic field B. Show that Lagrange's equations applied to the Lagrangian

  $$L=\frac{1}{2} m \dot{\mathbf{r}}^{2}+q \dot{\mathbf{r}} \cdot \mathbf{A}(\mathbf{r})$$

  where $\mathbf{A}$ is the vector potential such that $\mathbf{B}=\operatorname{curl} \mathbf{A}$, lead to the correct Lorentz force law. Compute the canonical momentum $\mathbf{p}$, and show that the Hamiltonian is $H=\frac{1}{2} m \dot{\mathbf{r}}^{2}$.

  (ii) Expressing the velocity components $\dot{r}_{i}$ in terms of the canonical momenta and co-ordinates for the above system, derive the following formulae for Poisson brackets: (b) $\left\{m \dot{r}_{i}, m \dot{r}_{j}\right\}=q \epsilon_{i j k} B_{k}$; (c) $\left\{m \dot{r}_{i}, r_{j}\right\}=-\delta_{i j}$; (d) $\left\{m \dot{r}_{i}, f\left(r_{j}\right)\right\}=-\frac{\partial}{\partial r_{i}} f\left(r_{j}\right)$.

  (a) $\{F G, H\}=F\{G, H\}+\{F, H\} G$, for any functions $F, G, H$;

  Now consider a particle moving in the field of a magnetic monopole,

  $$B_{i}=g \frac{r_{i}}{r^{3}} .$$

  Show that $\{H, \mathbf{J}\}=0$, where $\mathbf{J}=m \mathbf{r} \wedge \dot{\mathbf{r}}-g q \hat{\mathbf{r}}$. Explain why this means that $\mathbf{J}$ is conserved.

  Show that, if $g=0$, conservation of $\mathbf{J}$ implies that the particle moves in a plane perpendicular to $\mathbf{J}$. What type of surface does the particle move on if $g \neq 0$ ?

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Principles of Statistics

• A4.13 B4.15

  Principles of Statistics | Part II, 2001

  Write an account, with appropriate examples, of one of the following:

  (a) Inference in multi-parameter exponential families;

  (b) Asymptotic properties of maximum-likelihood estimators and their use in hypothesis testing;

  (c) Bootstrap inference.

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Probability and Measure

• B4.11

  Probability and Measure | Part II, 2001

  State the first and second Borel-Cantelli Lemmas and the Kolmogorov 0-1 law.

  Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of independent random variables with distribution given

  by

  $$\mathbb{P}\left(X_{n}=n\right)=\frac{1}{n}=1-\mathbb{P}\left(X_{n}=0\right)$$

  and set $S_{n}=\sum_{i=1}^{n} X_{i}$.

  (a) Show that there exist constants $0 \leqslant c_{1} \leqslant c_{2} \leqslant \infty$ such that $\lim \inf _{n}\left(S_{n} / n\right)=c_{1}$, almost surely and $\lim \sup _{n}\left(S_{n} / n\right)=c_{2}$ almost surely.

  (b) Let $Y_{k}=\sum_{i=k+1}^{2 k} X_{i}$ and $\tilde{Y}_{k}=\sum_{i=1}^{k} Z_{i}^{(k)}$, where $\left(Z_{i}^{(k)}\right)_{i=1}^{k}$ are independent with

  $$\mathbb{P}\left(Z_{i}^{(k)}=k\right)=\frac{1}{2 k}=1-\mathbb{P}\left(Z_{i}^{(k)}=0\right), \quad 1 \leqslant i \leqslant k,$$

  and suppose that $\alpha \in \mathbb{Z}^{+}$.

  Use the fact that $\mathbb{P}\left(Y_{k} \geqslant \alpha k\right) \geqslant \mathbb{P}\left(\tilde{Y}_{k} \geqslant \alpha k\right)$ to show that there exists $p_{\alpha}>0$ such that $\mathbb{P}\left(Y_{k} \geqslant \alpha k\right) \geqslant p_{\alpha}$ for all sufficiently large $k$.

  [You may use the Poisson approximation to the binomial distribution without proof.]

  By considering a suitable subsequence of $\left(Y_{k}\right)$, or otherwise, show that $c_{2}=\infty$.

  (c) Show that $c_{1} \leqslant 1$. Consider an appropriately chosen sequence of random times $T_{i}$, with $2 T_{i} \leqslant T_{i+1}$, for which $\left(S_{T_{i}} / T_{i}\right) \leqslant 3 c_{1} / 2$. Using the fact that the random variables $\left(Y_{T_{i}}\right)$ are independent, and by considering the events $\left\{Y_{T_{i}}=0\right\}$, or otherwise, show that $c_{1}=0$.

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Quantum Physics

• A4.16

  Quantum Physics | Part II, 2001

  A harmonic oscillator of frequency $\omega$ is in thermal equilibrium with a heat bath at temperature $T$. Show that the mean number of quanta $n$ in the oscillator is

  $$n=\frac{1}{e^{\hbar \omega / k T}-1} .$$

  Use this result to show that the density of photons of frequency $\omega$ for cavity radiation at temperature $T$ is

  $$n(\omega)=\frac{\omega^{2}}{\pi^{2} c^{3}} \frac{1}{e^{\hbar \omega / k T}-1}$$

  By considering this system in thermal equilibrium with a set of distinguishable atoms, derive formulae for the Einstein $A$ and $B$ coefficients.

  Give a brief description of the operation of a laser.

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Representation Theory

• B4.2

  Representation Theory | Part II, 2001

  Let $G$ be the Heisenberg group of order $p^{3}$. This is the subgroup

  $$G=\left\{\left(\begin{array}{ccc} 1 & a & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right) \mid a, b, x \in \mathbf{F}_{p}\right\}$$

  of $3 \times 3$ matrices over the finite field $\mathbf{F}_{p}$ ( $p$ prime). Let $H$ be the subgroup of $G$ of such matrices with $a=0$.

  (i) Find all one dimensional representations of $G$.

  [You may assume without proof that $[G, G]$ is equal to the set of matrices in $G$ with $a=b=0 .]$

  (ii) Let $\psi: \mathbf{F}_{p}=\mathbf{Z} / p \mathbf{Z} \longrightarrow \mathbf{C}^{*}$ be a non-trivial one dimensional representation of $\mathbf{F}_{p}$, and define a one dimensional representation $\rho$ of $H$ by

  $$\rho\left(\begin{array}{lll} 1 & 0 & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right)=\psi(x)$$

  Show that $V_{\psi}=\operatorname{Ind}_{H}^{G}(\rho)$ is irreducible.

  (iii) List all the irreducible representations of $G$ and explain why your list is complete.

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Riemann Surfaces

• B4.8

  Riemann Surfaces | Part II, 2001

  Let $\lambda$ and $\mu$ be fixed, non-zero complex numbers, with $\lambda / \mu \notin \mathbb{R}$, and let $\Lambda=\mathbb{Z} \mu+\mathbb{Z} \lambda$ be the lattice they generate in $\mathbb{C}$. The series

  $$\wp(z)=\frac{1}{z^{2}}+\sum_{m, n}\left[\frac{1}{(z-m \lambda-n \mu)^{2}}-\frac{1}{(m \lambda+n \mu)^{2}}\right]$$

  with the sum taken over all pairs $(m, n) \in \mathbb{Z} \times \mathbb{Z}$ other than $(0,0)$, is known to converge to an elliptic function, meaning a meromorphic function $\wp: \mathbb{C} \rightarrow \mathbb{C} \cup\{\infty\}$ satisfying $\wp(z)=\wp(z+\mu)=\wp(z+\lambda)$ for all $z \in \mathbb{C}$. ( $\wp$ is called the Weierstrass function.)

  (a) Find the three zeros of $\wp^{\prime}$ modulo $\Lambda$, explaining why there are no others.

  (b) Show that, for any number $a \in \mathbb{C}$, other than the three values $\wp(\lambda / 2), \wp(\mu / 2)$ and $\wp((\lambda+\mu) / 2)$, the equation $\wp(z)=a$ has exactly two solutions, modulo $\Lambda$; whereas, for each of the specified values, it has a single solution.

  [In (a) and (b), you may use, without proof, any known results about valencies and degrees of holomorphic maps between compact Riemann surfaces, provided you state them correctly.]

  (c) Prove that every even elliptic function $\phi(z)$ is a rational function of $\wp(z)$; that is, there exists a rational function $R$ for which $\phi(z)=R(\wp(z))$.

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Statistical Physics

• B4.23

  Statistical Physics | Part II, 2001

  Given that the free energy $F$ can be written in terms of the partition function $Z$ as $F=-k T \log Z$ show that the entropy $S$ and internal energy $E$ are given by

  $$S=k \frac{\partial(T \log Z)}{\partial T}, \quad E=k T^{2} \frac{\partial \log Z}{\partial T}$$

  A system of particles has Hamiltonian $H(\mathbf{p}, \mathbf{q})$ where $\mathbf{p}$ is the set of particle momenta and $\mathbf{q}$ the set of particle coordinates. Write down the expression for the classical partition function $Z_{C}$ for this system in equilibrium at temperature $T$. In a particular case $H$ is given by

  $$H(\mathbf{p}, \mathbf{q})=p_{\alpha} A_{\alpha \beta}(\mathbf{q}) p_{\beta}+q_{\alpha} B_{\alpha \beta}(\mathbf{q}) q_{\beta}$$

  Let $H$ be a homogeneous function in all the $p_{\alpha}, 1 \leq \alpha \leq N$, and in a subset of the $q_{\alpha}, 1 \leq \alpha \leq M(M \leq N)$. Derive the principle of equipartition for this system.

  A system consists of $N$ weakly interacting harmonic oscillators each with Hamiltonian

  $$h(p, q)=\frac{1}{2} p^{2}+\frac{1}{2} \omega^{2} q^{2} .$$

  Using equipartition calculate the classical specific heat of the system, $C_{C}(T)$. Also calculate the classical entropy $S_{C}(T)$.

  Write down the expression for the quantum partition function of the system and derive expressions for the specific heat $C(T)$ and the entropy $S(T)$ in terms of $N$ and the parameter $\theta=\hbar \omega / k T$. Show for $\theta \ll 1$ that

  $$C(T)=C_{C}(T)+O(\theta), \quad S(T)=S_{C}(T)+S_{0}+O(\theta)$$

  where $S_{0}$ should be calculated. Comment briefly on the physical significance of the constant $S_{0}$ and why it is non-zero.

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Statistical Physics and Cosmology

• A4.18

  Statistical Physics and Cosmology | Part II, 2001

  (i) Given that $g(p) d p$ is the number of eigenstates of a gas particle with momentum between $p$ and $p+d p$, write down the Bose-Einstein distribution $\bar{n}(p)$ for the average number of particles with momentum between $p$ and $p+d p$, as a function of temperature $T$ and chemical potential $\mu$.

  Given that $\mu=0$ and $g(p)=8 \pi \frac{V p^{2}}{h^{3}}$ for a gas of photons, obtain a formula for the energy density $\rho_{T}$ at temperature $T$ in the form

  $$\rho_{T}=\int_{0}^{\infty} \epsilon_{T}(\nu) d \nu,$$

  where $\epsilon_{T}(\nu)$ is a function of the photon frequency $\nu$ that you should determine. Hence show that the value $\nu_{\text {peak }}$ of $\nu$ at the maximum of $\epsilon_{T}(\nu)$ is proportional to $T$.

  A thermally isolated photon gas undergoes a slow change of its volume $V$. Why is $\bar{n}(p)$ unaffected by this change? Use this fact to show that $V T^{3}$ remains constant.

  (ii) According to the "Hot Big Bang" theory, the Universe evolved by expansion from an earlier state in which it was filled with a gas of electrons, protons and photons (with $n_{e}=n_{p}$ ) at thermal equilibrium at a temperature $T$ such that

  $$2 m_{e} c^{2} \gg k T \gg B$$

  where $m_{e}$ is the electron mass and $B$ is the binding energy of a hydrogen atom. Why must the composition have been different when $k T \gg 2 m_{e} c^{2}$ ? Why must it change as the temperature falls to $k T \ll B$ ? Why does this lead to a thermal decoupling of radiation from matter?

  The baryon number of the Universe can be taken to be the number of protons, either as free particles or as hydrogen atom nuclei. Let $n_{b}$ be the baryon number density and $n_{\gamma}$ the photon number density. Why is the ratio $\eta=n_{b} / n_{\gamma}$ unchanged by the expansion of the universe? Given that $\eta \ll 1$, obtain an estimate of the temperature $T_{D}$ at which decoupling occurs, as a function of $\eta$ and $B$. How does this decoupling lead to the concept of a "surface of last scattering" and a prediction of a Cosmic Microwave Background Radiation (CMBR)?

  Part II

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Stochastic Financial Models

• A4.12 B4.16

  Stochastic Financial Models | Part II, 2001

  Write an essay on the mean-variance approach to portfolio selection in a one-period model. Your essay should contrast the solution in the case when all the assets are risky with that for the case when there is a riskless asset.

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Theoretical Geophysics

• A4.20

  Theoretical Geophysics | Part II, 2001

  Write down expressions for the phase speed $c$ and group velocity $c_{g}$ in one dimension for general waves of the form $A \exp [i(k x-\omega t)]$ with dispersion relation $\omega(k)$. Briefly indicate the physical significance of $c$ and $c_{g}$ for a wavetrain of finite size.

  The dispersion relation for internal gravity waves with wavenumber $\mathbf{k}=(k, 0, m)$ in an incompressible stratified fluid with constant buoyancy frequency $N$ is

  $$\omega=\frac{\pm N k}{\left(k^{2}+m^{2}\right)^{1 / 2}} .$$

  Calculate the group velocity $\mathbf{c}_{g}$ and show that it is perpendicular to $\mathbf{k}$. Show further that the horizontal components of $\mathbf{k} / \omega$ and $\mathbf{c}_{g}$ have the same sign and that the vertical components have the opposite sign.

  The vertical velocity $w$ of small-amplitude internal gravity waves is governed by

  $$\frac{\partial^{2}}{\partial t^{2}}\left(\nabla^{2} w\right)+N^{2} \nabla_{h}^{2} w=0$$

  where $\nabla_{h}^{2}$ is the horizontal part of the Laplacian and $N$ is constant.

  Find separable solutions to $(*)$ of the form $w(x, z, t)=X(x-U t) Z(z)$ corresponding to waves with constant horizontal phase speed $U>0$. Comment on the nature of these solutions for $0<k<N / U$ and for $k>N / U$.

  A semi-infinite stratified fluid occupies the region $z>h(x, t)$ above a moving lower boundary $z=h(x, t)$. Construct the solution to $(*)$ for the case $h=\epsilon \sin [k(x-U t)]$, where $\epsilon$ and $k$ are constants and $\epsilon \ll 1$.

  Sketch the orientation of the wavecrests, the propagation direction and the group velocity for the case $0<k<N / U$.

  Part II

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Transport Processes

• A4.19

  Transport Processes | Part II, 2001

  Fluid flows in the $x$-direction past the infinite plane $y=0$ with uniform but timedependent velocity $U(t)=U_{0} G\left(t / t_{0}\right)$, where $G$ is a positive function with timescale $t_{0}$. A long region of the plane, $0<x<L$, is heated and has temperature $T_{0}\left(1+\gamma(x / L)^{n}\right)$, where $T_{0}, \gamma, n$ are constants $[\gamma=O(1)]$; the remainder of the plane is insulating $\left(T_{y}=0\right)$. The fluid temperature far from the heated region is $T_{0}$. A thermal boundary layer is formed over the heated region. The full advection-diffusion equation for temperature $T(x, y, t)$ is

  $$T_{t}+U(t) T_{x}=D\left(T_{y y}+T_{x x}\right),$$

  where $D$ is the thermal diffusivity. By considering the steady case $(G \equiv 1)$, derive a scale for the thickness of the boundary layer, and explain why the term $T_{x x}$ in (1) can be neglected if $U_{0} L / D \gg 1$.

  Neglecting $T_{x x}$, use the change of variables

  $$\tau=\frac{t}{t_{0}}, \quad \xi=\frac{x}{L}, \quad \eta=y\left[\frac{U(t)}{D x}\right]^{1 / 2}, \quad \frac{T-T_{0}}{T_{0}}=\gamma\left(\frac{x}{L}\right)^{n} f(\xi, \eta, \tau)$$

  to transform the governing equation to

  $$f_{\eta \eta}+\frac{1}{2} \eta f_{\eta}-n f=\xi f_{\xi}+\frac{L \xi}{t_{0} U_{0}}\left(\frac{G_{\tau}}{2 G^{2}} \eta f_{\eta}+\frac{1}{G} f_{\tau}\right)$$

  Write down the boundary conditions to be satisfied by $f$ in the region $0<\xi<1$.

  In the case in which $U$ is slowly-varying, so $\epsilon=\frac{L}{t_{0} U_{0}} \ll 1$, consider a solution for $f$ in the form

  $$f=f_{0}(\eta)+\epsilon f_{1}(\xi, \eta, \tau)+O\left(\epsilon^{2}\right) .$$

  Explain why $f_{0}$ is independent of $\xi$ and $\tau$.

  Henceforth take $n=\frac{1}{2}$. Calculate $f_{0}(\eta)$ and show that

  $$f_{1}=\frac{G_{\tau} \xi}{G^{2}} g(\eta)$$

  where $g$ satisfies the ordinary differential equation

  $$g^{\prime \prime}+\frac{1}{2} \eta g^{\prime}-\frac{3}{2} g=\frac{-\eta}{4} \int_{\eta}^{\infty} e^{-u^{2} / 4} d u .$$

  State the boundary conditions on $g(\eta)$.

  The heat transfer per unit length of the heated region is $-\left.D T_{y}\right|_{y=0}$. Use the above results to show that the total rate of heat transfer is

  $$T_{0}[D L U(t)]^{1 / 2} \frac{\gamma}{2}\left\{\sqrt{\pi}-\frac{\epsilon G_{\tau}}{G^{2}} g^{\prime}(0)+O\left(\epsilon^{2}\right)\right\}$$

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Waves in Fluid and Solid Media

• B4.27

  Waves in Fluid and Solid Media | Part II, 2001

  Derive the ray-tracing equations governing the evolution of a wave packet $\phi(\mathbf{x}, t)=$ $A(\mathbf{x}, t) \exp \{i \psi(\mathbf{x}, t)\}$ in a slowly varying medium, stating the conditions under which the equations are valid.

  Consider now a stationary obstacle in a steadily moving homogeneous two-dimensional medium which has the dispersion relation

  $$\omega\left(k_{1}, k_{2}\right)=\alpha\left(k_{1}^{2}+k_{2}^{2}\right)^{1 / 4}-V k_{1}$$

  where $(V, 0)$ is the velocity of the medium. The obstacle generates a steady wave system. Writing $\left(k_{1}, k_{2}\right)=\kappa(\cos \phi, \sin \phi)$, show that the wave satisfies

  $$\kappa=\frac{\alpha^{2}}{V^{2} \cos ^{2} \phi}$$

  Show that the group velocity of these waves can be expressed as

  $$\mathbf{c}_{g}=V\left(\frac{1}{2} \cos ^{2} \phi-1, \frac{1}{2} \cos \phi \sin \phi\right) .$$

  Deduce that the waves occupy a wedge of semi-angle $\sin ^{-1} \frac{1}{3}$ about the negative $x_{1}$-axis.

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