Part II, 2012, Paper 1

# Part II, 2012, Paper 1

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Paper 1, Section II, I

comment(a) Let $X$ be an affine variety, $k[X]$ its ring of functions, and let $p \in X$. Assume $k$ is algebraically closed. Define the tangent space $T_{p} X$ at $p$. Prove the following assertions.

(i) A morphism of affine varieties $f: X \rightarrow Y$ induces a linear map

$d f: T_{p} X \rightarrow T_{f(p)} Y$

(ii) If $g \in k[X]$ and $U:=\{x \in X \mid g(x) \neq 0\}$, then $U$ has the natural structure of an affine variety, and the natural morphism of $U$ into $X$ induces an isomorphism $T_{p} U \rightarrow T_{p} X$ for all $p \in U$.

(iii) For all $s \geqslant 0$, the subset $\left\{x \in X \mid \operatorname{dim} T_{x} X \geqslant s\right\}$ is a Zariski-closed subvariety of $X$.

(b) Show that the set of nilpotent $2 \times 2$ matrices

$X=\left\{x \in \operatorname{Mat}_{2}(k) \mid x^{2}=0\right\}$

may be realised as an affine surface in $\mathbf{A}^{3}$, and determine its tangent space at all points $x \in X$.

Define what it means for two varieties $Y_{1}$ and $Y_{2}$ to be birationally equivalent, and show that the variety $X$ of nilpotent $2 \times 2$ matrices is birationally equivalent to $\mathbf{A}^{2}$.

Paper 1, Section II, G

commentDefine the notions of covering projection and of locally path-connected space. Show that a locally path-connected space is path-connected if it is connected.

Suppose $f: Y \rightarrow X$ and $g: Z \rightarrow X$ are continuous maps, the space $Y$ is connected and locally path-connected and that $g$ is a covering projection. Suppose also that we are given base-points $x_{0}, y_{0}, z_{0}$ satisfying $f\left(y_{0}\right)=x_{0}=g\left(z_{0}\right)$. Show that there is a continuous $\tilde{f}: Y \rightarrow Z$ satisfying $\tilde{f}\left(y_{0}\right)=z_{0}$ and $g \tilde{f}=f$ if and only if the image of $f_{*}: \Pi_{1}\left(Y, y_{0}\right) \rightarrow \Pi_{1}\left(X, x_{0}\right)$ is contained in that of $g_{*}: \Pi_{1}\left(Z, z_{0}\right) \rightarrow \Pi_{1}\left(X, x_{0}\right)$. [You may assume the path-lifting and homotopy-lifting properties of covering projections.]

Now suppose $X$ is locally path-connected, and both $f: Y \rightarrow X$ and $g: Z \rightarrow X$ are covering projections with connected domains. Show that $Y$ and $Z$ are homeomorphic as spaces over $X$ if and only if the images of their fundamental groups under $f_{*}$ and $g_{*}$ are conjugate subgroups of $\Pi_{1}\left(X, x_{0}\right)$.

Paper 1, Section II, E

commentGive an account of the variational principle for establishing an upper bound on the ground-state energy $E_{0}$ of a particle moving in a potential $V(x)$ in one dimension.

A particle of unit mass moves in the potential

$V(x)= \begin{cases}\infty & x \leqslant 0 \\ \lambda x & x>0\end{cases}$

with $\lambda$ a positive constant. Explain why it is important that any trial wavefunction used to derive an upper bound on $E_{0}$ should be chosen to vanish for $x \leqslant 0$.

Use the trial wavefunction

$\psi(x)= \begin{cases}0 & x \leqslant 0 \\ x e^{-a x} & x>0\end{cases}$

where $a$ is a positive real parameter, to establish an upper bound $E_{0} \leqslant E(a, \lambda)$ for the energy of the ground state, and hence derive the lowest upper bound on $E_{0}$ as a function of $\lambda$.

Explain why the variational method cannot be used in this case to derive an upper bound for the energy of the first excited state.

Paper 1, Section II, $27 \mathrm{~K}$

comment(a) Give the definition of a Poisson process $\left(N_{t}, t \geqslant 0\right)$ with rate $\lambda$, using its transition rates. Show that for each $t \geqslant 0$, the distribution of $N_{t}$ is Poisson with a parameter to be specified.

Let $J_{0}=0$ and let $J_{1}, J_{2}, \ldots$ denote the jump times of $\left(N_{t}, t \geqslant 0\right)$. What is the distribution of $\left(J_{n+1}-J_{n}, n \geqslant 0\right) ?$ (You do not need to justify your answer.)

(b) Let $n \geqslant 1$. Compute the joint probability density function of $\left(J_{1}, J_{2}, \ldots, J_{n}\right)$ given $\left\{N_{t}=n\right\}$. Deduce that, given $\left\{N_{t}=n\right\},\left(J_{1}, \ldots, J_{n}\right)$ has the same distribution as the nondecreasing rearrangement of $n$ independent uniform random variables on $[0, t]$.

(c) Starting from time 0, passengers arrive on platform $9 \mathrm{~B}$ at King's Cross station, with constant rate $\lambda>0$, in order to catch a train due to depart at time $t>0$. Using the above results, or otherwise, find the expected total time waited by all passengers (the sum of all passengers' waiting times).

Paper 1, Section II, B

commentWhat precisely is meant by the statement that

$f(x) \sim \sum_{n=0}^{\infty} d_{n} x^{n}$

as $x \rightarrow 0 ?$

Consider the Stieltjes integral

$I(x)=\int_{1}^{\infty} \frac{\rho(t)}{1+x t} d t$

where $\rho(t)$ is bounded and decays rapidly as $t \rightarrow \infty$, and $x>0$. Find an asymptotic series for $I(x)$ of the form $(*)$, as $x \rightarrow 0$, and prove that it has the asymptotic property.

In the case that $\rho(t)=e^{-t}$, show that the coefficients $d_{n}$ satisfy the recurrence relation

$d_{n}=(-1)^{n} \frac{1}{e}-n d_{n-1} \quad(n \geqslant 1)$

and that $d_{0}=\frac{1}{e}$. Hence find the first three terms in the asymptotic series.

Paper 1, Section I, A

commentConsider a heavy symmetric top of mass $M$, pinned at point $P$, which is a distance $l$ from the centre of mass.

(a) Working in the body frame $\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right)$ (where $\mathbf{e}_{3}$ is the symmetry axis of the top) define the Euler angles $(\psi, \theta, \phi)$ and show that the components of the angular velocity can be expressed in terms of the Euler angles as

$\boldsymbol{\omega}=(\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi, \dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi, \dot{\psi}+\dot{\phi} \cos \theta)$

(b) Write down the Lagrangian of the top in terms of the Euler angles and the principal moments of inertia $I_{1}, I_{3}$.

(c) Find the three constants of motion.

Paper 1, Section I, G

commentLet $\mathcal{A}$ and $\mathcal{B}$ be alphabets of sizes $m$ and a respectively. What does it mean to say that $c: \mathcal{A} \rightarrow \mathcal{B}^{*}$ is a decodable code? State Kraft's inequality.

Suppose that a source emits letters from the alphabet $\mathcal{A}=\{1,2, \ldots, m\}$, each letter $j$ occurring with (known) probability $p_{j}>0$. Let $S$ be the codeword-length random variable for a decodable code $c: \mathcal{A} \rightarrow \mathcal{B}^{*}$, where $|\mathcal{B}|=a$. It is desired to find a decodable code that minimizes the expected value of $a^{S}$. Establish the lower bound $\mathbb{E}\left(a^{S}\right) \geqslant\left(\sum_{j=1}^{m} \sqrt{p_{j}}\right)^{2}$, and characterise when equality occurs. [Hint. You may use without proof the Cauchy-Schwarz inequality, that (for positive $x_{i}, y_{i}$ )

$\sum_{i=1}^{m} x_{i} y_{i} \leqslant\left(\sum_{i=1}^{m} x_{i}^{2}\right)^{1 / 2}\left(\sum_{i=1}^{m} y_{i}^{2}\right)^{1 / 2}$

with equality if and only if $x_{i}=\lambda y_{i}$ for all i.]

Paper 1, Section II, 12G

commentDefine a cyclic binary code of length $n$.

Show how codewords can be identified with polynomials in such a way that cyclic binary codes correspond to ideals in the polynomial ring with a suitably chosen multiplication rule.

Prove that any cyclic binary code $C$ has a unique generator, that is, a polynomial $c(X)$ of minimum degree, such that the code consists of the multiples of this polynomial. Prove that the rank of the code equals $n-\operatorname{deg} c(X)$, and show that $c(X)$ divides $X^{n}-1$.

Show that the repetition and parity check codes are cyclic, and determine their generators.

Paper 1, Section I, E

commentThe number density of photons in equilibrium at temperature $T$ is given by

$n=\frac{8 \pi}{(h c)^{3}} \int_{0}^{\infty} \frac{\nu^{2} d \nu}{e^{\beta h \nu}-1}$

where $\beta=1 /\left(k_{B} T\right)\left(k_{B}\right.$ is Boltzmann's constant). Show that $n \propto T^{3}$. Show further that $\epsilon \propto T^{4}$, where $\epsilon$ is the photon energy density.

Write down the Friedmann equation for the scale factor $a(t)$ of a flat homogeneous and isotropic universe. State the relation between $a$ and the mass density $\rho$ for a radiation-dominated universe and hence deduce the time-dependence of $a$. How does the temperature $T$ depend on time?

Paper 1, Section II, E

commentThe Friedmann equation for the scale factor $a(t)$ of a homogeneous and isotropic universe of mass density $\rho$ is

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

where $\dot{a}=d a / d t$. Explain how the value of the constant $k$ affects the late-time $(t \rightarrow \infty)$ behaviour of $a$.

Explain briefly why $\rho \propto 1 / a^{3}$ in a matter-dominated (zero-pressure) universe. By considering the scale factor $a$ of a closed universe as a function of conformal time $\tau$, defined by $d \tau=a^{-1} d t$, show that

$a(\tau)=\frac{\Omega_{0}}{2\left(\Omega_{0}-1\right)}[1-\cos (\sqrt{k} c \tau)]$

where $\Omega_{0}$ is the present $\left(\tau=\tau_{0}\right)$ density parameter, with $a\left(\tau_{0}\right)=1$. Use this result to show that

$t(\tau)=\frac{\Omega_{0}}{2 H_{0}\left(\Omega_{0}-1\right)^{3 / 2}}[\sqrt{k} c \tau-\sin (\sqrt{k} c \tau)],$

where $H_{0}$ is the present Hubble parameter. Find the time $t_{B C}$ at which this model universe ends in a "big crunch".

Given that $\sqrt{k} c \tau_{0} \ll 1$, obtain an expression for the present age of the universe in terms of $H_{0}$ and $\Omega_{0}$, according to this model. How does it compare with the age of a flat universe?

Paper 1, Section II, I

commentDefine the geodesic curvature $k_{g}$ of a regular curve in an oriented surface $S \subset \mathbb{R}^{3}$. When is $k_{g}=0$ along a curve?

Explain briefly what is meant by the Euler characteristic $\chi$ of a compact surface $S \subset \mathbb{R}^{3}$. State the global Gauss-Bonnet theorem with boundary terms.

Let $S$ be a surface with positive Gaussian curvature that is diffeomorphic to the sphere $S^{2}$ and let $\gamma_{1}, \gamma_{2}$ be two disjoint simple closed curves in $S$. Can both $\gamma_{1}$ and $\gamma_{2}$ be geodesics? Can both $\gamma_{1}$ and $\gamma_{2}$ have constant geodesic curvature? Justify your answers.

[You may assume that the complement of a simple closed curve in $S^{2}$ consists of two open connected regions.]

Paper 1, Section I, $7 \mathrm{D}$

commentState the Poincaré-Bendixson theorem.

A model of a chemical process obeys the second-order system

$\dot{x}=1-x(1+a)+x^{2} y, \quad \dot{y}=a x-x^{2} y$

where $a>0$. Show that there is a unique fixed point at $(x, y)=(1, a)$ and that it is unstable if $a>2$. Show that trajectories enter the region bounded by the lines $x=1 / q$, $y=0, y=a q$ and $x+y=1+a q$, provided $q>(1+a)$. Deduce that there is a periodic orbit when $a>2$.

Paper 1, Section II, B

commentA particle of mass $m$ and charge $q$ moves relativistically under the influence of a constant electric field $E$ in the positive $z$-direction, and a constant magnetic field $B$ also in the positive $z$-direction.

In some inertial observer's coordinate system, the particle starts at

$x=R, \quad y=0, \quad z=0, \quad t=0$

with velocity given by

$\dot{x}=0, \quad \dot{y}=u, \quad \dot{z}=0,$

where the dot indicates differentiation with respect to the proper time of the particle. Show that the subsequent motion of the particle, as seen by the inertial observer, is a helix.

a) What is the radius of the helix as seen by the inertial observer?

b) What are the $x$ and $y$ coordinates of the axis of the helix?

c) What is the $z$ coordinate of the particle after a proper time $\tau$ has elapsed, as measured by the particle?

Paper 1, Section II, C

commentDefine the strain-rate tensor $e_{i j}$ in terms of the velocity components $u_{i}$. Write down the relation between $e_{i j}$, the pressure $p$ and the stress $\sigma_{i j}$ in an incompressible Newtonian fluid of viscosity $\mu$. Show that the local rate of stress-working $\sigma_{i j} \partial u_{i} / \partial x_{j}$ is equal to the local rate of dissipation $2 \mu e_{i j} e_{i j}$.

An incompressible fluid of density $\rho$ and viscosity $\mu$ occupies the semi-infinite region $y>0$ above a rigid plane boundary $y=0$ which oscillates with velocity $(V \cos \omega t, 0,0)$. The fluid is at rest at infinity. Determine the velocity field produced by the boundary motion after any transients have decayed.

Show that the time-averaged rate of dissipation is

$\frac{1}{4} \sqrt{2} V^{2}(\mu \rho \omega)^{1 / 2}$

per unit area of the boundary. Verify that this is equal to the time average of the rate of working by the boundary on the fluid per unit area.

Paper 1, Section I, E

commentRecall that if $f(z)$ is analytic in a neighbourhood of $z_{0} \neq 0$, then

$f(z)+\overline{f\left(z_{0}\right)}=2 u\left(\frac{z+\overline{z_{0}}}{2}, \frac{z-\overline{z_{0}}}{2 i}\right)$

where $u(x, y)$ is the real part of $f(z)$. Use this fact to construct the imaginary part of an analytic function whose real part is given by

$u(x, y)=y \int_{-\infty}^{\infty} \frac{g(t) d t}{(t-x)^{2}+y^{2}}, \quad x, y \in \mathbb{R}, y \neq 0$

where $g(t)$ is real and has sufficient smoothness and decay.

Paper 1, Section II, E

comment(a) Suppose that $F(z), z=x+i y, x, y \in \mathbb{R}$, is analytic in the upper-half complex $z$-plane and $O(1 / z)$ as $z \rightarrow \infty, y \geqslant 0$. Show that the real and imaginary parts of $F(x)$, denoted by $U(x)$ and $V(x)$ respectively, satisfy the so-called Kramers-Kronig formulae:

$U(x)=H V(x), \quad V(x)=-H U(x), \quad x \in \mathbb{R} .$

Here, $H$ denotes the Hilbert transform, i.e.,

$(H f)(x)=\frac{1}{\pi} \mathrm{PV} \int_{-\infty}^{\infty} \frac{f(\xi)}{\xi-x} d \xi$

where $\mathrm{PV}$ denotes the principal value integral.

(b) Let the real function $u(x, y)$ satisfy the Laplace equation in the upper-half complex z-plane, i.e.,

$\frac{\partial^{2} u(x, y)}{\partial x^{2}}+\frac{\partial^{2} u(x, y)}{\partial y^{2}}=0, \quad-\infty<x<\infty, \quad y>0$

Assuming that $u(x, y)$ decays for large $|x|$ and for large $y$, show that $F=u_{z}$ is an analytic function for $\operatorname{Im} z>0, z=x+i y$. Then, find an expression for $u_{y}(x, 0)$ in terms of $u_{x}(x, 0)$.

Paper 1, Section II, 18H

commentList all subfields of the cyclotomic field $\mathbb{Q}\left(\boldsymbol{\mu}_{20}\right)$ obtained by adjoining all 20 th roots of unity to $\mathbb{Q}$, and draw the lattice diagram of inclusions among them. Write all the subfields in the form $\mathbb{Q}(\alpha)$ or $\mathbb{Q}(\alpha, \beta)$. Briefly justify your answer.

[The description of the Galois group of cyclotomic fields and the fundamental theorem of Galois theory can be used freely without proof.]

Paper 1, Section II, B

comment(i) Using the condition that the metric tensor $g_{a b}$ is covariantly constant, derive an expression for the Christoffel symbol $\Gamma_{b c}^{a}=\Gamma_{c b}^{a}$.

(ii) Show that

$\Gamma_{b a}^{a}=\frac{1}{2} g^{a c} g_{a c, b}$

Hence establish the covariant divergence formula

$V_{; a}^{a}=\frac{1}{\sqrt{-g}} \frac{\partial}{\partial x^{a}}\left(\sqrt{-g} V^{a}\right)$

where $g$ is the determinant of the metric tensor.

[It may be assumed that $\partial_{a}(\log \operatorname{det} M)=\operatorname{trace}\left(M^{-1} \partial_{a} M\right)$ for any invertible matrix $M$ ].

(iii) The Kerr-Newman metric, describing the spacetime outside a rotating black hole of mass $M$, charge $Q$ and angular momentum per unit mass $a$, is given in appropriate units by

$\begin{aligned} d s^{2}=&-\left(d t-a \sin ^{2} \theta d \phi\right)^{2} \frac{\Delta}{\rho^{2}} \\ &+\left(\left(r^{2}+a^{2}\right) d \phi-a d t\right)^{2} \frac{\sin ^{2} \theta}{\rho^{2}}+\left(\frac{d r^{2}}{\Delta}+d \theta^{2}\right) \rho^{2} \end{aligned}$

where $\rho^{2}=r^{2}+a^{2} \cos ^{2} \theta$ and $\Delta=r^{2}-2 M r+a^{2}+Q^{2}$. Explain why this metric is stationary, and make a choice of one of the parameters which reduces it to a static metric.

Show that, in the static metric obtained, the equation

$\left(g^{a b} \Phi_{, b}\right)_{; a}=0$

for a function $\Phi=\Phi(t, r)$ admits solutions of the form

$\Phi=\sin (\omega t) R(r)$

where $\omega$ is constant and $R(r)$ satisfies an ordinary differential equation which should be found.

Paper 1, Section I, G

commentLet $G$ be a crystallographic group of the Euclidean plane. Define the lattice and the point group of $G$. Suppose that the lattice for $G$ is $\{(k, 0): k \in \mathbb{Z}\}$. Show that there are five different possibilities for the point group. Show that at least one of these point groups can arise from two groups $G$ that are not conjugate in the group of all isometries of the Euclidean plane.

Paper 1, Section II, G

commentDefine the axis of a loxodromic Möbius transformation acting on hyperbolic 3-space.

When do two loxodromic transformations commute? Justify your answer.

Let $G$ be a Kleinian group that contains a loxodromic transformation. Show that the fixed point of any loxodromic transformation in $G$ lies in the limit set of $G$. Prove that the set of such fixed points is dense in the limit set. Give examples to show that the set of such fixed points can be equal to the limit set or a proper subset.

Paper 1, Section II, F

commentState Markov's inequality and Chebyshev's inequality.

Let $\mathcal{G}^{(2)}(n, p)$ denote the probability space of bipartite graphs with vertex classes $U=\{1,2, \ldots, n\}$ and $V=\{-1,-2, \ldots,-n\}$, with each possible edge $u v(u \in U,$, $v \in V)$ present, independently, with probability $p$. Let $X$ be the number of subgraphs of $G \in \mathcal{G}^{(2)}(n, p)$ that are isomorphic to the complete bipartite graph $K_{2,2}$. Write down $\mathbb{E} X$ and $\operatorname{Var}(X)$. Hence show that $p=1 / n$ is a threshold for $G \in \mathcal{G}^{(2)}(n, p)$ to contain $K_{2,2}$, in the sense that if $n p \rightarrow \infty$ then a. e. $G \in \mathcal{G}^{(2)}(n, p)$ contains a $K_{2,2}$, whereas if $n p \rightarrow 0$ then a. e. $G \in \mathcal{G}^{(2)}(n, p)$ does not contain a $K_{2,2}$.

By modifying a random $G \in \mathcal{G}^{(2)}(n, p)$ for suitably chosen $p$, show that, for each $n$, there exists a bipartite graph $H$ with $n$ vertices in each class such that $K_{2,2} \not \subset H$ but $e(H) \geqslant \frac{3}{4}\left(\frac{n}{\sqrt[3]{n-1}}\right)^{2} .$

Paper 1, Section II, D

commentState the Arnold-Liouville theorem.

Consider an integrable system with six-dimensional phase space, and assume that $\nabla \wedge \mathbf{p}=0$ on any Liouville tori $p_{i}=p_{i}\left(q_{j}, c_{j}\right)$, where $\nabla=\left(\partial / \partial q_{1}, \partial / \partial q_{2}, \partial / \partial q_{3}\right)$.

(a) Define the action variables and use Stokes' theorem to show that the actions are independent of the choice of the cycles.

(b) Define the generating function, and show that the angle coordinates are periodic with period $2 \pi$.

Paper 1, Section II, G

commentWhat is meant by the dual $X^{*}$ of a normed space $X$ ? Show that $X^{*}$ is a Banach space.

Let $X=C^{1}(0,1)$, the space of functions $f:(0,1) \rightarrow \mathbb{R}$ possessing a bounded, continuous first derivative. Endow $X$ with the sup norm $\|f\|_{\infty}=\sup _{x \in(0,1)}|f(x)|$. Which of the following maps $T: X \rightarrow \mathbb{R}$ are elements of $X^{*}$ ? Give brief justifications or counterexamples as appropriate.

- $T f=f\left(\frac{1}{2}\right)$;
- $T f=\|f\|_{\infty}$
- $T f=\int_{0}^{1} f(x) d x$
- $T f=f^{\prime}\left(\frac{1}{2}\right)$.

Now suppose that $X$ is a (real) Hilbert space. State and prove the Riesz representation theorem. Describe the natural $\operatorname{map} X \rightarrow X^{* *}$ and show that it is surjective.

[All normed spaces are over $\mathbb{R}$. You may assume that if $Y$ is a closed subspace of a Hilbert space $X$ then $\left.X=Y \oplus Y^{\perp} .\right]$

Paper 1, Section II, H

commentState Zorn's lemma, and show how it may be deduced from the Axiom of Choice using the Bourbaki-Witt theorem (which should be clearly stated but not proved).

Show that, if $a$ and $b$ are distinct elements of a distributive lattice $L$, there is a lattice homomorphism $f: L \rightarrow\{0,1\}$ with $f(a) \neq f(b)$. Indicate briefly how this result may be used to prove the completeness theorem for propositional logic.

Paper 1, Section I, $\mathbf{6 C}$

commentKrill is the main food source for baleen whales. The following model has been proposed for the coupled evolution of populations of krill and whales, with $x(t)$ being the number of krill and $y(t)$ being the number of whales:

$\begin{array}{r} \frac{d x}{d t}=r x\left(1-\frac{x}{K}\right)-a x y \\ \frac{d y}{d t}=s y\left(1-\frac{y}{b x}\right) \end{array}$

where $r, s, a, b$ and $K$ are positive constants.

Give a biological interpretation for the form of the two differential equations.

Show that a steady state is possible with $x>0$ and $y>0$ and write down expressions for the steady-state values of $x$ and $y$.

Determine whether this steady state is stable.

Paper 1, Section II, F

commentLet $K$ be a number field, and $\mathcal{O}_{K}$ its ring of integers. Write down a characterisation of the units in $\mathcal{O}_{K}$ in terms of the norm. Without assuming Dirichlet's units theorem, prove that for $K$ a quadratic field the quotient of the unit group by $\{\pm 1\}$ is cyclic (i.e. generated by one element). Find a generator in the cases $K=\mathbb{Q}(\sqrt{-3})$ and $K=\mathbb{Q}(\sqrt{11})$.

Determine all integer solutions of the equation $x^{2}-11 y^{2}=n$ for $n=-1,5,14$.

Paper 1, Section I, I

commentShow that the continued fraction for $\sqrt{13}$ is $[3 ; \overline{1,1,1,1,6}]$.

Hence, or otherwise, find a solution to the equation $x^{2}-13 y^{2}=1$ in positive integers $x$ and $y$. Write down an expression for another solution.

Paper 1, Section II, D

commentThe Poisson equation $u_{x x}=f$ in the unit interval $\Omega=[0,1], u=0$ on $\partial \Omega$ is discretised with the formula

$u_{i-1}+u_{i+1}-2 u_{i}=h^{2} f_{i}$

where $1 \leqslant i \leqslant n, u_{i} \approx u(i h)$ and $i h$ are the grid points.

(i) Define the above system of equations in vector form $A \mathbf{u}=\mathbf{b}$ and describe the relaxed Jacobi method with relaxation parameter $\omega$ for solving this linear system. For $\mathbf{x}^{*}$ and $\mathbf{x}^{(\nu)}$ being the exact solution and the iterated solution respectively, let $\mathbf{e}^{(\nu)}=\mathbf{x}^{(\nu)}-\mathbf{x}^{*}$ be the error and $H_{\omega}$ the iteration matrix, so that

$\mathbf{e}^{(\nu+1)}=H_{\omega} \mathbf{e}^{(\nu)}$

Express $H_{\omega}$ in terms of the matrix $A$, the diagonal part $D$ of $A$ and $\omega$, and find the eigenvectors $\mathbf{v}_{k}$ and the eigenvalues $\lambda_{k}(\omega)$ of $H_{\omega}$.

(ii) For $A$ as above, let

$\mathbf{e}^{(\nu)}=\sum_{k=1}^{n} a_{k}^{(\nu)} \mathbf{v}_{k}$

be the expansion of the error with respect to the eigenvectors of $H_{\omega}$. Derive conditions on $\omega$ such that the method converges for any $n$, and prove that, for any such $\omega$, the rate of convergence of $\mathbf{e}^{(\nu)} \rightarrow 0$ is not faster than $\left(1-c / n^{2}\right)^{\nu}$.

(iii) Show that, for some $\omega$, the high frequency components $\left(\frac{n+1}{2} \leqslant k \leqslant n\right)$ of the error $\mathbf{e}^{(\nu)}$ tend to zero much faster than $\left(1-c / n^{2}\right)^{\nu}$. Determine the optimal parameter $\omega_{*}$ which provides the largest suppression of the high frequency components per iteration, and find the corresponding attenuation factor $\mu_{*}$ (i.e., the least $\mu_{\omega}$ such that $\left|a_{k}^{(\nu+1)}\right| \leqslant \mu_{\omega}\left|a_{k}^{(\nu)}\right|$ for $\left.\frac{n+1}{2} \leqslant k \leqslant n\right)$.

Paper 1, Section II, B

commentLet $u_{0}: \mathbb{R} \rightarrow \mathbb{R}, u_{0} \in C^{1}(\mathbb{R}), u_{0}(x) \geqslant 0$ for all $x \in \mathbb{R}$. Consider the partial differential equation for $u=u(x, y)$,

$4 y u_{x}+3 u_{y}=u^{2}, \quad(x, y) \in \mathbb{R}^{2}$

subject to the Cauchy condition $u(x, 0)=u_{0}(x)$.

i) Compute the solution of the Cauchy problem by the method of characteristics.

ii) Prove that the domain of definition of the solution contains

$(x, y) \in \mathbb{R} \times\left(-\infty, \frac{3}{\sup _{x \in \mathbb{R}}\left(u_{0}(x)\right)}\right)$

Paper 1, Section II, A

commentLet $a$ and $a^{\dagger}$ be the simple harmonic oscillator annihilation and creation operators, respectively. Write down the commutator $\left[a, a^{\dagger}\right]$.

Consider a new operator $b=c a+s a^{\dagger}$, where $c \equiv \cosh \theta, s \equiv \sinh \theta$ with $\theta$ a real constant. Show that

$\left[b, b^{\dagger}\right]=1$

Consider the Hamiltonian

$H=\epsilon a^{\dagger} a+\frac{1}{2} \lambda\left(a^{\dagger^{2}}+a^{2}\right),$

where $\epsilon$ and $\lambda$ are real and such that $\epsilon>\lambda>0$. Assuming that $\epsilon c-\lambda s=E c$ and $\lambda c-\epsilon s=E s$, with $E$ a real constant, show that

$[b, H]=E b$

Thus, calculate the energy of $b\left|E_{a}\right\rangle$ in terms of $E$ and $E_{a}$, where $E_{a}$ is an eigenvalue of $H$.

Assuming that $b\left|E_{\min }\right\rangle=0$, calculate $E_{\min }$ in terms of $\lambda, s$ and $c$. Find the possible values of $x=s / c$. Finally, show that the energy eigenvalues of the system are

$E_{n}=-\frac{\epsilon}{2}+\left(n+\frac{1}{2}\right) \sqrt{\epsilon^{2}-\lambda^{2}}$

Paper 1, Section II, $28 \mathrm{~K}$

commentProve that, if $T$ is complete sufficient for $\Theta$, and $S$ is a function of $T$, then $S$ is the minimum variance unbiased estimator of $\mathbb{E}(S \mid \Theta)$.

When the parameter $\Theta$ takes a value $\theta>0$, observables $\left(X_{1}, \ldots, X_{n}\right)$ arise independently from the exponential distribution $\mathcal{E}(\theta)$, having probability density function

$p(x \mid \theta)=\theta e^{-\theta x} \quad(x>0) .$

Show that the family of distributions

$\Theta \sim \operatorname{Gamma}(\alpha, \beta) \quad(\alpha>0, \beta>0),$

with probability density function

$\pi(\theta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} \theta^{\alpha-1} e^{-\beta \theta} \quad(\theta>0),$

is a conjugate family for Bayesian inference about $\Theta$ (where $\Gamma(\alpha)$ is the Gamma function).

Show that the expectation of $\Lambda:=\log \Theta$, under prior distribution (1), is $\psi(\alpha)-\log \beta$, where $\psi(\alpha):=(\mathrm{d} / \mathrm{d} \alpha) \log \Gamma(\alpha)$. What is the prior variance of $\Lambda$ ? Deduce the posterior expectation and variance of $\Lambda$, given $\left(X_{1}, \ldots, X_{n}\right)$.

Let $\tilde{\Lambda}$ denote the limiting form of the posterior expectation of $\Lambda$ as $\alpha, \beta \downarrow 0$. Show that $\tilde{\Lambda}$ is the minimum variance unbiased estimator of $\Lambda$. What is its variance?

Paper 1, Section II, J

commentCarefully state and prove Jensen's inequality for a convex function $c: I \rightarrow \mathbb{R}$, where $I \subseteq \mathbb{R}$ is an interval. Assuming that $c$ is strictly convex, give necessary and sufficient conditions for the inequality to be strict.

Let $\mu$ be a Borel probability measure on $\mathbb{R}$, and suppose $\mu$ has a strictly positive probability density function $f_{0}$ with respect to Lebesgue measure. Let $\mathcal{P}$ be the family of all strictly positive probability density functions $f$ on $\mathbb{R}$ with respect to Lebesgue measure such that $\log \left(f / f_{0}\right) \in L^{1}(\mu)$. Let $X$ be a random variable with distribution $\mu$. Prove that the mapping

$f \mapsto \mathbb{E}\left[\log \frac{f}{f_{0}}(X)\right]$

has a unique maximiser over $\mathcal{P}$, attained when $f=f_{0}$ almost everywhere.

Paper 1, Section II, 19H

commentWrite down the character table of $D_{10}$.

Suppose that $G$ is a group of order 60 containing 24 elements of order 5,20 elements of order 3 and 15 elements of order 2 . Calculate the character table of $G$, justifying your answer.

[You may assume the formula for induction of characters, provided you state it clearly.]

Paper 1, Section II, I

comment(i) Let $f(z)=\sum_{n=1}^{\infty} z^{2^{n}}$. Show that the unit circle is the natural boundary of the function element $(D(0,1), f)$.

(ii) Let $U=\{z \in \mathbf{C}: \operatorname{Re}(z)>0\} \subset \mathbf{C}$; explain carefully how a holomorphic function $f$ may be defined on $U$ satisfying the equation

$\left(f(z)^{2}-1\right)^{2}=z$

Let $\mathcal{F}$ denote the connected component of the space of germs $\mathcal{G}$ (of holomorphic functions on $\mathbf{C} \backslash\{0\})$ corresponding to the function element $(U, f)$, with associated holomorphic $\operatorname{map} \pi: \mathcal{F} \rightarrow \mathbf{C} \backslash\{0\}$. Determine the number of points of $\mathcal{F}$ in $\pi^{-1}(w)$ when (a) $w=\frac{1}{2}$, and (b) $w=1$.

[You may assume any standard facts about analytic continuations that you may need.]

Paper 1, Section I, K

commentLet $Y_{1}, \ldots, Y_{n}$ be independent with $Y_{i} \sim \frac{1}{n_{i}} \operatorname{Bin}\left(n_{i}, \mu_{i}\right), i=1, \ldots, n$, and

$\log \left(\frac{\mu_{i}}{1-\mu_{i}}\right)=x_{i}^{\top} \beta$

where $x_{i}$ is a $p \times 1$ vector of regressors and $\beta$ is a $p \times 1$ vector of parameters. Write down the likelihood of the data $Y_{1}, \ldots, Y_{n}$ as a function of $\mu=\left(\mu_{1}, \ldots, \mu_{n}\right)$. Find the unrestricted maximum likelihood estimator of $\mu$, and the form of the maximum likelihood estimator $\hat{\mu}=\left(\hat{\mu}_{1}, \ldots, \hat{\mu}_{n}\right)$ under the logistic model (1).

Show that the deviance for a comparison of the full (saturated) model to the generalised linear model with canonical link (1) using the maximum likelihood estimator $\hat{\beta}$ can be simplified to

$D(y ; \hat{\mu})=-2 \sum_{i=1}^{n}\left[n_{i} y_{i} x_{i}^{\top} \hat{\beta}-n_{i} \log \left(1-\hat{\mu}_{i}\right)\right]$

Finally, obtain an expression for the deviance residual in this generalised linear model.

Paper 1, Section II, K

commentThe treatment for a patient diagnosed with cancer of the prostate depends on whether the cancer has spread to the surrounding lymph nodes. It is common to operate on the patient to obtain samples from the nodes which can then be analysed under a microscope. However it would be preferable if an accurate assessment of nodal involvement could be made without surgery. For a sample of 53 prostate cancer patients, a number of possible predictor variables were measured before surgery. The patients then had surgery to determine nodal involvement. We want to see if nodal involvement can be accurately predicted from the available variables and determine which ones are most important. The variables take the values 0 or 1 .

$r$ An indicator $0=$ no $/ 1=$ yes of nodal involvement.

aged The patient's age, split into less than $60(=0)$ and 60 or over $(=1)$.

stage A measurement of the size and position of the tumour observed by palpation with the fingers. A serious case is coded as 1 and a less serious case as 0 .

grade Another indicator of the seriousness of the cancer which is determined by a pathology reading of a biopsy taken by needle before surgery. A value of 1 indicates a more serious case of cancer.

xray Another measure of the seriousness of the cancer taken from an X-ray reading. A value of 1 indicates a more serious case of cancer.

acid The level of acid phosphatase in the blood serum where $1=$ high and $0=$ low.

A binomial generalised linear model with a logit link was fitted to the data to predict nodal involvement and the following output obtained:

Part II, 2012 List of Questions

[TURN OVER (a) Give an interpretation of the coefficient of xray.

(b) Give the numerical value of the sum of the squared deviance residuals.

(c) Suppose that the predictors, stage, grade and xray are positively correlated. Describe the effect that this correlation is likely to have on our ability to determine the strength of these predictors in explaining the response.

(d) The probability of observing a value of $70.252$ under a Chi-squared distribution with 52 degrees of freedom is $0.047$. What does this information tell us about the null model for this data? Justify your answer.

(e) What is the lowest predicted probability of the nodal involvement for any future patient?

(f) The first plot in Figure 1 shows the (Pearson) residuals and the fitted values. Explain why the points lie on two curves.

(g) The second plot in Figure 1 shows the value of $\hat{\beta}-\hat{\beta}_{(i)}$ where $(i)$ indicates that patient $i$ was dropped in computing the fit. The values for each predictor, including the intercept, are shown. Could a single case change our opinion of which predictors are important in predicting the response?

Figure 1: The plot on the left shows the Pearson residuals and the fitted values. The plot on the right shows the changes in the regression coefficients when a single point is omitted for each predictor.

Paper 1, Section II, C

commentA meson consists of two quarks, attracted by a linear potential energy

$V=\alpha x$

where $x$ is the separation between the quarks and $\alpha$ is a constant. Treating the quarks classically, compute the vibrational partition function that arises from the separation of quarks. What is the average separation of the quarks at temperature $T$ ?

Consider an ideal gas of these mesons that have the orientation of the quarks fixed so the mesons do not rotate. Compute the total partition function of the gas. What is its heat capacity $C_{V}$ ?

[Note: $\int_{-\infty}^{+\infty} d x e^{-a x^{2}}=\sqrt{\pi / a} .$ ]

Paper 1, Section II, J

commentConsider a multi-period binomial model with a risky asset $\left(S_{0}, \ldots, S_{T}\right)$ and a riskless asset $\left(B_{0}, \ldots, B_{T}\right)$. In each period, the value of the risky asset $S$ is multiplied by $u$ if the period was good, and by $d$ otherwise. The riskless asset is worth $B_{t}=(1+r)^{t}$ at time $0 \leqslant t \leqslant T$, where $r \geqslant 0$.

(i) Assuming that $T=1$ and that

$d<1+r<u$

show how any contingent claim to be paid at time 1 can be priced and exactly replicated. Briefly explain the significance of the condition (1), and indicate how the analysis of the single-period model extends to many periods.

(ii) Now suppose that $T=2$. We assume that $u=2, d=1 / 3, r=1 / 2$, and that the risky asset is worth $S_{0}=27$ at time zero. Find the time- 0 value of an American put option with strike price $K=28$ and expiry at time $T=2$, and find the optimal exercise policy. (Assume that the option cannot be exercised immediately at time zero.)

Paper 1, Section I, F

commentState a version of the Baire category theorem for a complete metric space. Let $T$ be the set of real numbers $x$ with the property that, for each positive integer $n$, there exist integers $p$ and $q$ with $q \geqslant 2$ such that

$0<\left|x-\frac{p}{q}\right|<\frac{1}{q^{n}} .$

Is $T$ an open subset of $\mathbb{R}$ ? Is $T$ a dense subset of $\mathbb{R}$ ? Justify your answers.

Paper 1, Section II, 39D

commentWrite down the linearized equations governing motion in an inviscid compressible fluid and, assuming an adiabatic relationship $p=p(\rho)$, derive the wave equation for the velocity potential $\phi(\mathbf{x}, t)$. Obtain from these linearized equations the energy equation

$\frac{\partial E}{\partial t}+\nabla \cdot \mathbf{I}=0$

and give expressions for the acoustic energy density $E$ and the acoustic intensity, or energyflux vector, I.

An inviscid compressible fluid occupies the half-space $y>0$, and is bounded by a very thin flexible membrane of negligible mass at an undisturbed position $y=0$. Small acoustic disturbances with velocity potential $\phi(x, y, t)$ in the fluid cause the membrane to be deflected to $y=\eta(x, t)$. The membrane is supported by springs that, in the deflected state, exert a restoring force $K \eta \delta x$ on an element $\delta x$ of the membrane. Show that the dispersion relation for waves proportional to $\exp (i k x-i \omega t)$ propagating freely along the membrane is

$\left(k^{2}-\frac{\omega^{2}}{c_{0}^{2}}\right)^{1 / 2}-\frac{\rho_{0} \omega^{2}}{K}=0$

where $\rho_{0}$ is the density of the fluid and $c_{0}$ is the sound speed. Show that in such a wave the component $\left\langle I_{y}\right\rangle$ of mean acoustic intensity perpendicular to the membrane is zero.