• Paper 2, Section I, E

Define differentiability of a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. Let $a>0$ be a constant. For which points $(x, y) \in \mathbb{R}^{2}$ is

$f(x, y)=|x|^{a}+|x-y|$

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• Paper 2, Section II, E

What is meant by saying that two norms on a real vector space are Lipschitz equivalent?

Show that any two norms on $\mathbb{R}^{n}$ are Lipschitz equivalent. [You may assume that a continuous function on a closed bounded set in $\mathbb{R}^{n}$ has closed bounded image.]

Show that $\|f\|_{1}=\int_{-1}^{1}|f(x)| d x$ defines a norm on the space $C[-1,1]$ of continuous real-valued functions on $[-1,1]$. Is it Lipschitz equivalent to the uniform norm? Justify your answer. Prove that the normed space $\left(C[-1,1],\|\cdot\|_{1}\right)$ is not complete.

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• Paper 2, Section II, A

(i) Let $C$ be an anticlockwise contour defined by a square with vertices at $z=x+i y$ where

$|x|=|y|=\left(2 N+\frac{1}{2}\right) \pi$

for large integer $N$. Let

$I=\oint_{C} \frac{\pi \cot z}{(z+\pi a)^{4}} d z$

Assuming that $I \rightarrow 0$ as $N \rightarrow \infty$, prove that, if $a$ is not an integer, then

$\sum_{n=-\infty}^{\infty} \frac{1}{(n+a)^{4}}=\frac{\pi^{4}}{3 \sin ^{2}(\pi a)}\left(\frac{3}{\sin ^{2}(\pi a)}-2\right) .$

(ii) Deduce the value of

$\sum_{n=-\infty}^{\infty} \frac{1}{\left(n+\frac{1}{2}\right)^{4}}$

(iii) Briefly justify the assumption that $I \rightarrow 0$ as $N \rightarrow \infty$.

[Hint: For part (iii) it is sufficient to consider, at most, one vertical side of the square and one horizontal side and to use a symmetry argument for the remaining sides.]

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• Paper 2, Section I, $\mathbf{6 C}$

Maxwell's equations are

$\begin{gathered} \boldsymbol{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}, \quad \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0, \quad \nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\epsilon_{0} \mu_{0} \frac{\partial \mathbf{E}}{\partial t} \end{gathered}$

Find the equation relating $\rho$ and $\mathbf{J}$ that must be satisfied for consistency, and give the interpretation of this equation.

Now consider the "magnetic limit" where $\rho=0$ and the term $\epsilon_{0} \mu_{0} \frac{\partial \mathbf{E}}{\partial t}$ is neglected. Let $\mathbf{A}$ be a vector potential satisfying the gauge condition $\boldsymbol{\nabla} \cdot \mathbf{A}=0$, and assume the scalar potential vanishes. Find expressions for $\mathbf{E}$ and $\mathbf{B}$ in terms of $\mathbf{A}$ and show that Maxwell's equations are all satisfied provided $\mathbf{A}$ satisfies the appropriate Poisson equation.

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• Paper 2, Section II, C

(i) Consider an infinitely long solenoid parallel to the $z$-axis whose cross section is a simple closed curve of arbitrary shape. A current $I$, per unit length of the solenoid, flows around the solenoid parallel to the $x-y$ plane. Show using the relevant Maxwell equation that the magnetic field $\mathbf{B}$ inside the solenoid is uniform, and calculate its magnitude.

(ii) A wire loop in the shape of a regular hexagon of side length $a$ carries a current $I$. Use the Biot-Savart law to calculate $\mathbf{B}$ at the centre of the loop.

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• Paper 2, Section I, D

A body of volume $V$ lies totally submerged in a motionless fluid of uniform density $\rho$. Show that the force $\mathbf{F}$ on the body is given by

$\mathbf{F}=-\int_{S}\left(p-p_{0}\right) \mathbf{n} d S$

where $p$ is the pressure in the fluid and $p_{0}$ is atmospheric pressure. You may use without proof the generalised divergence theorem in the form

$\int_{S} \phi \mathbf{n} d S=\int_{V} \boldsymbol{\nabla} \phi d V$

Deduce that

$\mathbf{F}=\rho g V \hat{\mathbf{z}},$

where $\hat{\mathbf{z}}$ is the vertically upward unit vector. Interpret this result.

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• Paper 2, Section II, F

Suppose that $\pi: S^{2} \rightarrow \mathbb{C}_{\infty}$ is stereographic projection. Show that, via $\pi$, every rotation of $S^{2}$ corresponds to a Möbius transformation in $P S U(2)$.

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• Paper 2, Section I, F

Show that the quaternion group $Q_{8}=\{\pm 1, \pm i, \pm j, \pm k\}$, with $i j=k=-j i$, $i^{2}=j^{2}=k^{2}=-1$, is not isomorphic to the symmetry group $D_{8}$ of the square.

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• Paper 2, Section II, F

Define the notion of a Euclidean domain and show that $\mathbb{Z}[i]$ is Euclidean.

Is $4+i$ prime in $\mathbb{Z}[i]$ ?

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• Paper 2, Section $I$, G

Let $V$ be an $n$-dimensional $\mathbb{R}$-vector space with an inner product. Let $W$ be an $m$-dimensional subspace of $V$ and $W^{\perp}$ its orthogonal complement, so that every element $v \in V$ can be uniquely written as $v=w+w^{\prime}$ for $w \in W$ and $w^{\prime} \in W^{\perp}$.

The reflection map with respect to $W$ is defined as the linear map

$f_{W}: V \ni w+w^{\prime} \longmapsto w-w^{\prime} \in V$

Show that $f_{W}$ is an orthogonal transformation with respect to the inner product, and find its determinant.

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• Paper 2, Section II, G

Let $n$ be a positive integer, and let $V$ be a $\mathbb{C}$-vector space of complex-valued functions on $\mathbb{R}$, generated by the set $\{\cos k x, \sin k x ; k=0,1, \ldots, n-1\}$.

(i) Let $\langle f, g\rangle=\int_{0}^{2 \pi} f(x) \overline{g(x)} d x$ for $f, g \in V$. Show that this is a positive definite Hermitian form on $V$.

(ii) Let $\Delta(f)=\frac{d^{2}}{d x^{2}} f(x)$. Show that $\Delta$ is a self-adjoint linear transformation of $V$ with respect to the form defined in (i).

(iii) Find an orthonormal basis of $V$ with respect to the form defined in (i), which consists of eigenvectors of $\Delta$.

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• Paper 2, Section II, H

(i) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain on the finite state space $S$ with transition matrix $P$. Fix a subset $A \subseteq S$, and let

$H=\inf \left\{n \geqslant 0: X_{n} \in A\right\} .$

Fix a function $g$ on $S$ such that $0 for all $i \in S$, and let

$V_{i}=\mathbb{E}\left[\prod_{n=0}^{H-1} g\left(X_{n}\right) \mid X_{0}=i\right]$

where $\prod_{n=0}^{-1} a_{n}=1$ by convention. Show that

$V_{i}= \begin{cases}1 & \text { if } i \in A \\ g(i) \sum_{j \in S} P_{i j} V_{j} & \text { otherwise. }\end{cases}$

(ii) A flea lives on a polyhedron with $N$ vertices, labelled $1, \ldots, N$. It hops from vertex to vertex in the following manner: if one day it is on vertex $i>1$, the next day it hops to one of the vertices labelled $1, \ldots, i-1$ with equal probability, and it dies upon reaching vertex 1. Let $X_{n}$ be the position of the flea on day $n$. What are the transition probabilities for the Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ ?

(iii) Let $H$ be the number of days the flea is alive, and let

$V_{i}=\mathbb{E}\left(s^{H} \mid X_{0}=i\right)$

where $s$ is a real number such that $0. Show that $V_{1}=1$ and

$\frac{i}{s} V_{i+1}=V_{i}+\frac{i-1}{s} V_{i}$

for $i \geqslant 1$. Conclude that

$\mathbb{E}\left(s^{H} \mid X_{0}=N\right)=\prod_{i=1}^{N-1}\left(1+\frac{s-1}{i}\right)$

[Hint. Use part (i) with $A=\{1\}$ and a well-chosen function $g$. ]

(iv) Show that

$\mathbb{E}\left(H \mid X_{0}=N\right)=\sum_{i=1}^{N-1} \frac{1}{i}$

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• Paper 2, Section I, A

The Legendre equation is

$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+n(n+1) y=0$

for $-1 \leqslant x \leqslant 1$ and non-negative integers $n$.

Write the Legendre equation as an eigenvalue equation for an operator $L$ in SturmLiouville form. Show that $L$ is self-adjoint and find the orthogonality relation between the eigenfunctions.

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• Paper 2, Section II, A

Use a Green's function to find an integral expression for the solution of the equation

$\frac{d^{2} \theta}{d t^{2}}+4 \frac{d \theta}{d t}+29 \theta=f(t)$

for $t \geqslant 0$ subject to the initial conditions

$\theta(0)=0 \quad \text { and } \quad \frac{d \theta}{d t}(0)=0$

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• Paper 2, Section I, 4G

(i) Let $t>0$. For $\mathbf{x}=(x, y), \mathbf{x}^{\prime}=\left(x^{\prime}, y^{\prime}\right) \in \mathbb{R}^{2}$, let

$\begin{gathered} d\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left|x^{\prime}-x\right|+t\left|y^{\prime}-y\right|, \\ \delta\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\sqrt{\left(x^{\prime}-x\right)^{2}+\left(y^{\prime}-y\right)^{2}} \end{gathered}$

( $\delta$ is the usual Euclidean metric on $\mathbb{R}^{2}$.) Show that $d$ is a metric on $\mathbb{R}^{2}$ and that the two metrics $d, \delta$ give rise to the same topology on $\mathbb{R}^{2}$.

(ii) Give an example of a topology on $\mathbb{R}^{2}$, different from the one in (i), whose induced topology (subspace topology) on the $x$-axis is the usual topology (the one defined by the metric $\left.d\left(x, x^{\prime}\right)=\left|x^{\prime}-x\right|\right)$. Justify your answer.

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• Paper 2, Section II, B

What is the $Q R$-decomposition of a matrix A? Explain how to construct the matrices $Q$ and $R$ by the Gram-Schmidt procedure, and show how the decomposition can be used to solve the matrix equation $A \mathbf{x}=\mathbf{b}$ when $A$ is a square matrix.

Why is this procedure not useful for numerical decomposition of large matrices? Give a brief description of an alternative procedure using Givens rotations.

Find a $Q R$-decomposition for the matrix

$\mathrm{A}=\left[\begin{array}{rrrr} 3 & 4 & 7 & 13 \\ -6 & -8 & -8 & -12 \\ 3 & 4 & 7 & 11 \\ 0 & 2 & 5 & 7 \end{array}\right]$

Is your decomposition unique? Use the decomposition you have found to solve the equation

$A x=\left[\begin{array}{c} 4 \\ 6 \\ 2 \\ 9 \end{array}\right]$

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• Paper 2, Section I, H

Let $N=\{1, \ldots, n\}$ be the set of nodes of a network, where 1 is the source and $n$ is the $\operatorname{sink}$. Let $c_{i j}$ denote the capacity of the arc from node $i$ to node $j$.

(i) In the context of maximising the flow through this network, define the following terms: feasible flow, flow value, cut, cut capacity.

(ii) State and prove the max-flow min-cut theorem for network flows.

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• Paper 2, Section II, C

The quantum mechanical angular momentum operators are

$L_{i}=-i \hbar \epsilon_{i j k} x_{j} \frac{\partial}{\partial x_{k}} \quad(i=1,2,3)$

Show that each of these is hermitian.

The total angular momentum operator is defined as $\mathbf{L}^{2}=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}$. Show that $\left\langle\mathbf{L}^{2}\right\rangle \geqslant\left\langle L_{3}^{2}\right\rangle$ in any state, and show that the only states where $\left\langle\mathbf{L}^{2}\right\rangle=\left\langle L_{3}^{2}\right\rangle$ are those with no angular dependence. Verify that the eigenvalues of the operators $\mathbf{L}^{2}$ and $L_{3}^{2}$ (whose values you may quote without proof) are consistent with these results.

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• Paper 2, Section I, H

Let $X_{1}, \ldots, X_{n}$ be random variables with joint density function $f\left(x_{1}, \ldots, x_{n} ; \theta\right)$, where $\theta$ is an unknown parameter. The null hypothesis $H_{0}: \theta=\theta_{0}$ is to be tested against the alternative hypothesis $H_{1}: \theta=\theta_{1}$.

(i) Define the following terms: critical region, Type I error, Type II error, size, power.

(ii) State and prove the Neyman-Pearson lemma.

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• Paper 2, Section II, D

(i) Let $I[y]=\int_{0}^{1}\left(\left(y^{\prime}\right)^{2}-y^{2}\right) d x$, where $y$ is twice differentiable and $y(0)=y(1)=0$. Write down the associated Euler-Lagrange equation and show that the only solution is $y(x)=0$.

(ii) Let $J[y]=\int_{0}^{1}\left(y^{\prime}+y \tan x\right)^{2} d x$, where $y$ is twice differentiable and $y(0)=y(1)=$ 0 . Show that $J[y]=0$ only if $y(x)=0$.

(iii) Show that $I[y]=J[y]$ and deduce that the extremal value of $I[y]$ is a global minimum.

(iv) Use the second variation of $I[y]$ to verify that the extremal value of $I[y]$ is a local minimum.

(v) How would your answers to part (i) differ in the case $I[y]=\int_{0}^{2 \pi}\left(\left(y^{\prime}\right)^{2}-y^{2}\right) d x$, where $y(0)=y(2 \pi)=0$ ? Show that the solution $y(x)=0$ is not a global minimizer in this case. (You may use without proof the result $I[x(2 \pi-x)]=-\frac{8}{15}\left(2 \pi^{2}-5\right)$.) Explain why the arguments of parts (iii) and (iv) cannot be used.

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