# Part IB, 2005

### Jump to course

1.II.11B

commentLet $\left(f_{n}\right)_{n \geqslant 1}$ be a sequence of continuous real-valued functions defined on a set $E \subset \mathbf{R}$. Suppose that the functions $f_{n}$ converge uniformly to a function $f$. Prove that $f$ is continuous on $E$.

Show that the series $\sum_{n=1}^{\infty} 1 / n^{1+x}$ defines a continuous function on the half-open interval $(0,1]$.

[Hint: You may assume the convergence of standard series.]

2.I.3B

commentDefine uniform continuity for a real-valued function defined on an interval in $\mathbf{R}$.

Is a uniformly continuous function on the interval $(0,1)$ necessarily bounded?

Is $1 / x$ uniformly continuous on $(0,1)$ ?

Is $\sin (1 / x)$ uniformly continuous on $(0,1)$ ?

Justify your answers.

2.II.13B

commentUse the standard metric on $\mathbf{R}^{n}$ in this question.

(i) Let $A$ be a nonempty closed subset of $\mathbf{R}^{n}$ and $y$ a point in $\mathbf{R}^{n}$. Show that there is a point $x \in A$ which minimizes the distance to $y$, in the sense that $d(x, y) \leqslant d(a, y)$ for all $a \in A$.

(ii) Suppose that the set $A$ in part (i) is convex, meaning that $A$ contains the line segment between any two of its points. Show that point $x \in A$ described in part (i) is unique.

3.I.3B

commentLet $f: \mathbf{R}^{2} \rightarrow \mathbf{R}$ be a function. What does it mean to say that $f$ is differentiable at a point $(a, b)$ in $\mathbf{R}^{2}$ ? Show that if $f$ is differentiable at $(a, b)$, then $f$ is continuous at $(a, b)$.

For each of the following functions, determine whether or not it is differentiable at $(0,0)$. Justify your answers.

(i)

$f(x, y)= \begin{cases}x^{2} y^{2}\left(x^{2}+y^{2}\right)^{-1} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$

(ii)

$f(x, y)= \begin{cases}x^{2}\left(x^{2}+y^{2}\right)^{-1} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$

3.II.13B

commentLet $f$ be a real-valued differentiable function on an open subset $U$ of $\mathbf{R}^{n}$. Assume that $0 \notin U$ and that for all $x \in U$ and $\lambda>0, \lambda x$ is also in $U$. Suppose that $f$ is homogeneous of degree $c \in \mathbf{R}$, in the sense that $f(\lambda x)=\lambda^{c} f(x)$ for all $x \in U$ and $\lambda>0$. By means of the Chain Rule or otherwise, show that

$\left.D f\right|_{x}(x)=c f(x)$

for all $x \in U$. (Here $\left.D f\right|_{x}$ denotes the derivative of $f$ at $x$, viewed as a linear map $\mathbf{R}^{n} \rightarrow \mathbf{R}$.)

Conversely, show that any differentiable function $f$ on $U$ with $\left.D f\right|_{x}(x)=c f(x)$ for all $x \in U$ must be homogeneous of degree $c$.

4.I 3 B

commentLet $V$ be the vector space of continuous real-valued functions on $[0,1]$. Show that the function

$\|f\|=\int_{0}^{1}|f(x)| d x$

defines a norm on $V$.

For $n=1,2, \ldots$, let $f_{n}(x)=e^{-n x}$. Is $f_{n}$ a convergent sequence in the space $V$ with this norm? Justify your answer.

4.II.13B

commentLet $F:[-a, a] \times\left[x_{0}-r, x_{0}+r\right] \rightarrow \mathbf{R}$ be a continuous function. Let $C$ be the maximum value of $|F(t, x)|$. Suppose there is a constant $K$ such that

$|F(t, x)-F(t, y)| \leqslant K|x-y|$

for all $t \in[-a, a]$ and $x, y \in\left[x_{0}-r, x_{0}+r\right]$. Let $b<\min (a, r / C, 1 / K)$. Show that there is a unique $C^{1}$ function $x:[-b, b] \rightarrow\left[x_{0}-r, x_{0}+r\right]$ such that

$x(0)=x_{0}$

and

$\frac{d x}{d t}=F(t, x(t)) .$

[Hint: First show that the differential equation with its initial condition is equivalent to the integral equation

$\left.x(t)=x_{0}+\int_{0}^{t} F(s, x(s)) d s .\right]$

3.II.14A

commentState the Cauchy integral formula, and use it to deduce Liouville's theorem.

Let $f$ be a meromorphic function on the complex plane such that $\left|f(z) / z^{n}\right|$ is bounded outside some disc (for some fixed integer $n$ ). By considering Laurent expansions, or otherwise, show that $f$ is a rational function in $z$.

4.I.4A

commentLet $\gamma:[0,1] \rightarrow \mathbf{C}$ be a closed path, where all paths are assumed to be piecewise continuously differentiable, and let $a$ be a complex number not in the image of $\gamma$. Write down an expression for the winding number $n(\gamma, a)$ in terms of a contour integral. From this characterization of the winding number, prove the following properties:

(a) If $\gamma_{1}$ and $\gamma_{2}$ are closed paths not passing through zero, and if $\gamma:[0,1] \rightarrow \mathbf{C}$ is defined by $\gamma(t)=\gamma_{1}(t) \gamma_{2}(t)$ for all $t$, then

$n(\gamma, 0)=n\left(\gamma_{1}, 0\right)+n\left(\gamma_{2}, 0\right)$

(b) If $\eta:[0,1] \rightarrow \mathbf{C}$ is a closed path whose image is contained in $\{\operatorname{Re}(z)>0\}$, then $n(\eta, 0)=0$.

(c) If $\gamma_{1}$ and $\gamma_{2}$ are closed paths and $a$ is a complex number, not in the image of either path, such that

$\left|\gamma_{1}(t)-\gamma_{2}(t)\right|<\left|\gamma_{1}(t)-a\right|$

for all $t$, then $n\left(\gamma_{1}, a\right)=n\left(\gamma_{2}, a\right)$.

[You may wish here to consider the path defined by $\eta(t)=1-\left(\gamma_{1}(t)-\gamma_{2}(t)\right) /\left(\gamma_{1}(t)-a\right)$.]

1.I.3F

commentState the Cauchy integral formula.

Using the Cauchy integral formula, evaluate

$\int_{|z|=2} \frac{z^{3}}{z^{2}+1} d z$

1.II.13F

commentDetermine a conformal mapping from $\Omega_{0}=\mathbf{C} \backslash[-1,1]$ to the complex unit disc $\Omega_{1}=\{z \in \mathbf{C}:|z|<1\} .$

[Hint: A standard method is first to map $\Omega_{0}$ to $\mathbf{C} \backslash(-\infty, 0]$, then to the complex right half-plane $\{z \in \mathbf{C}: \operatorname{Re} z>0\}$ and, finally, to $\left.\Omega_{1} .\right]$

2.II.14F

commentLet $F=P / Q$ be a rational function, where $\operatorname{deg} Q \geqslant \operatorname{deg} P+2$ and $Q$ has no real zeros. Using the calculus of residues, write a general expression for

$\int_{-\infty}^{\infty} F(x) e^{i x} d x$

in terms of residues and briefly sketch its proof.

Evaluate explicitly the integral

$\int_{-\infty}^{\infty} \frac{\cos x}{4+x^{4}} d x$

3.I.5F

commentDefine a harmonic function and state when the harmonic functions $f$ and $g$ are conjugate

Let $\{u, v\}$ and $\{p, q\}$ be two pairs of harmonic conjugate functions. Prove that $\{p(u, v), q(u, v)\}$ are also harmonic conjugate.

4.II.15F

commentDetermine the Fourier expansion of the function $f(x)=\sin \lambda x$, where $-\pi \leqslant x \leqslant \pi$, in the two cases where $\lambda$ is an integer and $\lambda$ is a real non-integer.

Using the Parseval identity in the case $\lambda=\frac{1}{2}$, find an explicit expression for the sum

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

1.II.16H

commentFor a static charge density $\rho(\mathbf{x})$ show that the energy may be expressed as

$E=\frac{1}{2} \int \rho \phi \mathrm{d}^{3} x=\frac{\epsilon_{0}}{2} \int \mathbf{E}^{2} \mathrm{~d}^{3} x,$

where $\phi(\mathbf{x})$ is the electrostatic potential and $\mathbf{E}(\mathbf{x})$ is the electric field.

Determine the scalar potential and electric field for a sphere of radius $R$ with a constant charge density $\rho$. Also determine the total electrostatic energy.

In a nucleus with $Z$ protons the volume is proportional to $Z$. Show that we may expect the electric contribution to energy to be proportional to $Z^{\frac{5}{3}}$.

2.I.6H

commentWrite down Maxwell's equations in the presence of a charge density $\rho$ and current density $\mathbf{J}$. Show that it is necessary that $\rho, \mathbf{J}$ satisfy a conservation equation.

If $\rho, \mathbf{J}$ are zero outside a fixed region $V$ show that the total charge inside $V$ is a constant and also that

$\frac{\mathrm{d}}{\mathrm{d} t} \int_{V} \mathbf{x} \rho \mathrm{d}^{3} x=\int_{V} \mathbf{J ~}^{3} x .$

2.II.17H

commentAssume the magnetic field

$\mathbf{B}(\mathbf{x})=b(\mathbf{x}-3 \hat{\mathbf{z}} \hat{\mathbf{z}} \cdot \mathbf{x}),$

where $\hat{\mathbf{z}}$ is a unit vector in the vertical direction. Show that this satisfies the expected equations for a static magnetic field in vacuum.

A circular wire loop, of radius $a$, mass $m$ and resistance $R$, lies in a horizontal plane with its centre on the $z$-axis at a height $z$ and there is a magnetic field given by $(*)$. Calculate the magnetic flux arising from this magnetic field through the loop and also the force acting on the loop when a current $I$ is flowing around the loop in a clockwise direction about the $z$-axis.

Obtain an equation of motion for the height $z(t)$ when the wire loop is falling under gravity. Show that there is a solution in which the loop falls with constant speed $v$ which should be determined. Verify that in this situation the rate at which heat is generated by the current flowing in the loop is equal to the rate of loss of gravitational potential energy. What happens when $R \rightarrow 0$ ?

3.II.17H

commentIf $\mathbf{E}(\mathbf{x}, t), \mathbf{B}(\mathbf{x}, t)$ are solutions of Maxwell's equations in a region without any charges or currents show that $\mathbf{E}^{\prime}(\mathbf{x}, t)=c \mathbf{B}(\mathbf{x}, t), \mathbf{B}^{\prime}(\mathbf{x}, t)=-\mathbf{E}(\mathbf{x}, t) / c$ are also solutions.

At the boundary of a perfect conductor with normal $\mathbf{n}$ briefly explain why

$\mathbf{n} \cdot \mathbf{B}=0, \quad \mathbf{n} \times \mathbf{E}=0$

Electromagnetic waves inside a perfectly conducting tube with axis along the $z$-axis are given by the real parts of complex solutions of Maxwell's equations of the form

$\mathbf{E}(\mathbf{x}, t)=\mathbf{e}(x, y) e^{i(k z-\omega t)}, \quad \mathbf{B}(\mathbf{x}, t)=\mathbf{b}(x, y) e^{i(k z-\omega t)} .$

Suppose $b_{z}=0$. Show that we can find a solution in this case in terms of a function $\psi(x, y)$ where

$\left(e_{x}, e_{y}\right)=\left(\frac{\partial}{\partial x} \psi, \frac{\partial}{\partial y} \psi\right), \quad e_{z}=i\left(k-\frac{\omega^{2}}{k c^{2}}\right) \psi,$

so long as $\psi$ satisfies

$\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\gamma^{2}\right) \psi=0$

for suitable $\gamma$. Show that the boundary conditions are satisfied if $\psi=0$ on the surface of the tube.

Obtain a similar solution with $e_{z}=0$ but show that the boundary conditions are now satisfied if the normal derivative $\partial \psi / \partial n=0$ on the surface of the tube.

4.I.7H

commentFor a static current density $\mathbf{J}(\mathbf{x})$ show that we may choose the vector potential $\mathbf{A}(\mathbf{x})$ so that

$-\nabla^{2} \mathbf{A}=\mu_{0} \mathbf{J} .$

For a loop $L$, centred at the origin, carrying a current $I$ show that

$\mathbf{A}(\mathbf{x})=\frac{\mu_{0} I}{4 \pi} \oint_{L} \frac{1}{|\mathbf{x}-\mathbf{r}|} \mathrm{d} \mathbf{r} \sim-\frac{\mu_{0} I}{4 \pi} \frac{1}{|\mathbf{x}|^{3}} \oint_{L} \frac{1}{2} \mathbf{x} \times(\mathbf{r} \times \mathrm{d} \mathbf{r}) \quad \text { as } \quad|\mathbf{x}| \rightarrow \infty$

[You may assume

$-\nabla^{2} \frac{1}{4 \pi|\mathbf{x}|}=\delta^{3}(\mathbf{x})$

and for fixed vectors $\mathbf{a}, \mathbf{b}$

$\left.\oint_{L} \mathbf{a} \cdot \mathrm{d} \mathbf{r}=0, \quad \oint_{L}(\mathbf{a} \cdot \mathbf{r} \mathbf{b} \cdot \mathrm{d} \mathbf{r}+\mathbf{b} \cdot \mathbf{r} \mathbf{a} \cdot \mathrm{d} \mathbf{r})=0 .\right]$

1.I.5E

commentExplain how a streamfunction $\psi$ can be used to represent in Cartesian Coordinates an incompressible flow in two dimensions. Show that the streamlines are given by $\psi=$ const.

Consider the two-dimensional incompressible flow

$\mathbf{u}(x, y, t)=(x+\sin t,-y)$

(a) Find the streamfunction, and hence the streamlines at $t=\frac{\pi}{2}$.

(b) Find the path of a fluid particle released at $t=0$ from $\left(x_{0}, 1\right)$. For what value of $x_{0}$ does the particle not tend to infinity?

1.II.17E

commentState Bernoulli's expression for the pressure in an unsteady potential flow with conservative force $-\nabla \chi$.

A spherical bubble in an incompressible liquid of density $\rho$ has radius $R(t)$. If the pressure far from the bubble is $p_{\infty}$ and inside the bubble is $p_{b}$, show that

$p_{b}-p_{\infty}=\rho\left(\frac{3}{2} \dot{R}^{2}+R \ddot{R}\right)$

Calculate the kinetic energy $K(t)$ in the flow outside the bubble, and hence show that

$\dot{K}=\left(p_{b}-p_{\infty}\right) \dot{V}$

where $V(t)$ is the volume of the bubble.

If $p_{b}(t)=p_{\infty} V_{0} / V$, show that

$K=K_{0}+p_{\infty}\left(V_{0} \ln \frac{V}{V_{0}}-V+V_{0}\right)$

where $K=K_{0}$ when $V=V_{0}$.

2.I.8E

commentFor a steady flow of an incompressible fluid of density $\rho$, show that

$\mathbf{u} \times \boldsymbol{\omega}=\nabla H,$

where $\boldsymbol{\omega}=\nabla \times \mathbf{u}$ is the vorticity and $H$ is to be found. Deduce that $H$ is constant along streamlines.

Now consider a flow in the $x y$-plane described by a streamfunction $\psi(x, y)$. Evaluate $\mathbf{u} \times \boldsymbol{\omega}$ and deduce from $H=H(\psi)$ that

$\frac{d H}{d \psi}+\omega=0$

3.II.18E

commentConsider the velocity potential in plane polar coordinates

$\phi(r, \theta)=U\left(r+\frac{a^{2}}{r}\right) \cos \theta+\frac{\kappa \theta}{2 \pi}$

Find the velocity field and show that it corresponds to flow past a cylinder $r=a$ with circulation $\kappa$ and uniform flow $U$ at large distances.

Find the distribution of pressure $p$ over the surface of the cylinder. Hence find the $x$ and $y$ components of the force on the cylinder

$\left(F_{x}, F_{y}\right)=\int(\cos \theta, \sin \theta) p a d \theta .$

4.II.18E

commentA fluid of density $\rho_{1}$ occupies the region $z>0$ and a second fluid of density $\rho_{2}$ occupies the region $z<0$. State the equations and boundary conditions that are satisfied by the corresponding velocity potentials $\phi_{1}$ and $\phi_{2}$ and pressures $p_{1}$ and $p_{2}$ when the system is perturbed so that the interface is at $z=\zeta(x, t)$ and the motion is irrotational.

Seek a set of linearised equations and boundary conditions when the disturbances are proportional to $e^{i(k x-\omega t)}$, and derive the dispersion relation

$\omega^{2}=\frac{\rho_{2}-\rho_{1}}{\rho_{2}+\rho_{1}} g k,$

where $g$ is the gravitational acceleration.

1.I $2 \mathrm{~A} \quad$

commentLet $\sigma: \mathbf{R}^{2} \rightarrow \mathbf{R}^{3}$ be the map defined by

$\sigma(u, v)=((a+b \cos u) \cos v,(a+b \cos u) \sin v, b \sin u)$

where $0<b<a$. Describe briefly the image $T=\sigma\left(\mathbf{R}^{2}\right) \subset \mathbf{R}^{3}$. Let $V$ denote the open subset of $\mathbf{R}^{2}$ given by $0<u<2 \pi, 0<v<2 \pi$; prove that the restriction $\sigma_{V}$ defines a smooth parametrization of a certain open subset (which you should specify) of $T$. Hence, or otherwise, prove that $T$ is a smooth embedded surface in $\mathbf{R}^{3}$.

[You may assume that the image under $\sigma$ of any open set $B \subset \mathbf{R}^{2}$ is open in $T$.]

2.II.12A

commentLet $U$ be an open subset of $\mathbf{R}^{2}$ equipped with a Riemannian metric. For $\gamma:[0,1] \rightarrow U$ a smooth curve, define what is meant by its length and energy. Prove that length $(\gamma)^{2} \leq \operatorname{energy}(\gamma)$, with equality if and only if $\dot{\gamma}$ has constant norm with respect to the metric.

Suppose now $U$ is the upper half plane model of the hyperbolic plane, and $P, Q$ are points on the positive imaginary axis. Show that a smooth curve $\gamma$ joining $P$ and $Q$ represents an absolute minimum of the length of such curves if and only if $\gamma(t)=i v(t)$, with $v$ a smooth monotonic real function.

Suppose that a smooth curve $\gamma$ joining the above points $P$ and $Q$ represents a stationary point for the energy under proper variations; deduce from an appropriate form of the Euler-Lagrange equations that $\gamma$ must be of the above form, with $\dot{v} / v$ constant.

3.I.2A

commentWrite down the Riemannian metric on the disc model $\Delta$ of the hyperbolic plane. Given that the length minimizing curves passing through the origin correspond to diameters, show that the hyperbolic circle of radius $\rho$ centred on the origin is just the Euclidean circle centred on the origin with Euclidean $\operatorname{radius~} \tanh (\rho / 2)$. Prove that the hyperbolic area is $2 \pi(\cosh \rho-1)$.

State the Gauss-Bonnet theorem for the area of a hyperbolic triangle. Given a hyperbolic triangle and an interior point $P$, show that the distance from $P$ to the nearest side is at most $\cosh ^{-1}(3 / 2)$.

3.II.12A

commentDescribe geometrically the stereographic projection map $\pi$ from the unit sphere $S^{2}$ to the extended complex plane $\mathbf{C}_{\infty}=\mathbf{C} \cup\{\infty\}$, positioned equatorially, and find a formula for $\pi$.

Show that any Möbius transformation $T \neq 1$ on $\mathbf{C}_{\infty}$ has one or two fixed points. Show that the Möbius transformation corresponding (under the stereographic projection map) to a rotation of $S^{2}$ through a non-zero angle has exactly two fixed points $z_{1}$ and $z_{2}$, where $z_{2}=-1 / \bar{z}_{1}$. If now $T$ is a Möbius transformation with two fixed points $z_{1}$ and $z_{2}$ satisfying $z_{2}=-1 / \bar{z}_{1}$, prove that either $T$ corresponds to a rotation of $S^{2}$, or one of the fixed points, say $z_{1}$, is an attractive fixed point, i.e. for $z \neq z_{2}, T^{n} z \rightarrow z_{1}$ as $n \rightarrow \infty$.

[You may assume the fact that any rotation of $S^{2}$ corresponds to some Möbius transformation of $\mathbf{C}_{\infty}$ under the stereographic projection map.]

4.II.12A

commentGiven a parametrized smooth embedded surface $\sigma: V \rightarrow U \subset \mathbf{R}^{3}$, where $V$ is an open subset of $\mathbf{R}^{2}$ with coordinates $(u, v)$, and a point $P \in U$, define what is meant by the tangent space at $P$, the unit normal $\mathbf{N}$ at $P$, and the first fundamental form

$E d u^{2}+2 F d u d v+G d v^{2} .$

[You need not show that your definitions are independent of the parametrization.]

The second fundamental form is defined to be

$L d u^{2}+2 M d u d v+N d v^{2},$

where $L=\sigma_{u u} \cdot \mathbf{N}, M=\sigma_{u v} \cdot \mathbf{N}$ and $N=\sigma_{v v} \cdot \mathbf{N}$. Prove that the partial derivatives of $\mathbf{N}$ (considered as a vector-valued function of $u, v$ ) are of the form $\mathbf{N}_{u}=a \sigma_{u}+b \sigma_{v}$, $\mathbf{N}_{v}=c \sigma_{u}+d \sigma_{v}$, where

$-\left(\begin{array}{cc} L & M \\ M & N \end{array}\right)=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)\left(\begin{array}{ll} E & F \\ F & G \end{array}\right)$

Explain briefly the significance of the determinant $a d-b c$.

$3 . \mathrm{II} . 11 \mathrm{C}$

comment(i) Define a primitive polynomial in $\mathbb{Z}[x]$, and prove that the product of two primitive polynomials is primitive. Deduce that $\mathbb{Z}[x]$ is a unique factorization domain.

(ii) Prove that

$\mathbb{Q}[x] /\left(x^{5}-4 x+2\right)$

is a field. Show, on the other hand, that

$\mathbb{Z}[x] /\left(x^{5}-4 x+2\right)$

is an integral domain, but is not a field.

1.II.10C

commentLet $G$ be a group, and $H$ a subgroup of finite index. By considering an appropriate action of $G$ on the set of left cosets of $H$, prove that $H$ always contains a normal subgroup $K$ of $G$ such that the index of $K$ in $G$ is finite and divides $n$ !, where $n$ is the index of $H$ in $G$.

Now assume that $G$ is a finite group of order $p q$, where $p$ and $q$ are prime numbers with $p<q$. Prove that the subgroup of $G$ generated by any element of order $q$ is necessarily normal.

2.I.2C

commentDefine an automorphism of a group $G$, and the natural group law on the set $\operatorname{Aut}(G)$ of all automorphisms of $G$. For each fixed $h$ in $G$, put $\psi(h)(g)=h g h^{-1}$ for all $g$ in $G$. Prove that $\psi(h)$ is an automorphism of $G$, and that $\psi$ defines a homomorphism from $G$ into $\operatorname{Aut}(G)$.

2.II.11C

commentLet $A$ be the abelian group generated by two elements $x, y$, subject to the relation $6 x+9 y=0$. Give a rigorous explanation of this statement by defining $A$ as an appropriate quotient of a free abelian group of rank 2. Prove that $A$ itself is not a free abelian group, and determine the exact structure of $A$.

3.I.1C

commentDefine what is meant by two elements of a group $G$ being conjugate, and prove that this defines an equivalence relation on $G$. If $G$ is finite, sketch the proof that the cardinality of each conjugacy class divides the order of $G$.

4.I.2C

commentState Eisenstein's irreducibility criterion. Let $n$ be an integer $>1$. Prove that $1+x+\ldots+x^{n-1}$ is irreducible in $\mathbb{Z}[x]$ if and only if $n$ is a prime number.

4.II.11C

commentLet $R$ be the ring of Gaussian integers $\mathbb{Z}[i]$, where $i^{2}=-1$, which you may assume to be a unique factorization domain. Prove that every prime element of $R$ divides precisely one positive prime number in $\mathbb{Z}$. List, without proof, the prime elements of $R$, up to associates.

Let $p$ be a prime number in $\mathbb{Z}$. Prove that $R / p R$ has cardinality $p^{2}$. Prove that $R / 2 R$ is not a field. If $p \equiv 3 \bmod 4$, show that $R / p R$ is a field. If $p \equiv 1 \bmod 4$, decide whether $R / p R$ is a field or not, justifying your answer.

1.I.1C

commentLet $V$ be an $n$-dimensional vector space over $\mathbf{R}$, and let $\beta: V \rightarrow V$ be a linear map. Define the minimal polynomial of $\beta$. Prove that $\beta$ is invertible if and only if the constant term of the minimal polynomial of $\beta$ is non-zero.

1.II.9C

commentLet $V$ be a finite dimensional vector space over $\mathbf{R}$, and $V^{*}$ be the dual space of $V$.

If $W$ is a subspace of $V$, we define the subspace $\alpha(W)$ of $V^{*}$ by

$\alpha(W)=\left\{f \in V^{*}: f(w)=0 \text { for all } w \text { in } W\right\}$

Prove that $\operatorname{dim}(\alpha(W))=\operatorname{dim}(V)-\operatorname{dim}(W)$. Deduce that, if $A=\left(a_{i j}\right)$ is any real $m \times n$-matrix of rank $r$, the equations

$\sum_{j=1}^{n} a_{i j} x_{j}=0 \quad(i=1, \ldots, m)$

have $n-r$ linearly independent solutions in $\mathbf{R}^{n}$.

2.I.1C

commentLet $\Omega$ be the set of all $2 \times 2$ matrices of the form $\alpha=a I+b J+c K+d L$, where $a, b, c, d$ are in $\mathbf{R}$, and

$I=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right), J=\left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right), K=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right), L=\left(\begin{array}{cc} 0 & i \\ i & 0 \end{array}\right) \quad\left(i^{2}=-1\right) .$

Prove that $\Omega$ is closed under multiplication and determine its dimension as a vector space over $\mathbf{R}$. Prove that

$(a I+b J+c K+d L)(a I-b J-c K-d L)=\left(a^{2}+b^{2}+c^{2}+d^{2}\right) I$

and deduce that each non-zero element of $\Omega$ is invertible.

2.II.10C

comment(i) Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix with entries in C. Define the determinant of $A$, the cofactor of each $a_{i j}$, and the adjugate matrix $\operatorname{adj}(A)$. Assuming the expansion of the determinant of a matrix in terms of its cofactors, prove that

$\operatorname{adj}(A) A=\operatorname{det}(A) I_{n}$

where $I_{n}$ is the $n \times n$ identity matrix.

(ii) Let

$A=\left(\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right)$

Show the eigenvalues of $A$ are $\pm 1, \pm i$, where $i^{2}=-1$, and determine the diagonal matrix to which $A$ is similar. For each eigenvalue, determine a non-zero eigenvector.

3.II.10B

commentLet $S$ be the vector space of functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that the $n$th derivative of $f$ is defined and continuous for every $n \geqslant 0$. Define linear maps $A, B: S \rightarrow S$ by $A(f)=d f / d x$ and $B(f)(x)=x f(x)$. Show that

$[A, B]=1_{S},$

where in this question $[A, B]$ means $A B-B A$ and $1_{S}$ is the identity map on $S$.

Now let $V$ be any real vector space with linear maps $A, B: V \rightarrow V$ such that $[A, B]=1_{V}$. Suppose that there is a nonzero element $y \in V$ with $A y=0$. Let $W$ be the subspace of $V$ spanned by $y, B y, B^{2} y$, and so on. Show that $A(B y)$ is in $W$ and give a formula for it. More generally, show that $A\left(B^{i} y\right)$ is in $W$ for each $i \geqslant 0$, and give a formula for it.

Show, using your formula or otherwise, that $\left\{y, B y, B^{2} y, \ldots\right\}$ are linearly independent. (Or, equivalently: show that $y, B y, B^{2} y, \ldots, B^{n} y$ are linearly independent for every $n \geqslant 0$.)

4.I.1B

commentDefine what it means for an $n \times n$ complex matrix to be unitary or Hermitian. Show that every eigenvalue of a Hermitian matrix is real. Show that every eigenvalue of a unitary matrix has absolute value 1 .

Show that two eigenvectors of a Hermitian matrix that correspond to different eigenvalues are orthogonal, using the standard inner product on $\mathbf{C}^{n}$.

4.II.10B

comment(i) Let $V$ be a finite-dimensional real vector space with an inner product. Let $e_{1}, \ldots, e_{n}$ be a basis for $V$. Prove by an explicit construction that there is an orthonormal basis $f_{1}, \ldots, f_{n}$ for $V$ such that the span of $e_{1}, \ldots, e_{i}$ is equal to the span of $f_{1}, \ldots, f_{i}$ for every $1 \leqslant i \leqslant n$.

(ii) For any real number $a$, consider the quadratic form

$q_{a}(x, y, z)=x y+y z+z x+a x^{2}$

on $\mathbf{R}^{3}$. For which values of $a$ is $q_{a}$ nondegenerate? When $q_{a}$ is nondegenerate, compute its signature in terms of $a$.

1.II.19D

commentEvery night Lancelot and Guinevere sit down with four guests for a meal at a circular dining table. The six diners are equally spaced around the table and just before each meal two individuals are chosen at random and they exchange places from the previous night while the other four diners stay in the same places they occupied at the last meal; the choices on successive nights are made independently. On the first night Lancelot and Guinevere are seated next to each other.

Find the probability that they are seated diametrically opposite each other on the $(n+1)$ th night at the round table, $n \geqslant 1$.

2.II.20D

commentConsider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ with state space $\{0,1,2, \ldots\}$ and transition probabilities given by

$P_{i, j}=p q^{i-j+1}, \quad 0<j \leqslant i+1, \quad \text { and } \quad P_{i, 0}=q^{i+1} \quad \text { for } \quad i \geqslant 0$

with $P_{i, j}=0$, otherwise, where $0<p<1$ and $q=1-p$.

For each $i \geqslant 1$, let

$h_{i}=\mathbb{P}\left(X_{n}=0, \text { for some } n \geqslant 0 \mid X_{0}=i\right),$

that is, the probability that the chain ever hits the state 0 given that it starts in state $i$. Write down the equations satisfied by the probabilities $\left\{h_{i}, i \geqslant 1\right\}$ and hence, or otherwise, show that they satisfy a second-order recurrence relation with constant coefficients. Calculate $h_{i}$ for each $i \geqslant 1$.

Determine for each value of $p, 0<p<1$, whether the chain is transient, null recurrent or positive recurrent and in the last case calculate the stationary distribution.

[Hint: When the chain is positive recurrent, the stationary distribution is geometric.]

3.I.9D

commentProve that if two states of a Markov chain communicate then they have the same period.

Consider a Markov chain with state space $\{1,2, \ldots, 7\}$ and transition probabilities determined by the matrix

$\left(\begin{array}{ccccccc} 0 & \frac{1}{4} & \frac{1}{4} & 0 & 0 & \frac{1}{4} & \frac{1}{4} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{2} & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & 0 & \frac{1}{6} & \frac{1}{6} \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$

Identify the communicating classes of the chain and for each class state whether it is open or closed and determine its period.

4.I.9D

commentProve that the simple symmetric random walk in three dimensions is transient.

[You may wish to recall Stirling's formula: $n ! \sim(2 \pi)^{\frac{1}{2}} n^{n+\frac{1}{2}} e^{-n} .$ ]

1.II.14E

commentFind the Fourier Series of the function

$f(\theta)= \begin{cases}1 & 0 \leq \theta<\pi \\ -1 & \pi \leq \theta<2 \pi\end{cases}$

Find the solution $\phi(r, \theta)$ of the Poisson equation in two dimensions inside the unit disk $r \leq 1$

$\nabla^{2} \phi=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}=f(\theta)$

subject to the boundary condition $\phi(1, \theta)=0$.

[Hint: The general solution of $r^{2} R^{\prime \prime}+r R^{\prime}-n^{2} R=r^{2}$ is $R=a r^{n}+b r^{-n}-r^{2} /\left(n^{2}-4\right) .$ ]

From the solution, show that

$\int_{r \leq 1} f \phi d A=-\frac{4}{\pi} \sum_{n \text { odd }} \frac{1}{n^{2}(n+2)^{2}}$

2.I.5E

commentConsider the differential equation for $x(t)$ in $t>0$

$\ddot{x}-k^{2} x=f(t),$

subject to boundary conditions $x(0)=0$, and $\dot{x}(0)=0$. Find the Green function $G\left(t, t^{\prime}\right)$ such that the solution for $x(t)$ is given by

$x(t)=\int_{0}^{t} G\left(t, t^{\prime}\right) f\left(t^{\prime}\right) d t^{\prime} .$

2.II.15E

commentWrite down the Euler-Lagrange equation for the variational problem for $r(z)$

$\delta \int_{-h}^{h} F\left(z, r, r^{\prime}\right) d z=0$

with boundary conditions $r(-h)=r(h)=R$, where $R$ is a given positive constant. Show that if $F$ does not depend explicitly on $z$, i.e. $F=F\left(r, r^{\prime}\right)$, then the equation has a first integral

$F-r^{\prime} \frac{\partial F}{\partial r^{\prime}}=\frac{1}{k},$

where $k$ is a constant.

An axisymmetric soap film $r(z)$ is formed between two circular rings $r=R$ at $z=\pm H$. Find the equation governing the shape which minimizes the surface area. Show that the shape takes the form

$r(z)=k^{-1} \cosh k z .$

Show that there exist no solution if $R / H<\sinh A$, where $A$ is the unique positive solution of $A=\operatorname{coth} A$.

3.I.6E

commentDescribe briefly the method of Lagrangian multipliers for finding the stationary points of a function $f(x, y)$ subject to a constraint $g(x, y)=0$.

Use the method to find the stationary values of $x y$ subject to the constraint $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 .$

3.II.15H

commentObtain the power series solution about $t=0$ of

$\left(1-t^{2}\right) \frac{\mathrm{d}^{2}}{\mathrm{~d} t^{2}} y-2 t \frac{\mathrm{d}}{\mathrm{d} t} y+\lambda y=0$

and show that regular solutions $y(t)=P_{n}(t)$, which are polynomials of degree $n$, are obtained only if $\lambda=n(n+1), n=0,1,2, \ldots$ Show that the polynomial must be even or odd according to the value of $n$.

Show that

$\int_{-1}^{1} P_{n}(t) P_{m}(t) \mathrm{d} t=k_{n} \delta_{n m}$

for some $k_{n}>0$.

Using the identity

$\left(x \frac{\partial^{2}}{\partial x^{2}} x+\frac{\partial}{\partial t}\left(1-t^{2}\right) \frac{\partial}{\partial t}\right) \frac{1}{\left(1-2 x t+x^{2}\right)^{\frac{1}{2}}}=0,$

and considering an expansion $\sum_{n} a_{n}(x) P_{n}(t)$ show that

$\frac{1}{\left(1-2 x t+x^{2}\right)^{\frac{1}{2}}}=\sum_{n=0}^{\infty} x^{n} P_{n}(t), \quad 0<x<1$

if we assume $P_{n}(1)=1$.

By considering

$\int_{-1}^{1} \frac{1}{1-2 x t+x^{2}} d t$

determine the coefficient $k_{n}$.

4.I.5H

commentShow how the general solution of the wave equation for $y(x, t)$,

$\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} y(x, t)-\frac{\partial^{2}}{\partial x^{2}} y(x, t)=0$

can be expressed as

$y(x, t)=f(c t-x)+g(c t+x) .$

Show that the boundary conditions $y(0, t)=y(L, t)=0$ relate the functions $f$ and $g$ and require them to be periodic with period $2 L$.

Show that, with these boundary conditions,

$\frac{1}{2} \int_{0}^{L}\left(\frac{1}{c^{2}}\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right) \mathrm{d} x=\int_{-L}^{L} g^{\prime}(c t+x)^{2} \mathrm{~d} x$

and that this is a constant independent of $t$.

4.II.16H

commentDefine an isotropic tensor and show that $\delta_{i j}, \epsilon_{i j k}$ are isotropic tensors.

For $\hat{\mathbf{x}}$ a unit vector and $\mathrm{d} S(\hat{\mathbf{x}})$ the area element on the unit sphere show that

$\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}_{i_{1}} \ldots \hat{x}_{i_{n}}$

is an isotropic tensor for any $n$. Hence show that

$\begin{aligned} &\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}_{i} \hat{x}_{j}=a \delta_{i j}, \quad \int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}_{i} \hat{x}_{j} \hat{x}_{k}=0 \\ &\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}_{i} \hat{x}_{j} \hat{x}_{k} \hat{x}_{l}=b\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right) \end{aligned}$

for some $a, b$ which should be determined.

Explain why

$\int_{V} \mathrm{~d}^{3} x\left(x_{1}+\sqrt{-1} x_{2}\right)^{n} f(|\mathbf{x}|)=0, \quad n=2,3,4$

where $V$ is the region inside the unit sphere.

[The general isotropic tensor of rank 4 has the form $a \delta_{i j} \delta_{k l}+b \delta_{i k} \delta_{j l}+c \delta_{i l} \delta_{j k} .$ ]

1.II.12A

commentSuppose that $\left(X, d_{X}\right)$ and $\left(Y, d_{Y}\right)$ are metric spaces. Show that the definition

$d\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=d_{X}\left(x_{1}, x_{2}\right)+d_{Y}\left(y_{1}, y_{2}\right)$

defines a metric on the product $X \times Y$, under which the projection map $\pi: X \times Y \rightarrow Y$ is continuous.

If $\left(X, d_{X}\right)$ is compact, show that every sequence in $X$ has a subsequence converging to a point of $X$. Deduce that the projection map $\pi$ then has the property that, for any closed subset $F \subset X \times Y$, the image $\pi(F)$ is closed in $Y$. Give an example to show that this fails if $\left(X, d_{X}\right)$ is not assumed compact.

2.I.4A

commentLet $X$ be a topological space. Suppose that $U_{1}, U_{2}, \ldots$ are connected subsets of $X$ with $U_{j} \cap U_{1}$ non-empty for all $j>0$. Prove that

$W=\bigcup_{j>0} U_{j}$

is connected. If each $U_{j}$ is path-connected, prove that $W$ is path-connected.

3.I.4A

commentShow that a topology $\tau_{1}$ is determined on the real line $\mathbf{R}$ by specifying that a nonempty subset is open if and only if it is a union of half-open intervals $\{a \leq x<b\}$, where $a<b$ are real numbers. Determine whether $\left(\mathbf{R}, \tau_{1}\right)$ is Hausdorff.

Let $\tau_{2}$ denote the cofinite topology on $\mathbf{R}$ (that is, a non-empty subset is open if and only if its complement is finite). Prove that the identity map induces a continuous $\operatorname{map}\left(\mathbf{R}, \tau_{1}\right) \rightarrow\left(\mathbf{R}, \tau_{2}\right)$.

4.II.14A

commentLet $(M, d)$ be a metric space, and $F$ a non-empty closed subset of $M$. For $x \in M$, set

$d(x, F)=\inf _{z \in F} d(x, z)$

Prove that $d(x, F)$ is a continuous function of $x$, and that it is strictly positive for $x \notin F$.

A topological space is called normal if for any pair of disjoint closed subsets $F_{1}, F_{2}$, there exist disjoint open subsets $U_{1} \supset F_{1}, U_{2} \supset F_{2}$. By considering the function

$d\left(x, F_{1}\right)-d\left(x, F_{2}\right)$

or otherwise, deduce that any metric space is normal.

Suppose now that $X$ is a normal topological space, and that $F_{1}, F_{2}$ are disjoint closed subsets in $X$. Prove that there exist open subsets $W_{1} \supset F_{1}, W_{2} \supset F_{2}$, whose closures are disjoint. In the case when $X=\mathbf{R}^{2}$ with the standard metric topology, $F_{1}=\{(x,-1 / x): x<0\}$ and $F_{2}=\{(x, 1 / x): x>0\}$, find explicit open subsets $W_{1}, W_{2}$ with the above property.

1.I.6F

commentDetermine the Cholesky factorization (without pivoting) of the matrix

$A=\left[\begin{array}{ccc} 2 & -4 & 2 \\ -4 & 10+\lambda & 2+3 \lambda \\ 2 & 2+3 \lambda & 23+9 \lambda \end{array}\right]$

where $\lambda$ is a real parameter. Hence, find the range of values of $\lambda$ for which the matrix $A$ is positive definite.

2.II.18F

comment(a) Let $\left\{Q_{n}\right\}_{n \geqslant 0}$ be a set of polynomials orthogonal with respect to some inner product $(\cdot, \cdot)$ in the interval $[a, b]$. Write explicitly the least-squares approximation to $f \in C[a, b]$ by an $n$ th-degree polynomial in terms of the polynomials $\left\{Q_{n}\right\}_{n \geqslant 0}$.

(b) Let an inner product be defined by the formula

$(g, h)=\int_{-1}^{1}\left(1-x^{2}\right)^{-\frac{1}{2}} g(x) h(x) d x$

Determine the $n$th degree polynomial approximation of $f(x)=\left(1-x^{2}\right)^{\frac{1}{2}}$ with respect to this inner product as a linear combination of the underlying orthogonal polynomials.

3.II.19F

commentGiven real $\mu \neq 0$, we consider the matrix

$A=\left[\begin{array}{cccc} \frac{1}{\mu} & 1 & 0 & 0 \\ -1 & \frac{1}{\mu} & 1 & 0 \\ 0 & -1 & \frac{1}{\mu} & 1 \\ 0 & 0 & -1 & \frac{1}{\mu} \end{array}\right]$

Construct the Jacobi and Gauss-Seidel iteration matrices originating in the solution of the linear system $A x=b$.

Determine the range of real $\mu \neq 0$ for which each iterative procedure converges.

4.I.8F

commentDefine Gaussian quadrature.

Evaluate the coefficients of the Gaussian quadrature of the integral

$\int_{-1}^{1}\left(1-x^{2}\right) f(x) d x$

which uses two function evaluations.

1.I.8D

commentConsider the problem:

$\begin{array}{ll} \text { Minimize } & \sum_{i=1}^{m} \sum_{j=1}^{n} c_{i j} x_{i j} \\ \text { subject to } & \sum_{j=1}^{n} x_{i j}=a_{i}, \quad i=1, \ldots, m, \\ & \sum_{i=1}^{m} x_{i j}=b_{j}, \quad j=1, \ldots, n \\ & x_{i j} \geqslant 0, \quad \text { for all } i, j \end{array}$

where $a_{i} \geqslant 0, b_{j} \geqslant 0$ satisfy $\sum_{i=1}^{m} a_{i}=\sum_{j=1}^{n} b_{j}$.

Formulate the dual of this problem and state necessary and sufficient conditions for optimality.

2.I.9D

commentExplain what is meant by a two-person zero-sum game with payoff matrix $A=\left(a_{i j}\right)$.

Show that the problems of the two players may be expressed as a dual pair of linear programming problems. State without proof a set of sufficient conditions for a pair of strategies for the two players to be optimal.

3.II.20D

Consider the linear programming problem

$\begin{aligned} \operatorname{maximize} \quad 4 x_{1}+x_{2}-9 x_{3} & \\ \text { subject to } \quad x_{2}-11 x_{3} & \leqslant 11 \\ -3 x_{1}+2 x_{2}-7 x_{3} & \leqslant 16 \\ 9 x_{1}-2 x_{2}+10 x_{3} & \leqslant 29, \quad x_{i} \geqslant 0, \quad i=1,2,3 . \end{aligned}$

(a) After adding slack variables $z_{1}, z_{2}$