Part II, 2016, Paper 1

# Part II, 2016, Paper 1

### Jump to course

Paper 1, Section II, H

commentLet $k$ be an algebraically closed field.

(a) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

(b) Let $f: \mathbb{A}^{1} \rightarrow \mathbb{A}^{3}$ be the map given by

$f(t)=\left(t, t^{2}, t^{3}\right)$

Show that $f$ is a morphism. Show that the image of $f$ is a closed subvariety of $\mathbb{A}^{3}$ and determine its ideal.

(c) Let $g: \mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1} \rightarrow \mathbb{P}^{7}$ be the map given by

$g\left(\left(s_{1}, t_{1}\right),\left(s_{2}, t_{2}\right),\left(s_{3}, t_{3}\right)\right)=\left(s_{1} s_{2} s_{3}, s_{1} s_{2} t_{3}, s_{1} t_{2} s_{3}, s_{1} t_{2} t_{3}, t_{1} s_{2} s_{3}, t_{1} s_{2} t_{3}, t_{1} t_{2} s_{3}, t_{1} t_{2} t_{3}\right) .$

Show that the image of $g$ is a closed subvariety of $\mathbb{P}^{7}$.

Paper 1, Section II, G

commentLet $T=S^{1} \times S^{1}$ be the 2-dimensional torus. Let $\alpha: S^{1} \rightarrow T$ be the inclusion of the coordinate circle $S^{1} \times\{1\}$, and let $X$ be the result of attaching a 2-cell along $\alpha$.

(a) Write down a presentation for the fundamental group of $X$ (with respect to some basepoint), and identify it with a well-known group.

(b) Compute the simplicial homology of any triangulation of $X$.

(c) Show that $X$ is not homotopy equivalent to any compact surface.

Paper 1, Section II, A

commentA particle in one dimension of mass $m$ and energy $E=\hbar^{2} k^{2} / 2 m(k>0)$ is incident from $x=-\infty$ on a potential $V(x)$ with $V(x) \rightarrow 0$ as $x \rightarrow-\infty$ and $V(x)=\infty$ for $x>0$. The relevant solution of the time-independent Schrödinger equation has the asymptotic form

$\psi(x) \sim \exp (i k x)+r(k) \exp (-i k x), \quad x \rightarrow-\infty$

Explain briefly why a pole in the reflection amplitude $r(k)$ at $k=i \kappa$ with $\kappa>0$ corresponds to the existence of a stable bound state in this potential. Indicate why a pole in $r(k)$ just below the real $k$-axis, at $k=k_{0}-i \rho$ with $k_{0} \gg \rho>0$, corresponds to a quasi-stable bound state. Find an approximate expression for the lifetime $\tau$ of such a quasi-stable state.

Now suppose that

$V(x)= \begin{cases}\left(\hbar^{2} U / 2 m\right) \delta(x+a) & \text { for } x<0 \\ \infty & \text { for } x>0\end{cases}$

where $U>0$ and $a>0$ are constants. Compute the reflection amplitude $r(k)$ in this case and deduce that there are quasi-stable bound states if $U$ is large. Give expressions for the wavefunctions and energies of these states and compute their lifetimes, working to leading non-vanishing order in $1 / U$ for each expression.

[ You may assume $\psi=0$ for $x \geqslant 0$ and $\lim _{\epsilon \rightarrow 0+}\left\{\psi^{\prime}(-a+\epsilon)-\psi^{\prime}(-a-\epsilon)\right\}=U \psi(-a)$.]

Paper 1, Section II, J

comment(a) Define a continuous-time Markov chain $X$ with infinitesimal generator $Q$ and jump chain $Y$.

(b) Prove that if a state $x$ is transient for $Y$, then it is transient for $X$.

(c) Prove or provide a counterexample to the following: if $x$ is positive recurrent for $X$, then it is positive recurrent for $Y$.

(d) Consider the continuous-time Markov chain $\left(X_{t}\right)_{t \geqslant 0}$ on $\mathbb{Z}$ with non-zero transition rates given by

$q(i, i+1)=2 \cdot 3^{|i|}, \quad q(i, i)=-3^{|i|+1} \quad \text { and } \quad q(i, i-1)=3^{|i|}$

Determine whether $X$ is transient or recurrent. Let $T_{0}=\inf \left\{t \geqslant J_{1}: X_{t}=0\right\}$, where $J_{1}$ is the first jump time. Does $X$ have an invariant distribution? Justify your answer. Calculate $\mathbb{E}_{0}\left[T_{0}\right]$.

(e) Let $X$ be a continuous-time random walk on $\mathbb{Z}^{d}$ with $q(x)=\|x\|^{\alpha} \wedge 1$ and $q(x, y)=q(x) /(2 d)$ for all $y \in \mathbb{Z}^{d}$ with $\|y-x\|=1$. Determine for which values of $\alpha$ the walk is transient and for which it is recurrent. In the recurrent case, determine the range of $\alpha$ for which it is also positive recurrent. [Here $\|x\|$ denotes the Euclidean norm of $x$.]

Paper 1, Section I, $4 \mathrm{~F}$

commentState the pumping lemma for context-free languages (CFLs). Which of the following are CFLs? Justify your answers.

(i) $\left\{a^{2 n} b^{3 n} \mid n \geqslant 3\right\}$.

(ii) $\left\{a^{2 n} b^{3 n} c^{5 n} \mid n \geqslant 0\right\}$.

(iii) $\left\{a^{p} \mid p\right.$ is a prime $\}$.

Let $L, M$ be CFLs. Show that $L \cup M$ is also a CFL.

Paper 1, Section II, F

comment(a) Define a recursive set and a recursively enumerable (r.e.) set. Prove that $E \subseteq \mathbb{N}$ is recursive if and only if both $E$ and $\mathbb{N} \backslash E$ are r.e.

(b) Define the halting set $\mathbb{K}$. Prove that $\mathbb{K}$ is r.e. but not recursive.

(c) Let $E_{1}, E_{2}, \ldots, E_{n}$ be r.e. sets. Prove that $\bigcup_{i=1}^{n} E_{i}$ and $\bigcap_{i=1}^{n} E_{i}$ are r.e. Show by an example that the union of infinitely many r.e. sets need not be r.e.

(d) Let $E$ be a recursive set and $f: \mathbb{N} \rightarrow \mathbb{N}$ a (total) recursive function. Prove that the set $\{f(n) \mid n \in E\}$ is r.e. Is it necessarily recursive? Justify your answer.

[Any use of Church's thesis in your answer should be explicitly stated.]

Paper 1, Section I, E

commentConsider a one-parameter family of transformations $q_{i}(t) \mapsto Q_{i}(s, t)$ such that $Q_{i}(0, t)=q_{i}(t)$ for all time $t$, and

$\frac{\partial}{\partial s} L\left(Q_{i}, \dot{Q}_{i}, t\right)=0$

where $L$ is a Lagrangian and a dot denotes differentiation with respect to $t$. State and prove Noether's theorem.

Consider the Lagrangian

$L=\frac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)-V(x+y, y+z),$

where the potential $V$ is a function of two variables. Find a continuous symmetry of this Lagrangian and construct the corresponding conserved quantity. Use the Euler-Lagrange equations to explicitly verify that the function you have constructed is independent of $t$.

Paper 1, Section I, G

commentFind the average length of an optimum decipherable binary code for a source that emits five words with probabilities

$0.25,0.15,0.15,0.2,0.25$

Show that, if a source emits $N$ words (with $N \geqslant 2$ ), and if $l_{1}, \ldots, l_{N}$ are the lengths of the codewords in an optimum encoding over the binary alphabet, then

$l_{1}+\cdots+l_{N} \leqslant \frac{1}{2}\left(N^{2}+N-2\right) .$

[You may assume that an optimum encoding can be given by a Huffman encoding.]

Paper 1, Section II, G

commentWhat does it mean to say a binary code $C$ has length $n$, size $m$ and minimum distance d?

Let $A(n, d)$ be the largest value of $m$ for which there exists an $[n, m, d]$-code. Prove that

$\frac{2^{n}}{V(n, d-1)} \leqslant A(n, d) \leqslant \frac{2^{n}}{V(n,\lfloor(d-1) / 2\rfloor)}$

where

$V(n, r)=\sum_{j=0}^{r}\left(\begin{array}{l} n \\ j \end{array}\right)$

Another bound for $A(n, d)$ is the Singleton bound given by

$A(n, d) \leqslant 2^{n-d+1}$

Prove the Singleton bound and give an example of a linear code of length $n>1$ that satisfies $A(n, d)=2^{n-d+1}$.

Paper 1, Section I, C

commentThe expansion scale factor, $a(t)$, for an isotropic and spatially homogeneous universe containing material with pressure $p$ and mass density $\rho$ obeys the equations

$\begin{gathered} \dot{\rho}+3(\rho+p) \frac{\dot{a}}{a}=0, \\ \left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G \rho}{3}-\frac{k}{a^{2}}+\frac{\Lambda}{3}, \end{gathered}$

where the speed of light is set equal to unity, $G$ is Newton's constant, $k$ is a constant equal to 0 or $\pm 1$, and $\Lambda$ is the cosmological constant. Explain briefly the interpretation of these equations.

Show that these equations imply

$\frac{\ddot{a}}{a}=-\frac{4 \pi G(\rho+3 p)}{3}+\frac{\Lambda}{3} .$

Hence show that a static solution with constant $a=a_{\mathrm{s}}$ exists when $p=0$ if

$\Lambda=4 \pi G \rho=\frac{k}{a_{\mathrm{s}}^{2}}$

What must the value of $k$ be, if the density $\rho$ is non-zero?

Paper 1, Section II, C

commentThe distribution function $f(\mathbf{x}, \mathbf{p}, t)$ gives the number of particles in the universe with position in $(\mathbf{x}, \mathbf{x}+\delta \mathbf{x})$ and momentum in $(\mathbf{p}, \mathbf{p}+\delta \mathbf{p})$ at time $t$. It satisfies the boundary condition that $f \rightarrow 0$ as $|\mathbf{x}| \rightarrow \infty$ and as $|\mathbf{p}| \rightarrow \infty$. Its evolution obeys the Boltzmann equation

$\frac{\partial f}{\partial t}+\frac{\partial f}{\partial \mathbf{p}} \cdot \frac{d \mathbf{p}}{d t}+\frac{\partial f}{\partial \mathbf{x}} \cdot \frac{d \mathbf{x}}{d t}=\left[\frac{d f}{d t}\right]_{\mathrm{col}}$

where the collision term $\left[\frac{d f}{d t}\right]_{c o l}$ describes any particle production and annihilation that occurs.

The universe expands isotropically and homogeneously with expansion scale factor $a(t)$, so the momenta evolve isotropically with magnitude $p \propto a^{-1}$. Show that the Boltzmann equation simplifies to

$\frac{\partial f}{\partial t}-\frac{\dot{a}}{a} \mathbf{p} \cdot \frac{\partial f}{\partial \mathbf{p}}=\left[\frac{d f}{d t}\right]_{\mathrm{col}}$

The number densities $n$ of particles and $\bar{n}$ of antiparticles are defined in terms of their distribution functions $f$ and $\bar{f}$, and momenta $p$ and $\bar{p}$, by

$n=\int_{0}^{\infty} f 4 \pi p^{2} d p \quad \text { and } \quad \bar{n}=\int_{0}^{\infty} \bar{f} 4 \pi \bar{p}^{2} d \bar{p}$

and the collision term may be assumed to be of the form

$\left[\frac{d f}{d t}\right]_{\mathrm{col}}=-\langle\sigma v\rangle \int_{0}^{\infty} \bar{f} f 4 \pi \bar{p}^{2} d \bar{p}+R$

where $\langle\sigma v\rangle$ determines the annihilation cross-section of particles by antiparticles and $R$ is the production rate of particles.

By integrating equation $(*)$ with respect to the momentum $\mathbf{p}$ and assuming that $\langle\sigma v\rangle$ is a constant, show that

$\frac{d n}{d t}+3 \frac{\dot{a}}{a} n=-\langle\sigma v\rangle n \bar{n}+Q$

where $Q=\int_{0}^{\infty} R 4 \pi p^{2} d p$. Assuming the same production rate $R$ for antiparticles, write down the corresponding equation satisfied by $\bar{n}$ and show that

$(n-\bar{n}) a^{3}=\mathrm{constant}$

Paper 1, Section II, G

commentDefine what is meant by the regular values and critical values of a smooth map $f: X \rightarrow Y$ of manifolds. State the Preimage Theorem and Sard's Theorem.

Suppose now that $\operatorname{dim} X=\operatorname{dim} Y$. If $X$ is compact, prove that the set of regular values is open in $Y$, but show that this may not be the case if $X$ is non-compact.

Construct an example with $\operatorname{dim} X=\operatorname{dim} Y$ and $X$ compact for which the set of critical values is not a submanifold of $Y$.

[Hint: You may find it helpful to consider the case $X=S^{1}$ and $Y=\mathbf{R}$. Properties of bump functions and the function $e^{-1 / x^{2}}$ may be assumed in this question.]

Paper 1, Section II, E

commentConsider the dynamical system

$\begin{aligned} &\dot{x}=x(y-a), \\ &\dot{y}=1-x-y^{2}, \end{aligned}$

where $-1<a<1$. Find and classify the fixed points of the system.

Use Dulac's criterion with a weighting function of the form $\phi=x^{p}$ and a suitable choice of $p$ to show that there are no periodic orbits for $a \neq 0$. For the case $a=0$ use the same weighting function to find a function $V(x, y)$ which is constant on trajectories. [Hint: $\phi \dot{\mathbf{x}}$ is Hamiltonian.]

Calculate the stable manifold at $(0,-1)$ correct to quadratic order in $x$.

Sketch the phase plane for the cases (i) $a=0$ and (ii) $a=\frac{1}{2}$.

Paper 1, Section II, E

commentA point particle of charge $q$ and mass $m$ moves in an electromagnetic field with 4 -vector potential $A_{\mu}(x)$, where $x^{\mu}$ is position in spacetime. Consider the action

$S=-m c \int\left(-\eta_{\mu \nu} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\nu}}{d \lambda}\right)^{1 / 2} d \lambda+q \int A_{\mu} \frac{d x^{\mu}}{d \lambda} d \lambda$

where $\lambda$ is an arbitrary parameter along the particle's worldline and $\eta_{\mu \nu}=\operatorname{diag}(-1,+1,+1,+1)$ is the Minkowski metric.

(a) By varying the action with respect to $x^{\mu}(\lambda)$, with fixed endpoints, obtain the equation of motion

$m \frac{d u^{\mu}}{d \tau}=q F_{\nu}^{\mu} u^{\nu} \text {, }$

where $\tau$ is the proper time, $u^{\mu}=d x^{\mu} / d \tau$ is the velocity 4-vector, and $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ is the field strength tensor.

(b) This particle moves in the field generated by a second point charge $Q$ that is held at rest at the origin of some inertial frame. By choosing a suitable expression for $A_{\mu}$ and expressing the first particle's spatial position in spherical polar coordinates $(r, \theta, \phi)$, show from the action $(*)$ that

$\begin{aligned} \mathcal{E} & \equiv \dot{t}-\Gamma / r \\ \ell c & \equiv r^{2} \dot{\phi} \sin ^{2} \theta \end{aligned}$

are constants, where $\Gamma=-q Q /\left(4 \pi \epsilon_{0} m c^{2}\right)$ and overdots denote differentiation with respect to $\tau$.

(c) Show that when the motion is in the plane $\theta=\pi / 2$,

$\mathcal{E}+\frac{\Gamma}{r}=\sqrt{1+\frac{\dot{r}^{2}}{c^{2}}+\frac{\ell^{2}}{r^{2}}}$

Hence show that the particle's orbit is bounded if $\mathcal{E}<1$, and that the particle can reach the origin in finite proper time if $\Gamma>|\ell|$.

Paper 1, Section II, B

commentState the vorticity equation and interpret the meaning of each term.

A planar vortex sheet is diffusing in the presence of a perpendicular straining flow. The flow is everywhere of the form $\mathbf{u}=(U(y, t),-E y, E z)$, where $U \rightarrow \pm U_{0}$ as $y \rightarrow \pm \infty$, and $U_{0}$ and $E>0$ are constants. Show that the vorticity has the form $\boldsymbol{\omega}=\omega(y, t) \mathbf{e}_{z}$, and obtain a scalar equation describing the evolution of $\omega$.

Explain physically why the solution approaches a steady state in which the vorticity is concentrated near $y=0$. Use scaling to estimate the thickness $\delta$ of the steady vorticity layer as a function of $E$ and the kinematic viscosity $\nu$.

Determine the steady vorticity profile, $\omega(y)$, and the steady velocity profile, $U(y)$.

$\left[\right.$ Hint: $\left.\quad \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-u^{2}} \mathrm{~d} u .\right]$

State, with a brief physical justification, why you might expect this steady flow to be unstable to long-wavelength perturbations, defining what you mean by long.

Paper 1, Section I, A

commentEvaluate the integral

$f(p)=\mathcal{P} \int_{-\infty}^{\infty} d x \frac{e^{i p x}}{x^{4}-1}$

where $p$ is a real number, for (i) $p>0$ and (ii) $p<0$.

Paper 1, Section II, A

comment(a) Legendre's equation for $w(z)$ is

$\left(z^{2}-1\right) w^{\prime \prime}+2 z w^{\prime}-\ell(\ell+1) w=0, \quad \text { where } \quad \ell=0,1,2, \ldots$

Let $\mathcal{C}$ be a closed contour. Show by direct substitution that for $z$ within $\mathcal{C}$

$\int_{\mathcal{C}} d t \frac{\left(t^{2}-1\right)^{\ell}}{(t-z)^{\ell+1}}$

is a non-trivial solution of Legendre's equation.

(b) Now consider

$Q_{\nu}(z)=\frac{1}{4 i \sin \nu \pi} \int_{\mathcal{C}^{\prime}} d t \frac{\left(t^{2}-1\right)^{\nu}}{(t-z)^{\nu+1}}$

for real $\nu>-1$ and $\nu \neq 0,1,2, \ldots$. The closed contour $\mathcal{C}^{\prime}$ is defined to start at the origin, wind around $t=1$ in a counter-clockwise direction, then wind around $t=-1$ in a clockwise direction, then return to the origin, without encircling the point $z$. Assuming that $z$ does not lie on the real interval $-1 \leqslant x \leqslant 1$, show by deforming $\mathcal{C}^{\prime}$ onto this interval that functions $Q_{\ell}(z)$ may be defined as limits of $Q_{\nu}(z)$ with $\nu \rightarrow \ell=0,1,2, \ldots$.

Find an explicit expression for $Q_{0}(z)$ and verify that it satisfies Legendre's equation with $\ell=0$.

Paper 1, Section II, H

comment(a) Prove that if $K$ is a field and $f \in K[t]$, then there exists a splitting field $L$ of $f$ over $K$. [You do not need to show uniqueness of $L$.]

(b) Let $K_{1}$ and $K_{2}$ be algebraically closed fields of the same characteristic. Show that either $K_{1}$ is isomorphic to a subfield of $K_{2}$ or $K_{2}$ is isomorphic to a subfield of $K_{1}$. [For subfields $F_{i}$ of $K_{1}$ and field homomorphisms $\psi_{i}: F_{i} \rightarrow K_{2}$ with $i=1$, 2, we say $\left(F_{1}, \psi_{1}\right) \leqslant\left(F_{2}, \psi_{2}\right)$ if $F_{1}$ is a subfield of $F_{2}$ and $\left.\psi_{2}\right|_{F_{1}}=\psi_{1}$. You may assume the existence of a maximal pair $(F, \psi)$ with respect to the partial order just defined.]

(c) Give an example of a finite field extension $K \subseteq L$ such that there exist $\alpha, \beta \in L \backslash K$ where $\alpha$ is separable over $K$ but $\beta$ is not separable over $K .$

Paper 1, Section II, D

commentConsider a family of geodesics with $s$ an affine parameter and $V^{a}$ the tangent vector on each curve. The equation of geodesic deviation for a vector field $W^{a}$ is

$\frac{D^{2} W^{a}}{D s^{2}}=R_{b c d}^{a} V^{b} V^{c} W^{d}$

where $\frac{D}{D s}$ denotes the directional covariant derivative $V^{b} \nabla_{b}$.

(i) Show that if

$V^{b} \frac{\partial W^{a}}{\partial x^{b}}=W^{b} \frac{\partial V^{a}}{\partial x^{b}}$

then $W^{a}$ satisfies $(*)$.

(ii) Show that $V^{a}$ and $s V^{a}$ satisfy $(*)$.

(iii) Show that if $W^{a}$ is a Killing vector field, meaning that $\nabla_{b} W_{a}+\nabla_{a} W_{b}=0$, then $W^{a}$ satisfies $(*)$.

(iv) Show that if $W^{a}=w U^{a}$ satisfies $(*)$, where $w$ is a scalar field and $U^{a}$ is a time-like unit vector field, then

$\begin{gathered} \frac{d^{2} w}{d s^{2}}=\left(\Omega^{2}-K\right) w \\ -\frac{D U^{a}}{D s} \frac{D U_{a}}{D s} \quad \text { and } \quad K=R_{a b c d} U^{a} V^{b} V^{c} U^{d} \end{gathered}$

$\begin{aligned} & \text { where } \quad \Omega^{2}=-\frac{D U^{a}}{D s} \frac{D U_{a}}{D s} \text { and } \quad K=R_{a b c d} U^{a} V^{b} V^{c} U^{d} \text {. } \end{aligned}$

[You may use: $\nabla_{b} \nabla_{c} X^{a}-\nabla_{c} \nabla_{b} X^{a}=R_{d b c}^{a} X^{d}$ for any vector field $\left.X^{a} .\right]$

Paper 1, Section II, 16G

comment(a) Show that if $G$ is a planar graph then $\chi(G) \leqslant 5$. [You may assume Euler's formula, provided that you state it precisely.]

(b) (i) Prove that if $G$ is a triangle-free planar graph then $\chi(G) \leqslant 4$.

(ii) Prove that if $G$ is a planar graph of girth at least 6 then $\chi(G) \leqslant 3$.

(iii) Does there exist a constant $g$ such that, if $G$ is a planar graph of girth at least $g$, then $\chi(G) \leqslant 2 ?$ Justify your answer.

Paper 1, Section II, D

commentWhat does it mean for an evolution equation $u_{t}=K\left(x, u, u_{x}, \ldots\right)$ to be in Hamiltonian form? Define the associated Poisson bracket.

An evolution equation $u_{t}=K\left(x, u, u_{x}, \ldots\right)$ is said to be bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways, i.e.

$K=\mathcal{J} \delta H_{0}=\mathcal{E} \delta H_{1}$

for Hamiltonian operators $\mathcal{J}, \mathcal{E}$ and functionals $H_{0}, H_{1}$. By considering the sequence $\left\{H_{m}\right\}_{m \geqslant 0}$ defined by the recurrence relation

$\mathcal{E} \delta H_{m+1}=\mathcal{J} \delta H_{m}$

show that bi-Hamiltonian systems possess infinitely many first integrals in involution. [You may assume that $(*)$ can always be solved for $H_{m+1}$, given $H_{m}$.]

The Harry Dym equation for the function $u=u(x, t)$ is

$u_{t}=\frac{\partial^{3}}{\partial x^{3}}\left(u^{-1 / 2}\right)$

This equation can be written in Hamiltonian form $u_{t}=\mathcal{E} \delta H_{1}$ with

$\mathcal{E}=2 u \frac{\partial}{\partial x}+u_{x}, \quad H_{1}[u]=\frac{1}{8} \int u^{-5 / 2} u_{x}^{2} \mathrm{~d} x$

Show that the Harry Dym equation possesses infinitely many first integrals in involution. [You need not verify the Jacobi identity if your argument involves a Hamiltonian operator.]

Paper 1, Section II, I

comment(a) State the closed graph theorem.

(b) Prove the closed graph theorem assuming the inverse mapping theorem.

(c) Let $X, Y, Z$ be Banach spaces and $T: X \rightarrow Y, S: Y \rightarrow Z$ be linear maps. Suppose that $S \circ T$ is bounded and $S$ is both bounded and injective. Show that $T$ is bounded.

Paper 1, Section II, F

commentWhich of the following statements are true? Justify your answers. (a) Every ordinal is of the form $\alpha+n$, where $\alpha$ is a limit ordinal and $n \in \omega$. (b) Every ordinal is of the form $\omega^{\alpha} \cdot m+n$, where $\alpha$ is an ordinal and $m, n \in \omega$. (c) If $\alpha=\omega \cdot \alpha$, then $\alpha=\omega^{\omega} \cdot \beta$ for some $\beta$. (d) If $\alpha=\omega^{\alpha}$, then $\alpha$ is uncountable. (e) If $\alpha>1$ and $\alpha=\alpha^{\omega}$, then $\alpha$ is uncountable.

[Standard laws of ordinal arithmetic may be assumed, but if you use the Division Algorithm you should prove it.]

Paper 1, Section I, B

commentConsider an epidemic model where susceptibles are vaccinated at per capita rate $v$, but immunity (from infection or vaccination) is lost at per capita rate $b$. The system is given by

$\begin{aligned} &\frac{d S}{d t}=-r I S+b(N-I-S)-v S \\ &\frac{d I}{d t}=r I S-a I \end{aligned}$

where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, $N$ is the total population size and all parameters are positive. The basic reproduction ratio $R_{0}=r N / a$ satisfies $R_{0}>1$.

Find the critical vaccination rate $v_{c}$, in terms of $b$ and $R_{0}$, such that the system has an equilibrium with the disease present if $v<v_{c}$. Show that this equilibrium is stable when it exists.

Find the long-term outcome for $S$ and $I$ if $v>v_{c}$.

Paper 1, Section II, F

comment(a) Let $f(X) \in \mathbb{Q}[X]$ be an irreducible polynomial of degree $n, \theta \in \mathbb{C}$ a root of $f$, and $K=\mathbb{Q}(\theta)$. Show that $\operatorname{disc}(f)=\pm N_{K / \mathbb{Q}}\left(f^{\prime}(\theta)\right)$.

(b) Now suppose $f(X)=X^{n}+a X+b$. Write down the matrix representing multiplication by $f^{\prime}(\theta)$ with respect to the basis $1, \theta, \ldots, \theta^{n-1}$ for $K$. Hence show that

$\operatorname{disc}(f)=\pm\left((1-n)^{n-1} a^{n}+n^{n} b^{n-1}\right)$

(c) Suppose $f(X)=X^{4}+X+1$. Determine $\mathcal{O}_{K}$. [You may quote any standard result, as long as you state it clearly.]

Paper 1, Section I, I

commentDefine the Riemann zeta function $\zeta(s)$ for $\operatorname{Re}(s)>1$. State and prove the alternative formula for $\zeta(s)$ as an Euler product. Hence or otherwise show that $\zeta(s) \neq 0$ for $\operatorname{Re}(s)>1$.

Paper 1, Section II, B

comment(a) Consider the periodic function

$f(x)=5+2 \cos \left(2 \pi x-\frac{\pi}{2}\right)+3 \cos (4 \pi x)$

on the interval $[0,1]$. The $N$-point discrete Fourier transform of $f$ is defined by

$F_{N}(n)=\frac{1}{N} \sum_{k=0}^{N-1} f_{k} \omega_{N}^{-n k}, \quad n=0,1, \ldots, N-1$

where $\omega_{N}=e^{2 \pi i / N}$ and $f_{k}=f(k / N)$. Compute $F_{4}(n), n=0, \ldots, 3$, using the Fast Fourier Transform (FFT). Compare your result with what you get by computing $F_{4}(n)$ directly from $(*)$. Carefully explain all your computations.

(b) Now let $f$ be any analytic function on $\mathbb{R}$ that is periodic with period 1 . The continuous Fourier transform of $f$ is defined by

$\hat{f}_{n}=\int_{0}^{1} f(\tau) e^{-2 \pi i n \tau} d \tau, \quad n \in \mathbb{Z}$

Use integration by parts to show that the Fourier coefficients $\hat{f}_{n}$ decay spectrally.

Explain what it means for the discrete Fourier transform of $f$ to approximate the continuous Fourier transform with spectral accuracy. Prove that it does so.

What can you say about the behaviour of $F_{N}(N-n)$ as $N \rightarrow \infty$ for fixed $n$ ?

Paper 1, Section II, A

commentA particle in one dimension has position and momentum operators $\hat{x}$ and $\hat{p}$ whose eigenstates obey

$\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right), \quad\left\langle p \mid p^{\prime}\right\rangle=\delta\left(p-p^{\prime}\right), \quad\langle x \mid p\rangle=(2 \pi \hbar)^{-1 / 2} e^{i x p / \hbar}$

For a state $|\psi\rangle$, define the position-space and momentum-space wavefunctions $\psi(x)$ and $\tilde{\psi}(p)$ and show how each of these can be expressed in terms of the other.

Write down the translation operator $U(\alpha)$ and check that your expression is consistent with the property $U(\alpha)|x\rangle=|x+\alpha\rangle$. For a state $|\psi\rangle$, relate the position-space and momentum-space wavefunctions for $U(\alpha)|\psi\rangle$ to $\psi(x)$ and $\tilde{\psi}(p)$ respectively.

Now consider a harmonic oscillator with mass $m$, frequency $\omega$, and annihilation and creation operators

$a=\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2}\left(\hat{x}+\frac{i}{m \omega} \hat{p}\right), \quad a^{\dagger}=\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2}\left(\hat{x}-\frac{i}{m \omega} \hat{p}\right)$

Let $\psi_{n}(x)$ and $\tilde{\psi}_{n}(p)$ be the wavefunctions corresponding to the normalised energy eigenstates $|n\rangle$, where $n=0,1,2, \ldots$.

(i) Express $\psi_{0}(x-\alpha)$ explicitly in terms of the wavefunctions $\psi_{n}(x)$.

(ii) Given that $\tilde{\psi}_{n}(p)=f_{n}(u) \tilde{\psi}_{0}(p)$, where the $f_{n}$ are polynomials and $u=(2 / \hbar m \omega)^{1 / 2} p$, show that

$e^{-i \gamma u}=e^{-\gamma^{2} / 2} \sum_{n=0}^{\infty} \frac{\gamma^{n}}{\sqrt{n !}} f_{n}(u) \text { for any real } \gamma$

[You may quote standard results for a harmonic oscillator. You may also use, without proof, $e^{A+B}=e^{A} e^{B} e^{-\frac{1}{2}[A, B]}$ for operators $A$ and $B$ which each commute with $\left.[A, B] .\right]$

Paper 1, Section II, $\mathbf{2 7 J}$

commentDerive the maximum likelihood estimator $\hat{\theta}_{n}$ based on independent observations $X_{1}, \ldots, X_{n}$ that are identically distributed as $N(\theta, 1)$, where the unknown parameter $\theta$ lies in the parameter space $\Theta=\mathbb{R}$. Find the limiting distribution of $\sqrt{n}\left(\widehat{\theta}_{n}-\theta\right)$ as $n \rightarrow \infty$.

Now define

$\begin{array}{rll} \tilde{\theta}_{n} & =\widehat{\theta}_{n} & \text { whenever }\left|\widehat{\theta}_{n}\right|>n^{-1 / 4}, \\ & =0 & \text { otherwise, } \end{array}$

$\begin{aligned} & =0 \text { otherwise, } \end{aligned}$

and find the limiting distribution of $\sqrt{n}\left(\tilde{\theta}_{n}-\theta\right)$ as $n \rightarrow \infty$.

Calculate

$\lim _{n \rightarrow \infty} \sup _{\theta \in \Theta} n E_{\theta}\left(T_{n}-\theta\right)^{2}$

for the choices $T_{n}=\widehat{\theta}_{n}$ and $T_{n}=\widetilde{\theta}_{n}$. Based on the above findings, which estimator $T_{n}$ of $\theta$ would you prefer? Explain your answer.

[Throughout, you may use standard facts of stochastic convergence, such as the central limit theorem, provided they are clearly stated.]

Paper 1, Section II, J

commentThroughout this question $(E, \mathcal{E}, \mu)$ is a measure space and $\left(f_{n}\right), f$ are measurable functions.

(a) Give the definitions of pointwise convergence, pointwise a.e. convergence, and convergence in measure.

(b) If $f_{n} \rightarrow f$ pointwise a.e., does $f_{n} \rightarrow f$ in measure? Give a proof or a counterexample.

(c) If $f_{n} \rightarrow f$ in measure, does $f_{n} \rightarrow f$ pointwise a.e.? Give a proof or a counterexample.

(d) Now suppose that $(E, \mathcal{E})=([0,1], \mathcal{B}([0,1]))$ and that $\mu$ is Lebesgue measure on $[0,1]$. Suppose $\left(f_{n}\right)$ is a sequence of Borel measurable functions on $[0,1]$ which converges pointwise a.e. to $f$.

(i) For each $n, k$ let $E_{n, k}=\bigcup_{m \geqslant n}\left\{x:\left|f_{m}(x)-f(x)\right|>1 / k\right\}$. Show that $\lim _{n \rightarrow \infty} \mu\left(E_{n, k}\right)=0$ for each $k \in \mathbb{N}$.

(ii) Show that for every $\epsilon>0$ there exists a set $A$ with $\mu(A)<\epsilon$ so that $f_{n} \rightarrow f$ uniformly on $[0,1] \backslash A$.

(iii) Does (ii) hold with $[0,1]$ replaced by $\mathbb{R}$ ? Give a proof or a counterexample.

Paper 1, Section II, I

commentLet $N$ be a normal subgroup of the finite group $G$. Explain how a (complex) representation of $G / N$ gives rise to an associated representation of $G$, and briefly describe which representations of $G$ arise this way.

Let $G$ be the group of order 54 which is given by

$G=\left\langle a, b: a^{9}=b^{6}=1, b^{-1} a b=a^{2}\right\rangle$

Find the conjugacy classes of $G$. By observing that $N_{1}=\langle a\rangle$ and $N_{2}=\left\langle a^{3}, b^{2}\right\rangle$ are normal in $G$, or otherwise, construct the character table of $G$.

Paper 1, Section II, H

comment(a) Let $f: R \rightarrow S$ be a non-constant holomorphic map between Riemann surfaces. Prove that $f$ takes open sets of $R$ to open sets of $S$.

(b) Let $U$ be a simply connected domain strictly contained in $\mathbb{C}$. Is there a conformal equivalence between $U$ and $\mathbb{C}$ ? Justify your answer.

(c) Let $R$ be a compact Riemann surface and $A \subset R$ a discrete subset. Given a non-constant holomorphic function $f: R \backslash A \rightarrow \mathbb{C}$, show that $f(R \backslash A)$ is dense in $\mathbb{C}$.

Paper 1, Section I, K

commentThe body mass index (BMI) of your closest friend is a good predictor of your own BMI. A scientist applies polynomial regression to understand the relationship between these two variables among 200 students in a sixth form college. The $R$ commands

$>$ fit. $1<-\operatorname{lm}(B M I \sim$ poly $($ friendBMI , 2, raw=T $))$

$>$ fit. $2<-\operatorname{lm}(B M I \sim$ poly $($ friendBMI, 3, raw $=\mathrm{T}))$

fit the models $Y=\beta_{0}+\beta_{1} X+\beta_{2} X^{2}+\varepsilon$ and $Y=\beta_{0}+\beta_{1} X+\beta_{2} X^{2}+\beta_{3} X^{3}+\varepsilon$, respectively, with $\varepsilon \sim N\left(0, \sigma^{2}\right)$ in each case.

Setting the parameters raw to FALSE:

$>$ fit. $3<-\operatorname{lm}(B M I \sim$ poly $($ friendBMI , 2, raw=F $)$ )

$>$ fit. $4<-\operatorname{lm}(\mathrm{BMI} \sim$ poly $($ friendBMI, 3, raw $=\mathrm{F}))$

fits the models $Y=\beta_{0}+\beta_{1} P_{1}(X)+\beta_{2} P_{2}(X)+\varepsilon$ and $Y=\beta_{0}+\beta_{1} P_{1}(X)+\beta_{2} P_{2}(X)+$ $\beta_{3} P_{3}(X)+\varepsilon$, with $\varepsilon \sim N\left(0, \sigma^{2}\right)$. The function $P_{i}$ is a polynomial of degree $i$. Furthermore, the design matrix output by the function poly with raw=F satisfies:

$>t($ poly $($ friendBMI, 3, raw $=F)) \% * \%$ poly $(a, 3$, raw $=F)$

$\begin{array}{rrrr}1 & 2 & 3 \\ 1 & 1.000000 e+00 & 1.288032 \mathrm{e}-16 & 3.187554 \mathrm{e}-17 \\ 2 & 1.288032 \mathrm{e}-16 & 1.000000 \mathrm{e}+00 & -6.201636 \mathrm{e}-17 \\ 3 & 3.187554 \mathrm{e}-17 & -6.201636 \mathrm{e}-17 & 1.000000 \mathrm{e}+00\end{array}$

How does the variance of $\hat{\beta}$ differ in the models $f i t .2$ and $f i t .4$ ? What about the variance of the fitted values $\hat{Y}=X \hat{\beta}$ ? Finally, consider the output of the commands

$>\operatorname{anova}$ (fit.1,fit.2)

anova(fit.3,fit.4)

Define the test statistic computed by this function and specify its distribution. Which command yields a higher statistic?

Paper 1, Section II, K

comment(a) Let $Y$ be an $n$-vector of responses from the linear model $Y=X \beta+\varepsilon$, with $\beta \in \mathbb{R}^{p}$. The internally studentized residual is defined by

$s_{i}=\frac{Y_{i}-x_{i}^{\top} \hat{\beta}}{\tilde{\sigma} \sqrt{1-p_{i}}},$

where $\hat{\beta}$ is the least squares estimate, $p_{i}$ is the leverage of sample $i$, and

$\tilde{\sigma}^{2}=\frac{\|Y-X \hat{\beta}\|_{2}^{2}}{(n-p)} .$

Prove that the joint distribution of $s=\left(s_{1}, \ldots, s_{n}\right)^{\top}$ is the same in the following two models: (i) $\varepsilon \sim N(0, \sigma I)$, and (ii) $\varepsilon \mid \sigma \sim N(0, \sigma I)$, with $1 / \sigma \sim \chi_{\nu}^{2}$ (in this model, $\varepsilon_{1}, \ldots, \varepsilon_{n}$ are identically $t_{\nu}$-distributed). [Hint: A random vector $Z$ is spherically symmetric if for any orthogonal matrix $H, H Z \stackrel{d}{=} Z$. If $Z$ is spherically symmetric and a.s. nonzero, then $Z /\|Z\|_{2}$ is a uniform point on the sphere; in addition, any orthogonal projection of $Z$ is also spherically symmetric. A standard normal vector is spherically symmetric.]

(b) A social scientist regresses the income of 120 Cambridge graduates onto 20 answers from a questionnaire given to the participants in their first year. She notices one questionnaire with very unusual answers, which she suspects was due to miscoding. The sample has a leverage of $0.8$. To check whether this sample is an outlier, she computes its externally studentized residual,

$t_{i}=\frac{Y_{i}-x_{i}^{\top} \hat{\beta}}{\tilde{\sigma}_{(i)} \sqrt{1-p_{i}}}=4.57,$

where $\tilde{\sigma}_{(i)}$ is estimated from a fit of all samples except the one in question, $\left(x_{i}, Y_{i}\right)$. Is this a high leverage point? Can she conclude this sample is an outlier at a significance level of $5 \%$ ?

(c) After examining the following plot of residuals against the response, the investigator calculates the externally studentized residual of the participant denoted by the black dot, which is $2.33$. Can she conclude this sample is an outlier with a significance level of $5 \%$ ?

Part II, $2016 \quad$ List of Questions

Paper 1, Section II, C

commentConsider an ideal quantum gas with one-particle states $|i\rangle$ of energy $\epsilon_{i}$. Let $p_{i}^{\left(n_{i}\right)}$ denote the probability that state $|i\rangle$ is occupied by $n_{i}$ particles. Here, $n_{i}$ can take the values 0 or 1 for fermions and any non-negative integer for bosons. The entropy of the gas is given by

$S=-k_{B} \sum_{i} \sum_{n_{i}} p_{i}^{\left(n_{i}\right)} \ln p_{i}^{\left(n_{i}\right)}$

(a) Write down the constraints that must be satisfied by the probabilities if the average energy $\langle E\rangle$ and average particle number $\langle N\rangle$ are kept at fixed values.

Show that if $S$ is maximised then

$p_{i}^{\left(n_{i}\right)}=\frac{1}{\mathcal{Z}_{i}} e^{-\left(\beta \epsilon_{i}+\gamma\right) n_{i}}$

where $\beta$ and $\gamma$ are Lagrange multipliers. What is $\mathcal{Z}_{i}$ ?

(b) Insert these probabilities $p_{i}^{\left(n_{i}\right)}$ into the expression for $S$, and combine the result with the first law of thermodynamics to find the meaning of $\beta$ and $\gamma$.

(c) Calculate the average occupation number $\left\langle n_{i}\right\rangle=\sum_{n_{i}} n_{i} p_{i}^{\left(n_{i}\right)}$ for a gas of fermions.

Paper 1, Section II, 28K

comment(a) What is a Brownian motion?

(b) State the Brownian reflection principle. State the Cameron-Martin theorem for Brownian motion with constant drift.

(c) Let $\left(W_{t}\right)_{t \geqslant 0}$ be a Brownian motion. Show that

$\mathbb{P}\left(\max _{0 \leqslant s \leqslant t}\left(W_{s}+a s\right) \leqslant b\right)=\Phi\left(\frac{b-a t}{\sqrt{t}}\right)-e^{2 a b} \Phi\left(\frac{-b-a t}{\sqrt{t}}\right)$

where $\Phi$ is the standard normal distribution function.

(d) Find

$\mathbb{P}\left(\max _{u \geqslant t}\left(W_{u}+a u\right) \leqslant b\right)$

Paper 1, Section I, H

commentBy considering the function $\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ defined by

$R\left(a_{0}, \ldots, a_{n}\right)=\sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right|$

or otherwise, show that there exist $K_{n}>0$ and $\delta_{n}>0$ such that

$K_{n} \sum_{j=0}^{n}\left|a_{j}\right| \geqslant \sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right| \geqslant \delta_{n} \sum_{j=0}^{n}\left|a_{j}\right|$

for all $a_{j} \in \mathbb{R}, 0 \leqslant j \leqslant n$.

Show, quoting carefully any theorems you use, that we must have $\delta_{n} \rightarrow 0$ as $n \rightarrow \infty$.

Paper 1, Section II, D

commentWrite down the linearised equations governing motion of an inviscid compressible fluid at uniform entropy. Assuming that the velocity is irrotational, show that it may be derived from a velocity potential $\phi(\mathbf{x}, t)$ satisfying the wave equation

$\frac{\partial^{2} \phi}{\partial t^{2}}=c_{0}^{2} \nabla^{2} \phi$

and identify the wave speed $c_{0}$. Obtain from these linearised equations the energyconservation equation

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

and give expressions for the acoustic-energy density $E$ and the acoustic-energy flux $\mathbf{I}$ in terms of $\phi$.

Such a fluid occupies a semi-infinite waveguide $x>0$ of square cross-section $0<y<a$, $0<z<a$ bounded by rigid walls. An impenetrable membrane closing the end $x=0$ makes prescribed small displacements to

$x=X(y, z, t) \equiv \operatorname{Re}\left[e^{-i \omega t} A(y, z)\right]$

where $\omega>0$ and $|A| \ll a, c_{0} / \omega$. Show that the velocity potential is given by

$\phi=\operatorname{Re}\left[e^{-i \omega t} \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \cos \left(\frac{m \pi y}{a}\right) \cos \left(\frac{n \pi z}{a}\right) f_{m n}(x)\right]$

where the functions $f_{m n}(x)$, including their amplitudes, are to be determined, with the sign of any square roots specified clearly.

If $0<\omega<\pi c_{0} / a$, what is the asymptotic behaviour of $\phi$ as $x \rightarrow+\infty$ ? Using this behaviour and the energy-conservation equation averaged over both time and the crosssection, or otherwise, determine the double-averaged energy flux along the waveguide,

$\left\langle\overline{I_{x}}\right\rangle(x) \equiv \frac{\omega}{2 \pi a^{2}} \int_{0}^{2 \pi / \omega} \int_{0}^{a} \int_{0}^{a} I_{x}(x, y, z, t) \mathrm{d} y \mathrm{~d} z \mathrm{~d} t$

explaining why this is independent of $x$.