Part II, 2009, Paper 3

Algebraic Geometry

• Paper 3, Section II, G

  Algebraic Geometry | Part II, 2009

  Let $V$ be a smooth projective curve, and let $D$ be an effective divisor on $V$. Explain how $D$ defines a morphism $\phi_{D}$ from $V$ to some projective space. State the necessary and sufficient conditions for $\phi_{D}$ to be finite. State the necessary and sufficient conditions for $\phi_{D}$ to be an isomorphism onto its image.

  Let $V$ have genus 2 , and let $K$ be an effective canonical divisor. Show that the morphism $\phi_{K}$ is a morphism of degree 2 from $V$ to $\mathbb{P}^{1}$.

  By considering the divisor $K+P_{1}+P_{2}$ for points $P_{i}$ with $P_{1}+P_{2} \nsim K$, show that there exists a birational morphism from $V$ to a singular plane quartic.

  [You may assume the Riemann-Roch Theorem.]

  comment

Algebraic Topology

• Paper 3, Section II, G

  Algebraic Topology | Part II, 2009

  (i) Suppose that $(C, d)$ and $\left(C^{\prime}, d^{\prime}\right)$ are chain complexes, and $f, g: C \rightarrow C^{\prime}$ are chain maps. Define what it means for $f$ and $g$ to be chain homotopic.

  Show that if $f$ and $g$ are chain homotopic, and $f_{*}, g_{*}: H_{*}(C) \rightarrow H_{*}\left(C^{\prime}\right)$ are the induced maps, then $f_{*}=g_{*}$.

  (ii) Define the Euler characteristic of a finite chain complex.

  Given that one of the sequences below is exact and the others are not, which is the exact one?

  $$\begin{aligned} &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{25} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \\ &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \\ &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{19} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{23} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \end{aligned}$$

  Justify your choice.

  comment

Applications of Quantum Mechanics

• Paper 3, Section II, D

  Applications of Quantum Mechanics | Part II, 2009

  An electron of charge $-e$ and mass $m$ is subject to a magnetic field of the form $\mathbf{B}=(0,0, B(y))$, where $B(y)$ is everywhere greater than some positive constant $B_{0}$. In a stationary state of energy $E$, the electron's wavefunction $\Psi$ satisfies

  $$-\frac{\hbar^{2}}{2 m}\left(\boldsymbol{\nabla}+\frac{i e}{\hbar} \mathbf{A}\right)^{2} \Psi+\frac{e \hbar}{2 m} \mathbf{B} \cdot \boldsymbol{\sigma} \Psi=E \Psi,$$

  where $\mathbf{A}$ is the vector potential and $\sigma_{1}, \sigma_{2}$ and $\sigma_{3}$ are the Pauli matrices.

  Assume that the electron is in a spin down state and has no momentum along the $z$-axis. Show that with a suitable choice of gauge, and after separating variables, equation (*) can be reduced to

  $$-\frac{d^{2} \chi}{d y^{2}}+(k+a(y))^{2} \chi-b(y) \chi=\epsilon \chi,$$

  where $\chi$ depends only on $y, \epsilon$ is a rescaled energy, and $b(y)$ a rescaled magnetic field strength. What is the relationship between $a(y)$ and $b(y)$ ?

  Show that $(* *)$ can be factorized in the form $M^{\dagger} M \chi=\epsilon \chi$ where

  $$M=\frac{d}{d y}+W(y)$$

  for some function $W(y)$, and deduce that $\epsilon$ is non-negative.

  Show that zero energy states exist for all $k$ and are therefore infinitely degenerate.

  comment

Applied Probability

• Paper 3, Section II, J

  Applied Probability | Part II, 2009

  (a) Define the Poisson process $\left(N_{t}, t \geqslant 0\right)$ with rate $\lambda>0$, in terms of its holding times. Show that for all times $t \geqslant 0, N_{t}$ has a Poisson distribution, with a parameter which you should specify.

  (b) Let $X$ be a random variable with probability density function

  $$f(x)=\frac{1}{2} \lambda^{3} x^{2} e^{-\lambda x} \mathbf{1}_{\{x>0\}} .$$

  Prove that $X$ is distributed as the sum $Y_{1}+Y_{2}+Y_{3}$ of three independent exponential random variables of rate $\lambda$. Calculate the expectation, variance and moment generating function of $X$.

  Consider a renewal process $\left(X_{t}, t \geqslant 0\right)$ with holding times having density $(*)$. Prove that the renewal function $m(t)=\mathbb{E}\left(X_{t}\right)$ has the form

  $$m(t)=\frac{\lambda t}{3}-\frac{1}{3} p_{1}(t)-\frac{2}{3} p_{2}(t)$$

  where $p_{1}(t)=\mathbb{P}\left(N_{t}=1 \bmod 3\right), p_{2}(t)=\mathbb{P}\left(N_{t}=2 \bmod 3\right)$ and $\left(N_{t}, t \geqslant 0\right)$ is the Poisson process of rate $\lambda$.

  (c) Consider the delayed renewal process $\left(X_{t}^{\mathrm{D}}, t \geqslant 0\right)$ with holding times $S_{1}^{\mathrm{D}}, S_{2}, S_{3}, \ldots$ where $\left(S_{n}, n \geqslant 1\right)$, are the holding times of $\left(X_{t}, t \geqslant 0\right)$ from (b). Specify the distribution of $S_{1}^{\mathrm{D}}$ for which the delayed process becomes the renewal process in equilibrium.

  [You may use theorems from the course provided that you state them clearly.]

  comment

Asymptotic Methods

• Paper 3, Section II, A

  Asymptotic Methods | Part II, 2009

  Consider the contour-integral representation

  $$J_{0}(x)=\operatorname{Re} \frac{1}{i \pi} \int_{C} e^{i x \cosh t} d t$$

  of the Bessel function $J_{0}$ for real $x$, where $C$ is any contour from $-\infty-\frac{i \pi}{2}$ to $+\infty+\frac{i \pi}{2}$.

  Writing $t=u+i v$, give in terms of the real quantities $u, v$ the equation of the steepest-descent contour from $-\infty-\frac{i \pi}{2}$ to $+\infty+\frac{i \pi}{2}$ which passes through $t=0$.

  Deduce the leading term in the asymptotic expansion of $J_{0}(x)$, valid as $x \rightarrow \infty$

  $$J_{0}(x) \sim \sqrt{\frac{2}{\pi x}} \cos \left(x-\frac{\pi}{4}\right)$$

  comment

Classical Dynamics

• Paper 3, Section I, E

  Classical Dynamics | Part II, 2009

  (a) Show that the principal moments of inertia of a uniform circular cylinder of radius $a$, length $h$ and mass $M$ about its centre of mass are $I_{1}=I_{2}=M\left(a^{2} / 4+h^{2} / 12\right)$ and $I_{3}=M a^{2} / 2$, with the $x_{3}$ axis being directed along the length of the cylinder.

  (b) Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of an arbitrary rigid body as viewed in the body frame are

  $$\begin{aligned} &I_{1} \frac{d \omega_{1}}{d t}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \frac{d \omega_{2}}{d t}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \end{aligned}$$

  and

  $$I_{3} \frac{d \omega_{3}}{d t}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}$$

  Show that, for the cylinder of part $(\mathrm{a}), \omega_{3}$ is constant. Show further that, when $\omega_{3} \neq 0$, the angular momentum vector precesses about the $x_{3}$ axis with angular velocity $\Omega$ given by

  $$\Omega=\left(\frac{3 a^{2}-h^{2}}{3 a^{2}+h^{2}}\right) \omega_{3}$$

  comment

Coding and Cryptography

• Paper 3, Section I, $4 \mathrm{H}$

  Coding and Cryptography | Part II, 2009

  Define a binary code of length 15 with information rate $11 / 15$ which will correct single errors. Show that it has the rate stated and give an explicit procedure for identifying the error. Show that the procedure works.

  [Hint: You may wish to imitate the corresponding discussion for a code of length 7 .]

  comment

Cosmology

• Paper 3, Section I, D

  Cosmology | Part II, 2009

  (a) Write down an expression for the total gravitational potential energy $E_{\text {grav }}$ of a spherically symmetric star of outer radius $R$ in terms of its mass density $\rho(r)$ and the total mass $m(r)$ inside a radius $r$, satisfying the relation $d m / d r=4 \pi r^{2} \rho(r)$.

  An isotropic mass distribution obeys the pressure-support equation,

  $$\frac{d P}{d r}=-\frac{G m \rho}{r^{2}}$$

  where $P(r)$ is the pressure. Multiply this expression by $4 \pi r^{3}$ and integrate with respect to $r$ to derive the virial theorem relating the kinetic and gravitational energy of the star

  $$E_{\mathrm{kin}}=-\frac{1}{2} E_{\mathrm{grav}}$$

  where you may assume for a non-relativistic ideal gas that $E_{\mathrm{kin}}=\frac{3}{2}\langle P\rangle V$, with $\langle P\rangle$ the average pressure.

  (b) Consider a white dwarf supported by electron Fermi degeneracy pressure $P \approx h^{2} n^{5 / 3} / m_{\mathrm{e}}$, where $m_{\mathrm{e}}$ is the electron mass and $n$ is the number density. Assume a uniform density $\rho(r)=m_{\mathrm{p}} n(r) \approx m_{\mathrm{p}}\langle n\rangle$, so the total mass of the star is given by $M=(4 \pi / 3)\langle n\rangle m_{\mathrm{p}} R^{3}$ where $m_{\mathrm{p}}$ is the proton mass. Show that the total energy of the white dwarf can be written in the form

  $$E_{\mathrm{total}}=E_{\mathrm{kin}}+E_{\mathrm{grav}}=\frac{\alpha}{R^{2}}-\frac{\beta}{R},$$

  where $\alpha, \beta$ are positive constants which you should specify. Deduce that the white dwarf has a stable radius $R_{\mathrm{WD}}$ at which the energy is minimized, that is,

  $$R_{\mathrm{WD}} \sim \frac{h^{2} M^{-1 / 3}}{G m_{\mathrm{e}} m_{\mathrm{p}}^{5 / 3}} .$$

  comment

• Paper 3, Section II, D

  Cosmology | Part II, 2009

  In the Zel'dovich approximation, particle trajectories in a flat expanding universe are described by $\mathbf{r}(\mathbf{q}, t)=a(t)[\mathbf{q}+\mathbf{\Psi}(\mathbf{q}, t)]$, where $a(t)$ is the scale factor of the universe, $\mathbf{q}$ is the unperturbed comoving trajectory and $\boldsymbol{\Psi}$ is the comoving displacement. The particle equation of motion is

  $$\ddot{\mathbf{r}}=-\nabla \Phi-\frac{1}{\rho} \nabla P$$

  where $\rho$ is the mass density, $P$ is the pressure $\left(P \ll \rho c^{2}\right)$ and $\Phi$ is the Newtonian potential which satisfies the Poisson equation $\nabla^{2} \Phi=4 \pi G \rho$.

  (i) Show that the fractional density perturbation and the pressure gradient are given by

  $$\delta \equiv \frac{\rho-\bar{\rho}}{\bar{\rho}} \approx-\nabla_{\mathbf{q}} \cdot \boldsymbol{\Psi}, \quad \nabla P \approx-\bar{\rho} \frac{c_{s}^{2}}{a} \nabla_{\mathbf{q}}^{2} \boldsymbol{\Psi}$$

  where $\nabla_{\mathbf{q}}$ has components $\partial / \partial q_{i}, \bar{\rho}=\bar{\rho}(t)$ is the homogeneous background density and $c_{s}^{2} \equiv \partial P / \partial \rho$ is the sound speed. [You may assume that the Jacobian $\left|\partial r_{i} / \partial q_{j}\right|^{-1}=$ $\left|a \delta_{i j}+a \partial \psi_{i} / \partial q_{j}\right|^{-1} \approx a^{-3}\left(1-\nabla_{\mathbf{q}} \cdot \boldsymbol{\Psi}\right)$ for $\left.|\boldsymbol{\Psi}| \ll|\mathbf{q}| .\right]$

  Use this result to integrate the Poisson equation once and obtain then the evolution equation for the comoving displacement:

  $$\ddot{\boldsymbol{\Psi}}+2 \frac{\dot{a}}{a} \dot{\boldsymbol{\Psi}}-4 \pi G \bar{\rho} \boldsymbol{\Psi}-\frac{c_{s}^{2}}{a^{2}} \nabla_{\mathbf{q}}^{2} \boldsymbol{\Psi}=0$$

  [You may assume that the integral of $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ is $\nabla \Phi=4 \pi G \bar{\rho} \mathbf{r} / 3$, that $\boldsymbol{\Psi}$ is irrotational and that the Raychaudhuri equation is $\ddot{a} / a \approx-4 \pi G \bar{\rho} / 3$ for $P \ll \rho c^{2}$.]

  Consider the Fourier expansion $\delta(\mathbf{x}, t)=\sum_{\mathbf{k}} \delta_{\mathbf{k}} \exp (i \mathbf{k} \cdot \mathbf{x})$ of the density perturbation using the comoving wavenumber $\mathbf{k}(k=|\mathbf{k}|)$ and obtain the evolution equation for the mode $\delta_{\mathbf{k}}$ :

  $$\ddot{\delta}_{\mathbf{k}}+2 \frac{\dot{a}}{a} \dot{\delta}_{\mathbf{k}}-\left(4 \pi G \bar{\rho}-c_{s}^{2} k^{2} / a^{2}\right) \delta_{\mathbf{k}}=0$$

  (ii) Consider a flat matter-dominated universe with $a(t)=\left(t / t_{0}\right)^{2 / 3}$ (background density $\left.\bar{\rho}=1 /\left(6 \pi G t^{2}\right)\right)$ and with an equation of state $P=\beta \rho^{4 / 3}$ to show that $(*)$ becomes

  $$\ddot{\delta}_{\mathbf{k}}+\frac{4}{3 t} \dot{\delta}_{\mathbf{k}}-\frac{1}{t^{2}}\left(\frac{2}{3}-\bar{v}_{s}^{2} k^{2}\right) \delta_{\mathbf{k}}=0$$

  where the constant $\bar{v}_{s}^{2} \equiv(4 \beta / 3)(6 \pi G)^{-1 / 3} t_{0}^{4 / 3}$. Seek power law solutions of the form $\delta_{\mathbf{k}} \propto t^{\alpha}$ to find the growing and decaying modes

  $$\delta_{\mathbf{k}}=A_{\mathbf{k}} t^{n+}+B_{\mathbf{k}} t^{n-} \quad \text { where } \quad n_{\pm}=-\frac{1}{6} \pm\left[\left(\frac{5}{6}\right)^{2}-\bar{v}_{s}^{2} k^{2}\right]^{1 / 2} .$$

  comment

Differential Geometry

• Paper 3, Section II, H

  Differential Geometry | Part II, 2009

  (a) State and prove the Theorema Egregium.

  (b) Let $X$ be a minimal surface without boundary in $\mathbb{R}^{3}$ which is closed as a subset of $\mathbb{R}^{3}$, and assume that $X$ is not contained in a closed ball. Let $\Pi$ be a plane in $\mathbb{R}^{3}$ with the property that $D_{n} \rightarrow \infty$ as $n \rightarrow \infty$, where for $n=0,1, \ldots$,

  $$D_{n}=\inf _{x \in X, d(x, 0) \geqslant n} d(x, \Pi)$$

  Here $d(x, y)$ denotes the Euclidean distance between $x$ and $y$ and $d(x, \Pi)=\inf _{y \in \Pi} d(x, y)$. Assume moreover that $X$ contains no planar points. Show that $X$ intersects $\Pi$.

  comment

Dynamical Systems

• Paper 3, Section I, E

  Dynamical Systems | Part II, 2009

  Consider the one-dimensional real map $x_{n+1}=F\left(x_{n}\right)=r x_{n}^{2}\left(1-x_{n}\right)$, where $r>0$. Locate the fixed points and explain for what ranges of the parameter $r$ each fixed point exists. For what range of $r$ does $F$ map the open interval $(0,1)$ into itself?

  Determine the location and type of all the bifurcations from the fixed points which occur. Sketch the location of the fixed points in the $(r, x)$ plane, indicating stability.

  comment

• Paper 3, Section II, E

  Dynamical Systems | Part II, 2009

  Consider the dynamical system

  $$\begin{aligned} &\dot{x}=-a x-2 x y, \\ &\dot{y}=x^{2}+y^{2}-b, \end{aligned}$$

  where $a \geqslant 0$ and $b>0$.

  (i) Find and classify the fixed points. Show that a bifurcation occurs when $4 b=a^{2}>0$.

  (ii) After shifting coordinates to move the relevant fixed point to the origin, and setting $a=2 \sqrt{b}-\mu$, carry out an extended centre manifold calculation to reduce the two-dimensional system to one of the canonical forms, and hence determine the type of bifurcation that occurs when $4 b=a^{2}>0$. Sketch phase portraits in the cases $0<a^{2}<4 b$ and $0<4 b<a^{2}$.

  (iii) Sketch the phase portrait in the case $a=0$. Prove that periodic orbits exist if and only if $a=0$.

  comment

Electrodynamics

• Paper 3, Section II, C

  Electrodynamics | Part II, 2009

  A particle of charge of $q$ moves along a trajectory $y^{a}(s)$ in spacetime where $s$ is the proper time on the particle's world-line.

  Explain briefly why, in the gauge $\partial_{a} A^{a}=0$, the potential at the spacetime point $x$ is given by

  $$A^{a}(x)=\frac{\mu_{0} q}{2 \pi} \int d s \frac{d y^{a}}{d s} \theta\left(x^{0}-y^{0}(s)\right) \delta\left(\left(x^{c}-y^{c}(s)\right)\left(x^{d}-y^{d}(s)\right) \eta_{c d}\right)$$

  Evaluate this integral for a point charge moving relativistically along the $z$-axis, $x=y=0$, at constant velocity $v$ so that $z=v t .$

  Check your result by starting from the potential of a point charge at rest

  $$\begin{aligned} \mathbf{A} &=0 \\ \phi &=\frac{\mu_{0} q}{4 \pi\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}} \end{aligned}$$

  and making an appropriate Lorentz transformation.

  comment

Fluid Dynamics II

• Paper 3, Section II, E

  Fluid Dynamics II | Part II, 2009

  An axisymmetric incompressible Stokes flow has the Stokes stream function $\Psi(R, \theta)$ in spherical polar coordinates $(R, \theta, \phi)$. Give expressions for the components $u_{R}$ and $u_{\theta}$ of the flow field in terms of $\Psi$, and show that

  $$\nabla \times \mathbf{u}=\left(0,0,-\frac{D^{2} \Psi}{R \sin \theta}\right)$$

  where

  $$D^{2} \Psi=\frac{\partial^{2} \Psi}{\partial R^{2}}+\frac{\sin \theta}{R^{2}} \frac{\partial}{\partial \theta}\left(\frac{1}{\sin \theta} \frac{\partial \Psi}{\partial \theta}\right)$$

  Write down the equation satisfied by $\Psi$.

  Verify that the Stokes stream function

  $$\Psi(R, \theta)=\frac{1}{2} U \sin ^{2} \theta\left(R^{2}-\frac{3}{2} a R+\frac{1}{2} \frac{a^{3}}{R}\right)$$

  represents the Stokes flow past a stationary sphere of radius $a$, when the fluid at large distance from the sphere moves at speed $U$ along the axis of symmetry.

  A sphere of radius a moves vertically upwards in the $z$ direction at speed $U$ through fluid of density $\rho$ and dynamic viscosity $\mu$, towards a free surface at $z=0$. Its distance $d$ from the surface is much greater than $a$. Assuming that the surface remains flat, show that the conditions of zero vertical velocity and zero tangential stress at $z=0$ can be approximately met for large $d / a$ by combining the Stokes flow for the sphere with that of an image sphere of the same radius located symmetrically above the free surface. Hence determine the leading-order behaviour of the horizontal flow on the free surface as a function of $r$, the horizontal distance from the sphere's centre line.

  What is the size of the next correction to your answer as a power of $a / d ?$ [Detailed calculation is not required.]

  [Hint: For an axisymmetric vector field $\mathbf{u}$,

  $$\nabla \times \mathbf{u}=\left(\frac{1}{R \sin \theta} \frac{\partial}{\partial \theta}\left(u_{\phi} \sin \theta\right),-\frac{1}{R} \frac{\partial}{\partial R}\left(R u_{\phi}\right), \frac{1}{R} \frac{\partial}{\partial R}\left(R u_{\theta}\right)-\frac{1}{R} \frac{\partial u_{R}}{\partial \theta}\right)$$

  comment

Further Complex Methods

• Paper 3, Section $\mathbf{I}$, B

  Further Complex Methods | Part II, 2009

  Suppose that the real function $u(x, y)$ satisfies Laplace's equation in the upper half complex $z$-plane, $z=x+i y, x \in \mathbb{R}, y>0$, where

  $$u(x, y) \rightarrow 0 \quad \text { as } \quad \sqrt{x^{2}+y^{2}} \rightarrow \infty, \quad u(x, 0)=g(x), \quad x \in \mathbb{R} .$$

  The function $u(x, y)$ can then be expressed in terms of the Poisson integral

  $$u(x, y)=\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{y g(\xi)}{(x-\xi)^{2}+y^{2}} d \xi, \quad x \in \mathbb{R}, y>0$$

  By employing the formula

  $$f(z)=2 u\left(\frac{z+\bar{a}}{2}, \frac{z-\bar{a}}{2 i}\right)-\overline{f(a)}$$

  where $a$ is a complex constant with $\operatorname{Im} a>0$, show that the analytic function whose real part is $u(x, y)$ is given by

  $$f(z)=\frac{1}{i \pi} \int_{-\infty}^{\infty} \frac{g(\xi)}{\xi-z} d \xi+i c, \quad \operatorname{Im} z>0$$

  where $c$ is a real constant.

  comment

Galois Theory

• Paper 3, Section II, H

  Galois Theory | Part II, 2009

  Let $K=\mathbb{F}_{p}(x)$, the function field in one variable, and let $G=\mathbb{F}_{p}$. The group $G$ acts as automorphisms of $K$ by $\sigma_{a}(x)=x+a$. Show that $K^{G}=\mathbb{F}_{p}(y)$, where $y=x^{p}-x$.

  [State clearly any theorems you use.]

  Is $K / K^{G}$ a separable extension?

  Now let

  $$H=\left\{\left(\begin{array}{ll} d & a \\ 0 & 1 \end{array}\right): a \in \mathbb{F}_{p}, d \in \mathbb{F}_{p}^{*}\right\}$$

  and let $H$ act on $K$ by $\left(\begin{array}{ll}d & a \\ 0 & 1\end{array}\right) x=d x+a$. (The group structure on $H$ is given by matrix multiplication.) Compute $K^{H}$. Describe your answer in the form $\mathbb{F}_{p}(z)$ for an explicit $z \in K$.

  Is $K^{G} / K^{H}$ a Galois extension? Find the minimum polynomial for $y$ over the field $K^{H}$.

  comment

Geometry of Group Actions

• Paper 3, Section I, F

  Geometry of Group Actions | Part II, 2009

  Explain why there are discrete subgroups of the Möbius group $\mathbb{P} S L_{2}(\mathbb{C})$ which abstractly are free groups of rank 2 .

  comment

Graph Theory

• Paper 3, Section II, F

  Graph Theory | Part II, 2009

  (a) State Brooks' theorem concerning the chromatic number $\chi(G)$ of a graph $G$. Prove it in the case when $G$ is 3-connected.

  [If you wish to assume that $G$ is regular, you should explain why this assumption is justified.]

  (b) State Vizing's theorem concerning the edge-chromatic number $\chi^{\prime}(G)$ of a graph $G$.

  (c) Are the following statements true or false? Justify your answers.

  (1) If $G$ is a connected graph on more than two vertices then $\chi(G) \leqslant \chi^{\prime}(G)$.

  (2) For every ordering of the vertices of a graph $G$, if we colour $G$ using the greedy algorithm (on this ordering) then the number of colours we use is at most $2 \chi(G)$.

  (3) For every ordering of the edges of a graph $G$, if we edge-colour $G$ using the greedy algorithm (on this ordering) then the number of colours we use is at most $2 \chi^{\prime}(G)$.

  comment

Integrable Systems

• Paper 3, Section II, B

  Integrable Systems | Part II, 2009

  Consider the partial differential equation

  $$\frac{\partial u}{\partial t}=u^{n} \frac{\partial u}{\partial x}+\frac{\partial^{2 k+1} u}{\partial x^{2 k+1}}$$

  where $u=u(x, t)$ and $k, n$ are non-negative integers.

  (i) Find a Lie point symmetry of $(*)$ of the form

  $$(x, t, u) \longrightarrow(\alpha x, \beta t, \gamma u),$$

  where $(\alpha, \beta, \gamma)$ are non-zero constants, and find a vector field generating this symmetry. Find two more vector fields generating Lie point symmetries of (*) which are not of the form $(* *)$ and verify that the three vector fields you have found form a Lie algebra.

  (ii) Put $(*)$ in a Hamiltonian form.

  comment

Linear Analysis

• Paper 3, Section II, H

  Linear Analysis | Part II, 2009

  (a) State the Arzela-Ascoli theorem, explaining the meaning of all concepts involved.

  (b) Prove the Arzela-Ascoli theorem.

  (c) Let $K$ be a compact topological space. Let $\left(f_{n}\right)_{n \in \mathbb{N}}$ be a sequence in the Banach space $C(K)$ of real-valued continuous functions over $K$ equipped with the supremum norm $\|\cdot\|$. Assume that for every $x \in K$, the sequence $f_{n}(x)$ is monotone increasing and that $f_{n}(x) \rightarrow f(x)$ for some $f \in C(K)$. Show that $\left\|f_{n}-f\right\| \rightarrow 0$ as $n \rightarrow \infty$.

  comment

Logic and Set Theory

• Paper 3, Section II, G

  Logic and Set Theory | Part II, 2009

  Let $x \subseteq \alpha$ be a subset of a (von Neumann) ordinal $\alpha$ taken with the induced ordering. Using the recursion theorem or otherwise show that $x$ is order isomorphic to a unique ordinal $\mu(x)$. Suppose that $x \subseteq y \subseteq \alpha$. Show that $\mu(x) \leqslant \mu(y) \leqslant \alpha$.

  Suppose that $x_{0} \subseteq x_{1} \subseteq x_{2} \subseteq \cdots$ is an increasing sequence of subsets of $\alpha$, with $x_{i}$ an initial segment of $x_{j}$ whenever $i<j$. Show that $\mu\left(\bigcup_{n} x_{n}\right)=\bigcup_{n} \mu\left(x_{n}\right)$. Does this result still hold if the condition on initial segments is omitted? Justify your answer.

  Suppose that $x_{0} \supseteq x_{1} \supseteq x_{2} \supseteq \cdots$ is a decreasing sequence of subsets of $\alpha$. Why is the sequence $\mu\left(x_{n}\right)$ eventually constant? Is it the case that $\mu\left(\bigcap_{n} x_{n}\right)=\bigcap_{n} \mu\left(x_{n}\right)$ ? Justify your answer.

  comment

Mathematical Biology

• Paper 3, Section I, A

  Mathematical Biology | Part II, 2009

  Consider an organism moving on a one-dimensional lattice of spacing $a$, taking steps either to the right or the left at regular time intervals $\tau$. In this random walk there is a slight bias to the right, that is the probabilities of moving to the right and left, $\alpha$ and $\beta$, are such that $\alpha-\beta=\epsilon$, where $0<\epsilon \ll 1$. Write down the appropriate master equation for this process. Show by taking the continuum limit in space and time that $p(x, t)$, the probability that an organism initially at $x=0$ is at $x$ after time $t$, obeys

  $$\frac{\partial p}{\partial t}+V \frac{\partial p}{\partial x}=D \frac{\partial^{2} p}{\partial x^{2}}$$

  Express the constants $V$ and $D$ in terms of $a, \tau, \alpha$ and $\beta$.

  comment

• Paper 3, Section II, A

  Mathematical Biology | Part II, 2009

  An activator-inhibitor reaction diffusion system in dimensionless form is given by

  $$u_{t}=u_{x x}+\frac{u^{2}}{v}-b u, \quad v_{t}=d v_{x x}+u^{2}-v$$

  where $b$ and $d$ are positive constants. Which is the activitor and which the inhibitor? Determine the positive steady states and show, by an examination of the eigenvalues in a linear stability analysis of the spatially uniform situation, that the reaction kinetics is stable if $b<1$.

  Determine the conditions for the steady state to be driven unstable by diffusion. Show that the parameter domain for diffusion-driven instability is given by $0<b<1$, $b d>3+2 \sqrt{2}$, and sketch the $(b, d)$ parameter space in which diffusion-driven instability occurs. Further show that at the bifurcation to such an instability the critical wave number $k_{c}$ is given by $k_{c}^{2}=(1+\sqrt{2}) / d$.

  comment

Number Theory

• Paper 3, Section I, G

  Number Theory | Part II, 2009

  For any integer $x \geqslant 2$, define $\theta(x)=\sum_{p \leqslant x} \log p$, where the sum is taken over all primes $p \leqslant x$. Put $\theta(1)=0$. By studying the integer

  $$\left(\begin{array}{c} 2 n \\ n \end{array}\right)$$

  where $n \geqslant 1$ is an integer, prove that

  $$\theta(2 n)-\theta(n)<2 n \log 2$$

  Deduce that

  $$\theta(x)<(4 \log 2) x$$

  for all $x \geqslant 1$.

  comment

• Paper 3, Section II, G

  Number Theory | Part II, 2009

  Let $p$ be an odd prime. Prove that there is an equal number of quadratic residues and non-residues in the set $\{1, \ldots, p-1\}$.

  If $n$ is an integer prime to $p$, let $m_{n}$ be an integer such that $n m_{n} \equiv 1 \bmod p$. Prove that

  $$n(n+1) \equiv n^{2}\left(1+m_{n}\right) \bmod p$$

  and deduce that

  $$\sum_{n=1}^{p-1}\left(\frac{n(n+1)}{p}\right)=-1$$

  comment

Numerical Analysis

• Paper 3, Section II, B

  Numerical Analysis | Part II, 2009

  Prove that all Toeplitz tridiagonal $M \times M$ matrices $A$ of the form

  $$A=\left[\begin{array}{rrrrr} a & b & & & \\ -b & a & b & & \\ & \ddots & \ddots & \ddots & \\ & & -b & a & b \\ & & & -b & a \end{array}\right]$$

  share the same eigenvectors $\left(\boldsymbol{v}^{(k)}\right)_{k=1}^{M}$, with the components $\boldsymbol{v}_{m}^{(k)}=i^{m} \sin \frac{k m \pi}{M+1}, m=$ $1, \ldots, M$, where $i=\sqrt{-1}$, and find their eigenvalues.

  The advection equation

  $$\frac{\partial u}{\partial t}=\frac{\partial u}{\partial x}, \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant t \leqslant T,$$

  is approximated by the Crank-Nicolson scheme

  $$u_{m}^{n+1}-u_{m}^{n}=\frac{1}{4} \mu\left(u_{m+1}^{n+1}-u_{m-1}^{n+1}\right)+\frac{1}{4} \mu\left(u_{m+1}^{n}-u_{m-1}^{n}\right)$$

  where $\mu=\frac{\Delta t}{(\Delta x)^{2}}, \Delta x=\frac{1}{M+1}$, and $u_{m}^{n}$ is an approximation to $u(m \Delta x, n \Delta t)$. Assuming that $u(0, t)=u(1, t)=0$, show that the above scheme can be written in the form

  $$B \boldsymbol{u}^{n+1}=C \boldsymbol{u}^{n}, \quad 0 \leqslant n \leqslant T / \Delta t-1$$

  where $\boldsymbol{u}^{n}=\left[u_{1}^{n}, \ldots, u_{M}^{n}\right]^{T}$ and the real matrices $B$ and $C$ should be found. Using matrix analysis, find the range of $\mu$ for which the scheme is stable. [Fourier analysis is not acceptable.]

  comment

Optimization and Control

• Paper 3, Section II, I

  Optimization and Control | Part II, 2009

  Two scalar systems have dynamics

  $$x_{t+1}=x_{t}+u_{t}+\epsilon_{t}, \quad y_{t+1}=y_{t}+w_{t}+\eta_{t},$$

  where $\left\{\epsilon_{t}\right\}$ and $\left\{\eta_{t}\right\}$ are independent sequences of independent and identically distributed random variables of mean 0 and variance 1 . Let

  $$F(x)=\inf _{\pi} \mathbb{E}\left[\sum_{t=0}^{\infty}\left(x_{t}^{2}+u_{t}^{2}\right)(2 / 3)^{t} \mid x_{0}=x\right]$$

  where $\pi$ is a policy in which $u_{t}$ depends on only $x_{0}, \ldots, x_{t}$.

  Show that $G(x)=P x^{2}+d$ is a solution to the optimality equation satisfied by $F(x)$, for some $P$ and $d$ which you should find.

  Find the optimal controls.

  State a theorem that justifies $F(x)=G(x)$.

  For each of the two cases (a) $\lambda=0$ and (b) $\lambda=1$, find controls $\left\{u_{t}, w_{t}\right\}$ which minimize

  $$\mathbb{E}\left[\sum_{t=0}^{\infty}\left(x_{t}^{2}+2 \lambda x_{t} y_{t}+y_{t}^{2}+u_{t}^{2}+w_{t}^{2}\right)(2 / 3+\lambda / 12)^{t} \mid x_{0}=x, y_{0}=y\right]$$

  comment

Partial Differential Equations

• Paper 3, Section II, B

  Partial Differential Equations | Part II, 2009

  (a) Consider the nonlinear elliptic problem

  $$\begin{cases}\Delta u=f(u, x), & x \in \Omega \subseteq \mathbb{R}^{d} \\ u=u_{D}, & x \in \partial \Omega\end{cases}$$

  Let $\frac{\partial f}{\partial u}(y, x) \geqslant 0$ for all $y \in \mathbb{R}, x \in \Omega$. Prove that there exists at most one classical solution.

  [Hint: Use the weak maximum principle.]

  (b) Let $\varphi \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ be a radial function. Prove that the Fourier transform of $\varphi$ is radial too.

  (c) Let $\varphi \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ be a radial function. Solve

  $$-\Delta u+u=\varphi(x), \quad x \in \mathbb{R}^{n}$$

  by Fourier transformation and prove that $u$ is a radial function.

  (d) State the Lax-Milgram lemma and explain its use in proving the existence and uniqueness of a weak solution of

  $$\begin{gathered} -\Delta u+a(x) u=f(x), x \in \Omega \\ u=0 \text { on } \partial \Omega \end{gathered}$$

  where $\Omega \subseteq \mathbb{R}^{d}$ bounded, $0 \leqslant \underline{a} \leqslant a(x) \leqslant \bar{a}<\infty$ for all $x \in \Omega$ and $f \in L^{2}(\Omega)$.

  comment

Principles of Quantum Mechanics

• Paper 3, Section II, C

  Principles of Quantum Mechanics | Part II, 2009

  (i) Consider two quantum systems with angular momentum states $|j m\rangle$ and $|1 q\rangle$. The eigenstates corresponding to their combined angular momentum can be written as

  $$|J M\rangle=\sum_{q m} C_{q m}^{J M}|1 q\rangle|j m\rangle,$$

  where $C_{q m}^{J M}$ are Clebsch-Gordan coefficients for addition of angular momenta one and $j$. What are the possible values of $J$ and how must $q, m$ and $M$ be related for $C_{q m}^{J} M \neq 0$ ?

  Construct all states $|J M\rangle$ in terms of product states in the case $j=\frac{1}{2}$.

  (ii) A general stationary state for an electron in a hydrogen atom $|n \ell m\rangle$ is specified by the principal quantum number $n$ in addition to the labels $\ell$ and $m$ corresponding to the total orbital angular momentum and its component in the 3-direction (electron spin is ignored). An oscillating electromagnetic field can induce a transition to a new state $\left|n^{\prime} \ell^{\prime} m^{\prime}\right\rangle$ and, in a suitable approximation, the amplitude for this to occur is proportional to

  $$\left\langle n^{\prime} \ell^{\prime} m^{\prime}\left|\hat{x}_{i}\right| n \ell m\right\rangle,$$

  where $\hat{x}_{i}(i=1,2,3)$ are components of the electron's position. Give clear but concise arguments based on angular momentum which lead to conditions on $\ell, m, \ell^{\prime}, m^{\prime}$ and $i$ for the amplitude to be non-zero.

  Explain briefly how parity can be used to obtain an additional selection rule.

  [Standard angular momentum states $|j m\rangle$ are joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, obeying

  $$J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle, \quad J_{3}|j m\rangle=m|j m\rangle .$$

  You may also assume that $X_{\pm 1}=\frac{1}{\sqrt{2}}\left(\mp \hat{x}_{1}-i \hat{x}_{2}\right)$ and $X_{0}=\hat{x}_{3}$ have commutation relations with orbital angular momentum $\mathbf{L}$ given by

  $$\left[L_{3}, X_{q}\right]=q X_{q}, \quad\left[L_{\pm}, X_{q}\right]=\sqrt{(1 \mp q)(2 \pm q)} X_{q \pm 1}$$

  Units in which $\hbar=1$ are to be used throughout. ]

  comment

Principles of Statistics

• Paper 3, Section II, I

  Principles of Statistics | Part II, 2009

  What is meant by an equaliser decision rule? What is meant by an extended Bayes rule? Show that a decision rule that is both an equaliser rule and extended Bayes is minimax.

  Let $X_{1}, \ldots, X_{n}$ be independent and identically distributed random variables with the normal distribution $\mathcal{N}\left(\theta, h^{-1}\right)$, and let $k>0$. It is desired to estimate $\theta$ with loss function $L(\theta, a)=1-\exp \left\{-\frac{1}{2} k(a-\theta)^{2}\right\}$.

  Suppose the prior distribution is $\theta \sim \mathcal{N}\left(m_{0}, h_{0}^{-1}\right)$. Find the Bayes act and the Bayes loss posterior to observing $X_{1}=x_{1}, \ldots, X_{n}=x_{n}$. What is the Bayes risk of the Bayes rule with respect to this prior distribution?

  Show that the rule that estimates $\theta$ by $\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}$ is minimax.

  comment

Probability and Measure

• Paper 3, Section II, J

  Probability and Measure | Part II, 2009

  State and prove the first and second Borel-Cantelli lemmas.

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent Cauchy random variables. Thus, each $X_{n}$ is real-valued, with density function

  $$f(x)=\frac{1}{\pi\left(1+x^{2}\right)} .$$

  Show that

  $$\limsup _{n \rightarrow \infty} \frac{\log X_{n}}{\log n}=c, \quad \text { almost surely, }$$

  for some constant $c$, to be determined.

  comment

Representation Theory

• Paper 3, Section II, F

  Representation Theory | Part II, 2009

  Let $G=\mathrm{SU}(2)$. Let $V_{n}$ be the complex vector space of homogeneous polynomials of degree $n$ in two variables $z_{1}, z_{2}$. Define the usual left action of $G$ on $V_{n}$ and denote by $\rho_{n}: G \rightarrow \operatorname{GL}\left(V_{n}\right)$ the representation induced by this action. Describe the character $\chi_{n}$ afforded by $\rho_{n}$.

  Quoting carefully any results you need, show that

  (i) The representation $\rho_{n}$ has dimension $n+1$ and is irreducible for $n \in \mathbb{Z}_{\geqslant 0}$;

  (ii) Every finite-dimensional continuous irreducible representation of $G$ is one of the $\rho_{n}$;

  (iii) $V_{n}$ is isomorphic to its dual $V_{n}^{*}$.

  comment

Riemann Surfaces

• Paper 3, Section II, G

  Riemann Surfaces | Part II, 2009

  (i) Let $f(z)=\sum_{n=1}^{\infty} z^{2^{n}}$. Show that the unit circle is the natural boundary of the function element $(D(0,1), f)$, where $D(0,1)=\{z \in \mathbb{C}:|z|<1\}$.

  (ii) Let $X$ be a connected Riemann surface and $(D, h)$ a function element on $X$ into $\mathbb{C}$. Define a germ of $(D, h)$ at a point $p \in D$. Let $\mathcal{G}$ be the set of all the germs of function elements on $X$ into $\mathbb{C}$. Describe the topology and the complex structure on $\mathcal{G}$, and show that $\mathcal{G}$ is a covering of $X$ (in the sense of complex analysis). Show that there is a oneto-one correspondence between complete holomorphic functions on $X$ into $\mathbb{C}$ and the connected components of $\mathcal{G}$. [You are not required to prove that the topology on $\mathcal{G}$ is secondcountable.]

  comment

Statistical Modelling

• Paper 3, Section $I$, I

  Statistical Modelling | Part II, 2009

  Consider the linear model $Y=X \beta+\varepsilon$, where $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$ and $X$ is an $n \times p$ matrix of full rank $p<n$. Suppose that the parameter $\beta$ is partitioned into $k$ sets as follows: $\beta^{\top}=\left(\beta_{1}^{\top} \cdots \beta_{k}^{\top}\right)$. What does it mean for a pair of sets $\beta_{i}, \beta_{j}, i \neq j$, to be orthogonal? What does it mean for all $k$ sets to be mutually orthogonal?

  In the model

  $$Y_{i}=\beta_{0}+\beta_{1} x_{i 1}+\beta_{2} x_{i 2}+\varepsilon_{i}$$

  where $\varepsilon_{i} \sim N\left(0, \sigma^{2}\right)$ are independent and identically distributed, find necessary and sufficient conditions on $x_{11}, \ldots, x_{n 1}, x_{12}, \ldots, x_{n 2}$ for $\beta_{0}, \beta_{1}$ and $\beta_{2}$ to be mutually orthogonal.

  If $\beta_{0}, \beta_{1}$ and $\beta_{2}$ are mutually orthogonal, what consequence does this have for the joint distribution of the corresponding maximum likelihood estimators $\hat{\beta}_{0}, \hat{\beta}_{1}$ and $\hat{\beta}_{2}$ ?

  comment

Statistical Physics

• Paper 3, Section II, D

  Statistical Physics | Part II, 2009

  Consider an ideal Bose gas in an external potential such that the resulting density of single particle states is given by

  $$g(\varepsilon)=B \varepsilon^{7 / 2}$$

  where $B$ is a positive constant.

  (i) Derive an expression for the critical temperature for Bose-Einstein condensation of a gas of $N$ of these atoms.

  [Recall

  $$\left.\frac{1}{\Gamma(n)} \int_{0}^{\infty} \frac{x^{n-1} \mathrm{~d} x}{z^{-1} e^{x}-1}=\sum_{\ell=1}^{\infty} \frac{z^{\ell}}{\ell^{n}}\right]$$

  (ii) What is the internal energy $E$ of the gas in the condensed state as a function of $N$ and $T$ ?

  (iii) Now consider the high temperature, classical limit instead. How does the internal energy $E$ depend on $N$ and $T$ ?

  comment

Stochastic Financial Models

• Paper 3, Section II, J

  Stochastic Financial Models | Part II, 2009

  What is a Brownian motion? State the assumptions of the Black-Scholes model of an asset price, and derive the time- 0 price of a European call option struck at $K$, and expiring at $T$.

  Find the time- 0 price of a European call option expiring at $T$, but struck at $S_{t}$, where $t \in(0, T)$, and $S_{t}$ is the price of the underlying asset at time $t$.

  comment

Topics in Analysis

• Paper 3, Section I, $2 F$

  Topics in Analysis | Part II, 2009

  (a) If $f:(0,1) \rightarrow \mathbb{R}$ is continuous, prove that there exists a sequence of polynomials $P_{n}$ such that $P_{n} \rightarrow f$ uniformly on compact subsets of $(0,1)$.

  (b) If $f:(0,1) \rightarrow \mathbb{R}$ is continuous and bounded, prove that there exists a sequence of polynomials $Q_{n}$ such that $Q_{n}$ are uniformly bounded on $(0,1)$ and $Q_{n} \rightarrow f$ uniformly on compact subsets of $(0,1)$.

  comment

• Paper 3, Section II, F

  Topics in Analysis | Part II, 2009

  (a) State Runge's theorem on uniform approximation of analytic functions by polynomials.

  (b) Let $\Omega$ be an unbounded, connected, proper open subset of $\mathbb{C}$. For any given compact set $K \subset \mathbb{C} \backslash \Omega$ and any $\zeta \in \Omega$, show that there exists a sequence of complex polynomials converging uniformly on $K$ to the function $f(z)=(z-\zeta)^{-1}$.

  (c) Give an example, with justification, of a connected open subset $\Omega$ of $\mathbb{C}$, a compact subset $K$ of $\mathbb{C} \backslash \Omega$ and a point $\zeta \in \Omega$ such that there is no sequence of complex polynomials converging uniformly on $K$ to the function $f(z)=(z-\zeta)^{-1}$.

  comment

Waves

• Paper 3, Section II, A

  Waves | Part II, 2009

  Starting from the equations of motion for an inviscid, incompressible, stratified fluid of density $\rho_{0}(z)$, where $z$ is the vertical coordinate, derive the dispersion relation

  $$\omega^{2}=\frac{N^{2}\left(k^{2}+\ell^{2}\right)}{\left(k^{2}+\ell^{2}+m^{2}\right)}$$

  for small amplitude internal waves of wavenumber $(k, \ell, m)$, where $N$ is the constant Brunt-Väisälä frequency (which should be defined), explaining any approximations you make. Describe the wave pattern that would be generated by a small body oscillating about the origin with small amplitude and frequency $\omega$, the fluid being otherwise at rest.

  The body continues to oscillate when the fluid has a slowly-varying velocity $[U(z), 0,0]$, where $U^{\prime}(z)>0$. Show that a ray which has wavenumber $\left(k_{0}, 0, m_{0}\right)$ with $m_{0}<0$ at $z=0$ will propagate upwards, but cannot go higher than $z=z_{c}$, where

  $$U\left(z_{c}\right)-U(0)=N\left(k_{0}^{2}+m_{0}^{2}\right)^{-1 / 2}$$

  Explain what happens to the disturbance as $z$ approaches $z_{c}$.

  comment