• # 1.I.1A

Define uniform continuity for functions defined on a (bounded or unbounded) interval in $\mathbb{R}$.

Is it true that a real function defined and uniformly continuous on $[0,1]$ is necessarily bounded?

Is it true that a real function defined and with a bounded derivative on $[1, \infty)$ is necessarily uniformly continuous there?

Which of the following functions are uniformly continuous on $[1, \infty)$ :

(i) $x^{2}$;

(ii) $\sin \left(x^{2}\right)$;

(iii) $\frac{\sin x}{x}$ ?

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• # 1.II.10A

Show that each of the functions below is a metric on the set of functions $x(t) \in$ $C[a, b]$ :

$\begin{gathered} d_{1}(x, y)=\sup _{t \in[a, b]}|x(t)-y(t)| \\ d_{2}(x, y)=\left\{\int_{a}^{b}|x(t)-y(t)|^{2} d t\right\}^{1 / 2} \end{gathered}$

Is the space complete in the $d_{1}$ metric? Justify your answer.

Show that the set of functions

$x_{n}(t)= \begin{cases}0, & -1 \leqslant t<0 \\ n t, & 0 \leqslant t<1 / n \\ 1, & 1 / n \leqslant t \leqslant 1\end{cases}$

is a Cauchy sequence with respect to the $d_{2}$ metric on $C[-1,1]$, yet does not tend to a limit in the $d_{2}$ metric in this space. Hence, deduce that this space is not complete in the $d_{2}$ metric.

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• # 1.I.7E

State the Cauchy integral formula.

Assuming that the function $f(z)$ is analytic in the disc $|z|<1$, prove that, for every $0, it is true that

$\frac{d^{n} f(0)}{d z^{n}}=\frac{n !}{2 \pi i} \int_{|\xi|=r} \frac{f(\xi)}{\xi^{n+1}} d \xi, \quad n=0,1, \ldots$

[Taylor's theorem may be used if clearly stated.]

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• # 1.II.16E

Let the function $F$ be integrable for all real arguments $x$, such that

$\int_{-\infty}^{\infty}|F(x)| d x<\infty$

and assume that the series

$f(\tau)=\sum_{n=-\infty}^{\infty} F(2 n \pi+\tau)$

converges uniformly for all $0 \leqslant \tau \leqslant 2 \pi$.

Prove the Poisson summation formula

$f(\tau)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} \hat{F}(n) e^{i n \tau}$

where $\hat{F}$ is the Fourier transform of $F$. [Hint: You may show that

$\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{-i m x} f(x) d x=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-i m x} F(x) d x$

or, alternatively, prove that $f$ is periodic and express its Fourier expansion coefficients explicitly in terms of $\hat{F}$.]

Letting $F(x)=e^{-|x|}$, use the Poisson summation formula to evaluate the sum

$\sum_{n=-\infty}^{\infty} \frac{1}{1+n^{2}}$

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• # 1.I.6G

Determine the pressure at a depth $z$ below the surface of a static fluid of density $\rho$ subject to gravity $g$. A rigid body having volume $V$ is fully submerged in such a fluid. Calculate the buoyancy force on the body.

An iceberg of uniform density $\rho_{I}$ is observed to float with volume $V_{I}$ protruding above a large static expanse of seawater of density $\rho_{w}$. What is the total volume of the iceberg?

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• # 1.II.15G

A fluid motion has velocity potential $\phi(x, y, t)$ given by

$\phi=\epsilon y \cos (x-t)$

where $\epsilon$ is a constant. Find the corresponding velocity field $\mathbf{u}(x, y, t)$. Determine $\nabla \cdot \mathbf{u}$.

The time-average of a quantity $\psi(x, y, t)$ is defined as $\frac{1}{2 \pi} \int_{0}^{2 \pi} \psi(x, y, t) d t$.

Show that the time-average of this velocity field at every point $(x, y)$ is zero.

Write down an expression for the fluid acceleration and find the time-average acceleration at $(x, y)$.

Suppose now that $|\epsilon| \ll 1$. The material particle at $(0,0)$ at time $t=0$ is marked with dye. Write down equations for its subsequent motion and verify that its position $(x, y)$ at time $t>0$ is given (correct to terms of order $\epsilon^{2}$ ) as

\begin{aligned} &x=\epsilon^{2}\left(\frac{1}{2} t-\frac{1}{4} \sin 2 t\right) \\ &y=\epsilon \sin t \end{aligned}

Deduce the time-average velocity of the dyed particle correct to this order.

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• # 1.I.4B

Write down the Riemannian metric on the disc model $\Delta$ of the hyperbolic plane. What are the geodesics passing through the origin? Show that the hyperbolic circle of radius $\rho$ centred on the origin is just the Euclidean circle centred on the origin with Euclidean radius $\tanh (\rho / 2)$.

Write down an isometry between the upper half-plane model $H$ of the hyperbolic plane and the disc model $\Delta$, under which $i \in H$ corresponds to $0 \in \Delta$. Show that the hyperbolic circle of radius $\rho$ centred on $i$ in $H$ is a Euclidean circle with centre $i \cosh \rho$ and of radius $\sinh \rho$.

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• # 1.II.13B

Describe geometrically the stereographic projection map $\phi$ from the unit sphere $S^{2}$ to the extended complex plane $\mathbb{C}_{\infty}=\mathbb{C} \cup \infty$, and find a formula for $\phi$. Show that any rotation of $S^{2}$ about the $z$-axis corresponds to a Möbius transformation of $\mathbb{C}_{\infty}$. You are given that the rotation of $S^{2}$ defined by the matrix

$\left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{array}\right)$

corresponds under $\phi$ to a Möbius transformation of $\mathbb{C}_{\infty}$; deduce that any rotation of $S^{2}$ about the $x$-axis also corresponds to a Möbius transformation.

Suppose now that $u, v \in \mathbb{C}$ correspond under $\phi$ to distinct points $P, Q \in S^{2}$, and let $d$ denote the angular distance from $P$ to $Q$ on $S^{2}$. Show that $-\tan ^{2}(d / 2)$ is the cross-ratio of the points $u, v,-1 / \bar{u},-1 / \bar{v}$, taken in some order (which you should specify). [You may assume that the cross-ratio is invariant under Möbius transformations.]

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• # 1.I $. 5 \mathrm{C} \quad$

Determine for which values of $x \in \mathbb{C}$ the matrix

$M=\left(\begin{array}{ccc} x & 1 & 1 \\ 1-x & 0 & -1 \\ 2 & 2 x & 1 \end{array}\right)$

is invertible. Determine the rank of $M$ as a function of $x$. Find the adjugate and hence the inverse of $M$ for general $x$.

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• # 1.II.14C

(a) Find a matrix $M$ over $\mathbb{C}$ with both minimal polynomial and characteristic polynomial equal to $(x-2)^{3}(x+1)^{2}$. Furthermore find two matrices $M_{1}$ and $M_{2}$ over $\mathbb{C}$ which have the same characteristic polynomial, $(x-3)^{5}(x-1)^{2}$, and the same minimal polynomial, $(x-3)^{2}(x-1)^{2}$, but which are not conjugate to one another. Is it possible to find a third such matrix, $M_{3}$, neither conjugate to $M_{1}$ nor to $M_{2}$ ? Justify your answer.

(b) Suppose $A$ is an $n \times n$ matrix over $\mathbb{R}$ which has minimal polynomial of the form $\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)$ for distinct roots $\lambda_{1} \neq \lambda_{2}$ in $\mathbb{R}$. Show that the vector space $V=\mathbb{R}^{n}$ on which $A$ defines an endomorphism $\alpha: V \rightarrow V$ decomposes as a direct sum into $V=\operatorname{ker}\left(\alpha-\lambda_{1} \iota\right) \oplus \operatorname{ker}\left(\alpha-\lambda_{2} \iota\right)$, where $\iota$ is the identity.

[Hint: Express $v \in V$ in terms of $\left(\alpha-\lambda_{1} \iota\right)(v)$ and $\left.\left(\alpha-\lambda_{2} \iota\right)(v) .\right]$

Now suppose that $A$ has minimal polynomial $\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right) \ldots\left(x-\lambda_{m}\right)$ for distinct $\lambda_{1}, \ldots, \lambda_{m} \in \mathbb{R}$. By induction or otherwise show that

$V=\operatorname{ker}\left(\alpha-\lambda_{1} \iota\right) \oplus \operatorname{ker}\left(\alpha-\lambda_{2} \iota\right) \oplus \ldots \oplus \operatorname{ker}\left(\alpha-\lambda_{m} \iota\right)$

Use this last statement to prove that an arbitrary matrix $A \in M_{n \times n}(\mathbb{R})$ is diagonalizable if and only if all roots of its minimal polynomial lie in $\mathbb{R}$ and have multiplicity $1 .$

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• # 1.I.2H

The even function $f(x)$ has the Fourier cosine series

$f(x)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \cos n x$

in the interval $-\pi \leqslant x \leqslant \pi$. Show that

$\frac{1}{\pi} \int_{-\pi}^{\pi}(f(x))^{2} d x=\frac{1}{2} a_{0}^{2}+\sum_{n=1}^{\infty} a_{n}^{2}$

Find the Fourier cosine series of $x^{2}$ in the same interval, and show that

$\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\frac{\pi^{4}}{90}$

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• # 1.II.11H

Use the substitution $y=x^{p}$ to find the general solution of

$\mathcal{L}_{x} y \equiv \frac{d^{2} y}{d x^{2}}-\frac{2}{x^{2}} y=0$

Find the Green's function $G(x, \xi), 0<\xi<\infty$, which satisfies

$\mathcal{L}_{x} G(x, \xi)=\delta(x-\xi)$

for $x>0$, subject to the boundary conditions $G(x, \xi) \rightarrow 0$ as $x \rightarrow 0$ and as $x \rightarrow \infty$, for each fixed $\xi$.

Hence, find the solution of the equation

$\mathcal{L}_{x} y= \begin{cases}1, & 0 \leqslant x<1, \\ 0, & x>1\end{cases}$

subject to the same boundary conditions.

Verify that both forms of your solution satisfy the appropriate equation and boundary conditions, and match at $x=1$.

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• # 1.I.8B

Let $q(x, y)=a x^{2}+b x y+c y^{2}$ be a binary quadratic form with integer coefficients. Define what is meant by the discriminant $d$ of $q$, and show that $q$ is positive-definite if and only if $a>0>d$. Define what it means for the form $q$ to be reduced. For any integer $d<0$, we define the class number $h(d)$ to be the number of positive-definite reduced binary quadratic forms (with integer coefficients) with discriminant $d$. Show that $h(d)$ is always finite (for negative $d)$. Find $h(-39)$, and exhibit the corresponding reduced forms.

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• # 1.II.17B

Let $\phi$ be a symmetric bilinear form on a finite dimensional vector space $V$ over a field $k$ of characteristic $\neq 2$. Prove that the form $\phi$ may be diagonalized, and interpret the rank $r$ of $\phi$ in terms of the resulting diagonal form.

For $\phi$ a symmetric bilinear form on a real vector space $V$ of finite dimension $n$, define the signature $\sigma$ of $\phi$, proving that it is well-defined. A subspace $U$ of $V$ is called null if $\left.\phi\right|_{U} \equiv 0$; show that $V$ has a null subspace of dimension $n-\frac{1}{2}(r+|\sigma|)$, but no null subspace of higher dimension.

Consider now the quadratic form $q$ on $\mathbb{R}^{5}$ given by

$2\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}+x_{5} x_{1}\right)$

Write down the matrix $A$ for the corresponding symmetric bilinear form, and calculate $\operatorname{det} A$. Hence, or otherwise, find the rank and signature of $q$.

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• # 1.I $. 9 \mathrm{~F} \quad$

A quantum mechanical particle of mass $m$ and energy $E$ encounters a potential step,

$V(x)= \begin{cases}0, & x<0 \\ V_{0}, & x \geqslant 0\end{cases}$

Calculate the probability $P$ that the particle is reflected in the case $E>V_{0}$.

If $V_{0}$ is positive, what is the limiting value of $P$ when $E$ tends to $V_{0}$ ? If $V_{0}$ is negative, what is the limiting value of $P$ as $V_{0}$ tends to $-\infty$ for fixed $E$ ?

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• # 1.II.18F

Consider a quantum-mechanical particle of mass $m$ moving in a potential well,

$V(x)= \begin{cases}0, & -a

(a) Verify that the set of normalised energy eigenfunctions are

$\psi_{n}(x)=\sqrt{\frac{1}{a}} \sin \left(\frac{n \pi(x+a)}{2 a}\right), \quad n=1,2, \ldots$

and evaluate the corresponding energy eigenvalues $E_{n}$.

(b) At time $t=0$ the wavefunction for the particle is only nonzero in the positive half of the well,

$\psi(x)= \begin{cases}\sqrt{\frac{2}{a}} \sin \left(\frac{\pi x}{a}\right), & 0

Evaluate the expectation value of the energy, first using

$\langle E\rangle=\int_{-a}^{a} \psi H \psi d x$

and secondly using

$\langle E\rangle=\sum_{n}\left|a_{n}\right|^{2} E_{n},$

where the $a_{n}$ are the expansion coefficients in

$\psi(x)=\sum_{n} a_{n} \psi_{n}(x)$

Hence, show that

$1=\frac{1}{2}+\frac{8}{\pi^{2}} \sum_{p=0}^{\infty} \frac{(2 p+1)^{2}}{\left[(2 p+1)^{2}-4\right]^{2}}$

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• # 1.I.3D

Let $X_{1}, \ldots, X_{n}$ be independent, identically distributed $N\left(\mu, \mu^{2}\right)$ random variables, $\mu>0$.

Find a two-dimensional sufficient statistic for $\mu$, quoting carefully, without proof, any result you use.

What is the maximum likelihood estimator of $\mu$ ?

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• # 1.II.12D

What is a simple hypothesis? Define the terms size and power for a test of one simple hypothesis against another.

State, without proof, the Neyman-Pearson lemma.

Let $X$ be a single random variable, with distribution $F$. Consider testing the null hypothesis $H_{0}: F$ is standard normal, $N(0,1)$, against the alternative hypothesis $H_{1}: F$ is double exponential, with density $\frac{1}{4} e^{-|x| / 2}, x \in \mathbb{R}$.

Find the test of size $\alpha, \alpha<\frac{1}{4}$, which maximises power, and show that the power is $e^{-t / 2}$, where $\Phi(t)=1-\alpha / 2$ and $\Phi$ is the distribution function of $N(0,1)$.

[Hint: if $X \sim N(0,1), P(|X|>1)=0.3174 .]$

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