Part II, 2013, Paper 1

# Part II, 2013, Paper 1

### Jump to course

Paper 1, Section II, H

commentLet $V \subset \mathbb{A}^{n}$ be an affine variety over an algebraically closed field $k$. What does it mean to say that $V$ is irreducible? Show that any non-empty affine variety $V \subset \mathbb{A}^{n}$ is the union of a finite number of irreducible affine varieties $V_{j} \subset \mathbb{A}^{n}$.

Define the ideal $I(V)$ of $V$. Show that $I(V)$ is a prime ideal if and only if $V$ is irreducible.

Assume that the base field $k$ has characteristic zero. Determine the irreducible components of

$V\left(X_{1} X_{2}, X_{1} X_{3}+X_{2}^{2}-1, X_{1}^{2}\left(X_{1}-X_{3}\right)\right) \subset \mathbb{A}^{3}$

Paper 1, Section II, 21G

comment(i) Define the notion of the fundamental group $\pi_{1}\left(X, x_{0}\right)$ of a path-connected space $X$ with base point $x_{0}$.

(ii) Prove that if a group $G$ acts freely and properly discontinuously on a simply connected space $Z$, then $\pi_{1}\left(G \backslash Z, x_{0}\right)$ is isomorphic to $G$. [You may assume the homotopy lifting property, provided that you state it clearly.]

(iii) Suppose that $p, q$ are distinct points on the 2 -sphere $S^{2}$ and that $X=S^{2} /(p \sim q)$. Exhibit a simply connected space $Z$ with an action of a group $G$ as in (ii) such that $X=G \backslash Z$, and calculate $\pi_{1}\left(X, x_{0}\right)$.

Paper 1, Section II, D

commentConsider a quantum system with Hamiltonian $\widehat{H}$and energy levels

$E_{0}<E_{1}<E_{2}<\ldots$

For any state $|\psi\rangle$ define the Rayleigh-Ritz quotient $R[\psi]$ and show the following:

(i) the ground state energy $E_{0}$ is the minimum value of $R[\psi]$;

(ii) all energy eigenstates are stationary points of $R[\psi]$ with respect to variations of $|\psi\rangle$.

Under what conditions can the value of $R\left[\psi_{\alpha}\right]$ for a trial wavefunction $\psi_{\alpha}$ (depending on some parameter $\alpha$ ) be used as an estimate of the energy $E_{1}$ of the first excited state? Explain your answer.

For a suitably chosen trial wavefunction which is the product of a polynomial and a Gaussian, use the Rayleigh-Ritz quotient to estimate $E_{1}$ for a particle of mass $m$ moving in a potential $V(x)=g|x|$, where $g$ is a constant.

[You may use the integral formulae,

$\begin{aligned} \int_{0}^{\infty} x^{2 n} \exp \left(-p x^{2}\right) d x &=\frac{(2 n-1) ! !}{2(2 p)^{n}} \sqrt{\frac{\pi}{p}} \\ \int_{0}^{\infty} x^{2 n+1} \exp \left(-p x^{2}\right) d x &=\frac{n !}{2 p^{n+1}} \end{aligned}$

where $n$ is a non-negative integer and $p$ is a constant. ]

Paper 1, Section II, J

commentLet $\left(X_{t}, t \geqslant 0\right)$ be a Markov chain on $\{0,1, \ldots\}$ with $Q$-matrix given by

$\begin{aligned} q_{n, n+1} &=\lambda_{n} \\ q_{n, 0} &=\lambda_{n} \varepsilon_{n} \quad(n>0) \\ q_{n, m} &=0 \quad \text { if } m \notin\{0, n, n+1\} \end{aligned}$

where $\varepsilon_{n}, \lambda_{n}>0$.

(i) Show that $X$ is transient if and only if $\sum_{n} \varepsilon_{n}<\infty$. [You may assume without proof that $x(1-\delta) \leqslant \log (1+x) \leqslant x$ for all $\delta>0$ and all sufficiently small positive $x$.]

(ii) Assume that $\sum_{n} \varepsilon_{n}<\infty$. Find a necessary and sufficient condition for $X$ to be almost surely explosive. [You may assume without proof standard results about pure birth processes, provided that they are stated clearly.]

(iii) Find a stationary measure for $X$. For the case $\lambda_{n}=\lambda$ and $\varepsilon_{n}=\alpha /(n+1)(\lambda, \alpha>0)$, show that $X$ is positive recurrent if and only if $\alpha>1$.

Paper 1, Section II, B

commentSuppose $\alpha>0$. Define what it means to say that

$F(x) \sim \frac{1}{\alpha x} \sum_{n=0}^{\infty} n !\left(\frac{-1}{\alpha x}\right)^{n}$

is an asymptotic expansion of $F(x)$ as $x \rightarrow \infty$. Show that $F(x)$ has no other asymptotic expansion in inverse powers of $x$ as $x \rightarrow \infty$.

To estimate the value of $F(x)$ for large $x$, one may use an optimal truncation of the asymptotic expansion. Explain what is meant by this, and show that the error is an exponentially small quantity in $x$.

Derive an integral respresentation for a function $F(x)$ with the above asymptotic expansion.

Paper 1, Section I, B

commentConsider an $n$-dimensional dynamical system with generalized coordinates and momenta $\left(q_{i}, p_{i}\right), i=1,2, \ldots, n$.

(a) Define the Poisson bracket $\{f, g\}$ of two functions $f\left(q_{i}, p_{i}, t\right)$ and $g\left(q_{i}, p_{i}, t\right)$.

(b) Assuming Hamilton's equations of motion, prove that if a function $G\left(q_{i}, p_{i}\right)$ Poisson commutes with the Hamiltonian, that is $\{G, H\}=0$, then $G$ is a constant of the motion.

(c) Assume that $q_{j}$ is an ignorable coordinate, that is the Hamiltonian does not depend on it explicitly. Using the formalism of Poisson brackets prove that the conjugate momentum $p_{j}$ is conserved.

Paper 1, Section I, H

commentA binary Huffman code is used for encoding symbols $1, \ldots, m$ occurring with respective probabilities $p_{1} \geqslant \cdots \geqslant p_{m}>0$ where $\sum_{1 \leqslant j \leqslant m} p_{j}=1$. Let $s_{1}$ be the length of a shortest codeword and $s_{m}$ the length of a longest codeword. Determine the maximal and minimal values of each of $s_{1}$ and $s_{m}$, and find binary trees for which they are attained.

Paper 1, Section II, H

commentDefine the bar product $C_{1} \mid C_{2}$ of binary linear codes $C_{1}$ and $C_{2}$, where $C_{2}$ is a subcode of $C_{1}$. Relate the rank and minimum distance of $C_{1} \mid C_{2}$ to those of $C_{1}$ and $C_{2}$ and justify your answer. Show that if $C^{\perp}$ denotes the dual code of $C$, then

$\left(C_{1} \mid C_{2}\right)^{\perp}=C_{2}^{\perp} \mid C_{1}^{\perp}$

Using the bar product construction, or otherwise, define the Reed-Muller code $\mathrm{RM}(d, r)$ for $0 \leqslant r \leqslant d$. Show that if $0 \leqslant r \leqslant d-1$, then the dual of $\mathrm{RM}(d, r)$ is again a Reed-Muller code.

Paper 1, Section I, D

commentThe Friedmann equation and the fluid conservation equation for a closed isotropic and homogeneous cosmology are given by

$\begin{aligned} &\frac{\dot{a}^{2}}{a^{2}}=\frac{8 \pi G \rho}{3}-\frac{1}{a^{2}} \\ &\dot{\rho}+3 \frac{\dot{a}}{a}(\rho+P)=0 \end{aligned}$

where the speed of light is set equal to unity, $G$ is the gravitational constant, $a(t)$ is the expansion scale factor, $\rho$ is the fluid mass density and $P$ is the fluid pressure, and overdots denote differentiation with respect to the time coordinate $t$.

If the universe contains only blackbody radiation and $a=0$ defines the zero of time $t$, show that

$a^{2}(t)=t\left(t_{*}-t\right)$

where $t_{*}$ is a constant. What is the physical significance of the time $t_{*}$ ? What is the value of the ratio $a(t) / t$ at the time when the scale factor is largest? Sketch the curve of $a(t)$ and identify its geometric shape.

Briefly comment on whether this cosmological model is a good description of the observed universe at any time in its history.

Paper 1, Section II, D

commentA spherically symmetric star of total mass $M_{s}$ has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is the radial distance from its centre. These quantities are related by the equations of hydrostatic equilibrium and mass conservation:

$\begin{aligned} \frac{d P}{d r} &=-\frac{G M(r) \rho}{r^{2}} \\ \frac{d M}{d r} &=4 \pi \rho r^{2} \end{aligned}$

where $M(r)$ is the mass inside radius $r$.

By integrating from the centre of the star at $r=0$, where $P=P_{c}$, to the surface of the star at $r=R_{s}$, where $P=P_{s}$, show that

$4 \pi R_{s}^{3} P_{s}=\Omega+3 \int_{0}^{M_{s}} \frac{P}{\rho} d M,$

where $\Omega$ is the total gravitational potential energy. Show that

$-\Omega>\frac{G M_{s}^{2}}{2 R_{s}}$

If the surface pressure is negligible and the star is a perfect gas of particles of mass $m$ with number density $n$ and $P=n k_{B} T$ at temperature $T$, and radiation pressure can be ignored, then show that

$3 \int_{0}^{M_{s}} \frac{P}{\rho} d M=\frac{3 k_{B}}{m} \bar{T},$

where $\bar{T}$ is the mean temperature of the star, which you should define.

Hence, show that the mean temperature of the star satisfies the inequality

$\bar{T}>\frac{G M_{s} m}{6 k_{B} R_{s}}$

Paper 1, Section II, H

commentFor $f: X \rightarrow Y$ a smooth map of manifolds, define the concepts of critical point, critical value and regular value.

With the obvious identification of $\mathbf{C}$ with $\mathbf{R}^{2}$, and hence also of $\mathbf{C}^{3}$ with $\mathbf{R}^{6}$, show that the complex-valued polynomial $z_{1}^{3}+z_{2}^{2}+z_{3}^{2}$ determines a smooth map $f: \mathbf{R}^{6} \rightarrow \mathbf{R}^{2}$ whose only critical point is at the origin. Hence deduce that $V:=f^{-1}((0,0)) \backslash\{\mathbf{0}\} \subset \mathbf{R}^{6}$ is a 4-dimensional manifold, and find the equations of its tangent space at any given point $\left(z_{1}, z_{2}, z_{3}\right) \in V$.

Now let $S^{5} \subset \mathbf{C}^{3}=\mathbf{R}^{6}$ be the unit 5 -sphere, defined by $\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}=1$. Given a point $P=\left(z_{1}, z_{2}, z_{3}\right) \in S^{5} \cap V$, by considering the vector $\left(2 z_{1}, 3 z_{2}, 3 z_{3}\right) \in \mathbf{C}^{3}=\mathbf{R}^{6}$ or otherwise, show that not all tangent vectors to $V$ at $P$ are tangent to $S^{5}$. Deduce that $S^{5} \cap V \subset \mathbf{R}^{6}$ is a compact three-dimensional manifold.

[Standard results may be quoted without proof if stated carefully.]

Paper 1, Section I, C

commentConsider the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$ which has a hyperbolic fixed point at the origin.

Define the stable and unstable invariant subspaces of the system linearised about the origin. Give a constraint on the dimensions of these two subspaces.

Define the local stable and unstable manifolds of the origin for the system. How are these related to the invariant subspaces of the linearised system?

For the system

$\begin{aligned} &\dot{x}=-x+x^{2}+y^{2} \\ &\dot{y}=y+y^{2}-x^{2} \end{aligned}$

calculate the stable and unstable manifolds of the origin, each correct up to and including cubic order.

Paper 1, Section II, 36B

comment(i) Starting from

$F^{\mu \nu}=\left(\begin{array}{cccc} 0 & E_{1} / c & E_{2} / c & E_{3} / c \\ -E_{1} / c & 0 & B_{3} & -B_{2} \\ -E_{2} / c & -B_{3} & 0 & B_{1} \\ -E_{3} / c & B_{2} & -B_{1} & 0 \end{array}\right)$

and performing a Lorentz transformation with $\gamma=1 / \sqrt{1-u^{2} / c^{2}}$, using

$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma u / c & 0 & 0 \\ -\gamma u / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

show how $\mathbf{E}$ and $\mathbf{B}$ transform under a Lorentz transformation.

(ii) By taking the limit $c \rightarrow \infty$, obtain the behaviour of $\mathbf{E}$ and $\mathbf{B}$ under a Galilei transfomation and verify the invariance under Galilei transformations of the nonrelativistic equation

$m \frac{d \mathbf{v}}{d t}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})$

(iii) Show that Maxwell's equations admit solutions of the form

$\mathbf{E}=\mathbf{E}_{0} f(t-\mathbf{n} \cdot \mathbf{x} / c), \quad \mathbf{B}=\mathbf{B}_{0} f(t-\mathbf{n} \cdot \mathbf{x} / c)$

where $f$ is an arbitrary function, $\mathbf{n}$ is a unit vector, and the constant vectors $\mathbf{E}_{0}$ and $\mathbf{B}_{0}$ are subject to restrictions which should be stated.

(iv) Perform a Galilei transformation of a solution $(\star)$, with $\mathbf{n}=(1,0,0)$. Show that, by a particular choice of $u$, the solution may brought to the form

$\tilde{\mathbf{E}}=\tilde{\mathbf{E}}_{0} g(\tilde{x}), \quad \tilde{\mathbf{B}}=\tilde{\mathbf{B}}_{0} g(\tilde{x}),$

where $g$ is an arbitrary function and $\tilde{x}$ is a spatial coordinate in the rest frame. By showing that $(t)$ is not a solution of Maxwell's equations in the boosted frame, conclude that Maxwell's equations are not invariant under Galilei transformations.

Paper 1, Section II, A

commentThe velocity field $\mathbf{u}$ and stress tensor $\sigma$ satisfy the Stokes equations in a volume $V$ bounded by a surface $S$. Let $\hat{\mathbf{u}}$ be another solenoidal velocity field. Show that

$\int_{S} \sigma_{i j} n_{j} \hat{u}_{i} d S=\int_{V} 2 \mu e_{i j} \hat{e}_{i j} d V$

where $\mathbf{e}$ and $\hat{\mathbf{e}}$ are the strain-rates corresponding to the velocity fields $\mathbf{u}$ and $\hat{\mathbf{u}}$ respectively, and $\mathbf{n}$ is the unit normal vector out of $V$. Hence, or otherwise, prove the minimum dissipation theorem for Stokes flow.

A particle moves at velocity $\mathbf{U}$ through a highly viscous fluid of viscosity $\mu$ contained in a stationary vessel. As the particle moves, the fluid exerts a drag force $\mathbf{F}$ on it. Show that

$-\mathbf{F} \cdot \mathbf{U}=\int_{V} 2 \mu e_{i j} e_{i j} d V .$

Consider now the case when the particle is a small cube, with sides of length $\ell$, moving in a very large vessel. You may assume that

$\mathbf{F}=-k \mu \ell \mathbf{U}$

for some constant $k$. Use the minimum dissipation theorem, being careful to declare the domain(s) involved, to show that

$3 \pi \leqslant k \leqslant 3 \sqrt{3} \pi .$

[You may assume Stokes' result for the drag on a sphere of radius $a, \mathbf{F}=-6 \pi \mu a \mathbf{U}$.]

Paper 1, Section I, E

commentProve that there are no second order linear ordinary homogeneous differential equations for which all points in the extended complex plane are analytic.

Find all such equations which have one regular singular point at $z=0$.

Paper 1, Section II, E

commentShow that the equation

$(z-1) w^{\prime \prime}-z w^{\prime}+(4-2 z) w=0$

has solutions of the form $w(z)=\int_{\gamma} \exp (z t) f(t) d t$, where

$f(t)=\frac{\exp (-t)}{(t-a)(t-b)^{2}}$

and the contour $\gamma$ is any closed curve in the complex plane, where $a$ and $b$ are real constants which should be determined.

Use this to find the general solution, evaluating the integrals explicitly.

Paper 1, Section II, I

comment(i) Give an example of a field $F$, contained in $\mathbb{C}$, such that $X^{4}+1$ is a product of two irreducible quadratic polynomials in $F[X]$. Justify your answer.

(ii) Let $F$ be any extension of degree 3 over $\mathbb{Q}$. Prove that the polynomial $X^{4}+1$ is irreducible over $F$.

(iii) Give an example of a prime number $p$ such that $X^{4}+1$ is a product of two irreducible quadratic polynomials in $\mathbb{F}_{p}[X]$. Justify your answer.

[Standard facts on fields, extensions, and finite fields may be quoted without proof, as long as they are stated clearly.]

Paper 1, Section II, 37D

commentThe curve $\gamma, x^{a}=x^{a}(\lambda)$, is a geodesic with affine parameter $\lambda$. Write down the geodesic equation satisfied by $x^{a}(\lambda)$.

Suppose the parameter is changed to $\mu(\lambda)$, where $d \mu / d \lambda>0$. Obtain the corresponding equation and find the condition for $\mu$ to be affine. Deduce that, whatever parametrization $\nu$ is used along the curve $\gamma$, the tangent vector $K^{a}$ to $\gamma$ satisfies

$\left(\nabla_{\nu} K\right)^{[a} K^{b]}=0 \text {. }$

Now consider a spacetime with metric $g_{a b}$, and conformal transformation

$\tilde{g}_{a b}=\Omega^{2}\left(x^{c}\right) g_{a b}$

The curve $\gamma$ is a geodesic of the metric connection of $g_{a b}$. What further restriction has to be placed on $\gamma$ so that it is also a geodesic of the metric connection of $\tilde{g}_{a b}$ ? Justify your answer.

Paper 1, Section I, G

commentShow that any pair of lines in hyperbolic 3-space that does not have a common endpoint must have a common normal. Is this still true when the pair of lines does have a common endpoint?

Paper 1, Section II, G

commentDefine the modular group $\Gamma$ acting on the upper half-plane.

Describe the set $S$ of points $z$ in the upper half-plane that have $\operatorname{Im}(T(z)) \leqslant \operatorname{Im}(z)$ for each $T \in \Gamma$. Hence find a fundamental set for $\Gamma$ acting on the upper half-plane.

Let $A$ and $J$ be the two Möbius transformations

$A: z \mapsto z+1 \quad \text { and } \quad J: z \mapsto-1 / z$

When is $\operatorname{Im}(J(z))>\operatorname{Im}(z) ?$

For any point $z$ in the upper half-plane, show that either $z \in S$ or else there is an integer $k$ with

$\operatorname{Im}\left(J\left(A^{k}(z)\right)\right)>\operatorname{Im}(z)$

Deduce that the modular group is generated by $A$ and $J$.

Paper 1, Section II, F

commentState and prove Hall's theorem about matchings in bipartite graphs.

Show that a regular bipartite graph has a matching meeting every vertex.

A graph is almost r-regular if each vertex has degree $r-1$ or $r$. Show that, if $r \geqslant 2$, an almost $r$-regular graph $G$ must contain an almost $(r-1)$-regular graph $H$ with $V(H)=V(G)$.

[Hint: First, if possible, remove edges from $G$ whilst keeping it almost r-regular.]

Paper 1, Section II, C

commentQuoting carefully all necessary results, use the theory of inverse scattering to derive the 1-soliton solution of the $\mathrm{KdV}$ equation

$u_{t}=6 u u_{x}-u_{x x x}$

Paper 1, Section II, F

commentState and prove the Closed Graph Theorem. [You may assume any version of the Baire Category Theorem provided it is clearly stated. If you use any other result from the course, then you must prove it.]

Let $X$ be a closed subspace of $\ell_{\infty}$ such that $X$ is also a subset of $\ell_{1}$. Show that the left-shift $L: X \rightarrow \ell_{1}$, given by $L\left(x_{1}, x_{2}, x_{3}, \ldots\right)=\left(x_{2}, x_{3}, \ldots\right)$, is bounded when $X$ is equipped with the sup-norm.

Paper 1, Section II, G

commentWrite down the recursive definitions of ordinal addition, multiplication and exponentiation.

Given that $F: \mathbf{O n} \rightarrow \mathbf{O n}$ is a strictly increasing function-class (i.e. $\alpha<\beta$ implies $F(\alpha)<F(\beta))$, show that $\alpha \leqslant F(\alpha)$ for all $\alpha$.

Show that every ordinal $\alpha$ has a unique representation in the form

$\alpha=\omega^{\alpha_{1}} \cdot a_{1}+\omega^{\alpha_{2}} \cdot a_{2}+\cdots+\omega^{\alpha_{n}} \cdot a_{n}$

where $n \in \omega, \alpha \geqslant \alpha_{1}>\alpha_{2}>\cdots>\alpha_{n}$, and $a_{1}, a_{2}, \ldots, a_{n} \in \omega \backslash\{0\}$.

Under what conditions can an ordinal $\alpha$ be represented in the form

$\omega^{\beta_{1}} \cdot b_{1}+\omega^{\beta_{2}} \cdot b_{2}+\cdots+\omega^{\beta_{m}} \cdot b_{m}$

where $\beta_{1}<\beta_{2}<\cdots<\beta_{m}$ and $b_{1}, b_{2}, \ldots, b_{m} \in \omega \backslash\{0\} ?$ Justify your answer.

[The laws of ordinal arithmetic (associative, distributive, etc.) may be assumed without proof.]

Paper 1, Section I, A

commentIn a discrete-time model, a proportion $\mu$ of mature bacteria divides at each time step. When a mature bacterium divides it is destroyed and two new immature bacteria are produced. A proportion $\lambda$ of the immature bacteria matures at each time step, and a proportion $k$ of mature bacteria dies at each time step. Show that this model may be represented by the equations

$\begin{aligned} &a_{t+1}=a_{t}+2 \mu b_{t}-\lambda a_{t} \\ &b_{t+1}=b_{t}-\mu b_{t}+\lambda a_{t}-k b_{t} \end{aligned}$

Give an expression for the general solution to these equations and show that the population may grow if $\mu>k$.

At time $T$, the population is treated with an antibiotic that completely stops bacteria from maturing, but otherwise has no direct effects. Explain what will happen to the population of bacteria afterwards, and give expressions for $a_{t}$ and $b_{t}$ for $t>T$ in terms of $a_{T}, b_{T}, \mu$ and $k$.

Paper 1, Section II, H

commentLet $f \in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Let $K=\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $f$.

(i) Show that if $\operatorname{disc}(f)$ is square-free then $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.

(ii) In the case $f(X)=X^{3}-3 X-25$ find the minimal polynomial of $\beta=3 /(1-\alpha)$ and hence compute the discriminant of $K$. What is the index of $\mathbb{Z}[\alpha]$ in $\mathcal{O}_{K}$ ?

[Recall that the discriminant of $X^{3}+p X+q$ is $-4 p^{3}-27 q^{2}$.]

Paper 1, Section I, I

commentState and prove Gauss's Lemma for the Legendre symbol $\left(\frac{a}{p}\right)$. For which odd primes $p$ is 2 a quadratic residue modulo $p ?$ Justify your answer.

Paper 1, Section II, 40C

commentLet

$A(\alpha)=\left(\begin{array}{ccc} 1 & \alpha & \alpha \\ \alpha & 1 & \alpha \\ \alpha & \alpha & 1 \end{array}\right), \quad \alpha \in \mathbb{R}$

(i) For which values of $\alpha$ is $A(\alpha)$ positive definite?

(ii) Formulate the Gauss-Seidel method for the solution $\mathbf{x} \in \mathbb{R}^{3}$ of a system

$A(\alpha) \mathbf{x}=\mathbf{b}$

with $A(\alpha)$ as defined above and $\mathbf{b} \in \mathbb{R}^{3}$. Prove that the Gauss-Seidel method converges to the solution of the above system whenever $A$ is positive definite. [You may state and use the Householder-John theorem without proof.]

(iii) For which values of $\alpha$ does the Jacobi iteration applied to the solution of the above system converge?

Paper 1, Section II, C

comment(i) Discuss briefly the concept of well-posedness of a Cauchy problem for a partial differential equation.

Solve the Cauchy problem

$\partial_{2} u+x_{1} \partial_{1} u=a u^{2}, \quad u\left(x_{1}, 0\right)=\phi\left(x_{1}\right),$

where $a \in \mathbb{R}, \phi \in C^{1}(\mathbb{R})$ and $\partial_{i}$ denotes the partial derivative with respect to $x_{i}$ for $i=1,2$.

For the case $a=0$ show that the solution satisfies $\max _{x_{1} \in \mathbb{R}}\left|u\left(x_{1}, x_{2}\right)\right|=\|\phi\|_{C^{0}}$, where the $C^{r}$ norm on functions $\phi=\phi\left(x_{1}\right)$ of one variable is defined by

$\|\phi\|_{C^{r}}=\sum_{i=0}^{r} \max _{x \in \mathbb{R}}\left|\partial_{1}^{i} \phi\left(x_{1}\right)\right|$

Deduce that the Cauchy problem is then well-posed in the uniform metric (i.e. the metric determined by the $C^{0}$ norm).

(ii) State the Cauchy-Kovalevskaya theorem and deduce that the following Cauchy problem for the Laplace equation,

$\partial_{1}^{2} u+\partial_{2}^{2} u=0, \quad u\left(x_{1}, 0\right)=0, \partial_{2} u\left(x_{1}, 0\right)=\phi\left(x_{1}\right)$

has a unique analytic solution in some neighbourhood of $x_{2}=0$ for any analytic function $\phi=\phi\left(x_{1}\right)$. Write down the solution for the case $\phi\left(x_{1}\right)=\sin \left(n x_{1}\right)$, and hence give a sequence of initial data $\left\{\phi_{n}\left(x_{1}\right)\right\}_{n=1}^{\infty}$ with the property that

$\left\|\phi_{n}\right\|_{C^{r}} \rightarrow 0, \quad \text { as } n \rightarrow \infty, \text { for each } r \in \mathbb{N},$

whereas $u_{n}$, the corresponding solution of $(*)$, satisfies

$\max _{x_{1} \in \mathbb{R}}\left|u_{n}\left(x_{1}, x_{2}\right)\right| \rightarrow+\infty, \quad \text { as } n \rightarrow \infty$

for any $x_{2} \neq 0$.

Paper 1, Section II, E

commentConsider a composite system of several identical particles. Describe how the multiparticle state is constructed from single-particle states. For the case of two identical particles, describe how considering the interchange symmetry leads to the definition of bosons and fermions.

Consider two non-interacting, identical particles, each with spin 1 . The singleparticle, spin-independent Hamiltonian $H\left(\hat{\mathbf{x}}_{i}, \hat{\mathbf{p}}_{i}\right)$ has non-degenerate eigenvalues $E_{n}$ and wavefunctions $\psi_{n}\left(\mathbf{x}_{i}\right)$ where $i=1,2$ labels the particle and $n=0,1,2,3, \ldots$ In terms of these single-particle wavefunctions and single-particle spin states $|1\rangle,|0\rangle$ and $|-1\rangle$, write down all of the two-particle states and energies for:

(i) the ground state;

(ii) the first excited state.

Assume now that $E_{n}$ is a linear function of $n$. Find the degeneracy of the $N^{\text {th }}$energy level of the two-particle system for:

(iii) $N$ even;

(iv) $N$ odd.

Paper 1, Section II, K

commentWhen the real parameter $\Theta$ takes value $\theta$, variables $X_{1}, X_{2}, \ldots$ arise independently from a distribution $P_{\theta}$ having density function $p_{\theta}(x)$ with respect to an underlying measure $\mu$. Define the score variable $U_{n}(\theta)$ and the information function $I_{n}(\theta)$ for estimation of $\Theta$ based on $\boldsymbol{X}^{n}:=\left(X_{1}, \ldots, X_{n}\right)$, and relate $I_{n}(\theta)$ to $i(\theta):=I_{1}(\theta)$.

State and prove the Cramér-Rao inequality for the variance of an unbiased estimator of $\Theta$. Under what conditions does this inequality become an equality? What is the form of the estimator in this case? [You may assume $\mathbb{E}_{\theta}\left\{U_{n}(\theta)\right\}=0, \operatorname{var}_{\theta}\left\{U_{n}(\theta)\right\}=I_{n}(\theta)$, and any further required regularity conditions, without comment.]

Let $\widehat{\Theta}_{n}$ be the maximum likelihood estimator of $\Theta$ based on $\boldsymbol{X}^{n}$. What is the asymptotic distribution of $n^{\frac{1}{2}}\left(\widehat{\Theta}_{n}-\Theta\right)$ when $\Theta=\theta$ ?

Suppose that, for each $n, \widehat{\Theta}_{n}$ is unbiased for $\Theta$, and the variance of $n^{\frac{1}{2}}\left(\widehat{\Theta}_{n}-\Theta\right)$ is exactly equal to its asymptotic variance. By considering the estimator $\alpha \widehat{\Theta}_{k}+(1-\alpha) \widehat{\Theta}_{n}$, or otherwise, show that, for $k<n, \operatorname{cov}_{\theta}\left(\widehat{\Theta}_{k}, \widehat{\Theta}_{n}\right)=\operatorname{var}_{\theta}\left(\widehat{\Theta}_{n}\right)$.

Paper 1, Section II, $26 \mathrm{~K}$

commentState Dynkin's $\pi$-system $/ d$-system lemma.

Let $\mu$ and $\nu$ be probability measures on a measurable space $(E, \mathcal{E})$. Let $\mathcal{A}$ be a $\pi$-system on $E$ generating $\mathcal{E}$. Suppose that $\mu(A)=\nu(A)$ for all $A \in \mathcal{A}$. Show that $\mu=\nu$.

What does it mean to say that a sequence of random variables is independent?

Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables, all uniformly distributed on $[0,1]$. Let $Y$ be another random variable, independent of $\left(X_{n}: n \in \mathbb{N}\right)$. Define random variables $Z_{n}$ in $[0,1]$ by $Z_{n}=\left(X_{n}+Y\right) \bmod 1$. What is the distribution of $Z_{1}$ ? Justify your answer.

Show that the sequence of random variables $\left(Z_{n}: n \in \mathbb{N}\right)$ is independent.

Paper 1, Section II, 19G

commentState and prove Maschke's Theorem for complex representations of finite groups.

Without using character theory, show that every irreducible complex representation of the dihedral group of order $10, D_{10}$, has dimension at most two. List the irreducible complex representations of $D_{10}$ up to isomorphism.

Let $V$ be the set of vertices of a regular pentagon with the usual action of $D_{10}$. Explicitly decompose the permutation representation $\mathbb{C} V$ into a direct sum of irreducible subrepresentations.

Paper 1, Section II, I

comment(i) Let $f(z)=\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series with radius of convergence $r$ in $(0, \infty)$. Show that there is at least one point $a$ on the circle $C=\{z \in \mathbb{C}:|z|=r\}$ which is a singular point of $f$, that is, there is no direct analytic continuation of $f$ in any neighbourhood of $a$.

(ii) Let $X$ and $Y$ be connected Riemann surfaces. Define the space $\mathcal{G}$ of germs of function elements of $X$ into $Y$. Define the natural topology on $\mathcal{G}$ and the natural $\operatorname{map} \pi: \mathcal{G} \rightarrow X$. [You may assume without proof that the topology on $\mathcal{G}$ is Hausdorff.] Show that $\pi$ is continuous. Define the natural complex structure on $\mathcal{G}$ which makes it into a Riemann surface. Finally, show that there is a bijection between the connected components of $\mathcal{G}$ and the complete holomorphic functions of $X$ into $Y$.

Paper 1, Section I, J

commentVariables $Y_{1}, \ldots, Y_{n}$ are independent, with $Y_{i}$ having a density $p\left(y \mid \mu_{i}\right)$ governed by an unknown parameter $\mu_{i}$. Define the deviance for a model $M$ that imposes relationships between the $\left(\mu_{i}\right)$.

From this point on, suppose $Y_{i} \sim \operatorname{Poisson}\left(\mu_{i}\right)$. Write down the log-likelihood of data $y_{1}, \ldots, y_{n}$ as a function of $\mu_{1}, \ldots, \mu_{n}$.

Let $\widehat{\mu}_{i}$ be the maximum likelihood estimate of $\mu_{i}$ under model $M$. Show that the deviance for this model is given by

$2 \sum_{i=1}^{n}\left\{y_{i} \log \frac{y_{i}}{\widehat{\mu}_{i}}-\left(y_{i}-\widehat{\mu}_{i}\right)\right\}$

Now suppose that, under $M, \log \mu_{i}=\beta^{\mathrm{T}} x_{i}, i=1, \ldots, n$, where $x_{1}, \ldots, x_{n}$ are known $p$-dimensional explanatory variables and $\beta$ is an unknown $p$-dimensional parameter. Show that $\widehat{\mu}:=\left(\widehat{\mu}_{1}, \ldots, \widehat{\mu}_{n}\right)^{\mathrm{T}}$ satisfies $X^{\mathrm{T}} y=X^{\mathrm{T}} \widehat{\mu}$, where $y=\left(y_{1}, \ldots, y_{n}\right)^{\mathrm{T}}$ and $X$ is the $(n \times p)$ matrix with rows $x_{1}^{\mathrm{T}}, \ldots, x_{n}^{\mathrm{T}}$, and express this as an equation for the maximum likelihood estimate $\widehat{\beta}$ of $\beta$. [You are not required to solve this equation.]

Paper 1, Section II, J

commentA cricket ball manufacturing company conducts the following experiment. Every day, a bowling machine is set to one of three levels, "Medium", "Fast" or "Spin", and then bowls 100 balls towards the stumps. The number of times the ball hits the stumps and the average wind speed (in kilometres per hour) during the experiment are recorded, yielding the following data (abbreviated):

$\begin{array}{llll}\text { Day } & \text { Wind } & \text { Level } & \text { Stumps } \\ 1 & 10 & \text { Medium } & 26 \\ 2 & 8 & \text { Medium } & 37 \\ \vdots & \vdots & \vdots & \vdots \\ 50 & 12 & \text { Medium } & 32 \\ 51 & 7 & \text { Fast } & 31 \\ \vdots & \vdots & \vdots & \vdots \\ 120 & 3 & \text { Fast } & 28 \\ 121 & 5 & \text { Spin } & 35 \\ \vdots & \vdots & \vdots & \vdots \\ 150 & 6 & \text { Spin } & 31\end{array}$

Write down a reasonable model for $Y_{1}, \ldots, Y_{150}$, where $Y_{i}$ is the number of times the ball hits the stumps on the $i^{t h}$ day. Explain briefly why we might want to include interactions between the variables. Write $R$ code to fit your model.

The company's statistician fitted her own generalized linear model using $\mathrm{R}$, and obtained the following summary (abbreviated):

$\begin{array}{lrrrrr}\text { >summary(ball) } & & & & \\ \text { Coefficients: } & & & & & \\ & \text { Estimate } & \text { Std. Error } & \text { z value } & \operatorname{Pr}(>|z|) & \\ \text { (Intercept) } & -0.37258 & 0.05388 & -6.916 & 4.66 \mathrm{e}-12 & * * * \\ \text { Wind } & 0.09055 & 0.01595 & 5.676 & 1.38 \mathrm{e}-08 & * * * \\ \text { LevelFast } & -0.10005 & 0.08044 & -1.244 & 0.213570 & \\ \text { LevelSpin } & 0.29881 & 0.08268 & 3.614 & 0.000301 & * * * \\ \text { Wind: LevelFast } & 0.03666 & 0.02364 & 1.551 & 0.120933 & \\ \text { Wind: LevelSpin } & -0.07697 & 0.02845 & -2.705 & 0.006825 & * *\end{array}$

Why are LevelMedium and Wind: LevelMedium not listed?

Suppose that, on another day, the bowling machine is set to "Spin", and the wind speed is 5 kilometres per hour. What linear function of the parameters should the statistician use in constructing a predictor of the number of times the ball hits the stumps that day?

Based on the above output, how might you improve the model? How could you fit your new model in $R$ ?

Paper 1, Section II, 35A

comment(i) What is the occupation number of a state $i$ with energy $E_{i}$ according to the Fermi-Dirac statistics for a given chemical potential $\mu$ ?

(ii) Assuming that the energy $E$ is spin independent, what is the number $g_{s}$ of electrons which can occupy an energy level?

(iii) Consider a semi-infinite metal slab occupying $z \leqslant 0$ (and idealized to have infinite extent in the $x y$ plane) and a vacuum environment at $z>0$. An electron with momentum $\left(p_{x}, p_{y}, p_{z}\right)$ inside the slab will escape the metal in the $+z$ direction if it has a sufficiently large momentum $p_{z}$ to overcome a potential barrier $V_{0}$ relative to the Fermi energy $\epsilon_{\mathrm{F}}$, i.e. if

$\frac{p_{z}^{2}}{2 m} \geqslant \epsilon_{\mathrm{F}}+V_{0}$

where $m$ is the electron mass.

At fixed temperature $T$, some fraction of electrons will satisfy this condition, which results in a current density $j_{z}$ in the $+z$ direction (an electron having escaped the metal once is considered lost, never to return). Each electron escaping provides a contribution $\delta j_{z}=-e v_{z}$ to this current density, where $v_{z}$ is the velocity and $e$ the elementary charge.

(a) Briefly describe the Fermi-Dirac distribution as a function of energy in the limit $k_{\mathrm{B}} T \ll \epsilon_{\mathrm{F}}$, where $k_{\mathrm{B}}$ is the Boltzmann constant. What is the chemical potential $\mu$ in this limit?

(b) Assume that the electrons behave like an ideal, non-relativistic Fermi gas and that $k_{\mathrm{B}} T \ll V_{0}$ and $k_{\mathrm{B}} T \ll \epsilon_{\mathrm{F}}$. Calculate the current density $j_{z}$ associated with the electrons escaping the metal in the $+z$ direction. How could we easily increase the strength of the current?

Paper 1, Section II, 29J

comment(i) Suppose that the price $S_{t}$ of an asset at time $t$ is given by

$S_{t}=S_{0} \exp \left\{\sigma B_{t}+\left(r-\frac{1}{2} \sigma^{2}\right) t\right\}$

where $B$ is a Brownian motion, $S_{0}$ and $\sigma$ are positive constants, and $r$ is the riskless rate of interest, assumed constant. In this model, explain briefly why the time-0 price of a derivative which delivers a bounded random variable $Y$ at time $T$ should be given by $E\left(e^{-r T} Y\right)$. What feature of this model ensures that the price is unique?

Derive an expression $C\left(S_{0}, K, T, r, \sigma\right)$ for the time- 0 price of a European call option with strike $K$ and expiry $T$. Explain the italicized terms.

(ii) Suppose now that the price $X_{t}$ of an asset at time $t$ is given by

$X_{t}=\sum_{j=1}^{n} w_{j} \exp \left\{\sigma_{j} B_{t}+\left(r-\frac{1}{2} \sigma_{j}^{2}\right) t\right\}$

where the $w_{j}$ and $\sigma_{j}$ are positive constants, and the other notation is as in part (i) above. Show that the time-0 price of a European call option with strike $K$ and expiry $T$ written on this asset can be expressed as

$\sum_{j=1}^{n} C\left(w_{j}, k_{j}, T, r, \sigma_{j}\right)$

where the $k_{j}$ are constants. Explain how the $k_{j}$ are characterized.

Paper 1, Section I, $2 F$

commentShow that $\sin (1)$ is irrational. [The angle is measured in radians.]

Paper 1, Section II, 39C

commentStarting from the equations for the one-dimensional unsteady flow of a perfect gas of uniform entropy, show that the Riemann invariants

$R_{\pm}=u \pm \frac{2}{\gamma-1}\left(c-c_{0}\right)$

are constant on characteristics $C_{\pm}$given by $d x / d t=u \pm c$, where $u(x, t)$ is the velocity of the gas, $c(x, t)$ is the local speed of sound, $c_{0}$ is a constant and $\gamma$ is the ratio of specific heats.

Such a gas initially occupies the region $x>0$ to the right of a piston in an infinitely long tube. The gas and the piston are initially at rest with $c=c_{0}$. At time $t=0$ the piston starts moving to the left at a constant velocity $V$. Find $u(x, t)$ and $c(x, t)$ in the three regions

$\begin{array}{cc} \text { (i) } & c_{0} t \leqslant x \\ \text { (ii) } & a t \leqslant x \leqslant c_{0} t \\ \text { (iii) } & -V t \leqslant x \leqslant a t, \end{array}$

where $a=c_{0}-\frac{1}{2}(\gamma+1) V$. What is the largest value of $V$ for which $c$ is positive throughout region (iii)? What happens if $V$ exceeds this value?