# Part IB, 2012

### Jump to course

Paper 1, Section II, E

commentState the inverse function theorem for a function $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$. Suppose $F$ is a differentiable bijection with $F^{-1}$ also differentiable. Show that the derivative of $F$ at any point in $\mathbb{R}^{n}$ is a linear isomorphism.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function such that the partial derivatives $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$ exist and are continuous. Assume there is a point $(a, b) \in \mathbb{R}^{2}$ for which $f(a, b)=0$ and $\frac{\partial f}{\partial x}(a, b) \neq 0$. Prove that there exist open sets $U \subset \mathbb{R}^{2}$ and $W \subset \mathbb{R}$ containing $(a, b)$ and $b$, respectively, such that for every $y \in W$ there exists a unique $x$ such that $(x, y) \in U$ and $f(x, y)=0$. Moreover, if we define $g: W \rightarrow \mathbb{R}$ by $g(y)=x$, prove that $g$ is differentiable with continuous derivative. Find the derivative of $g$ at $b$ in terms of $\frac{\partial f}{\partial x}(a, b)$ and $\frac{\partial f}{\partial y}(a, b)$.

Paper 2, Section I, E

commentLet $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function. What does it mean to say that $f$ is differentiable at a point $(x, y) \in \mathbb{R}^{2} ?$ Prove directly from this definition, that if $f$ is differentiable at $(x, y)$, then $f$ is continuous at $(x, y)$.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be the function:

$f(x, y)= \begin{cases}x^{2}+y^{2} & \text { if } x \text { and } y \text { are rational } \\ 0 & \text { otherwise. }\end{cases}$

For which points $(x, y) \in \mathbb{R}^{2}$ is $f$ differentiable? Justify your answer.

Paper 2, Section II, E

commentLet $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a mapping. Fix $a \in \mathbb{R}^{n}$ and prove that the following two statements are equivalent:

(i) Given $\varepsilon>0$ there is $\delta>0$ such that $\|f(x)-f(a)\|<\varepsilon$ whenever $\|x-a\|<\delta$ (we use the standard norm in Euclidean space).

(ii) $f\left(x_{n}\right) \rightarrow f(a)$ for any sequence $x_{n} \rightarrow a$.

We say that $f$ is continuous if (i) (or equivalently (ii)) holds for every $a \in \mathbb{R}^{n}$.

Let $E$ and $F$ be subsets of $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ respectively. For $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ as above, determine which of the following statements are always true and which may be false, giving a proof or a counterexample as appropriate.

(a) If $f^{-1}(F)$ is closed whenever $F$ is closed, then $f$ is continuous.

(b) If $f$ is continuous, then $f^{-1}(F)$ is closed whenever $F$ is closed.

(c) If $f$ is continuous, then $f(E)$ is open whenever $E$ is open.

(d) If $f$ is continuous, then $f(E)$ is bounded whenever $E$ is bounded.

(e) If $f$ is continuous and $f^{-1}(F)$ is bounded whenever $F$ is bounded, then $f(E)$ is closed whenever $E$ is closed.

Paper 3, Section I, $2 \mathbf{E}$

commentLet $C[0,1]$ be the set of continuous real-valued functions on $[0,1]$ with the uniform norm. Suppose $T: C[0,1] \rightarrow C[0,1]$ is defined by

$T(f)(x)=\int_{0}^{x} f\left(t^{3}\right) d t$

for all $x \in[0,1]$ and $f \in C[0,1]$. Is $T$ a contraction mapping? Does $T$ have a unique fixed point? Justify your answers.

Paper 3, Section II, E

commentLet $f_{n}$ be a sequence of continuous functions on the interval $[0,1]$ such that $f_{n}(x) \rightarrow f(x)$ for each $x$. For the three statements:

(a) $f_{n} \rightarrow f$ uniformly on $[0,1]$;

(b) $f$ is a continuous function;

(c) $\int_{0}^{1} f_{n}(x) d x \rightarrow \int_{0}^{1} f(x) d x$ as $n \rightarrow \infty ;$

say which of the six possible implications $(a) \Rightarrow(b),(a) \Rightarrow(c),(b) \Rightarrow(a),(b) \Rightarrow(c)$, $(c) \Rightarrow(a),(c) \Rightarrow(b)$ are true and which false, giving in each case a proof or counterexample.

Paper 4, Section I, E

commentLet $f: \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}$ be a bilinear function. Show that $f$ is differentiable at any point in $\mathbb{R}^{n} \times \mathbb{R}^{m}$ and find its derivative.

Paper 4, Section II, E

commentState and prove the Bolzano-Weierstrass theorem in $\mathbb{R}^{n}$. [You may assume the Bolzano-Weierstrass theorem in $\mathbb{R}$.]

Let $X \subset \mathbb{R}^{n}$ be a subset and let $f: X \rightarrow X$ be a mapping such that $d(f(x), f(y))=d(x, y)$ for all $x, y \in X$, where $d$ is the Euclidean distance in $\mathbb{R}^{n}$. Prove that if $X$ is closed and bounded, then $f$ is a bijection. Is this result still true if we drop the boundedness assumption on $X$ ? Justify your answer.

Paper 3, Section II, E

commentLet $D(a, R)$ denote the disc $|z-a|<R$ and let $f: D(a, R) \rightarrow \mathbb{C}$ be a holomorphic function. Using Cauchy's integral formula show that for every $r \in(0, R)$

$f(a)=\int_{0}^{1} f\left(a+r e^{2 \pi i t}\right) d t$

Deduce that if for every $z \in D(a, R),|f(z)| \leqslant|f(a)|$, then $f$ is constant.

Let $f: D(0,1) \rightarrow D(0,1)$ be holomorphic with $f(0)=0$. Show that $|f(z)| \leqslant|z|$ for all $z \in D(0,1)$. Moreover, show that if $|f(w)|=|w|$ for some $w \neq 0$, then there exists $\lambda$ with $|\lambda|=1$ such that $f(z)=\lambda z$ for all $z \in D(0,1)$.

Paper 4, Section I, $4 \mathrm{E}$

commentLet $h: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function with $h(i) \neq h(-i)$. Does there exist a holomorphic function $f$ defined in $|z|<1$ for which $f^{\prime}(z)=\frac{h(z)}{1+z^{2}}$ ? Does there exist a holomorphic function $f$ defined in $|z|>1$ for which $f^{\prime}(z)=\frac{h(z)}{1+z^{2}}$ ? Justify your answers.

Paper 1, Section I, A

commentFind a conformal transformation $\zeta=\zeta(z)$ that maps the domain $D, 0<\arg z<\frac{3 \pi}{2}$, on to the strip $0<\operatorname{Im}(\zeta)<1$.

Hence find a bounded harmonic function $\phi$ on $D$ subject to the boundary conditions $\phi=0, A$ on $\arg z=0, \frac{3 \pi}{2}$, respectively, where $A$ is a real constant.

Paper 1, Section II, 13A

commentUsing Cauchy's integral theorem, write down the value of a holomorphic function $f(z)$ where $|z|<1$ in terms of a contour integral around the unit circle, $\zeta=e^{i \theta} .$

By considering the point $1 / \bar{z}$, or otherwise, show that

$f(z)=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(\zeta) \frac{1-|z|^{2}}{|\zeta-z|^{2}} \mathrm{~d} \theta$

By setting $z=r e^{i \alpha}$, show that for any harmonic function $u(r, \alpha)$,

$u(r, \alpha)=\frac{1}{2 \pi} \int_{0}^{2 \pi} u(1, \theta) \frac{1-r^{2}}{1-2 r \cos (\alpha-\theta)+r^{2}} \mathrm{~d} \theta$

if $r<1$.

Assuming that the function $v(r, \alpha)$, which is the conjugate harmonic function to $u(r, \alpha)$, can be written as

$v(r, \alpha)=v(0)+\frac{1}{\pi} \int_{0}^{2 \pi} u(1, \theta) \frac{r \sin (\alpha-\theta)}{1-2 r \cos (\alpha-\theta)+r^{2}} \mathrm{~d} \theta$

deduce that

$f(z)=i v(0)+\frac{1}{2 \pi} \int_{0}^{2 \pi} u(1, \theta) \frac{\zeta+z}{\zeta-z} \mathrm{~d} \theta$

[You may use the fact that on the unit circle, $\zeta=1 / \bar{\zeta}$, and hence

$\left.\frac{\zeta}{\zeta-1 / \bar{z}}=-\frac{\bar{z}}{\bar{\zeta}-\bar{z}} \cdot\right]$

Paper 2, Section II, A

commentBy a suitable choice of contour show that, for $-1<\alpha<1$,

$\int_{0}^{\infty} \frac{x^{\alpha}}{1+x^{2}} \mathrm{~d} x=\frac{\pi}{2 \cos (\alpha \pi / 2)}$

Paper 3, Section I, A

commentState the formula for the Laplace transform of a function $f(t)$, defined for $t \geqslant 0$.

Let $f(t)$ be periodic with period $T$ (i.e. $f(t+T)=f(t)$ ). If $g(t)$ is defined to be equal to $f(t)$ in $[0, T]$ and zero elsewhere and its Laplace transform is $G(s)$, show that the Laplace transform of $f(t)$ is given by

$F(s)=\frac{G(s)}{1-e^{-s T}}$

Hence, or otherwise, find the inverse Laplace transform of

$F(s)=\frac{1}{s} \frac{1-e^{-s T / 2}}{1-e^{-s T}}$

Paper 4, Section II, A

commentState the convolution theorem for Fourier transforms.

The function $\phi(x, y)$ satisfies

$\nabla^{2} \phi=0$

on the half-plane $y \geqslant 0$, subject to the boundary conditions

$\begin{gathered} \phi \rightarrow 0 \text { as } y \rightarrow \infty \text { for all } x \\ \phi(x, 0)= \begin{cases}1, & |x| \leqslant 1 \\ 0, & |x|>1\end{cases} \end{gathered}$

Using Fourier transforms, show that

$\phi(x, y)=\frac{y}{\pi} \int_{-1}^{1} \frac{1}{y^{2}+(x-t)^{2}} \mathrm{~d} t$

and hence that

$\phi(x, y)=\frac{1}{\pi}\left[\tan ^{-1}\left(\frac{1-x}{y}\right)+\tan ^{-1}\left(\frac{1+x}{y}\right)\right]$

Paper 1, Section II, B

commentA sphere of radius a carries an electric charge $Q$ uniformly distributed over its surface. Calculate the electric field outside and inside the sphere. Also calculate the electrostatic potential outside and inside the sphere, assuming it vanishes at infinity. State the integral formula for the energy $U$ of the electric field and use it to evaluate $U$ as a function of $Q$

Relate $\frac{d U}{d Q}$ to the potential on the surface of the sphere and explain briefly the physical interpretation of the relation.

Paper 2, Section I, B

commentWrite down the expressions for a general, time-dependent electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ in terms of a vector potential $\mathbf{A}$ and scalar potential $\phi$. What is meant by a gauge transformation of $\mathbf{A}$ and $\phi$ ? Show that $\mathbf{E}$ and $\mathbf{B}$ are unchanged under a gauge transformation.

A plane electromagnetic wave has vector and scalar potentials

$\begin{aligned} \mathbf{A}(\mathbf{x}, t) &=\mathbf{A}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)} \\ \phi(\mathbf{x}, t) &=\phi_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)} \end{aligned}$

where $\mathbf{A}_{0}$ and $\phi_{0}$ are constants. Show that $\left(\mathbf{A}_{0}, \phi_{0}\right)$ can be modified to $\left(\mathbf{A}_{0}+\mu \mathbf{k}, \phi_{0}+\mu \omega\right)$ by a gauge transformation. What choice of $\mu$ leads to the modified $\mathbf{A}(\mathbf{x}, t)$ satisfying the Coulomb gauge condition $\boldsymbol{\nabla} \cdot \mathbf{A}=0$ ?

Paper 2, Section II, B

commentA straight wire has $n$ mobile, charged particles per unit length, each of charge $q$. Assuming the charges all move with velocity $v$ along the wire, show that the current is $I=n q v$.

Using the Lorentz force law, show that if such a current-carrying wire is placed in a uniform magnetic field of strength $B$ perpendicular to the wire, then the force on the wire, per unit length, is $B I$.

Consider two infinite parallel wires, with separation $L$, carrying (in the same sense of direction) positive currents $I_{1}$ and $I_{2}$, respectively. Find the force per unit length on each wire, determining both its magnitude and direction.

Paper 3, Section II, B

commentUsing the Maxwell equations

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

show that in vacuum, E satisfies the wave equation

$\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}=0$

where $c^{2}=\left(\epsilon_{0} \mu_{0}\right)^{-1}$, as well as $\nabla \cdot \mathbf{E}=0$. Also show that at a planar boundary between two media, $\mathbf{E}_{t}$ (the tangential component of $\mathbf{E}$ ) is continuous. Deduce that if one medium is of negligible resistance, $\mathbf{E}_{t}=0$.

Consider an empty cubic box with walls of negligible resistance on the planes $x=0$, $x=a, y=0, y=a, z=0, z=a$, where $a>0$. Show that an electric field in the interior of the form

$\begin{aligned} &E_{x}=f(x) \sin \left(\frac{m \pi y}{a}\right) \sin \left(\frac{n \pi z}{a}\right) e^{-i \omega t} \\ &E_{y}=g(y) \sin \left(\frac{l \pi x}{a}\right) \sin \left(\frac{n \pi z}{a}\right) e^{-i \omega t} \\ &E_{z}=h(z) \sin \left(\frac{l \pi x}{a}\right) \sin \left(\frac{m \pi y}{a}\right) e^{-i \omega t} \end{aligned}$

with $l, m$ and $n$ positive integers, satisfies the boundary conditions on all six walls. Now suppose that

$f(x)=f_{0} \cos \left(\frac{l \pi x}{a}\right), \quad g(y)=g_{0} \cos \left(\frac{m \pi y}{a}\right), \quad h(z)=h_{0} \cos \left(\frac{n \pi z}{a}\right)$

where $f_{0}, g_{0}$ and $h_{0}$ are constants. Show that the wave equation $(*)$ is satisfied, and determine the frequency $\omega$. Find the further constraint on $f_{0}, g_{0}$ and $h_{0}$ ?

Paper 4, Section I, B

commentDefine the notions of magnetic flux, electromotive force and resistance, in the context of a single closed loop of wire. Use the Maxwell equation

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

to derive Faraday's law of induction for the loop, assuming the loop is at rest.

Suppose now that the magnetic field is $\mathbf{B}=(0,0, B \tanh t)$ where $B$ is a constant, and that the loop of wire, with resistance $R$, is a circle of radius a lying in the $(x, y)$ plane. Calculate the current in the wire as a function of time.

Explain briefly why, even in a time-independent magnetic field, an electromotive force may be produced in a loop of wire that moves through the field, and state the law of induction in this situation.

Paper 1, Section I, A

commentViscous fluid, with viscosity $\mu$ and density $\rho$ flows along a straight circular pipe of radius $R$. The average velocity of the flow is $U$. Define a Reynolds number for the flow.

The flow is driven by a constant pressure gradient $-G>0$ along the pipe and the velocity is parallel to the axis of the pipe with magnitude $u(r)$ that satisfies

$\frac{\mu}{r} \frac{\mathrm{d}}{\mathrm{d} r}\left(r \frac{\mathrm{d} u}{\mathrm{~d} r}\right)=-G,$

where $r$ is the radial distance from the axis.

State the boundary conditions on $u$ and find the velocity as a function of $r$ assuming that it is finite on the axis $r=0$. Hence, show that the shear stress $\tau$ at the pipe wall is independent of the viscosity. Why is this the case?

Paper 1, Section II, A

commentConsider inviscid, incompressible fluid flow confined to the $(x, y)$ plane. The fluid has density $\rho$, and gravity can be neglected. Using the conservation of volume flux, determine the velocity potential $\phi(r)$ of a point source of strength $m$, in terms of the distance $r$ from the source.

Two point sources each of strength $m$ are located at $\boldsymbol{x}_{+}=(0, a)$ and $\boldsymbol{x}_{-}=(0,-a)$. Find the velocity potential of the flow.

Show that the flow in the region $y \geqslant 0$ is equivalent to the flow due to a source at $\boldsymbol{x}_{+}$and a fixed boundary at $y=0 .$

Find the pressure on the boundary $y=0$ and hence determine the force on the boundary.

[Hint: you may find the substitution $x=a \tan \theta$ useful for the calculation of the pressure.]

Paper 2, Section I, A

commentStarting from Euler's equation for the motion of an inviscid fluid, derive the vorticity equation in the form

$\frac{D \boldsymbol{\omega}}{D t}=\boldsymbol{\omega} \cdot \nabla \boldsymbol{u}$

Deduce that an initially irrotational flow remains irrotational.

Consider a plane flow that at time $t=0$ is described by the streamfunction

$\psi=x^{2}+y^{2} .$

Calculate the vorticity everywhere at times $t>0$.

Paper 3, Section II, A

commentA rigid circular cylinder of radius $a$ executes small amplitude oscillations with velocity $U(t)$ in a direction perpendicular to its axis, while immersed in an inviscid fluid of density $\rho$ contained within a larger concentric fixed cylinder of radius $b$. Gravity is negligible. Neglecting terms quadratic in the amplitude, determine the boundary condition on the velocity on the inner cylinder, and calculate the velocity potential of the induced flow.

With the same approximations show that the difference in pressures on the surfaces of the two cylinders has magnitude

$\rho \frac{\mathrm{d} U}{\mathrm{~d} t} \frac{a(b-a)}{b+a} \cos \theta$

where $\theta$ is the azimuthal angle measured from the direction of $U$.

Paper 4, Section II, A

commentThe equations governing the flow of a shallow layer of inviscid liquid of uniform depth $H$ rotating with angular velocity $\frac{1}{2} f$ about the vertical $z$-axis are

$\begin{aligned} \frac{\partial u}{\partial t}-f v &=-g \frac{\partial \eta}{\partial x} \\ \frac{\partial v}{\partial t}+f u &=-g \frac{\partial \eta}{\partial y} \\ \frac{\partial \eta}{t}+H\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) &=0 \end{aligned}$

where $u, v$ are the $x$ - and $y$-components of velocity, respectively, and $\eta$ is the elevation of the free surface. Show that these equations imply that

$\frac{\partial q}{\partial t}=0, \quad \text { where } \quad q=\omega-\frac{f \eta}{H} \text { and } \omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$

Consider an initial state where there is flow in the $y$-direction given by

$\begin{aligned} u &=\eta=0, \quad-\infty<x<\infty \\ v &= \begin{cases}\frac{g}{2 f} e^{2 x}, & x<0 \\ -\frac{g}{2 f} e^{-2 x}, & x>0\end{cases} \end{aligned}$

Find the initial potential vorticity.

Show that when this initial state adjusts, there is a final steady state independent of $y$ in which $\eta$ satisfies

$\frac{\partial^{2} \eta}{\partial x^{2}}-\frac{\eta}{a^{2}}= \begin{cases}e^{2 x}, & x<0 \\ e^{-2 x}, & x>0\end{cases}$

where $a^{2}=g H / f^{2}$.

In the case $a=1$, find the final free surface elevation that is finite at large $|x|$ and which is continuous and has continuous slope at $x=0$, and show that it is negative for all $x$.

Paper 1, Section I, G

commentDescribe a collection of charts which cover a circular cylinder of radius $R$. Compute the first fundamental form, and deduce that the cylinder is locally isometric to the plane.

Paper 2, Section II, G

commentLet $S$ be a closed surface, equipped with a triangulation. Define the Euler characteristic $\chi(S)$ of $S$. How does $\chi(S)$ depend on the triangulation?

Let $V, E$ and $F$ denote the number of vertices, edges and faces of the triangulation. Show that $2 E=3 F$.

Suppose now the triangulation is tidy, meaning that it has the property that no two vertices are joined by more than one edge. Deduce that $V$ satisfies

$V \geqslant \frac{7+\sqrt{49-24 \chi(S)}}{2} .$

Hence compute the minimal number of vertices of a tidy triangulation of the real projective plane. [Hint: it may be helpful to consider the icosahedron as a triangulation of the sphere $\left.S^{2} .\right]$

Paper 3, Section I, $5 G$

commentState a formula for the area of a hyperbolic triangle.

Hence, or otherwise, prove that if $l_{1}$ and $l_{2}$ are disjoint geodesics in the hyperbolic plane, there is at most one geodesic which is perpendicular to both $l_{1}$ and $l_{2}$.

Paper 3, Section II, G

commentDefine the first and second fundamental forms of a smooth surface $\Sigma \subset \mathbb{R}^{3}$, and explain their geometrical significance.

Write down the geodesic equations for a smooth curve $\gamma:[0,1] \rightarrow \Sigma$. Prove that $\gamma$ is a geodesic if and only if the derivative of the tangent vector to $\gamma$ is always orthogonal to $\Sigma$.

A plane $\Pi \subset \mathbb{R}^{3}$ cuts $\Sigma$ in a smooth curve $C \subset \Sigma$, in such a way that reflection in the plane $\Pi$ is an isometry of $\Sigma$ (in particular, preserves $\Sigma$ ). Prove that $C$ is a geodesic.

Paper 4, Section II, G

commentLet $\Sigma \subset \mathbb{R}^{3}$ be a smooth closed surface. Define the principal curvatures $\kappa_{\max }$ and $\kappa_{\min }$ at a point $p \in \Sigma$. Prove that the Gauss curvature at $p$ is the product of the two principal curvatures.

A point $p \in \Sigma$ is called a parabolic point if at least one of the two principal curvatures vanishes. Suppose $\Pi \subset \mathbb{R}^{3}$ is a plane and $\Sigma$ is tangent to $\Pi$ along a smooth closed curve $C=\Pi \cap \Sigma \subset \Sigma$. Show that $C$ is composed of parabolic points.

Can both principal curvatures vanish at a point of $C$ ? Briefly justify your answer.

Paper 1, Section II, G

commentLet $G$ be a finite group. What is a Sylow $p$-subgroup of $G$ ?

Assuming that a Sylow $p$-subgroup $H$ exists, and that the number of conjugates of $H$ is congruent to $1 \bmod p$, prove that all Sylow $p$-subgroups are conjugate. If $n_{p}$ denotes the number of Sylow $p$-subgroups, deduce that

$n_{p} \equiv 1 \quad \bmod p \quad \text { and } \quad n_{p}|| G \mid$

If furthermore $G$ is simple prove that either $G=H$ or

$|G| \mid n_{p} \text { ! }$

Deduce that a group of order $1,000,000$ cannot be simple.

Paper 2, Section I, $2 G$

commentWhat does it mean to say that the finite group $G$ acts on the set $\Omega$ ?

By considering an action of the symmetry group of a regular tetrahedron on a set of pairs of edges, show there is a surjective homomorphism $S_{4} \rightarrow S_{3}$.

[You may assume that the symmetric group $S_{n}$ is generated by transpositions.]

Paper 2, Section II, G

commentState Gauss's Lemma. State Eisenstein's irreducibility criterion.

(i) By considering a suitable substitution, show that the polynomial $1+X^{3}+X^{6}$ is irreducible over $\mathbb{Q}$.

(ii) By working in $\mathbb{Z}_{2}[X]$, show that the polynomial $1-X^{2}+X^{5}$ is irreducible over $\mathbb{Q}$.

Paper 3, Section I, $1 \mathbf{G}$

commentWhat is a Euclidean domain?

Giving careful statements of any general results you use, show that in the ring $\mathbb{Z}[\sqrt{-3}], 2$ is irreducible but not prime.

Paper 3, Section II, G

commentFor each of the following assertions, provide either a proof or a counterexample as appropriate:

(i) The ring $\mathbb{Z}_{2}[X] /\left\langle X^{2}+X+1\right\rangle$ is a field.

(ii) The ring $\mathbb{Z}_{3}[X] /\left\langle X^{2}+X+1\right\rangle$ is a field.

(iii) If $F$ is a finite field, the ring $F[X]$ contains irreducible polynomials of arbitrarily large degree.

(iv) If $R$ is the ring $C[0,1]$ of continuous real-valued functions on the interval $[0,1]$, and the non-zero elements $f, g \in R$ satisfy $f \mid g$ and $g \mid f$, then there is some unit $u \in R$ with $f=u \cdot g$.

Paper 4, Section I, $2 G$

commentAn idempotent element of a ring $R$ is an element $e$ satisfying $e^{2}=e$. A nilpotent element is an element e satisfying $e^{N}=0$ for some $N \geqslant 0$.

Let $r \in R$ be non-zero. In the ring $R[X]$, can the polynomial $1+r X$ be (i) an idempotent, (ii) a nilpotent? Can $1+r X$ satisfy the equation $(1+r X)^{3}=(1+r X)$ ? Justify your answers.

Paper 4, Section II, G

commentLet $R$ be a commutative ring with unit 1. Prove that an $R$-module is finitely generated if and only if it is a quotient of a free module $R^{n}$, for some $n>0$.

Let $M$ be a finitely generated $R$-module. Suppose now $I$ is an ideal of $R$, and $\phi$ is an $R$-homomorphism from $M$ to $M$ with the property that

$\phi(M) \subset I \cdot M=\left\{m \in M \mid m=r m^{\prime} \quad \text { with } \quad r \in I, m^{\prime} \in M\right\}$

Prove that $\phi$ satisfies an equation

$\phi^{n}+a_{n-1} \phi^{n-1}+\cdots+a_{1} \phi+a_{0}=0$

where each $a_{j} \in I$. [You may assume that if $T$ is a matrix over $R$, then $\operatorname{adj}(T) T=$ $\operatorname{det} T$ (id), with id the identity matrix.]

Deduce that if $M$ satisfies $I \cdot M=M$, then there is some $a \in R$ satisfying

$a-1 \in I \quad \text { and } \quad a M=0 .$

Give an example of a finitely generated $\mathbb{Z}$-module $M$ and a proper ideal $I$ of $\mathbb{Z}$ satisfying the hypothesis $I \cdot M=M$, and for your example, give an explicit such element $a$.

Paper 1, Section I, F

commentDefine the notions of basis and dimension of a vector space. Prove that two finitedimensional real vector spaces with the same dimension are isomorphic.

In each case below, determine whether the set $S$ is a basis of the real vector space $V:$

(i) $V=\mathbb{C}$ is the complex numbers; $S=\{1, i\}$.

(ii) $V=\mathbb{R}[x]$ is the vector space of all polynomials in $x$ with real coefficients; $S=\{1,(x-1),(x-1)(x-2),(x-1)(x-2)(x-3), \ldots\} .$

(iii) $V=\{f:[0,1] \rightarrow \mathbb{R}\} ; S=\left\{\chi_{p} \mid p \in[0,1]\right\}$, where

$\chi_{p}(x)= \begin{cases}1 & x=p \\ 0 & x \neq p\end{cases}$

Paper 1, Section II, F

commentDefine what it means for two $n \times n$ matrices to be similar to each other. Show that if two $n \times n$ matrices are similar, then the linear transformations they define have isomorphic kernels and images.

If $A$ and $B$ are $n \times n$ real matrices, we define $[A, B]=A B-B A$. Let

$\begin{aligned} K_{A} &=\left\{X \in M_{n \times n}(\mathbb{R}) \mid[A, X]=0\right\} \\ L_{A} &=\left\{[A, X] \mid X \in M_{n \times n}(\mathbb{R})\right\} \end{aligned}$

Show that $K_{A}$ and $L_{A}$ are linear subspaces of $M_{n \times n}(\mathbb{R})$. If $A$ and $B$ are similar, show that $K_{A} \cong K_{B}$ and $L_{A} \cong L_{B}$.

Suppose that $A$ is diagonalizable and has characteristic polynomial

$\left(x-\lambda_{1}\right)^{m_{1}}\left(x-\lambda_{2}\right)^{m_{2}}$

where $\lambda_{1} \neq \lambda_{2}$. What are $\operatorname{dim} K_{A}$ and $\operatorname{dim} L_{A} ?$

Paper 2, Section I, $1 F$

commentDefine the determinant $\operatorname{det} A$ of an $n \times n$ real matrix $A$. Suppose that $X$ is a matrix with block form

$X=\left(\begin{array}{cc} A & B \\ 0 & C \end{array}\right) \text {, }$

where $A, B$ and $C$ are matrices of dimensions $n \times n, n \times m$ and $m \times m$ respectively. Show that $\operatorname{det} X=(\operatorname{det} A)(\operatorname{det} C)$.

Paper 2, Section II, F

comment(i) Define the transpose of a matrix. If $V$ and $W$ are finite-dimensional real vector spaces, define the dual of a linear map $T: V \rightarrow W$. How are these two notions related?

Now suppose $V$ and $W$ are finite-dimensional inner product spaces. Use the inner product on $V$ to define a linear map $V \rightarrow V^{*}$ and show that it is an isomorphism. Define the adjoint of a linear map $T: V \rightarrow W$. How are the adjoint of $T$ and its dual related? If $A$ is a matrix representing $T$, under what conditions is the adjoint of $T$ represented by the transpose of $A$ ?

(ii) Let $V=C[0,1]$ be the vector space of continuous real-valued functions on $[0,1]$, equipped with the inner product

$\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t$

Let $T: V \rightarrow V$ be the linear map

$T f(t)=\int_{0}^{t} f(s) d s$

What is the adjoint of $T ?$

Paper 3, Section II, F

commentWhat is meant by the Jordan normal form of an $n \times n$ complex matrix?

Find the Jordan normal forms of the following matrices:

$\left(\begin{array}{llll} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \quad\left(\begin{array}{cccc} -1 & -1 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \end{array}\right), \quad\left(\begin{array}{cccc} 3 & 0 & 0 & 0 \\ 3 & 3 & 0 & 0 \\ 9 & 6 & 3 & 0 \\ 15 & 12 & 9 & 3 \end{array}\right)$

Suppose $A$ is an invertible $n \times n$ complex matrix. Explain how to derive the characteristic and minimal polynomials of $A^{n}$ from the characteristic and minimal polynomials of $A$. Justify your answer. [Hint: write each polynomial as a product of linear factors.]

Paper 4, Section I, F

commentLet $V$ be a complex vector space with basis $\left\{e_{1}, \ldots, e_{n}\right\}$. Define $T: V \rightarrow V$ by $T\left(e_{i}\right)=e_{i}-e_{i+1}$ for $i<n$ and $T\left(e_{n}\right)=e_{n}-e_{1}$. Show that $T$ is diagonalizable and find its eigenvalues. [You may use any theorems you wish, as long as you state them clearly.]

Paper 4, Section II, F

commentLet $V$ be a finite-dimensional real vector space of dimension $n$. A bilinear form $B: V \times V \rightarrow \mathbb{R}$ is nondegenerate if for all $\mathbf{v} \neq 0$ in $V$, there is some $\mathbf{w} \in V$ with $B(\mathbf{v}, \mathbf{w}) \neq 0$. For $\mathbf{v} \in V$, define $\langle\mathbf{v}\rangle^{\perp}=\{\mathbf{w} \in V \mid B(\mathbf{v}, \mathbf{w})=0\}$. Assuming $B$ is nondegenerate, show that $V=\langle\mathbf{v}\rangle \oplus\langle\mathbf{v}\rangle^{\perp}$ whenever $B(\mathbf{v}, \mathbf{v}) \neq 0$.

Suppose that $B$ is a nondegenerate, symmetric bilinear form on $V$. Prove that there is a basis $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\}$ of $V$ with $B\left(\mathbf{v}_{i}, \mathbf{v}_{j}\right)=0$ for $i \neq j$. [If you use the fact that symmetric matrices are diagonalizable, you must prove it.]

Define the signature of a quadratic form. Explain how to determine the signature of the quadratic form associated to $B$ from the basis you constructed above.

A linear subspace $V^{\prime} \subset V$ is said to be isotropic if $B(\mathbf{v}, \mathbf{w})=0$ for all $\mathbf{v}, \mathbf{w} \in V^{\prime}$. Show that if $B$ is nondegenerate, the maximal dimension of an isotropic subspace of $V$ is $(n-|\sigma|) / 2$, where $\sigma$ is the signature of the quadratic form associated to $B$.

Paper 1, Section II, 20H

commentA Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ has as its state space the integers, with

$p_{i, i+1}=p, \quad p_{i, i-1}=q=1-p$

and $p_{i j}=0$ otherwise. Assume $p>q$.

Let $T_{j}=\inf \left\{n \geqslant 1: X_{n}=j\right\}$ if this is finite, and $T_{j}=\infty$ otherwise. Let $V_{0}$ be the total number of hits on 0 , and let $V_{0}(n)$ be the total number of hits on 0 within times $0, \ldots, n-1$. Let

$\begin{aligned} h_{i} &=P\left(V_{0}>0 \mid X_{0}=i\right) \\ r_{i}(n) &=E\left[V_{0}(n) \mid X_{0}=i\right] \\ r_{i} &=E\left[V_{0} \mid X_{0}=i\right] \end{aligned}$

(i) Quoting an appropriate theorem, find, for every $i$, the value of $h_{i}$.

(ii) Show that if $\left(x_{i}, i \in \mathbb{Z}\right)$ is any non-negative solution to the system of equations

$\begin{aligned} x_{0} &=1+q x_{1}+p x_{-1}, \\ x_{i} &=q x_{i-1}+p x_{i+1}, \quad \text { for all } i \neq 0 \end{aligned}$

then $x_{i} \geqslant r_{i}(n)$ for all $i$ and $n$.

(iii) Show that $P\left(V_{0}\left(T_{1}\right) \geqslant k \mid X_{0}=1\right)=q^{k}$ and $E\left[V_{0}\left(T_{1}\right) \mid X_{0}=1\right]=q / p$.

(iv) Explain why $r_{i+1}=(q / p) r_{i}$ for $i>0$.

(v) Find $r_{i}$ for all $i$.

Paper 2, Section II, H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be the symmetric random walk on vertices of a connected graph. At each step this walk jumps from the current vertex to a neighbouring vertex, choosing uniformly amongst them. Let $T_{i}=\inf \left\{n \geqslant 1: X_{n}=i\right\}$. For each $i \neq j$ let $q_{i j}=P\left(T_{j}<T_{i} \mid X_{0}=i\right)$ and $m_{i j}=E\left(T_{j} \mid X_{0}=i\right)$. Stating any theorems that you use:

(i) Prove that the invariant distribution $\pi$ satisfies detailed balance.

(ii) Use reversibility to explain why $\pi_{i} q_{i j}=\pi_{j} q_{j i}$ for all $i, j$.

Consider a symmetric random walk on the graph shown below.

(iii) Find $m_{33}$.

(iv) The removal of any edge $(i, j)$ leaves two disjoint components, one which includes $i$ and one which includes $j$. Prove that $m_{i j}=1+2 e_{i j}(i)$, where $e_{i j}(i)$ is the number of edges in the component that contains $i$.

(v) Show that $m_{i j}+m_{j i} \in\{18,36,54,72\}$ for all $i \neq j$.

Paper 3, Section I, H

commentA runner owns $k$ pairs of running shoes and runs twice a day. In the morning she leaves her house by the front door, and in the evening she leaves by the back door. On starting each run she looks for shoes by the door through which she exits, and runs barefoot if none are there. At the end of each run she is equally likely to return through the front or back doors. She removes her shoes (if any) and places them by the door. In the morning of day 1 all shoes are by the back door so she must run barefoot.

Let $p_{00}^{(n)}$ be the probability that she runs barefoot on the morning of day $n+1$. What conditions are satisfied in this problem which ensure $\lim _{n \rightarrow \infty} p_{00}^{(n)}$ exists? Show that its value is $1 / 2 k$.

Find the expected number of days that will pass until the first morning that she finds all $k$ pairs of shoes at her front door.

Paper 4, Section I, H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be an irreducible Markov chain with $p_{i j}^{(n)}=P\left(X_{n}=j \mid X_{0}=i\right)$. Define the meaning of the statements:

(i) state $i$ is transient,

(ii) state $i$ is aperiodic.

Give a criterion for transience that can be expressed in terms of the probabilities $\left(p_{i i}^{(n)}, n=0,1, \ldots\right)$.

Prove that if a state $i$ is transient then all states are transient.

Prove that if a state $i$ is aperiodic then all states are aperiodic.

Suppose that $p_{i i}^{(n)}=0$ unless $n$ is divisible by 3 . Given any other state $j$, prove that $p_{j j}^{(n)}=0$ unless $n$ is divisible by 3 .

Paper 1, Section II, C

commentConsider the regular Sturm-Liouville (S-L) system

$(\mathcal{L} y)(x)-\lambda \omega(x) y(x)=0, \quad a \leqslant x \leqslant b$

where

$(\mathcal{L} y)(x):=-\left[p(x) y^{\prime}(x)\right]^{\prime}+q(x) y(x)$

with $\omega(x)>0$ and $p(x)>0$ for all $x$ in $[a, b]$, and the boundary conditions on $y$ are

$\left\{\begin{array}{l} A_{1} y(a)+A_{2} y^{\prime}(a)=0 \\ B_{1} y(b)+B_{2} y^{\prime}(b)=0 \end{array}\right.$

Show that with these boundary conditions, $\mathcal{L}$ is self-adjoint. By considering $y \mathcal{L} y$, or otherwise, show that the eigenvalue $\lambda$ can be written as

$\lambda=\frac{\int_{a}^{b}\left(p y^{\prime 2}+q y^{2}\right) d x-\left[p y y^{\prime}\right]_{a}^{b}}{\int_{a}^{b} \omega y^{2} d x}$

Now suppose that $a=0$ and $b=\ell$, that $p(x)=1, q(x) \geqslant 0$ and $\omega(x)=1$ for all $x \in[0, \ell]$, and that $A_{1}=1, A_{2}=0, B_{1}=k \in \mathbb{R}^{+}$and $B_{2}=1$. Show that the eigenvalues of this regular S-L system are strictly positive. Assuming further that $q(x)=0$, solve the system explicitly, and with the aid of a graph, show that there exist infinitely many eigenvalues $\lambda_{1}<\lambda_{2}<\cdots<\lambda_{n}<\cdots$. Describe the behaviour of $\lambda_{n}$ as $n \rightarrow \infty$.

Paper 2, Section I, C

commentUsing the method of characteristics, obtain a solution to the equation

$u_{x}+2 x u_{y}=y$

subject to the Cauchy data $u(0, y)=1+y^{2}$ for $-\frac{1}{2}<y<\frac{1}{2}$.

Sketch the characteristics and specify the greatest region of the plane in which a unique solution exists.

Paper 2, Section II, C

commentConsider the linear differential operator $\mathcal{L}$ defined by

$\mathcal{L} y:=-\frac{d^{2} y}{d x^{2}}+y$

on the interval $0 \leqslant x<\infty$. Given the boundary conditions $y(0)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$, find the Green's function $G(x, \xi)$ for $\mathcal{L}$ with these boundary conditions. Hence, or otherwise, obtain the solution of

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

subject to the above boundary conditions, where $\mu$ is a positive constant. Show that your piecewise solution is continuous at $x=\mu$ and has the value

$y(\mu)=\frac{1}{2}\left(1+e^{-2 \mu}-2 e^{-\mu}\right) .$

Paper 3, Section I, D

commentFor the step-function

$F(x)= \begin{cases}1, & |x| \leqslant 1 / 2 \\ 0, & \text { otherwise }\end{cases}$

its convolution with itself is the hat-function

$G(x)=[F * F](x)= \begin{cases}1-|x|, & |x| \leqslant 1 \\ 0, & \text { otherwise }\end{cases}$

Find the Fourier transforms of $F$ and $G$, and hence find the values of the integrals

$I_{1}=\int_{-\infty}^{\infty} \frac{\sin ^{2} y}{y^{2}} d y, \quad I_{2}=\int_{-\infty}^{\infty} \frac{\sin ^{4} y}{y^{4}} d y$

Paper 3, Section II, D

commentConsider Legendre's equation

$\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\lambda y=0 .$

Show that if $\lambda=n(n+1)$, with $n$ a non-negative integer, this equation has a solution $y=P_{n}(x)$, a polynomial of degree $n$. Find $P_{0}, P_{1}$ and $P_{2}$ explicitly, subject to the condition $P_{n}(1)=1$.

The general solution of Laplace's equation $\nabla^{2} \psi=0$ in spherical polar coordinates, in the axisymmetric case, has the form

$\psi(r, \theta)=\sum_{n=0}^{\infty}\left(A_{n} r^{n}+B_{n} r^{-(n+1)}\right) P_{n}(\cos \theta)$

Hence, find the solution of Laplace's equation in the region $a \leqslant r \leqslant b$ satisfying the boundary conditions

$\begin{cases}\psi(r, \theta)=1, & r=a \\ \psi(r, \theta)=3 \cos ^{2} \theta, & r=b\end{cases}$

Paper 4, Section I, D

commentShow that the general solution of the wave equation

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

can be written in the form

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

Hence derive the solution $y(x, t)$ subject to the initial conditions

$y(x, 0)=0, \quad \frac{\partial y}{\partial t}(x, 0)=\psi(x)$

Paper 4, Section II, D

commentLet $D \subset \mathbb{R}^{2}$ be a two-dimensional domain with boundary $S=\partial D$, and let

$G_{2}=G_{2}\left(\mathbf{r}, \mathbf{r}_{0}\right)=\frac{1}{2 \pi} \log \left|\mathbf{r}-\mathbf{r}_{0}\right|$

where $\mathbf{r}_{0}$ is a point in the interior of $D$. From Green's second identity,

$\int_{S}\left(\phi \frac{\partial \psi}{\partial n}-\psi \frac{\partial \phi}{\partial n}\right) d \ell=\int_{D}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d a$

derive Green's third identity

$u\left(\mathbf{r}_{0}\right)=\int_{D} G_{2} \nabla^{2} u d a+\int_{S}\left(u \frac{\partial G_{2}}{\partial n}-G_{2} \frac{\partial u}{\partial n}\right) d \ell$

[Here $\frac{\partial}{\partial n}$ denotes the normal derivative on $S$.]

Consider the Dirichlet problem on the unit $\operatorname{disc} D_{1}=\left\{\mathbf{r} \in \mathbb{R}^{2}:|\mathbf{r}| \leqslant 1\right\}$ :

$\begin{aligned} \nabla^{2} u=0, & \mathbf{r} \in D_{1} \\ u(\mathbf{r})=f(\mathbf{r}), & \mathbf{r} \in S_{1}=\partial D_{1} \end{aligned}$

Show that, with an appropriate function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$, the solution can be obtained by the formula

$u\left(\mathbf{r}_{0}\right)=\int_{S_{1}} f(\mathbf{r}) \frac{\partial}{\partial n} G\left(\mathbf{r}, \mathbf{r}_{0}\right) d \ell$

State the boundary conditions on $G$ and explain how $G$ is related to $G_{2}$.

For $\mathbf{r}, \mathbf{r}_{0} \in \mathbb{R}^{2}$, prove the identity

$\left|\frac{\mathbf{r}}{|\mathbf{r}|}-\mathbf{r}_{0}\right| \mathbf{r}||=\left|\frac{\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|}-\mathbf{r}\right| \mathbf{r}_{0}|| \text {, }$

and deduce that if the point $\mathbf{r}$ lies on the unit circle, then

$\left|\mathbf{r}-\mathbf{r}_{0}\right|=\left|\mathbf{r}_{0}\right|\left|\mathbf{r}-\mathbf{r}_{0}^{*}\right|, \text { where } \mathbf{r}_{0}^{*}=\frac{\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|^{2}}$

Hence, using the method of images, or otherwise, find an expression for the function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$. [An expression for $\frac{\partial}{\partial n} G$ is not required.]

Paper 1, Section II, F

A topological space $X$ is said to be normal if each point of $X$ is a closed subset of $X$ and for each pair of closed sets $C_{1}, C_{2} \subset X$ with $C_{1} \cap C_{2}=\emptyset$ there are open sets $U_{1}, U_{2} \subset X$ so that $C_{i} \subset U_{i}$ and $U_{1} \cap U_{2}=\emptyset$. In this case we say that the $U_{i}$ separate the $C_{i}$.

Show that a compact Hausdorff space is normal. [Hint: first consider the case where $C_{2}$ is a point.]

For $C \subset X$ we define an equivalence relation $\sim_{C}$