# Part IB, 2010

### Jump to course

Paper 1, Section II, G

commentState and prove the contraction mapping theorem. Demonstrate its use by showing that the differential equation $f^{\prime}(x)=f\left(x^{2}\right)$, with boundary condition $f(0)=1$, has a unique solution on $[0,1)$, with one-sided derivative $f^{\prime}(0)=1$ at zero.

Paper 2, Section I, G

commentLet $c>1$ be a real number, and let $F_{c}$ be the space of sequences $\mathbf{a}=\left(a_{1}, a_{2}, \ldots\right)$ of real numbers $a_{i}$ with $\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ convergent. Show that $\|\mathbf{a}\|_{c}=\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ defines a norm on $F_{c}$.

Let $F$ denote the space of sequences a with $\left|a_{i}\right|$ bounded; show that $F \subset F_{c}$. If $c^{\prime}>c$, show that the norms on $F$ given by restricting to $F$ the norms $\|\cdot\|_{c}$ on $F_{c}$ and $\|\cdot\|_{c^{\prime}}$ on $F_{c^{\prime}}$ are not Lipschitz equivalent.

By considering sequences of the form $\mathbf{a}^{(n)}=\left(a, a^{2}, \ldots, a^{n}, 0,0, \ldots\right)$ in $F$, for $a$ an appropriate real number, or otherwise, show that $F$ (equipped with the norm $\|.\|_{c}$ ) is not complete.

Paper 2, Section II, G

commentSuppose the functions $f_{n}(n=1,2, \ldots)$ are defined on the open interval $(0,1)$ and that $f_{n}$ tends uniformly on $(0,1)$ to a function $f$. If the $f_{n}$ are continuous, show that $f$ is continuous. If the $f_{n}$ are differentiable, show by example that $f$ need not be differentiable.

Assume now that each $f_{n}$ is differentiable and the derivatives $f_{n}^{\prime}$ converge uniformly on $(0,1)$. For any given $c \in(0,1)$, we define functions $g_{c, n}$ by

$g_{c, n}(x)= \begin{cases}\frac{f_{n}(x)-f_{n}(c)}{x-c} & \text { for } x \neq c, \\ f_{n}^{\prime}(c) & \text { for } x=c .\end{cases}$

Show that each $g_{c, n}$ is continuous. Using the general principle of uniform convergence (the Cauchy criterion) and the Mean Value Theorem, or otherwise, prove that the functions $g_{c, n}$ converge uniformly to a continuous function $g_{c}$ on $(0,1)$, where

$g_{c}(x)=\frac{f(x)-f(c)}{x-c} \quad \text { for } x \neq c$

Deduce that $f$ is differentiable on $(0,1)$.

Paper 3, Section I, G

commentConsider the map $f: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ given by

$f(x, y, z)=(x+y+z, x y+y z+z x, x y z)$

Show that $f$ is differentiable everywhere and find its derivative.

Stating carefully any theorem that you quote, show that $f$ is locally invertible near a point $(x, y, z)$ unless $(x-y)(y-z)(z-x)=0$.

Paper 3, Section II, G

commentLet $f: U \rightarrow \mathbf{R}^{n}$ be a map on an open subset $U \subset \mathbf{R}^{m}$. Explain what it means for $f$ to be differentiable on $U$. If $g: V \rightarrow \mathbf{R}^{m}$ is a differentiable map on an open subset $V \subset \mathbf{R}^{p}$ with $g(V) \subset U$, state and prove the Chain Rule for the derivative of the composite $f g$.

Suppose now $F: \mathbf{R}^{n} \rightarrow \mathbf{R}$ is a differentiable function for which the partial derivatives $D_{1} F(\mathbf{x})=D_{2} F(\mathbf{x})=\ldots=D_{n} F(\mathbf{x})$ for all $\mathbf{x} \in \mathbf{R}^{n}$. By considering the function $G: \mathbf{R}^{n} \rightarrow \mathbf{R}$ given by

$G\left(y_{1}, \ldots, y_{n}\right)=F\left(y_{1}, \ldots, y_{n-1}, y_{n}-\sum_{i=1}^{n-1} y_{i}\right)$

or otherwise, show that there exists a differentiable function $h: \mathbf{R} \rightarrow \mathbf{R}$ with $F\left(x_{1}, \ldots, x_{n}\right)=$ $h\left(x_{1}+\cdots+x_{n}\right)$ at all points of $\mathbf{R}^{n} .$

Paper 4, Section I, G

commentLet $S$ denote the set of continuous real-valued functions on the interval $[0,1]$. For $f, g \in S$, set

$d_{1}(f, g)=\sup \{|f(x)-g(x)|: x \in[0,1]\} \quad \text { and } \quad d_{2}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x$

Show that both $d_{1}$ and $d_{2}$ define metrics on $S$. Does the identity map on $S$ define a continuous map of metric spaces $\left(S, d_{1}\right) \rightarrow\left(S, d_{2}\right) ?$ Does the identity map define a continuous map of metric spaces $\left(S, d_{2}\right) \rightarrow\left(S, d_{1}\right)$ ?

Paper 4, Section II, G

commentWhat does it mean to say that a function $f$ on an interval in $\mathbf{R}$ is uniformly continuous? Assuming the Bolzano-Weierstrass theorem, show that any continuous function on a finite closed interval is uniformly continuous.

Suppose that $f$ is a continuous function on the real line, and that $f(x)$ tends to finite limits as $x \rightarrow \pm \infty$; show that $f$ is uniformly continuous.

If $f$ is a uniformly continuous function on $\mathbf{R}$, show that $f(x) / x$ is bounded as $x \rightarrow \pm \infty$. If $g$ is a continuous function on $\mathbf{R}$ for which $g(x) / x \rightarrow 0$ as $x \rightarrow \pm \infty$, determine whether $g$ is necessarily uniformly continuous, giving proof or counterexample as appropriate.

Paper 3, Section II, G

commentState Morera's theorem. Suppose $f_{n}(n=1,2, \ldots)$ are analytic functions on a domain $U \subset \mathbf{C}$ and that $f_{n}$ tends locally uniformly to $f$ on $U$. Show that $f$ is analytic on $U$. Explain briefly why the derivatives $f_{n}^{\prime}$ tend locally uniformly to $f^{\prime}$.

Suppose now that the $f_{n}$ are nowhere vanishing and $f$ is not identically zero. Let $a$ be any point of $U$; show that there exists a closed disc $\bar{\Delta} \subset U$ with centre $a$, on which the convergence of $f_{n}$ and $f_{n}^{\prime}$ are both uniform, and where $f$ is nowhere zero on $\bar{\Delta} \backslash\{a\}$. By considering

$\frac{1}{2 \pi i} \int_{C} \frac{f_{n}^{\prime}(w)}{f_{n}(w)} d w$

(where $C$ denotes the boundary of $\bar{\Delta}$ ), or otherwise, deduce that $f(a) \neq 0$.

Paper 4, Section I, G

commentState the principle of the argument for meromorphic functions and show how it follows from the Residue theorem.

Paper 1, Section I, A

comment(a) Write down the definition of the complex derivative of the function $f(z)$ of a single complex variable.

(b) Derive the Cauchy-Riemann equations for the real and imaginary parts $u(x, y)$ and $v(x, y)$ of $f(z)$, where $z=x+i y$ and

$f(z)=u(x, y)+i v(x, y)$

(c) State necessary and sufficient conditions on $u(x, y)$ and $v(x, y)$ for the function $f(z)$ to be complex differentiable.

Paper 1, Section II, A

commentCalculate the following real integrals by using contour integration. Justify your steps carefully.

(a)

$I_{1}=\int_{0}^{\infty} \frac{x \sin x}{x^{2}+a^{2}} d x, \quad a>0$

(b)

$I_{2}=\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x$

Paper 2, Section II, A

comment(a) Prove that a complex differentiable map, $f(z)$, is conformal, i.e. preserves angles, provided a certain condition holds on the first complex derivative of $f(z)$.

(b) Let $D$ be the region

$D:=\{z \in \mathbb{C}:|z-1|>1 \text { and }|z-2|<2\}$

Draw the region $D$. It might help to consider the two sets

$\begin{aligned} &C(1):=\{z \in \mathbb{C}:|z-1|=1\} \\ &C(2):=\{z \in \mathbb{C}:|z-2|=2\} \end{aligned}$

(c) For the transformations below identify the images of $D$.

Step 1: The first map is $f_{1}(z)=\frac{z-1}{z}$,

Step 2: The second map is the composite $f_{2} f_{1}$ where $f_{2}(z)=\left(z-\frac{1}{2}\right) i$,

Step 3: The third map is the composite $f_{3} f_{2} f_{1}$ where $f_{3}(z)=e^{2 \pi z}$.

(d) Write down the inverse map to the composite $f_{3} f_{2} f_{1}$, explaining any choices of branch.

[The composite $f_{2} f_{1}$ means $f_{2}\left(f_{1}(z)\right)$.]

Paper 3, Section I, A

comment(a) Prove that the real and imaginary parts of a complex differentiable function are harmonic.

(b) Find the most general harmonic polynomial of the form

$u(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3}$

where $a, b, c, d, x$ and $y$ are real.

(c) Write down a complex analytic function of $z=x+i y$ of which $u(x, y)$ is the real part.

Paper 4, Section II, A

commentA linear system is described by the differential equation

$y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)-2 y^{\prime}(t)+2 y(t)=f(t),$

with initial conditions

$y(0)=0, \quad y^{\prime}(0)=1, \quad y^{\prime \prime}(0)=1$

The Laplace transform of $f(t)$ is defined as

$\mathcal{L}[f(t)]=\tilde{f}(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$

You may assume the following Laplace transforms,

$\begin{aligned} \mathcal{L}[y(t)] &=\tilde{y}(s) \\ \mathcal{L}\left[y^{\prime}(t)\right] &=s \tilde{y}(s)-y(0) \\ \mathcal{L}\left[y^{\prime \prime}(t)\right] &=s^{2} \tilde{y}(s)-s y(0)-y^{\prime}(0) \\ \mathcal{L}\left[y^{\prime \prime \prime}(t)\right] &=s^{3} \tilde{y}(s)-s^{2} y(0)-s y^{\prime}(0)-y^{\prime \prime}(0) \end{aligned}$

(a) Use Laplace transforms to determine the response, $y_{1}(t)$, of the system to the signal

$f(t)=-2$

(b) Determine the response, $y_{2}(t)$, given that its Laplace transform is

$\tilde{y}_{2}(s)=\frac{1}{s^{2}(s-1)^{2}} .$

(c) Given that

$y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)-2 y^{\prime}(t)+2 y(t)=g(t)$

leads to the response with Laplace transform

$\tilde{y}(s)=\frac{1}{s^{2}(s-1)^{2}},$

determine $g(t)$.

Paper 1, Section II, C

commentA capacitor consists of three perfectly conducting coaxial cylinders of radii $a, b$ and $c$ where $0<a<b<c$, and length $L$ where $L \gg c$ so that end effects may be ignored. The inner and outer cylinders are maintained at zero potential, while the middle cylinder is held at potential $V$. Assuming its cylindrical symmetry, compute the electrostatic potential within the capacitor, the charge per unit length on the middle cylinder, the capacitance and the electrostatic energy, both per unit length.

Next assume that the radii $a$ and $c$ are fixed, as is the potential $V$, while the radius $b$ is allowed to vary. Show that the energy achieves a minimum when $b$ is the geometric mean of $a$ and $c$.

Paper 2, Section I, $\mathbf{6 C}$

commentWrite down Maxwell's equations for electromagnetic fields in a non-polarisable and non-magnetisable medium.

Show that the homogenous equations (those not involving charge or current densities) can be solved in terms of vector and scalar potentials $\mathbf{A}$ and $\phi$.

Then re-express the inhomogeneous equations in terms of $\mathbf{A}, \phi$ and $f=\nabla \cdot \mathbf{A}+c^{-2} \dot{\phi}$. Show that the potentials can be chosen so as to set $f=0$ and hence rewrite the inhomogeneous equations as wave equations for the potentials. [You may assume that the inhomogeneous wave equation $\nabla^{2} \psi-c^{-2} \ddot{\psi}=\sigma(\mathbf{x}, t)$ always has a solution $\psi(\mathbf{x}, t)$ for any given $\sigma(\mathbf{x}, t)$.]

Paper 2, Section II, C

commentA steady current $I_{2}$ flows around a loop $\mathcal{C}_{2}$ of a perfectly conducting narrow wire. Assuming that the gauge condition $\nabla \cdot \mathbf{A}=0$ holds, the vector potential at points away from the loop may be taken to be

$\mathbf{A}(\mathbf{r})=\frac{\mu_{0} I_{2}}{4 \pi} \oint_{\mathcal{C}_{2}} \frac{d \mathbf{r}_{2}}{\left|\mathbf{r}-\mathbf{r}_{2}\right|}$

First verify that the gauge condition is satisfied here. Then obtain the Biot-Savart formula for the magnetic field

$\mathbf{B}(\mathbf{r})=\frac{\mu_{0} I_{2}}{4 \pi} \oint_{\mathcal{C}_{2}} \frac{d \mathbf{r}_{2} \times\left(\mathbf{r}-\mathbf{r}_{2}\right)}{\left|\mathbf{r}-\mathbf{r}_{2}\right|^{3}}$

Next suppose there is a similar but separate loop $\mathcal{C}_{1}$ with current $I_{1}$. Show that the magnetic force exerted on loop $\mathcal{C}_{1}$ by loop $\mathcal{C}_{2}$ is

$\mathbf{F}_{12}=\frac{\mu_{0} I_{1} I_{2}}{4 \pi} \oint_{\mathcal{C}_{1}} \oint_{\mathcal{C}_{2}} d \mathbf{r}_{1} \times\left(d \mathbf{r}_{2} \times \frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|^{3}}\right)$

Is this consistent with Newton's third law? Justify your answer.

Paper 3, Section II, C

commentWrite down Maxwell's equations in a region with no charges and no currents. Show that if $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ is a solution then so is $\widetilde{\mathbf{E}}(\mathbf{x}, t)=c \mathbf{B}(\mathbf{x}, t)$ and $\widetilde{\mathbf{B}}(\mathbf{x}, t)=-\mathbf{E}(\mathbf{x}, t) / c$. Write down the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the boundary with unit normal $\mathbf{n}$ between a perfect conductor and a vacuum.

Electromagnetic waves propagate inside a tube of perfectly conducting material. The tube's axis is in the $z$-direction, and it is surrounded by a vacuum. The fields may be taken to be the real parts of

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

Write down Maxwell's equations in terms of $\mathbf{e}, \mathbf{b}, k$ and $\omega$.

Suppose first that $b_{z}(x, y)=0$. Show that the solution is determined by

$\mathbf{e}=\left(\frac{\partial \psi}{\partial x}, \frac{\partial \psi}{\partial y}, i k\left[1-\frac{\omega^{2}}{k^{2} c^{2}}\right] \psi\right)$

where the function $\psi(x, y)$ satisfies

$\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\gamma^{2} \psi=0$

and $\psi$ vanishes on the boundary of the tube. Here $\gamma^{2}$ is a constant whose value should be determined.

Obtain a similar condition for the case where $e_{z}(x, y)=0$. [You may find it useful to use a result from the first paragraph.] What is the corresponding boundary condition?

Paper 4, Section I, B

commentGive an expression for the force $\mathbf{F}$ on a charge $q$ moving at velocity $\mathbf{v}$ in electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$. Consider a stationary electric circuit $C$, and let $S$ be a stationary surface bounded by $C$. Derive from Maxwell's equations the result

$\mathcal{E}=-\frac{d \Phi}{d t}$

where the electromotive force $\mathcal{E}=\oint_{C} q^{-1} \mathbf{F} \cdot d \mathbf{r}$ and the flux $\Phi=\int_{S} \mathbf{B} \cdot d \mathbf{S}$.

Now assume that $(*)$ also holds for a moving circuit. A circular loop of wire of radius $a$ and total resistance $R$, whose normal is in the $z$-direction, moves at constant speed $v$ in the $x$-direction in the presence of a magnetic field $\mathbf{B}=\left(0,0, B_{0} x / d\right)$. Find the current in the wire.

Paper 1, Section I, B

commentA planar solenoidal velocity field has the velocity potential

$\phi(x, y, t)=x e^{-t}+y e^{t}$

Find and sketch (i) the streamlines at $t=0$; (ii) the pathline that passes through the origin at $t=0$; (iii) the locus at $t=0$ of points that pass through the origin at earlier times (streakline).

Paper 1, Section II, B

commentStarting with the Euler equations for an inviscid incompressible fluid, derive Bernoulli's theorem for unsteady irrotational flow.

Inviscid fluid of density $\rho$ is contained within a U-shaped tube with the arms vertical, of height $h$ and with the same (unit) cross-section. The ends of the tube are closed. In the equilibrium state the pressures in the two arms are $p_{1}$ and $p_{2}$ and the heights of the fluid columns are $\ell_{1}, \ell_{2}$.

The fluid in arm 1 is displaced upwards by a distance $\xi$ (and in the other arm downward by the same amount). In the subsequent evolution the pressure above each column may be taken as inversely proportional to the length of tube above the fluid surface. Using Bernoulli's theorem, show that $\xi(t)$ obeys the equation

$\rho\left(\ell_{1}+\ell_{2}\right) \ddot{\xi}+\frac{p_{1} \xi}{h-\ell_{1}-\xi}+\frac{p_{2} \xi}{h-\ell_{2}+\xi}+2 \rho g \xi=0$

Now consider the special case $\ell_{1}=\ell_{2}=\ell_{0}, p_{1}=p_{2}=p_{0}$. Construct a first integral of this equation and hence give an expression for the total kinetic energy $\rho \ell_{0} \dot{\xi}^{2}$ of the flow in terms of $\xi$ and the maximum displacement $\xi_{\max }$.

Paper 2, Section I, B

commentWrite down an expression for the velocity field of a line vortex of strength $\kappa$.

Consider $N$ identical line vortices of strength $\kappa$ arranged at equal intervals round a circle of radius $a$. Show that the vortices all move around the circle at constant angular velocity $(N-1) \kappa /\left(4 \pi a^{2}\right)$.

Paper 3, Section II, B

commentWrite down the exact kinematic and dynamic boundary conditions that apply at the free surface $z=\eta(x, t)$ of a fluid layer in the presence of gravity in the $z$-direction. Show how these may be approximated for small disturbances of a hydrostatic state about $z=0$. (The flow of the fluid is in the $(x, z)$-plane and may be taken to be irrotational, and the pressure at the free surface may be assumed to be constant.)

Fluid of density $\rho$ fills the region $0>z>-h$. At $z=-h$ the $z$-component of the velocity is $\epsilon \operatorname{Re}\left(e^{i \omega t} \cos k x\right)$, where $|\epsilon| \ll 1$. Find the resulting disturbance of the free surface, assuming this to be small. Explain physically why your answer has a singularity for a particular value of $\omega^{2}$.

Paper 4, Section II, B

commentWrite down the velocity potential for a line source flow of strength $m$ located at $(r, \theta)=(d, 0)$ in polar coordinates $(r, \theta)$ and derive the velocity components $u_{r}, u_{\theta}$.

A two-dimensional flow field consists of such a source in the presence of a circular cylinder of radius $a(a<d)$ centred at the origin. Show that the flow field outside the cylinder is the sum of the original source flow, together with that due to a source of the same strength at $\left(a^{2} / d, 0\right)$ and another at the origin, of a strength to be determined.

Use Bernoulli's law to find the pressure distribution on the surface of the cylinder, and show that the total force exerted on it is in the $x$-direction and of magnitude

$\frac{m^{2} \rho}{2 \pi^{2}} \int_{0}^{2 \pi} \frac{a d^{2} \sin ^{2} \theta \cos \theta}{\left(a^{2}+d^{2}-2 a d \cos \theta\right)^{2}} d \theta$

where $\rho$ is the density of the fluid. Without evaluating the integral, show that it is positive. Comment on the fact that the force on the cylinder is therefore towards the source.

Paper 1, Section I, F

comment(i) Define the notion of curvature for surfaces embedded in $\mathbb{R}^{3}$.

(ii) Prove that the unit sphere in $\mathbb{R}^{3}$ has curvature $+1$ at all points.

Paper 2, Section II, F

commentSuppose that $a>0$ and that $S \subset \mathbb{R}^{3}$ is the half-cone defined by $z^{2}=a\left(x^{2}+y^{2}\right)$, $z>0$. By using an explicit smooth parametrization of $S$, calculate the curvature of $S$.

Describe the geodesics on $S$. Show that for $a=3$, no geodesic intersects itself, while for $a>3$ some geodesic does so.

Paper 3, Section I, F

comment(i) Write down the Poincaré metric on the unit disc model $D$ of the hyperbolic plane. Compute the hyperbolic distance $\rho$ from $(0,0)$ to $(r, 0)$, with $0<r<1$.

(ii) Given a point $P$ in $D$ and a hyperbolic line $L$ in $D$ with $P$ not on $L$, describe how the minimum distance from $P$ to $L$ is calculated. Justify your answer.

Paper 3, Section II, F

commentDescribe the hyperbolic metric on the upper half-plane $H$. Show that any Möbius transformation that preserves $H$ is an isometry of this metric.

Suppose that $z_{1}, z_{2} \in H$ are distinct and that the hyperbolic line through $z_{1}$ and $z_{2}$ meets the real axis at $w_{1}, w_{2}$. Show that the hyperbolic distance $\rho\left(z_{1}, z_{2}\right)$ between $z_{1}$ and $z_{2}$ is given by $\rho\left(z_{1}, z_{2}\right)=\log r$, where $r$ is the cross-ratio of the four points $z_{1}, z_{2}, w_{1}, w_{2}$, taken in an appropriate order.

Paper 4, Section II, F

commentSuppose that $D$ is the unit disc, with Riemannian metric

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

[Note that this is not a multiple of the Poincaré metric.] Show that the diameters of $D$ are, with appropriate parametrization, geodesics.

Show that distances between points in $D$ are bounded, but areas of regions in $D$ are unbounded.

Paper 1, Section II, H

commentProve that the kernel of a group homomorphism $f: G \rightarrow H$ is a normal subgroup of the group $G$.

Show that the dihedral group $D_{8}$ of order 8 has a non-normal subgroup of order 2. Conclude that, for a group $G$, a normal subgroup of a normal subgroup of $G$ is not necessarily a normal subgroup of $G$.

Paper 2, Section I, $2 \mathrm{H}$

commentGive the definition of conjugacy classes in a group $G$. How many conjugacy classes are there in the symmetric group $S_{4}$ on four letters? Briefly justify your answer.

Paper 2, Section II, H

commentFor ideals $I, J$ of a ring $R$, their product $I J$ is defined as the ideal of $R$ generated by the elements of the form $x y$ where $x \in I$ and $y \in J$.

(1) Prove that, if a prime ideal $P$ of $R$ contains $I J$, then $P$ contains either $I$ or $J$.

(2) Give an example of $R, I$ and $J$ such that the two ideals $I J$ and $I \cap J$ are different from each other.

(3) Prove that there is a natural bijection between the prime ideals of $R / I J$ and the prime ideals of $R /(I \cap J)$.

Paper 3, Section I, H

commentLet $A$ be the ring of integers $\mathbb{Z}$ or the polynomial ring $\mathbb{C}[X]$. In each case, give an example of an ideal $I$ of $A$ such that the quotient ring $R=A / I$ has a non-trivial idempotent (an element $x \in R$ with $x \neq 0,1$ and $x^{2}=x$ ) and a non-trivial nilpotent element (an element $x \in R$ with $x \neq 0$ and $x^{n}=0$ for some positive integer $n$ ). Exhibit these elements and justify your answer.

Paper 3, Section II, H

commentLet $R$ be an integral domain and $R^{\times}$its group of units. An element of $S=R \backslash\left(R^{\times} \cup\{0\}\right)$ is irreducible if it is not a product of two elements in $S$. When $R$ is Noetherian, show that every element of $S$ is a product of finitely many irreducible elements of $S$.

Paper 4, Section I, H

commentLet $M$ be a free $\mathbb{Z}$-module generated by $e_{1}$ and $e_{2}$. Let $a, b$ be two non-zero integers, and $N$ be the submodule of $M$ generated by $a e_{1}+b e_{2}$. Prove that the quotient module $M / N$ is free if and only if $a, b$ are coprime.

Paper 4, Section II, $11 H$

commentLet $V=(\mathbb{Z} / 3 \mathbb{Z})^{2}$, a 2-dimensional vector space over the field $\mathbb{Z} / 3 \mathbb{Z}$, and let $e_{1}=\left(\begin{array}{c}1 \\ 0\end{array}\right), e_{2}=\left(\begin{array}{c}0 \\ 1\end{array}\right) \in V .$

(1) List all 1-dimensional subspaces of $V$ in terms of $e_{1}, e_{2}$. (For example, there is a subspace $\left\langle e_{1}\right\rangle$ generated by $\left.e_{1} .\right)$

(2) Consider the action of the matrix group

$G=G L_{2}(\mathbb{Z} / 3 \mathbb{Z})=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a, b, c, d \in \mathbb{Z} / 3 \mathbb{Z}, \quad a d-b c \neq 0\right\}$

on the finite set $X$ of all 1-dimensional subspaces of $V$. Describe the stabiliser group $K$ of $\left\langle e_{1}\right\rangle \in X$. What is the order of $K$ ? What is the order of $G$ ?

(3) Let $H \subset G$ be the subgroup of all elements of $G$ which act trivially on $X$. Describe $H$, and prove that $G / H$ is isomorphic to $S_{4}$, the symmetric group on four letters.

Paper 1, Section I, F

commentSuppose that $V$ is the complex vector space of polynomials of degree at most $n-1$ in the variable $z$. Find the Jordan normal form for each of the linear transformations $\frac{d}{d z}$ and $z \frac{d}{d z}$ acting on $V$.

Paper 1, Section II, F

commentLet $V$ denote the vector space of $n \times n$ real matrices.

(1) Show that if $\psi(A, B)=\operatorname{tr}\left(A B^{T}\right)$, then $\psi$ is a positive-definite symmetric bilinear form on $V$.

(2) Show that if $q(A)=\operatorname{tr}\left(A^{2}\right)$, then $q$ is a quadratic form on $V$. Find its rank and signature.

[Hint: Consider symmetric and skew-symmetric matrices.]

Paper 2, Section I, F

commentSuppose that $\phi$ is an endomorphism of a finite-dimensional complex vector space.

(i) Show that if $\lambda$ is an eigenvalue of $\phi$, then $\lambda^{2}$ is an eigenvalue of $\phi^{2}$.

(ii) Show conversely that if $\mu$ is an eigenvalue of $\phi^{2}$, then there is an eigenvalue $\lambda$ of $\phi$ with $\lambda^{2}=\mu$.

Paper 2, Section II, F

comment(i) Show that two $n \times n$ complex matrices $A, B$ are similar (i.e. there exists invertible $P$ with $A=P^{-1} B P$ ) if and only if they represent the same linear map $\mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ with respect to different bases.

(ii) Explain the notion of Jordan normal form of a square complex matrix.

(iii) Show that any square complex matrix $A$ is similar to its transpose.

(iv) If $A$ is invertible, describe the Jordan normal form of $A^{-1}$ in terms of that of $A$.

Justify your answers.

Paper 3, Section II, F

commentSuppose that $V$ is a finite-dimensional vector space over $\mathbb{C}$, and that $\alpha: V \rightarrow V$ is a $\mathbb{C}$-linear map such that $\alpha^{n}=1$ for some $n>1$. Show that if $V_{1}$ is a subspace of $V$ such that $\alpha\left(V_{1}\right) \subset V_{1}$, then there is a subspace $V_{2}$ of $V$ such that $V=V_{1} \oplus V_{2}$ and $\alpha\left(V_{2}\right) \subset V_{2}$.

[Hint: Show, for example by picking bases, that there is a linear map $\pi: V \rightarrow V_{1}$ with $\pi(x)=x$ for all $x \in V_{1}$. Then consider $\rho: V \rightarrow V_{1}$ with $\left.\rho(y)=\frac{1}{n} \sum_{i=0}^{n-1} \alpha^{i} \pi \alpha^{-i}(y) .\right]$

Paper 4, Section I, F

commentDefine the notion of an inner product on a finite-dimensional real vector space $V$, and the notion of a self-adjoint linear map $\alpha: V \rightarrow V$.

Suppose that $V$ is the space of real polynomials of degree at most $n$ in a variable $t$. Show that

$\langle f, g\rangle=\int_{-1}^{1} f(t) g(t) d t$

is an inner product on $V$, and that the map $\alpha: V \rightarrow V$ :

$\alpha(f)(t)=\left(1-t^{2}\right) f^{\prime \prime}(t)-2 t f^{\prime}(t)$

is self-adjoint.

Paper 4, Section II, F

comment(i) Show that the group $O_{n}(\mathbb{R})$ of orthogonal $n \times n$ real matrices has a normal subgroup $S O_{n}(\mathbb{R})=\left\{A \in O_{n}(\mathbb{R}) \mid \operatorname{det} A=1\right\}$.

(ii) Show that $O_{n}(\mathbb{R})=S O_{n}(\mathbb{R}) \times\left\{\pm I_{n}\right\}$ if and only if $n$ is odd.

(iii) Show that if $n$ is even, then $O_{n}(\mathbb{R})$ is not the direct product of $S O_{n}(\mathbb{R})$ with any normal subgroup.

[You may assume that the only elements of $O_{n}(\mathbb{R})$ that commute with all elements of $O_{n}(\mathbb{R})$ are $\pm I_{n}$.]

Paper 1, Section II, E

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain.

(a) What does it mean to say that a state $i$ is positive recurrent? How is this property related to the equilibrium probability $\pi_{i}$ ? You do not need to give a full proof, but you should carefully state any theorems you use.

(b) What is a communicating class? Prove that if states $i$ and $j$ are in the same communicating class and $i$ is positive recurrent then $j$ is positive recurrent also.

A frog is in a pond with an infinite number of lily pads, numbered $1,2,3, \ldots$ She hops from pad to pad in the following manner: if she happens to be on pad $i$ at a given time, she hops to one of pads $(1,2, \ldots, i, i+1)$ with equal probability.

(c) Find the equilibrium distribution of the corresponding Markov chain.

(d) Now suppose the frog starts on pad $k$ and stops when she returns to it. Show that the expected number of times the frog hops is $e(k-1)$ ! where $e=2.718 \ldots$ What is the expected number of times she will visit the lily pad $k+1$ ?

Paper 2, Section II, E

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a simple, symmetric random walk on the integers $\{\ldots,-1,0,1, \ldots\}$, with $X_{0}=0$ and $\mathbb{P}\left(X_{n+1}=i \pm 1 \mid X_{n}=i\right)=1 / 2$. For each integer $a \geqslant 1$, let $T_{a}=\inf \left\{n \geqslant 0: X_{n}=a\right\}$. Show that $T_{a}$ is a stopping time.

Define a random variable $Y_{n}$ by the rule

$Y_{n}= \begin{cases}X_{n} & \text { if } n<T_{a} \\ 2 a-X_{n} & \text { if } n \geqslant T_{a}\end{cases}$

Show that $\left(Y_{n}\right)_{n} \geqslant 0$ is also a simple, symmetric random walk.

Let $M_{n}=\max _{0 \leqslant i \leqslant n} X_{n}$. Explain why $\left\{M_{n} \geqslant a\right\}=\left\{T_{a} \leqslant n\right\}$ for $a \geqslant 0$. By using the process $\left(Y_{n}\right)_{n \geqslant 0}$ constructed above, show that, for $a \geqslant 0$,

$\mathbb{P}\left(M_{n} \geqslant a, X_{n} \leqslant a-1\right)=\mathbb{P}\left(X_{n} \geqslant a+1\right),$

and thus

$\mathbb{P}\left(M_{n} \geqslant a\right)=\mathbb{P}\left(X_{n} \geqslant a\right)+\mathbb{P}\left(X_{n} \geqslant a+1\right)$

Hence compute

$\mathbb{P}\left(M_{n}=a\right)$

when $a$ and $n$ are positive integers with $n \geqslant a$. [Hint: if $n$ is even, then $X_{n}$ must be even, and if $n$ is odd, then $X_{n}$ must be odd.]

Paper 3, Section I, E

commentAn intrepid tourist tries to ascend Springfield's famous infinite staircase on an icy day. When he takes a step with his right foot, he reaches the next stair with probability $1 / 2$, otherwise he falls down and instantly slides back to the bottom with probability $1 / 2$. Similarly, when he steps with his left foot, he reaches the next stair with probability $1 / 3$, or slides to the bottom with probability $2 / 3$. Assume that he always steps first with his right foot when he is at the bottom, and alternates feet as he ascends. Let $X_{n}$ be his position after his $n$th step, so that $X_{n}=i$ when he is on the stair $i, i=0,1,2, \ldots$, where 0 is the bottom stair.

(a) Specify the transition probabilities $p_{i j}$ for the Markov chain $\left(X_{n}\right)_{n} \geqslant 0$ for any $i, j \geqslant 0$.

(b) Find the equilibrium probabilities $\pi_{i}$, for $i \geqslant 0$. [Hint: $\left.\pi_{0}=5 / 9 .\right]$

(c) Argue that the chain is irreducible and aperiodic and evaluate the limit

$\lim _{n \rightarrow \infty} \mathbb{P}\left(X_{n}=i\right)$

for each $i \geqslant 0$.

Paper 4, Section I, E

commentConsider a Markov chain $\left(X_{n}\right)_{n} \geqslant 0$ with state space $\{a, b, c, d\}$ and transition probabilities given by the following table.

\begin{tabular}{c|cccc} & $a$ & $b$ & $c$ & $d$ \ \hline$a$ & $1 / 4$ & $1 / 4$ & $1 / 2$ & 0 \ $b$ & 0 & $1 / 4$ & 0 & $3 / 4$ \ $c$ & $1 / 2$ & 0 & $1 / 4$ & $1 / 4$ \ $d$ & 0 & $1 / 2$ & 0 & $1 / 2$ \end{tabular}

By drawing an appropriate diagram, determine the communicating classes of the chain, and classify them as either open or closed. Compute the following transition and hitting probabilities:

$\mathbb{P}\left(X_{n}=b \mid X_{0}=d\right)$ for a fixed $n \geqslant 0$

$\mathbb{P}\left(X_{n}=c\right.$ for some $\left.n \geqslant 1 \mid X_{0}=a\right)$.

Paper 1, Section II, A

comment(a) A function $f(t)$ is periodic with period $2 \pi$ and has continuous derivatives up to and including the $k$ th derivative. Show by integrating by parts that the Fourier coefficients of $f(t)$

$\begin{aligned} &a_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} f(t) \cos n t d t \\ &b_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} f(t) \sin n t d t \end{aligned}$

decay at least as fast as $1 / n^{k}$ as $n \rightarrow \infty$

(b) Calculate the Fourier series of $f(t)=|\sin t|$ on $[0,2 \pi]$.

(c) Comment on the decay rate of your Fourier series.

Paper 2, Section I, A

commentConsider the initial value problem

$\mathcal{L} x(t)=f(t), \quad x(0)=0, \quad \dot{x}(0)=0, \quad t \geqslant 0,$

where $\mathcal{L}$ is a second-order linear operator involving differentiation with respect to $t$. Explain briefly how to solve this by using a Green's function.

Now consider

$\ddot{x}(t)= \begin{cases}a & 0 \leqslant t \leqslant T \\ 0 & T<t<\infty\end{cases}$

where $a$ is a constant, subject to the same initial conditions. Solve this using the Green's function, and explain how your answer is related to a problem in Newtonian dynamics.

Paper 2, Section II, B

commentExplain briefly the use of the method of characteristics to solve linear first-order partial differential equations.

Use the method to solve the problem

$(x-y) \frac{\partial u}{\partial x}+(x+y) \frac{\partial u}{\partial y}=\alpha u$

where $\alpha$ is a constant, with initial condition $u(x, 0)=x^{2}, x \geqslant 0$.

By considering your solution explain:

(i) why initial conditions cannot be specified on the whole $x$-axis;

(ii) why a single-valued solution in the entire plane is not possible if $\alpha \neq 2$.

Paper 3, Section I, B

commentShow that Laplace's equation $\nabla^{2} \phi=0$ in polar coordinates $(r, \theta)$ has solutions proportional to $r^{\pm \alpha} \sin \alpha \theta, r^{\pm \alpha} \cos \alpha \theta$ for any constant $\alpha$.

Find the function $\phi$ satisfying Laplace's equation in the region $a<r<b, 0<\theta<\pi$, where $\phi(a, \theta)=\sin ^{3} \theta, \phi(b, \theta)=\phi(r, 0)=\phi(r, \pi)=0$.

[The Laplacian $\nabla^{2}$ in polar coordinates is

$\left.\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right]$

Paper 3, Section II, A

comment(a) Put the equation

$x \frac{d^{2} u}{d x^{2}}+\frac{d u}{d x}+\lambda x u=0, \quad 0 \leqslant x \leqslant 1$

into Sturm-Liouville form.

(b) Suppose $u_{n}(x)$ are eigenfunctions such that $u_{n}(x)$ are bounded as $x$ tends to zero and

$x \frac{d^{2} u_{n}}{d x^{2}}+\frac{d u_{n}}{d x}+\lambda_{n} x u_{n}=0, \quad 0 \leqslant x \leqslant 1$

Identify the weight function $w(x)$ and the most general boundary conditions on $u_{n}(x)$ which give the orthogonality relation

$\left(\lambda_{m}-\lambda_{n}\right) \int_{0}^{1} u_{m}(x) w(x) u_{n}(x) d x=0$

(c) The equation

$x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+x y=0, \quad x>0$

has a solution $J_{0}(x)$ and a second solution which is not bounded at the origin. The zeros of $J_{0}(x)$ arranged in ascending order are $j_{n}, n=1,2, \ldots$. Given that $u_{n}(1)=0$, show that the eigenvalues of the Sturm-Liouville problem in (b) are $\lambda=j_{n}^{2}, n=1,2, \ldots$

(d) Using the differential equations for $J_{0}(\alpha x)$ and $J_{0}(\beta x)$ and integration by parts, show that

$\int_{0}^{1} J_{0}(\alpha x) J_{0}(\beta x) x d x=\frac{\beta J_{0}(\alpha) J_{0}^{\prime}(\beta)-\alpha J_{0}(\beta) J_{0}^{\prime}(\alpha)}{\alpha^{2}-\beta^{2}} \quad(\alpha \neq \beta)$

Paper 4, Section I, A

comment(a) By considering strictly monotonic differentiable functions $\varphi(x)$, such that the zeros satisfy $\varphi(c)=0$ but $\varphi^{\prime}(c) \neq 0$, establish the formula

$\int_{-\infty}^{\infty} f(x) \delta(\varphi(x)) d x=\frac{f(c)}{\left|\varphi^{\prime}(c)\right|}$

Hence show that for a general differentiable function with only such zeros, labelled by $c$,

$\int_{-\infty}^{\infty} f(x) \delta(\varphi(x)) d x=\sum_{c} \frac{f(c)}{\left|\varphi^{\prime}(c)\right|}$

(b) Hence by changing to plane polar coordinates, or otherwise, evaluate,

$I=\int_{0}^{\infty} \int_{0}^{\infty}\left(x^{3}+y^{2} x\right) \delta\left(x^{2}+y^{2}-1\right) d y d x$

Paper 4, Section II, B

commentDefining the function $G_{f_{3}}\left(\mathbf{r} ; \mathbf{r}_{0}\right)=-1 /\left(4 \pi\left|\mathbf{r}-\mathbf{r}_{0}\right|\right)$, prove Green's third identity for functions $u(\mathbf{r})$ satisfying Laplace's equation in a volume $V$ with surface $S$, namely

$u\left(\mathbf{r}_{0}\right)=\int_{S}\left(u \frac{\partial G_{f_{3}}}{\partial n}-\frac{\partial u}{\partial n} G_{f_{3}}\right) d S$

A solution is sought to the Neumann problem for $\nabla^{2} u=0$ in the half plane $z>0$ :

$u=O\left(|\mathbf{x}|^{-a}\right), \quad \frac{\partial u}{\partial r}=O\left(|\mathbf{x}|^{-a-1}\right) \text { as }|\mathbf{x}| \rightarrow \infty, \quad \frac{\partial u}{\partial z}=p(x, y) \text { on } z=0$

where $a>0$. It is assumed that $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x, y) d x d y=0$. Explain why this condition is necessary.

Construct an appropriate Green's function $G\left(\mathbf{r} ; \mathbf{r}_{0}\right)$ satisfying $\partial G / \partial z=0$ at $z=0$, using the method of images or otherwise. Hence find the solution in the form

$u\left(x_{0}, y_{0}, z_{0}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x, y) f\left(x-x_{0}, y-y_{0}, z_{0}\right) d x d y$

where $f$ is to be determined.

Now let

$p(x, y)= \begin{cases}x & |x|,|y|<a \\ 0 & \text { otherwise }\end{cases}$

By expanding $f$ in inverse powers of $z_{0}$, show that

$u \rightarrow \frac{-2 a^{4} x_{0}}{3 \pi z_{0}^{3}} \quad \text { as } \quad z_{0} \rightarrow \infty .$

Paper 1, Section II, H

commentLet $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be continuous maps of topological spaces with $f \circ g=\mathrm{id}_{Y}$.

(1) Suppose that (i) $Y$ is path-connected, and (ii) for every $y \in Y$, its inverse image $f^{-1}(y)$ is path-connected. Prove that $X$ is path-connected.

(2) Prove the same statement when "path-connected" is everywhere replaced by "connected".

Paper 2, Section I, H

commentOn the set $\mathbb{Q}$ of rational numbers, the 3 -adic metric $d_{3}$ is defined as follows: for $x, y \in \mathbb{Q}$, define $d_{3}(x, x)=0$ and $d_{3}(x, y)=3^{-n}$, where $n$ is the integer satisfying $x-y=3^{n} u$ where $u$ is a rational number whose denominator and numerator are both prime to 3 .

(1) Show that this is indeed a metric on $\mathbb{Q}$.

(2) Show that in $\left(\mathbb{Q}, d_{3}\right)$, we have $3^{n} \rightarrow 0$ as $n \rightarrow \infty$ while $3^{-n} \nrightarrow \infty$ as $n \rightarrow \infty$. Let $d$ be the usual metric $d(x, y)=|x-y|$ on $\mathbb{Q}$. Show that neither the identity map $(\mathbb{Q}, d) \rightarrow\left(\mathbb{Q}, d_{3}\right)$ nor its inverse is continuous.

Paper 3, Section I, H

commentLet $X$ be a topological space and $Y$ be a set. Let $p: X \rightarrow Y$ be a surjection. The quotient topology on $Y$ is defined as follows: a subset $V \subset Y$ is open if and only if $p^{-1}(V)$ is open in $X$.

(1) Show that this does indeed define a topology on $Y$, and show that $p$ is continuous when we endow $Y$ with this topology.

(2) Let $Z$ be another topological space and $f: Y \rightarrow Z$ be a map. Show that $f$ is continuous if and only if $f \circ p: X \rightarrow Z$ is continuous.

Paper 4, Section II, H

comment(1) Prove that if $X$ is a compact topological space, then a closed subset $Y$ of $X$ endowed with the subspace topology is compact.

(2) Consider the following equivalence relation on $\mathbb{R}^{2}$ :

$\left(x_{1}, y_{1}\right) \sim\left(x_{2}, y_{2}\right) \Longleftrightarrow\left(x_{1}-x_{2}, y_{1}-y_{2}\right) \in \mathbb{Z}^{2}$

Let $X=\mathbb{R}^{2} / \sim$ be the quotient space endowed with the quotient topology, and let $p: \mathbb{R}^{2} \rightarrow X$ be the canonical surjection mapping each element to its equivalence class. Let $Z=\left\{(x, y) \in \mathbb{R}^{2} \mid y=\sqrt{2} x\right\} .$

(i) Show that $X$ is compact.

(ii) Assuming that $p(Z)$ is dense in $X$, show that $\left.p\right|_{Z}: Z \rightarrow p(Z)$ is bijective but not homeomorphic.

Paper 1, Section I, C

commentObtain the Cholesky decompositions of

$H_{3}=\left(\begin{array}{ccc} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \end{array}\right), \quad H_{4}=\left(\begin{array}{cccc} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \lambda \end{array}\right) .$

What is the minimum value of $\lambda$ for $H_{4}$ to be positive definite? Verify that if $\lambda=\frac{1}{7}$ then $H_{4}$ is positive definite.

Paper 1, Section II, 18C

Let

$\langle f, g\rangle=\int_{-\infty}^{\infty} e^{-x^{2}} f(x) g(x) d x$

be an inner product. The Hermite polynomials $H_{n}(x), n=0,1,2, \ldots$