• # Paper 3, Section II, F

Let $W \subseteq \mathbb{A}^{2}$ be the curve defined by the equation $y^{3}=x^{4}+1$ over the complex numbers $\mathbb{C}$, and let $X \subseteq \mathbb{P}^{2}$ be its closure.

(a) Show $X$ is smooth.

(b) Determine the ramification points of the $\operatorname{map} X \rightarrow \mathbb{P}^{1}$ defined by

$(x: y: z) \mapsto(x: z) .$

Using this, determine the Euler characteristic and genus of $X$, stating clearly any theorems that you are using.

(c) Let $\omega=\frac{d x}{y^{2}} \in \mathcal{K}_{X}$. Compute $\nu_{p}(\omega)$ for all $p \in X$, and determine a basis for $\mathcal{L}\left(\mathcal{K}_{X}\right)$

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• # Paper 3, Section II, F

Let $K$ be a simplicial complex, and $L$ a subcomplex. As usual, $C_{k}(K)$ denotes the group of $k$-chains of $K$, and $C_{k}(L)$ denotes the group of $k$-chains of $L$.

(a) Let

$C_{k}(K, L)=C_{k}(K) / C_{k}(L)$

for each integer $k$. Prove that the boundary map of $K$ descends to give $C_{\bullet}(K, L)$ the structure of a chain complex.

(b) The homology groups of $K$ relative to $L$, denoted by $H_{k}(K, L)$, are defined to be the homology groups of the chain complex $C_{\bullet}(K, L)$. Prove that there is a long exact sequence that relates the homology groups of $K$ relative to $L$ to the homology groups of $K$ and the homology groups of $L$.

(c) Let $D_{n}$ be the closed $n$-dimensional disc, and $S^{n-1}$ be the $(n-1)$-dimensional sphere. Exhibit simplicial complexes $K_{n}$ and subcomplexes $L_{n-1}$ such that $D_{n} \cong\left|K_{n}\right|$ in such a way that $\left|L_{n-1}\right|$ is identified with $S^{n-1}$.

(d) Compute the relative homology groups $H_{k}\left(K_{n}, L_{n-1}\right)$, for all integers $k \geqslant 0$ and $n \geqslant 2$ where $K_{n}$ and $L_{n-1}$ are as in (c).

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• # Paper 3, Section II, H

(a) Prove that in a finite-dimensional normed vector space the weak and strong topologies coincide.

(b) Prove that in a normed vector space $X$, a weakly convergent sequence is bounded. [Any form of the Banach-Steinhaus theorem may be used, as long as you state it clearly.]

(c) Let $\ell^{1}$ be the space of real-valued absolutely summable sequences. Suppose $\left(a^{k}\right)$ is a weakly convergent sequence in $\ell^{1}$ which does not converge strongly. Show there is a constant $\varepsilon>0$ and a sequence $\left(x^{k}\right)$ in $\ell^{1}$ which satisfies $x^{k} \rightarrow 0$ and $\left\|x^{k}\right\|_{\ell^{1}} \geqslant \varepsilon$ for all $k \geqslant 1$.

With $\left(x^{k}\right)$ as above, show there is some $y \in \ell^{\infty}$ and a subsequence $\left(x^{k_{n}}\right)$ of $\left(x^{k}\right)$ with $\left\langle x^{k_{n}}, y\right\rangle \geqslant \varepsilon / 3$ for all $n$. Deduce that every weakly convergent sequence in $\ell^{1}$ is strongly convergent.

[Hint: Define $y$ so that $y_{i}=\operatorname{sign} x_{i}^{k_{n}}$ for $b_{n-1}, where the sequence of integers $b_{n}$ should be defined inductively along with $\left.x^{k_{n}} .\right]$

(d) Is the conclusion of part (c) still true if we replace $\ell^{1}$ by $L^{1}([0,2 \pi]) ?$

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• # Paper 3, Section II, B

A Hamiltonian $H$ is invariant under the discrete translational symmetry of a Bravais lattice $\Lambda$. This means that there exists a unitary translation operator $T_{\mathbf{r}}$ such that $\left[H, T_{\mathbf{r}}\right]=0$ for all $\mathbf{r} \in \Lambda$. State and prove Bloch's theorem for $H$.

Consider the two-dimensional Bravais lattice $\Lambda$ defined by the basis vectors

$\mathbf{a}_{1}=\frac{a}{2}(\sqrt{3}, 1), \quad \mathbf{a}_{2}=\frac{a}{2}(\sqrt{3},-1)$

Find basis vectors $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ for the reciprocal lattice. Sketch the Brillouin zone. Explain why the Brillouin zone has only two physically distinct corners. Show that the positions of these corners may be taken to be $\mathbf{K}=\frac{1}{3}\left(2 \mathbf{b}_{1}+\mathbf{b}_{2}\right)$ and $\mathbf{K}^{\prime}=\frac{1}{3}\left(\mathbf{b}_{1}+2 \mathbf{b}_{2}\right)$.

The dynamics of a single electron moving on the lattice $\Lambda$ is described by a tightbinding model with Hamiltonian

$H=\sum_{\mathbf{r} \in \Lambda}\left[E_{0}|\mathbf{r}\rangle\langle\mathbf{r}|-\lambda\left(|\mathbf{r}\rangle\left\langle\mathbf{r}+\mathbf{a}_{1}|+| \mathbf{r}\right\rangle\left\langle\mathbf{r}+\mathbf{a}_{2}|+| \mathbf{r}+\mathbf{a}_{1}\right\rangle\left\langle\mathbf{r}|+| \mathbf{r}+\mathbf{a}_{2}\right\rangle\langle\mathbf{r}|\right)\right]$

where $E_{0}$ and $\lambda$ are real parameters. What is the energy spectrum as a function of the wave vector $\mathbf{k}$ in the Brillouin zone? How does the energy vary along the boundary of the Brillouin zone between $\mathbf{K}$ and $\mathbf{K}^{\prime}$ ? What is the width of the band?

In a real material, each site of the lattice $\Lambda$ contains an atom with a certain valency. Explain how the conducting properties of the material depend on the valency.

Suppose now that there is a second band, with minimum $E=E_{0}+\Delta$. For what values of $\Delta$ and the valency is the material an insulator?

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• # Paper 3, Section II, K

(a) What does it mean to say that a continuous-time Markov chain $X=\left(X_{t}: 0 \leqslant\right.$ $t \leqslant T$ ) with state space $S$ is reversible in equilibrium? State the detailed balance equations, and show that any probability distribution on $S$ satisfying them is invariant for the chain.

(b) Customers arrive in a shop in the manner of a Poisson process with rate $\lambda>0$. There are $s$ servers, and capacity for up to $N$ people waiting for service. Any customer arriving when the shop is full (in that the total number of customers present is $N+s$ ) is not admitted and never returns. Service times are exponentially distributed with parameter $\mu>0$, and they are independent of one another and of the arrivals process. Describe the number $X_{t}$ of customers in the shop at time $t$ as a Markov chain.

Calculate the invariant distribution $\pi$ of $X=\left(X_{t}: t \geqslant 0\right)$, and explain why $\pi$ is the unique invariant distribution. Show that $X$ is reversible in equilibrium.

[Any general result from the course may be used without proof, but must be stated clearly.]

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• # Paper 3, Section II, A

(a) State Watson's lemma for the case when all the functions and variables involved are real, and use it to calculate the asymptotic approximation as $x \rightarrow \infty$ for the integral $I$, where

$I=\int_{0}^{\infty} e^{-x t} \sin \left(t^{2}\right) d t$

(b) The Bessel function $J_{\nu}(z)$ of the first kind of order $\nu$ has integral representation

$J_{\nu}(z)=\frac{1}{\Gamma\left(\nu+\frac{1}{2}\right) \sqrt{\pi}}\left(\frac{z}{2}\right)^{\nu} \int_{-1}^{1} e^{i z t}\left(1-t^{2}\right)^{\nu-1 / 2} d t$

where $\Gamma$ is the Gamma function, $\operatorname{Re}(\nu)>1 / 2$ and $z$ is in general a complex variable. The complex version of Watson's lemma is obtained by replacing $x$ with the complex variable $z$, and is valid for $|z| \rightarrow \infty$ and $|\arg (z)| \leqslant \pi / 2-\delta<\pi / 2$, for some $\delta$ such that $0<\delta<\pi / 2$. Use this version to derive an asymptotic expansion for $J_{\nu}(z)$ as $|z| \rightarrow \infty$. For what values of $\arg (z)$ is this approximation valid?

[Hint: You may find the substitution $t=2 \tau-1$ useful.]

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• # Paper 3, Section I, $4 \mathrm{H}$

(a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form (CNF). Can a CFG in CNF ever define a language containing $\epsilon$ ? If $G_{\text {Chom }}$ denotes the result of converting an arbitrary CFG $G$ into one in CNF, state the relationship between $\mathcal{L}(G)$ and $\mathcal{L}\left(G_{\text {Chom }}\right)$.

(b) Let $G$ be a CFG in CNF. Give an algorithm that, on input of any word $v$ on the terminals of $G$, decides if $v \in \mathcal{L}(G)$ or not. Explain why your algorithm works.

(c) Convert the following CFG $G$ into a grammar in CNF:

\begin{aligned} S \rightarrow & S b b|a S| T \\ & T \rightarrow c c \end{aligned}

Does $\mathcal{L}(G)=\mathcal{L}\left(G_{\text {Chom }}\right)$ in this case? Justify your answer.

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• # Paper 3, Section II, 12H

(a) State the $s-m-n$ theorem and the recursion theorem.

(b) State and prove Rice's theorem.

(c) Show that if $g: \mathbb{N}_{0}^{2} \rightarrow \mathbb{N}_{0}$ is partial recursive, then there is some $e \in \mathbb{N}_{0}$ such that

$f_{e, 1}(y)=g(e, y) \quad \forall y \in \mathbb{N}_{0}$

(d) Show there exists some $m \in \mathbb{N}_{0}$ such that $W_{m}$ has exactly $m^{2}$ elements.

(e) Given $n \in \mathbb{N}_{0}$, is it possible to compute whether or not the number of elements of $W_{n}$ is a (finite) perfect square? Justify your answer.

[In this question $\mathbb{N}_{0}$ denotes the set of non-negative integers. Any use of Church's thesis in your answers should be explicitly stated.]

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• # Paper 3, Section I, E

A simple harmonic oscillator of mass $m$ and spring constant $k$ has the equation of motion

$m \ddot{x}=-k x .$

(a) Describe the orbits of the system in phase space. State how the action $I$ of the oscillator is related to a geometrical property of the orbits in phase space. Derive the action-angle variables $(\theta, I)$ and give the form of the Hamiltonian of the oscillator in action-angle variables.

(b) Suppose now that the spring constant $k$ varies in time. Under what conditions does the theory of adiabatic invariance apply? Assuming that these conditions hold, identify an adiabatic invariant and determine how the energy and amplitude of the oscillator vary with $k$ in this approximation.

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• # Paper 3, Section I, G

What does it mean to transmit reliably at rate $R$ through a binary symmetric channel $(\mathrm{BSC})$ with error probability $p$ ?

Assuming Shannon's second coding theorem (also known as Shannon's noisy coding theorem), compute the supremum of all possible reliable transmission rates of a BSC. Describe qualitatively the behaviour of the capacity as $p$ varies. Your answer should address the following cases,

(i) $p$ is small,

(ii) $p=1 / 2$,

(iii) $p>1 / 2$.

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• # Paper 3, Section I, B

Consider a spherically symmetric distribution of mass with density $\rho(r)$ at distance $r$ from the centre. Derive the pressure support equation that the pressure $P(r)$ has to satisfy for the system to be in static equilibrium.

Assume now that the mass density obeys $\rho(r)=A r^{2} P(r)$, for some positive constant A. Determine whether or not the system has a stable solution corresponding to a star of finite radius.

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• # Paper 3, Section II, B

[You may work in units of the speed of light, so that $c=1 .$ ]

Consider the process where protons and electrons combine to form neutral hydrogen atoms;

$p^{+}+e^{-} \leftrightarrow H^{0}+\gamma$

Let $n_{p}, n_{e}$ and $n_{H}$ denote the number densities for protons, electrons and hydrogen atoms respectively. The ionization energy of hydrogen is denoted $I$. State and derive $S a h a$ 's equation for the ratio $n_{e} n_{p} / n_{H}$, clearly describing the steps required.

[You may use without proof the following formula for the equilibrium number density of a non-relativistic species $a$ with $g_{a}$ degenerate states of mass $m$ at temperature $T$ such that $k_{B} T \ll m$,

$n_{a}=g_{a}\left(\frac{2 \pi m k_{B} T}{h^{2}}\right)^{3 / 2} \exp \left([\mu-m] / k_{B} T\right)$

where $\mu$ is the chemical potential and $k_{B}$ and $h$ are the Boltzmann and Planck constants respectively.]

The photon number density $n_{\gamma}$ is given as

$n_{\gamma}=\frac{16 \pi}{h^{3}} \zeta(3)\left(k_{B} T\right)^{3}$

where $\zeta(3) \simeq 1.20$. Consider now the fractional ionization $X_{e}=n_{e} /\left(n_{e}+n_{H}\right)$. In our universe $n_{e}+n_{H}=n_{p}+n_{H} \simeq \eta n_{\gamma}$ where $\eta$ is the baryon-to-photon number ratio. Find an expression for the ratio

$\frac{\left(1-X_{e}\right)}{X_{e}^{2}}$

in terms of $k_{B} T, \eta, I$ and the particle masses. One might expect neutral hydrogen to form at a temperature given by $k_{B} T \sim I \sim 13 \mathrm{eV}$, but instead in our universe it forms at the much lower temperature $k_{B} T \sim 0.3 \mathrm{eV}$. Briefly explain why this happens. Estimate the temperature at which neutral hydrogen would form in a hypothetical universe with $\eta=1$. Briefly explain your answer.

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• # Paper 3, Section II, H

(a) Let $\alpha:(a, b) \rightarrow \mathbb{R}^{2}$ be a regular curve without self intersection given by $\alpha(v)=(f(v), g(v))$ with $f(v)>0$ for $v \in(a, b)$.

Consider the local parametrisation given by

$\phi:(0,2 \pi) \times(a, b) \rightarrow \mathbb{R}^{3}$

where $\phi(u, v)=(f(v) \cos u, f(v) \sin u, g(v))$.

(i) Show that the image $\phi((0,2 \pi) \times(a, b))$ defines a regular surface $S$ in $\mathbb{R}^{3}$.

(ii) If $\gamma(s)=\phi(u(s), v(s))$ is a geodesic in $S$ parametrised by arc length, then show that $f(v(s))^{2} u^{\prime}(s)$ is constant in $s$. If $\theta(s)$ denotes the angle that the geodesic makes with the parallel $S \cap\{z=g(v(s))\}$, then show that $f(v(s)) \cos \theta(s)$ is constant in $s$.

(b) Now assume that $\alpha(v)=(f(v), g(v))$ extends to a smooth curve $\alpha:[a, b] \rightarrow \mathbb{R}^{2}$ such that $f(a)=0, f(b)=0, f^{\prime}(a) \neq 0, f^{\prime}(b) \neq 0$. Let $\bar{S}$ be the closure of $S$ in $\mathbb{R}^{3}$.

(i) State a necessary and sufficient condition on $\alpha(v)$ for $\bar{S}$ to be a compact regular surface. Justify your answer.

(ii) If $\bar{S}$ is a compact regular surface, and $\gamma:(-\infty, \infty) \rightarrow \bar{S}$ is a geodesic, show that there exists a non-empty open subset $U \subset \bar{S}$ such that $\gamma((-\infty, \infty)) \cap U=\emptyset$.

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• # Paper 3, Section II, E

Consider a dynamical system of the form

\begin{aligned} &\dot{x}=x(1-y+a x) \\ &\dot{y}=r y(-1+x-b y) \end{aligned}

on $\Lambda=\{(x, y): x>0$ and $y>0\}$, where $a, b$ and $r$ are real constants and $r>0$.

(a) For $a=b=0$, by considering a function of the form $V(x, y)=f(x)+g(y)$, show that all trajectories in $\Lambda$ are either periodic orbits or a fixed point.

(b) Using the same $V$, show that no periodic orbits in $\Lambda$ persist for small $a$ and $b$ if $a b<0$.

[Hint: for $a=b=0$ on the periodic orbits with period $T$, show that $\int_{0}^{T}(1-x) d t=0$ and hence that $\int_{0}^{T} x(1-x) d t=\int_{0}^{T}\left[-(1-x)^{2}+(1-x)\right] d t<0$.]

(c) By considering Dulac's criterion with $\phi=1 /(x y)$, show that there are no periodic orbits in $\Lambda$ if $a b<0$.

(d) Purely by consideration of the existence of fixed points in $\Lambda$ and their Poincaré indices, determine those $(a, b)$ for which the possibility of periodic orbits can be excluded.

(e) Combining the results above, sketch the $a-b$ plane showing where periodic orbits in $\Lambda$ might still be possible.

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• # Paper 3, Section II, E

A time-dependent charge distribution $\rho(t, \mathbf{x})$ localised in some region of size $a$ near the origin varies periodically in time with characteristic angular frequency $\omega$. Explain briefly the circumstances under which the dipole approximation for the fields sourced by the charge distribution is valid.

Far from the origin, for $r=|\mathbf{x}| \gg a$, the vector potential $\mathbf{A}(t, \mathbf{x})$ sourced by the charge distribution $\rho(t, \mathbf{x})$ is given by the approximate expression

$\mathbf{A}(t, \mathbf{x}) \simeq \frac{\mu_{0}}{4 \pi r} \int d^{3} \mathbf{x}^{\prime} \mathbf{J}\left(t-r / c, \mathbf{x}^{\prime}\right),$

where $\mathbf{J}(t, \mathbf{x})$ is the corresponding current density. Show that, in the dipole approximation, the large-distance behaviour of the magnetic field is given by,

$\mathbf{B}(t, \mathbf{x}) \simeq-\frac{\mu_{0}}{4 \pi r c} \hat{\mathbf{x}} \times \ddot{\mathbf{p}}(t-r / c)$

where $\mathbf{p}(t)$ is the electric dipole moment of the charge distribution. Assuming that, in the same approximation, the corresponding electric field is given as $\mathbf{E}=-c \hat{\mathbf{x}} \times \mathbf{B}$, evaluate the flux of energy through the surface element of a large sphere of radius $R$ centred at the origin. Hence show that the total power $P(t)$ radiated by the charge distribution is given by

$P(t)=\frac{\mu_{0}}{6 \pi c}|\ddot{\mathbf{p}}(t-R / c)|^{2}$

A particle of charge $q$ and mass $m$ undergoes simple harmonic motion in the $x$-direction with time period $T=2 \pi / \omega$ and amplitude $\mathcal{A}$ such that

$\mathbf{x}(t)=\mathcal{A} \sin (\omega t) \mathbf{i}_{x}$

Here $\mathbf{i}_{x}$ is a unit vector in the $x$-direction. Calculate the total power $P(t)$ radiated through a large sphere centred at the origin in the dipole approximation and determine its time averaged value,

$\langle P\rangle=\frac{1}{T} \int_{0}^{T} P(t) d t .$

For what values of the parameters $\mathcal{A}$ and $\omega$ is the dipole approximation valid?

Now suppose that the energy of the particle with trajectory $(\star)$ is given by the usual non-relativistic formula for a harmonic oscillator i.e. $E=m|\dot{\mathbf{x}}|^{2} / 2+m \omega^{2}|\mathbf{x}|^{2} / 2$, and that the particle loses energy due to the emission of radiation at a rate corresponding to the time-averaged power $\langle P\rangle$. Work out the half-life of this system (i.e. the time $t_{1 / 2}$ such that $\left.E\left(t_{1 / 2}\right)=E(0) / 2\right)$. Explain why the non-relativistic approximation for the motion of the particle is reliable as long as the dipole approximation is valid.

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• # Paper 3, Section II, A

For a fluid with kinematic viscosity $\nu$, the steady axisymmetric boundary-layer equations for flow primarily in the $z$-direction are

\begin{aligned} u \frac{\partial w}{\partial r}+w \frac{\partial w}{\partial z} &=\frac{\nu}{r} \frac{\partial}{\partial r}\left(r \frac{\partial w}{\partial r}\right) \\ \frac{1}{r} \frac{\partial(r u)}{\partial r}+\frac{\partial w}{\partial z} &=0 \end{aligned}

where $u$ is the fluid velocity in the $r$-direction and $w$ is the fluid velocity in the $z$-direction. A thin, steady, axisymmetric jet emerges from a point at the origin and flows along the $z$-axis in a fluid which is at rest far from the $z$-axis.

(a) Show that the momentum flux

$M:=\int_{0}^{\infty} r w^{2} d r$

is independent of the position $z$ along the jet. Deduce that the thickness $\delta(z)$ of the jet increases linearly with $z$. Determine the scaling dependence on $z$ of the centre-line velocity $W(z)$. Hence show that the jet entrains fluid.

(b) A similarity solution for the streamfunction,

$\psi(x, y, z)=\nu z g(\eta) \quad \text { with } \quad \eta:=r / z$

exists if $g$ satisfies the second order differential equation

$\eta g^{\prime \prime}-g^{\prime}+g g^{\prime}=0$

Using appropriate boundary and normalisation conditions (which you should state clearly) to solve this equation, show that

$g(\eta)=\frac{12 M \eta^{2}}{32 \nu^{2}+3 M \eta^{2}}$

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• # Paper 3, Section I, A

The equation

$z w^{\prime \prime}+w=0$

has solutions of the form

$w(z)=\int_{\gamma} e^{z t} f(t) d t$

for suitably chosen contours $\gamma$ and some suitable function $f(t)$.

(a) Find $f(t)$ and determine the required condition on $\gamma$, which you should express in terms of $z$ and $t$.

(b) Use the result of part (a) to specify a possible contour with the help of a clearly labelled diagram.

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• # Paper 3, Section II, F

Let $k$ be a field. For $m$ a positive integer, consider $X^{m}-1 \in k[X]$, where either char $k=0$, or char $k=p$ with $p$ not dividing $m$; explain why the polynomial has distinct roots in a splitting field.

For $m$ a positive integer, define the $m$ th cyclotomic polynomial $\Phi_{m} \in \mathbb{C}[X]$ and show that it is a monic polynomial in $\mathbb{Z}[X]$. Prove that $\Phi_{m}$ is irreducible over $\mathbb{Q}$ for all $m$. [Hint: If $\Phi_{m}=f g$, with $f, g \in \mathbb{Z}[X]$ and $f$ monic irreducible with $0<\operatorname{deg} f<\operatorname{deg} \Phi_{m}$, and $\varepsilon$ is a root of $f$, show first that $\varepsilon^{p}$ is a root of $f$ for any prime $p$ not dividing $m$.]

Let $F=X^{8}+X^{7}-X^{5}-X^{4}-X^{3}+X+1 \in \mathbb{Z}[X]$; by considering the product $\left(X^{2}-X+1\right) F$, or otherwise, show that $F$ is irreducible over $\mathbb{Q}$.

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• # Paper 3, Section II, D

(a) Let $\mathcal{M}$ be a manifold with coordinates $x^{\mu}$. The commutator of two vector fields $\boldsymbol{V}$ and $\boldsymbol{W}$ is defined as

$[\boldsymbol{V}, \boldsymbol{W}]^{\alpha}=V^{\nu} \partial_{\nu} W^{\alpha}-W^{\nu} \partial_{\nu} V^{\alpha}$

(i) Show that $[\boldsymbol{V}, \boldsymbol{W}]$ transforms like a vector field under a change of coordinates from $x^{\mu}$ to $\tilde{x}^{\mu}$.

(ii) Show that the commutator of any two basis vectors vanishes, i.e.

$\left[\frac{\partial}{\partial x^{\alpha}}, \frac{\partial}{\partial x^{\beta}}\right]=0$

(iii) Show that if $\boldsymbol{V}$ and $\boldsymbol{W}$ are linear combinations (not necessarily with constant coefficients) of $n$ vector fields $\boldsymbol{Z}_{(a)}, a=1, \ldots, n$ that all commute with one another, then the commutator $[\boldsymbol{V}, \boldsymbol{W}]$ is a linear combination of the same $n$ fields $Z_{(a)}$.

[You may use without proof the following relations which hold for any vector fields $\boldsymbol{V}_{1}, \boldsymbol{V}_{2}, \boldsymbol{V}_{3}$ and any function $f$ :

\begin{aligned} {\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right] } &=-\left[\boldsymbol{V}_{2}, \boldsymbol{V}_{1}\right] \\ {\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}+\boldsymbol{V}_{3}\right] } &=\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right]+\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{3}\right] \\ {\left[\boldsymbol{V}_{1}, f \boldsymbol{V}_{2}\right] } &=f\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right]+\boldsymbol{V}_{1}(f) \boldsymbol{V}_{2} \end{aligned}

but you should clearly indicate each time relation $(1),(2)$, or (3) is used.]

(b) Consider the 2-dimensional manifold $\mathbb{R}^{2}$ with Cartesian coordinates $\left(x^{1}, x^{2}\right)=$ $(x, y)$ carrying the Euclidean metric $g_{\alpha \beta}=\delta_{\alpha \beta}$.

(i) Express the coordinate basis vectors $\partial_{r}$ and $\partial_{\theta}$, where $r$ and $\theta$ denote the usual polar coordinates, in terms of their Cartesian counterparts.

(ii) Define the unit vectors

$\hat{\boldsymbol{r}}=\frac{\partial_{r}}{\left\|\partial_{r}\right\|}, \quad \hat{\boldsymbol{\theta}}=\frac{\partial_{\theta}}{\left\|\partial_{\theta}\right\|}$

and show that $(\hat{\boldsymbol{r}}, \hat{\boldsymbol{\theta}})$ are not a coordinate basis, i.e. there exist no coordinates $z^{\alpha}$ such that $\hat{\boldsymbol{r}}=\partial / \partial z^{1}$ and $\hat{\boldsymbol{\theta}}=\partial / \partial z^{2}$.

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• # Paper 3, Section II, G

(a) What does it mean to say that a graph is bipartite?

(b) Show that $G$ is bipartite if and only if it contains no cycles of odd length.

(c) Show that if $G$ is bipartite then

$\frac{\operatorname{ex}(n ; G)}{\left(\begin{array}{l} n \\ 2 \end{array}\right)} \rightarrow 0$

as $n \rightarrow \infty$.

[You may use without proof the Erdós-Stone theorem provided it is stated precisely.]

(d) Let $G$ be a graph of order $n$ with $m$ edges. Let $U$ be a random subset of $V(G)$ containing each vertex of $G$ independently with probability $\frac{1}{2}$. Let $X$ be the number of edges with precisely one vertex in $U$. Find, with justification, $\mathbb{E}(X)$, and deduce that $G$ contains a bipartite subgraph with at least $\frac{m}{2}$ edges.

By using another method of choosing a random subset of $V(G)$, or otherwise, show that if $n$ is even then $G$ contains a bipartite subgraph with at least $\frac{m n}{2(n-1)}$ edges.

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• # Paper 3, Section II, C

Suppose $\psi^{s}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ is a smooth one-parameter group of transformations acting on $\mathbb{R}^{2}$, with infinitesimal generator

$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$

(a) Define the $n^{\text {th }}$prolongation $\operatorname{Pr}^{(n)} V$ of $V$, and show that

$\operatorname{Pr}^{(n)} V=V+\sum_{i=1}^{n} \eta_{i} \frac{\partial}{\partial u^{(i)}}$

where you should give an explicit formula to determine the $\eta_{i}$ recursively in terms of $\xi$ and $\eta$.

(b) Find the $n^{t h}$ prolongation of each of the following generators:

$V_{1}=\frac{\partial}{\partial x}, \quad V_{2}=x \frac{\partial}{\partial x}, \quad V_{3}=x^{2} \frac{\partial}{\partial x}$

(c) Given a smooth, real-valued, function $u=u(x)$, the Schwarzian derivative is defined by,

$S=S[u]:=\frac{u_{x} u_{x x x}-\frac{3}{2} u_{x x}^{2}}{u_{x}^{2}}$

Show that,

$\operatorname{Pr}^{(3)} V_{i}(S)=c_{i} S,$

for $i=1,2,3$ where $c_{i}$ are real functions which you should determine. What can you deduce about the symmetries of the equations: (i) $S[u]=0$, (ii) $S[u]=1$, (iii) $S[u]=\frac{1}{x^{2}}$ ?

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• # Paper 3, Section II, H

(a) Let $X$ be a Banach space and consider the open unit ball $B=\{x \in X:\|x\|<1\}$. Let $T: X \rightarrow X$ be a bounded operator. Prove that $\overline{T(B)} \supset B \operatorname{implies} T(B) \supset B$.

(b) Let $P$ be the vector space of all polynomials in one variable with real coefficients. Let $\|\cdot\|$ be any norm on $P$. Show that $(P,\|\cdot\|)$ is not complete.

(c) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be entire, and assume that for every $z \in \mathbb{C}$ there is $n$ such that $f^{(n)}(z)=0$ where $f^{(n)}$ is the $n$-th derivative of $f$. Prove that $f$ is a polynomial.

[You may use that an entire function vanishing on an open subset of $\mathbb{C}$ must vanish everywhere.]

(d) A Banach space $X$ is said to be uniformly convex if for every $\varepsilon \in(0,2]$ there is $\delta>0$ such that for all $x, y \in X$ such that $\|x\|=\|y\|=1$ and $\|x-y\| \geqslant \varepsilon$, one has $\|(x+y) / 2\| \leqslant 1-\delta$. Prove that $\ell^{2}$ is uniformly convex.

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• # Paper 3, Section II, I

Define the von Neumann hierarchy of sets $V_{\alpha}$. Show that each $V_{\alpha}$ is transitive, and explain why $V_{\alpha} \subset V_{\beta}$ whenever $\alpha \leqslant \beta$. Prove that every set $x$ is a member of some $V_{\alpha}$.

Which of the following are true and which are false? Give proofs or counterexamples as appropriate. [You may assume standard properties of rank.]

(i) If the rank of a set $x$ is a (non-zero) limit then $x$ is infinite.

(ii) If the rank of a set $x$ is countable then $x$ is countable.

(iii) If every finite subset of a set $x$ has rank at most $\alpha$ then $x$ has rank at most $\alpha$.

(iv) For every ordinal $\alpha$ there exists a set of $\operatorname{rank} \alpha$.

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• # Paper 3, Section I, $\mathbf{6 C}$

A model of wound healing in one spatial dimension is given by

$\frac{\partial S}{\partial t}=r S(1-S)+D \frac{\partial^{2} S}{\partial x^{2}}$

where $S(x, t)$ gives the density of healthy tissue at spatial position $x$ at time $t$ and $r$ and $D$ are positive constants.

By setting $S(x, t)=f(\xi)$ where $\xi=x-c t$, seek a steady travelling wave solution where $f(\xi)$ tends to one for large negative $\xi$ and tends to zero for large positive $\xi$. By linearising around the leading edge, where $f \approx 1$, find the possible wave speeds $c$ of the system. Assuming that the full nonlinear system will settle to the slowest possible speed, express the wave speed as a function of $D$ and $r$.

Consider now a situation where the tissue is destroyed in some window of length $W$, i.e. $S(x, 0)=0$ for $0 for some constant $W>0$ and $S(x, 0)$ is equal to one elsewhere. Explain what will happen for subsequent times, illustrating your answer with sketches of $S(x, t)$. Determine approximately how long it will take for this wound to heal (in the sense that $S$ is close to one everywhere).

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• # Paper 3, Section II, C

(a) A stochastic birth-death process has a master equation given by

$\frac{d p_{n}}{d t}=\lambda\left(p_{n-1}-p_{n}\right)+\beta\left[(n+1) p_{n+1}-n p_{n}\right]$

where $p_{n}(t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p_{n}=0$ for $n<0$.

(i) Give a brief interpretation of $\lambda$ and $\beta$.

(ii) Derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$

(iii) Assuming that the generating function $\phi$ takes the form

$\phi(s, t)=e^{(s-1) f(t)}$

find $f(t)$ and hence show that, as $t \rightarrow \infty$, both the mean $\langle n\rangle$ and variance $\sigma^{2}$ of the population size tend to constant values, which you should determine.

(b) Now suppose an extra process is included: $k$ individuals are added to the population at rate $\epsilon(n)$.

(i) Write down the new master equation, and explain why, for $k>1$, the approach used in part (a) will fail.

(ii) By working with the master equation directly, find a differential equation for the rate of change of the mean population size $\langle n\rangle$.

(iii) Now take $\epsilon(n)=a n+b$ for positive constants $a$ and $b$. Show that for $\beta>a k$ the mean population size tends to a constant, which you should determine. Briefly describe what happens for $\beta.

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• # Paper 3, Section I, I

Let $f=(a, b, c)$ be a positive definite binary quadratic form with integer coefficients. What does it mean to say that $f$ is reduced? Show that if $f$ is reduced and has discriminant $d$, then $|b| \leqslant a \leqslant \sqrt{|d| / 3}$ and $b \equiv d(\bmod 2)$. Deduce that for fixed $d<0$, there are only finitely many reduced $f$ of discriminant $d$.

Find all reduced positive definite binary quadratic forms of discriminant $-15$.

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• # Paper 3, Section II, I

Let $p>2$ be a prime.

(a) What does it mean to say that an integer $g$ is a primitive root $\bmod p$ ?

(b) Let $k$ be an integer with $0 \leqslant k. Let

$S_{k}=\sum_{x=0}^{p-1} x^{k}$

Show that $S_{k} \equiv 0(\bmod p)$. [Recall that by convention $0^{0}=1$.]

(c) Let $f(X, Y, Z)=a X^{2}+b Y^{2}+c Z^{2}$ for some $a, b, c \in \mathbb{Z}$, and let $g=1-f^{p-1}$. Show that for any $x, y, z \in \mathbb{Z}, g(x, y, z) \equiv 0$ or $1(\bmod p)$, and that

$\sum_{x, y, z \in\{0,1, \ldots, p-1\}} g(x, y, z) \equiv 0 \quad(\bmod p) .$

Hence show that there exist integers $x, y, z$, not all divisible by $p$, such that $f(x, y, z) \equiv 0$ $(\bmod p)$.

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• # Paper 3, Section II, 40C

The diffusion equation

$u_{t}=u_{x x}, \quad 0 \leqslant x \leqslant 1, \quad t \geqslant 0,$

with the initial condition $u(x, 0)=\phi(x), 0 \leqslant x \leqslant 1$, and boundary conditions $u(0, t)=$ $u(1, t)=0$, is discretised by $u_{m}^{n} \approx u(m h, n k)$ with $k=\Delta t, h=\Delta x=1 /(1+M)$. The Courant number is given by $\mu=k / h^{2}$.

(a) The system is solved numerically by the method

$u_{m}^{n+1}=u_{m}^{n}+\mu\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right), \quad m=1,2, \ldots, M, \quad n \geqslant 0 .$

Prove directly that $\mu \leqslant 1 / 2$ implies convergence.

(b) Now consider the method

$a u_{m}^{n+1}-\frac{1}{4}(\mu-c)\left(u_{m-1}^{n+1}-2 u_{m}^{n+1}+u_{m+1}^{n+1}\right)=a u_{m}^{n}+\frac{1}{4}(\mu+c)\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right)$

where $a$ and $c$ are real constants. Using an eigenvalue analysis and carefully justifying each step, determine conditions on $\mu, a$ and $c$ for this method to be stable.

[You may use the notation $[\beta, \alpha, \beta]$ for the tridiagonal matrix with $\alpha$ along the diagonal, and $\beta$ along the sub-and super-diagonals and use without proof any relevant theorems about such matrices.]

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• # Paper 3, Section II, B

Consider the Hamiltonian $H=H_{0}+V$, where $V$ is a small perturbation. If $H_{0}|n\rangle=E_{n}|n\rangle$, write down an expression for the eigenvalues of $H$, correct to second order in the perturbation, assuming the energy levels of $H_{0}$ are non-degenerate.

In a certain three-state system, $H_{0}$ and $V$ take the form

$H_{0}=\left(\begin{array}{ccc} E_{1} & 0 & 0 \\ 0 & E_{2} & 0 \\ 0 & 0 & E_{3} \end{array}\right) \quad \text { and } \quad V=V_{0}\left(\begin{array}{ccc} 0 & \epsilon & \epsilon^{2} \\ \epsilon & 0 & 0 \\ \epsilon^{2} & 0 & 0 \end{array}\right)$

with $V_{0}$ and $\epsilon$ real, positive constants and $\epsilon \ll 1$.

(a) Consider first the case $E_{1}=E_{2} \neq E_{3}$ and $\left|\epsilon V_{0} /\left(E_{3}-E_{2}\right)\right| \ll 1$. Use the results of degenerate perturbation theory to obtain the energy eigenvalues correct to order $\epsilon$.

(b) Now consider the different case $E_{3}=E_{2} \neq E_{1}$ and $\left|\epsilon V_{0} /\left(E_{2}-E_{1}\right)\right| \ll 1$. Use the results of non-degenerate perturbation theory to obtain the energy eigenvalues correct to order $\epsilon^{2}$. Why is it not necessary to use degenerate perturbation theory in this case?

(c) Obtain the exact energy eigenvalues in case (b), and compare these to your perturbative results by expanding to second order in $\epsilon$.

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• # Paper 3, Section II, J

We consider the exponential model $\{f(\cdot, \theta): \theta \in(0, \infty)\}$, where

$f(x, \theta)=\theta e^{-\theta x} \quad \text { for } x \geqslant 0$

We observe an i.i.d. sample $X_{1}, \ldots, X_{n}$ from the model.

(a) Compute the maximum likelihood estimator $\hat{\theta}_{M L E}$ for $\theta$. What is the limit in distribution of $\sqrt{n}\left(\hat{\theta}_{M L E}-\theta\right)$ ?

(b) Consider the Bayesian setting and place a $\operatorname{Gamma}(\alpha, \beta), \alpha, \beta>0$, prior for $\theta$ with density

$\pi(\theta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} \theta^{\alpha-1} \exp (-\beta \theta) \quad \text { for } \theta>0$

where $\Gamma$ is the Gamma function satisfying $\Gamma(\alpha+1)=\alpha \Gamma(\alpha)$ for all $\alpha>0$. What is the posterior distribution for $\theta$ ? What is the Bayes estimator $\hat{\theta}_{\pi}$ for the squared loss?

(c) Show that the Bayes estimator is consistent. What is the limiting distribution of $\sqrt{n}\left(\hat{\theta}_{\pi}-\theta\right)$ ?

[You may use results from the course, provided you state them clearly.]

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• # Paper 3, Section II, K

(a) Let $X$ and $Y$ be real random variables such that $\mathbb{E}[f(X)]=\mathbb{E}[f(Y)]$ for every compactly supported continuous function $f$. Show that $X$ and $Y$ have the same law.

(b) Given a real random variable $Z$, let $\varphi_{Z}(s)=\mathbb{E}\left(e^{i s Z}\right)$ be its characteristic function. Prove the identity

$\iint g(\varepsilon s) f(x) e^{-i s x} \varphi_{Z}(s) d s d x=\int \hat{g}(t) \mathbb{E}[f(Z-\varepsilon t)] d t$

for real $\varepsilon>0$, where is $f$ is continuous and compactly supported, and where $g$ is a Lebesgue integrable function such that $\hat{g}$ is also Lebesgue integrable, where

$\hat{g}(t)=\int g(x) e^{i t x} d x$

is its Fourier transform. Use the above identity to derive a formula for $\mathbb{E}[f(Z)]$ in terms of $\varphi_{Z}$, and recover the fact that $\varphi_{Z}$ determines the law of $Z$ uniquely.

(c) Let $X$ and $Y$ be bounded random variables such that $\mathbb{E}\left(X^{n}\right)=\mathbb{E}\left(Y^{n}\right)$ for every positive integer $n$. Show that $X$ and $Y$ have the same law.

(d) The Laplace transform $\psi_{Z}(s)$ of a non-negative random variable $Z$ is defined by the formula

$\psi_{Z}(s)=\mathbb{E}\left(e^{-s Z}\right)$

for $s \geqslant 0$. Let $X$ and $Y$ be (possibly unbounded) non-negative random variables such that $\psi_{X}(s)=\psi_{Y}(s)$ for all $s \geqslant 0$. Show that $X$ and $Y$ have the same law.

(e) Let

$f(x ; k)=1_{\{x>0\}} \frac{1}{k !} x^{k} e^{-x}$

where $k$ is a non-negative integer and $1_{\{x>0\}}$ is the indicator function of the interval $(0,+\infty)$.

Given non-negative integers $k_{1}, \ldots, k_{n}$, suppose that the random variables $X_{1}, \ldots, X_{n}$ are independent with $X_{i}$ having density function $f\left(\cdot ; k_{i}\right)$. Find the density of the random variable $X_{1}+\cdots+X_{n}$.

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• # Paper 3, Section I, $10 D$

Let $B_{n}$ denote the set of all $n$-bit strings and write $N=2^{n}$. Let $I$ denote the identity operator on $n$ qubits and for $G=\left\{x_{1}, x_{2}, \ldots, x_{k}\right\} \subset B_{n}$ introduce the $n$-qubit operator

$Q=-H_{n} I_{0} H_{n} I_{G}$

where $H_{n}=H \otimes \ldots \otimes H$ is the Hadamard operation on each of the $n$ qubits, and $I_{0}$ and $I_{G}$ are given by

$I_{0}=I-2|00 \ldots 0\rangle\left\langle 00 \ldots 0\left|\quad I_{G}=I-2 \sum_{x \in G}\right| x\right\rangle\langle x|$

Also introduce the states

$\left|\psi_{0}\right\rangle=\frac{1}{\sqrt{N}} \sum_{x \in B_{n}}|x\rangle \quad\left|\psi_{G}\right\rangle=\frac{1}{\sqrt{k}} \sum_{x \in G}|x\rangle \quad\left|\psi_{B}\right\rangle=\frac{1}{\sqrt{N-k}} \sum_{x \notin G}|x\rangle$

Let $\mathcal{P}$ denote the real span of $\left|\psi_{0}\right\rangle$ and $\left|\psi_{G}\right\rangle$.

(a) Show that $Q$ maps $\mathcal{P}$ to itself, and derive a geometrical interpretation of the action of $Q$ on $\mathcal{P}$, stating clearly any results from Euclidean geometry that you use.

(b) Let $f: B_{n} \rightarrow B_{1}$ be the Boolean function such that $f(x)=1$ iff $x \in G$. Suppose that $k=N / 4$. Show that we can obtain an $x \in G$ with certainty by using just one application of the standard quantum oracle $U_{f}$ for $f$ (together with other operations that are independent of $f$ ).

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• # Paper 3, Section II, D

Let $\mathcal{H}_{d}$ denote a $d$-dimensional state space with orthonormal basis $\left\{|y\rangle: y \in \mathbb{Z}_{d}\right\}$. For any $f: \mathbb{Z}_{m} \rightarrow \mathbb{Z}_{n}$ let $U_{f}$ be the operator on $\mathcal{H}_{m} \otimes \mathcal{H}_{n}$ defined by

$U_{f}|x\rangle|y\rangle=|x\rangle|y+f(x) \bmod n\rangle$

for all $x \in \mathbb{Z}_{m}$ and $y \in \mathbb{Z}_{n}$.

(a) Define $Q F T$, the quantum Fourier transform $\bmod d$ (for any chosen $d)$.

(b) Let $S$ on $\mathcal{H}_{d}$ (for any chosen $d$ ) denote the operator defined by

$S|y\rangle=|y+1 \bmod d\rangle$

for $y \in \mathbb{Z}_{d}$. Show that the Fourier basis states $\left|\xi_{x}\right\rangle=Q F T|x\rangle$ for $x \in \mathbb{Z}_{d}$ are eigenstates of $S$. By expressing $U_{f}$ in terms of $S$ find a basis of eigenstates of $U_{f}$ and determine the corresponding eigenvalues.

(c) Consider the following oracle promise problem:

Input: an oracle for a function $f: \mathbb{Z}_{3} \rightarrow \mathbb{Z}_{3}$.

Promise: $f$ has the form $f(x)=s x+t$ where $s$ and $t$ are unknown coefficients (and with all arithmetic being $\bmod 3)$.

Problem: Determine $s$ with certainty.

Can this problem be solved by a single query to a classical oracle for $f$ (and possible further processing independent of $f)$ ? Give a reason for your answer.

Using the results of part (b) or otherwise, give a quantum algorithm for this problem that makes just one query to the quantum oracle $U_{f}$ for $f$.

(d) For any $f: \mathbb{Z}_{3} \rightarrow \mathbb{Z}_{3}$, let $f_{1}(x)=f(x+1)$ and $f_{2}(x)=-f(x)$ (all arithmetic being $\bmod 3$ ). Show how $U_{f_{1}}$ and $U_{f_{2}}$ can each be implemented with one use of $U_{f}$ together with other unitary gates that are independent of $f$.

(e) Consider now the oracle problem of the form in part (c) except that now $f$ is a quadratic function $f(x)=a x^{2}+b x+c$ with unknown coefficients $a, b, c$ (and all arithmetic being mod 3), and the problem is to determine the coefficient $a$ with certainty. Using the results of part (d) or otherwise, give a quantum algorithm for this problem that makes just two queries to the quantum oracle for $f$.

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• # Paper 3, Section II, I

In this question all representations are complex and $G$ is a finite group.

(a) State and prove Mackey's theorem. State the Frobenius reciprocity theorem.

(b) Let $X$ be a finite $G$-set and let $\mathbb{C} X$ be the corresponding permutation representation. Pick any orbit of $G$ on $X$ : it is isomorphic as a $G$-set to $G / H$ for some subgroup $H$ of $G$. Write down the character of $\mathbb{C}(G / H)$.

(i) Let $\mathbb{C}_{G}$ be the trivial representation of $G$. Show that $\mathbb{C} X$ may be written as a direct sum

$\mathbb{C} X=\mathbb{C}_{G} \oplus V$

for some representation $V$.

(ii) Using the results of (a) compute the character inner product $\left\langle 1_{H} \uparrow^{G}, 1_{H} \uparrow^{G}\right\rangle_{G}$ in terms of the number of $(H, H)$ double cosets.

(iii) Now suppose that $|X| \geqslant 2$, so that $V \neq 0$. By writing $\mathbb{C}(G / H)$ as a direct sum of irreducible representations, deduce from (ii) that the representation $V$ is irreducible if and only if $G$ acts 2 -transitively. In that case, show that $V$ is not the trivial representation.

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• # Paper 3, Section II, F

Let $\Lambda$ be a lattice in $\mathbb{C}$, and $f: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ a holomorphic map of complex tori. Show that $f$ lifts to a linear map $F: \mathbb{C} \rightarrow \mathbb{C}$.

Give the definition of $\wp(z):=\wp_{\Lambda}(z)$, the Weierstrass $\wp$-function for $\Lambda$. Show that there exist constants $g_{2}, g_{3}$ such that

$\wp^{\prime}(z)^{2}=4 \wp(z)^{3}-g_{2} \wp(z)-g_{3}$

Suppose $f \in \operatorname{Aut}(\mathbb{C} / \Lambda)$, that is, $f: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ is a biholomorphic group homomorphism. Prove that there exists a lift $F(z)=\zeta z$ of $f$, where $\zeta$ is a root of unity for which there exist $m, n \in \mathbb{Z}$ such that $\zeta^{2}+m \zeta+n=0$.

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• # Paper 3, Section I, J

(a) For a given model with likelihood $L(\beta), \beta \in \mathbb{R}^{p}$, define the Fisher information matrix in terms of the Hessian of the log-likelihood.

Consider a generalised linear model with design matrix $X \in \mathbb{R}^{n \times p}$, output variables $y \in \mathbb{R}^{n}$, a bijective link function, mean parameters $\mu=\left(\mu_{1}, \ldots, \mu_{n}\right)$ and dispersion parameters $\sigma_{1}^{2}=\ldots=\sigma_{n}^{2}=\sigma^{2}$. Assume $\sigma^{2}$ is known.

(b) State the form of the log-likelihood.

(c) For the canonical link, show that when the parameter