• # 1.II.11F

Let $a_{n}$ and $b_{n}$ be sequences of real numbers for $n \geqslant 1$ such that $\left|a_{n}\right| \leqslant c / n^{1+\epsilon}$ and $\left|b_{n}\right| \leqslant c / n^{1+\epsilon}$ for all $n \geqslant 1$, for some constants $c>0$ and $\epsilon>0$. Show that the series

$f(x)=\sum_{n \geqslant 1} a_{n} \cos n x+\sum_{n \geqslant 1} b_{n} \sin n x$

converges uniformly to a continuous function on the real line. Show that $f$ is periodic in the sense that $f(x+2 \pi)=f(x)$.

Now suppose that $\left|a_{n}\right| \leqslant c / n^{2+\epsilon}$ and $\left|b_{n}\right| \leqslant c / n^{2+\epsilon}$ for all $n \geqslant 1$, for some constants $c>0$ and $\epsilon>0$. Show that $f$ is differentiable on the real line, with derivative

$f^{\prime}(x)=-\sum_{n \geqslant 1} n a_{n} \sin n x+\sum_{n \geqslant 1} n b_{n} \cos n x .$

[You may assume the convergence of standard series.]

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• # 1.I.3D

Let $L$ be the Laplace operator, i.e., $L(g)=g_{x x}+g_{y y}$. Prove that if $f: \Omega \rightarrow \mathbf{C}$ is analytic in a domain $\Omega$, then

$L\left(|f(z)|^{2}\right)=4\left|f^{\prime}(z)\right|^{2}, \quad z \in \Omega .$

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• # 1.II.13D

By integrating round the contour involving the real axis and the line $\operatorname{Im}(z)=2 \pi$, or otherwise, evaluate

$\int_{-\infty}^{\infty} \frac{e^{a x}}{1+e^{x}} d x, \quad 0

Explain why the given restriction on the value $a$ is necessary.

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• # 1.II.16G

Three concentric conducting spherical shells of radii $a, b$ and $c(a carry charges $q,-2 q$ and $3 q$ respectively. Find the electric field and electric potential at all points of space.

Calculate the total energy of the electric field.

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• # 1.I.5A

Use the Euler equation for the motion of an inviscid fluid to derive the vorticity equation in the form

$D \boldsymbol{\omega} / D t=\boldsymbol{\omega} \cdot \nabla \mathbf{u} .$

Give a physical interpretation of the terms in this equation and deduce that irrotational flows remain irrotational.

In a plane flow the vorticity at time $t=0$ has the uniform value $\boldsymbol{\omega}_{0} \neq \mathbf{0}$. Find the vorticity everywhere at times $t>0$.

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• # 1.II.17A

A point source of fluid of strength $m$ is located at $\mathbf{x}_{s}=(0,0, a)$ in inviscid fluid of density $\rho$. Gravity is negligible. The fluid is confined to the region $z \geqslant 0$ by the fixed boundary $z=0$. Write down the equation and boundary conditions satisfied by the velocity potential $\phi$. Find $\phi$.

[Hint: consider the flow generated in unbounded fluid by the source $m$ together with an 'image source' of equal strength at $\overline{\mathbf{x}}_{s}=(0,0,-a)$.]

Use Bernoulli's theorem, which may be stated without proof, to find the fluid pressure everywhere on $z=0$. Deduce the magnitude of the hydrodynamic force on the boundary $z=0$. Determine whether the boundary is attracted toward the source or repelled from it.

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• # 1.I $2 \mathrm{H} \quad$

Define the hyperbolic metric in the upper half-plane model $H$ of the hyperbolic plane. How does one define the hyperbolic area of a region in $H$ ? State the Gauss-Bonnet theorem for hyperbolic triangles.

Let $R$ be the region in $H$ defined by

$0

Calculate the hyperbolic area of $R$.

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• # 1.II.10E

Find all subgroups of indices $2,3,4$ and 5 in the alternating group $A_{5}$ on 5 letters. You may use any general result that you choose, provided that you state it clearly, but you must justify your answers.

[You may take for granted the fact that $A_{4}$ has no subgroup of index 2.]

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• # 1.I.1H

Define what is meant by the minimal polynomial of a complex $n \times n$ matrix, and show that it is unique. Deduce that the minimal polynomial of a real $n \times n$ matrix has real coefficients.

For $n>2$, find an $n \times n$ matrix with minimal polynomial $(t-1)^{2}(t+1)$.

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• # 1.II.9H

Let $U, V$ be finite-dimensional vector spaces, and let $\theta$ be a linear map of $U$ into $V$. Define the rank $r(\theta)$ and the nullity $n(\theta)$ of $\theta$, and prove that

$r(\theta)+n(\theta)=\operatorname{dim} U$

Now let $\theta, \phi$ be endomorphisms of a vector space $U$. Define the endomorphisms $\theta+\phi$ and $\theta \phi$, and prove that

\begin{aligned} r(\theta+\phi) & \leqslant r(\theta)+r(\phi) \\ n(\theta \phi) & \leqslant n(\theta)+n(\phi) . \end{aligned}

Prove that equality holds in both inequalities if and only if $\theta+\phi$ is an isomorphism and $\theta \phi$ is zero.

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• # 1.II.19C

Explain what is meant by a stopping time of a Markov chain $\left(X_{n}\right)_{n \geq 0}$. State the strong Markov property.

Show that, for any state $i$, the probability, starting from $i$, that $\left(X_{n}\right)_{n \geq 0}$ makes infinitely many visits to $i$ can take only the values 0 or 1 .

$\sum_{n=0}^{\infty} \mathbb{P}_{i}\left(X_{n}=i\right)=\infty$

then $\left(X_{n}\right)_{n \geq 0}$ makes infinitely many visits to $i$ with probability $1 .$

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• # 1.II.14A

Define a second rank tensor. Show from your definition that if $M_{i j}$ is a second rank tensor then $M_{i i}$ is a scalar.

A rigid body consists of a thin flat plate of material having density $\rho(\mathbf{x})$ per unit area, where $\mathbf{x}$ is the position vector. The body occupies a region $D$ of the $(x, y)$-plane; its thickness in the $z$-direction is negligible. The moment of inertia tensor of the body is given as

$M_{i j}=\int_{D}\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \rho d S$

Show that the $z$-direction is an eigenvector of $M_{i j}$ and write down an integral expression for the corresponding eigenvalue $M_{\perp}$.

Hence or otherwise show that if the remaining eigenvalues of $M_{i j}$ are $M_{1}$ and $M_{2}$ then

$M_{\perp}=M_{1}+M_{2} .$

Find $M_{i j}$ for a circular disc of radius $a$ and uniform density having its centre at the origin.

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• # 1.II.12F

(i) Define the product topology on $X \times Y$ for topological spaces $X$ and $Y$, proving that your definition does define a topology.

(ii) Let $X$ be the logarithmic spiral defined in polar coordinates by $r=e^{\theta}$, where $-\infty<\theta<\infty$. Show that $X$ (with the subspace topology from $\mathbf{R}^{2}$ ) is homeomorphic to the real line.

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• # 1.I.6D

(a) Perform the LU-factorization with column pivoting of the matrix

$A=\left[\begin{array}{rrr} 2 & 1 & 1 \\ 4 & 1 & 0 \\ -2 & 2 & 1 \end{array}\right]$

(b) Explain briefly why every nonsingular matrix $A$ admits an LU-factorization with column pivoting.

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• # 1.I.8C

State the Lagrangian sufficiency theorem.

Let $p \in(1, \infty)$ and let $a_{1}, \ldots, a_{n} \in \mathbb{R}$. Maximize

$\sum_{i=1}^{n} a_{i} x_{i}$

subject to

$\sum_{i=1}^{n}\left|x_{i}\right|^{p} \leqslant 1, \quad x_{1}, \ldots, x_{n} \in \mathbb{R}$

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• # 1.II.15B

Let $V_{1}(x)$ and $V_{2}(x)$ be two real potential functions of one space dimension, and let $a$ be a positive constant. Suppose also that $V_{1}(x) \leqslant V_{2}(x) \leqslant 0$ for all $x$ and that $V_{1}(x)=V_{2}(x)=0$ for all $x$ such that $|x| \geqslant a$. Consider an incoming beam of particles described by the plane wave $\exp (i k x)$, for some $k>0$, scattering off one of the potentials $V_{1}(x)$ or $V_{2}(x)$. Let $p_{i}$ be the probability that a particle in the beam is reflected by the potential $V_{i}(x)$. Is it necessarily the case that $p_{1} \leqslant p_{2}$ ? Justify your answer carefully, either by giving a rigorous proof or by presenting a counterexample with explicit calculations of $p_{1}$ and $p_{2}$.

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• # 1.I.4B

A ball of clay of mass $m$ travels at speed $v$ in the laboratory frame towards an identical ball at rest. After colliding head-on, the balls stick together, moving in the same direction as the first ball was moving before the collision. Calculate the mass $m^{\prime}$ and speed $v^{\prime}$ of the combined lump, justifying your answers carefully.

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• # 1.II.18C

Let $X$ be a random variable whose distribution depends on an unknown parameter $\theta$. Explain what is meant by a sufficient statistic $T(X)$ for $\theta$.

In the case where $X$ is discrete, with probability mass function $f(x \mid \theta)$, explain, with justification, how a sufficient statistic may be found.

Assume now that $X=\left(X_{1}, \ldots, X_{n}\right)$, where $X_{1}, \ldots, X_{n}$ are independent nonnegative random variables with common density function

$f(x \mid \theta)= \begin{cases}\lambda e^{-\lambda(x-\theta)} & \text { if } x \geqslant \theta \\ 0 & \text { otherwise }\end{cases}$

Here $\theta \geq 0$ is unknown and $\lambda$ is a known positive parameter. Find a sufficient statistic for $\theta$ and hence obtain an unbiased estimator $\hat{\theta}$ for $\theta$ of variance $(n \lambda)^{-2}$.

[You may use without proof the following facts: for independent exponential random variables $X$ and $Y$, having parameters $\lambda$ and $\mu$ respectively, $X$ has mean $\lambda^{-1}$ and variance $\lambda^{-2}$ and $\min \{X, Y\}$ has exponential distribution of parameter $\lambda+\mu$.]

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