Part II, 2006, Paper 2

Algebraic Topology

• 2.II.21H

  Algebraic Topology | Part II, 2006

  State the simplicial approximation theorem. Compute the number of 0 -simplices (vertices) in the barycentric subdivision of an $n$-simplex and also compute the number of $n$-simplices. Finally, show that there are at most countably many homotopy classes of continuous maps from the 2-sphere to itself.

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Applications of Quantum Mechanics

• 2.II.33D

  Applications of Quantum Mechanics | Part II, 2006

  State and prove Bloch's theorem for the electron wave functions for a periodic potential $V(\mathbf{r})=V(\mathbf{r}+\mathbf{l})$ where $\mathbf{l}=\sum_{i} n_{i} \mathbf{a}_{i}$ is a lattice vector.

  What is the reciprocal lattice? Explain why the Bloch wave-vector $\mathbf{k}$ is arbitrary up to $\mathbf{k} \rightarrow \mathbf{k}+\mathbf{g}$, where $\mathbf{g}$ is a reciprocal lattice vector.

  Describe in outline why one can expect energy bands $E_{n}(\mathbf{k})=E_{n}(\mathbf{k}+\mathbf{g})$. Explain how $\mathbf{k}$ may be restricted to a Brillouin zone $B$ and show that the number of states in volume $d^{3} k$ is

  $$\frac{2}{(2 \pi)^{3}} \mathrm{~d}^{3} k$$

  Assuming that the velocity of an electron in the energy band with Bloch wave-vector $\mathbf{k}$ is

  $$\mathbf{v}(\mathbf{k})=\frac{1}{\hbar} \frac{\partial}{\partial \mathbf{k}} E_{n}(\mathbf{k})$$

  show that the contribution to the electric current from a full energy band is zero. Given that $n(\mathbf{k})=1$ for each occupied energy level, show that the contribution to the current density is then

  $$\mathbf{j}=-e \frac{2}{(2 \pi)^{3}} \int_{B} \mathrm{~d}^{3} k n(\mathbf{k}) \mathbf{v}(\mathbf{k})$$

  where $-e$ is the electron charge.

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Applied Probability

• 2.II.26J

  Applied Probability | Part II, 2006

  (a) Define a renewal process $\left(X_{t}\right)$ with independent, identically-distributed holding times $S_{1}, S_{2}, \ldots .$ State without proof the strong law of large numbers for $\left(X_{t}\right)$. State without proof the elementary renewal theorem for the mean value $m(t)=\mathbb{E} X_{t}$.

  (b) A circular bus route consists of ten bus stops. At exactly 5am, the bus starts letting passengers in at the main bus station (stop 1). It then proceeds to stop 2 where it stops to let passengers in and out. It continues in this fashion, stopping at stops 3 to 10 in sequence. After leaving stop 10, the bus heads to stop 1 and the cycle repeats. The travel times between stops are exponentially distributed with mean 4 minutes, and the time required to let passengers in and out at each stop are exponentially distributed with mean 1 minute. Calculate approximately the average number of times the bus has gone round its route by $1 \mathrm{pm}$.

  When the driver's shift finishes, at exactly $1 \mathrm{pm}$, he immediately throws all the passengers off the bus if the bus is already stopped, or otherwise, he drives to the next stop and then throws the passengers off. He then drives as fast as he can round the rest of the route to the main bus station. Giving reasons but not proofs, calculate approximately the average number of stops he will drive past at the end of his shift while on his way back to the main bus station, not including either the stop at which he throws off the passengers or the station itself.

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Classical Dynamics

• 2.I.9C

  Classical Dynamics | Part II, 2006

  Two point masses, each of mass $m$, are constrained to lie on a straight line and are connected to each other by a spring of force constant $k$. The left-hand mass is also connected to a wall on the left by a spring of force constant $j$. The right-hand mass is similarly connected to a wall on the right, by a spring of force constant $\ell$, so that the potential energy is

  $$V=\frac{1}{2} k\left(\eta_{1}-\eta_{2}\right)^{2}+\frac{1}{2} j \eta_{1}^{2}+\frac{1}{2} \ell \eta_{2}^{2}$$

  where $\eta_{i}$ is the distance from equilibrium of the $i^{\text {th }}$mass. Derive the equations of motion. Find the frequencies of the normal modes.

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Coding and Cryptography

• 2.I.4G

  Coding and Cryptography | Part II, 2006

  Let $\Sigma_{1}$ and $\Sigma_{2}$ be alphabets of sizes $m$ and $a$. What does it mean to say that an $a$-ary code $f: \Sigma_{1} \rightarrow \Sigma_{2}^{*}$ is decipherable? Show that if $f$ is decipherable then the word lengths $s_{1}, \ldots, s_{m}$ satisfy

  $$\sum_{i=1}^{m} a^{-s_{i}} \leqslant 1$$

  Find a decipherable binary code consisting of codewords 011, 0111, 01111, 11111, and three further codewords of length 2. How do you know the example you have given is decipherable?

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• 2.II.12G

  Coding and Cryptography | Part II, 2006

  Define a cyclic code. Show that there is a bijection between the cyclic codes of length $n$, and the factors of $X^{n}-1$ in $\mathbb{F}_{2}[X]$.

  If $n$ is an odd integer then we can find a finite extension $K$ of $\mathbb{F}_{2}$ that contains a primitive $n$th root of unity $\alpha$. Show that a cyclic code of length $n$ with defining set $\left\{\alpha, \alpha^{2}, \ldots, \alpha^{\delta-1}\right\}$ has minimum distance at least $\delta$. Show that if $n=7$ and $\delta=3$ then we obtain Hamming's original code.

  [You may quote a formula for the Vandermonde determinant without proof.]

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Cosmology

• 2.I.10D

  Cosmology | Part II, 2006

  The total energy of a gas can be expressed in terms of a momentum integral

  $$E=\int_{0}^{\infty} \mathcal{E}(p) \bar{n}(p) d p$$

  where $p$ is the particle momentum, $\mathcal{E}(p)=c \sqrt{p^{2}+m^{2} c^{2}}$ is the particle energy and $\bar{n}(p) d p$ is the average number of particles in the momentum range $p \rightarrow p+d p$. Consider particles in a cubic box of side $L$ with $p \propto L^{-1}$. Explain why the momentum varies as

  $$\frac{d p}{d V}=-\frac{p}{3 V}$$

  Consider the overall change in energy $d E$ due to the volume change $d V$. Given that the volume varies slowly, use the thermodynamic result $d E=-P d V$ (at fixed particle number $N$ and entropy $S$ ) to find the pressure

  $$P=\frac{1}{3 V} \int_{0}^{\infty} p \mathcal{E}^{\prime}(p) \bar{n}(p) d p .$$

  Use this expression to derive the equation of state for an ultrarelativistic gas.

  During the radiation-dominated era, photons remain in equilibrium with energy density $\epsilon_{\gamma} \propto T^{4}$ and number density $n_{\gamma} \propto T^{3}$. Briefly explain why the photon temperature falls inversely with the scale factor, $T \propto a^{-1}$. Discuss the implications for photon number and entropy conservation.

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• 2.II.15D

  Cosmology | Part II, 2006

  (a) Consider a homogeneous and isotropic universe filled with relativistic matter of mass density $\rho(t)$ and scale factor $a(t)$. Consider the energy $E(t) \equiv \rho(t) c^{2} V(t)$ of a small fluid element in a comoving volume $V_{0}$ where $V(t)=a^{3}(t) V_{0}$. Show that for slow (adiabatic) changes in volume, the density will satisfy the fluid conservation equation

  $$\dot{\rho}=-3 \frac{\dot{a}}{a}\left(\rho+P / c^{2}\right)$$

  where $P$ is the pressure.

  (b) Suppose that a flat $(k=0)$ universe is filled with two matter components:

  (i) radiation with an equation of state $P_{\mathrm{r}}=\frac{1}{3} \rho_{\mathrm{r}} c^{2}$.

  (ii) a gas of cosmic strings with an equation of state $P_{\mathrm{s}}=-\frac{1}{3} \rho_{\mathrm{s}} c^{2}$.

  Use the fluid conservation equation to show that the total relativistic mass density behaves as

  $$\rho=\frac{\rho_{\mathrm{r} 0}}{a^{4}}+\frac{\rho_{\mathrm{s} 0}}{a^{2}},$$

  where $\rho_{\mathrm{r} 0}$ and $\rho_{\mathrm{s} 0}$ are respectively the radiation and string densities today (that is, at $t=t_{0}$ when $a\left(t_{0}\right)=1$ ). Assuming that both the Hubble parameter today $H_{0}$ and the ratio $\beta \equiv \rho_{\mathrm{r} 0} / \rho_{\mathrm{s} 0}$ are known, show that the Friedmann equation can be rewritten as

  $$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{H_{0}^{2}}{a^{4}}\left(\frac{a^{2}+\beta}{1+\beta}\right) .$$

  Solve this equation to find the following solution for the scale factor

  $$a(t)=\frac{\left(H_{0} t\right)^{1 / 2}}{(1+\beta)^{1 / 2}}\left[H_{0} t+2 \beta^{1 / 2}(1+\beta)^{1 / 2}\right]^{1 / 2} .$$

  Show that the scale factor has the expected asymptotic behaviour at early times $t \rightarrow 0$.

  Hence show that the age of this universe today is

  $$t_{0}=H_{0}^{-1}(1+\beta)^{1 / 2}\left[(1+\beta)^{1 / 2}-\beta^{1 / 2}\right]$$

  and that the time $t_{\mathrm{eq}}$ of equal radiation and string densities $\left(\rho_{\mathrm{r}}=\rho_{\mathrm{s}}\right)$ is

  $$t_{\mathrm{eq}}=H_{0}^{-1}(\sqrt{2}-1) \beta^{1 / 2}(1+\beta)^{1 / 2}$$

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Differential Geometry

• 2.II $24 \mathrm{H} \quad$

  Differential Geometry | Part II, 2006

  Let $S \subset \mathbb{R}^{3}$ be a surface.

  (a) Define the exponential map $\exp _{p}$ at a point $p \in S$. Assuming that exp ${ }_{p}$ is smooth, show that $\exp _{p}$ is a diffeomorphism in a neighbourhood of the origin in $T_{p} S$.

  (b) Given a parametrization around $p \in S$, define the Christoffel symbols and show that they only depend on the coefficients of the first fundamental form.

  (c) Consider a system of normal co-ordinates centred at $p$, that is, Cartesian coordinates $(x, y)$ in $T_{p} S$ and parametrization given by $(x, y) \mapsto \exp _{p}\left(x e_{1}+y e_{2}\right)$, where $\left\{e_{1}, e_{2}\right\}$ is an orthonormal basis of $T_{p} S$. Show that all of the Christoffel symbols are zero at $p$.

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Dynamical Systems

• 2.I.7E

  Dynamical Systems | Part II, 2006

  Explain what is meant by a strict Lyapunov function on a domain $\mathcal{D}$ containing the origin for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$. Define the domain of stability of a fixed point $\mathbf{x}_{0}$.

  By considering the function $V=\frac{1}{2}\left(x^{2}+y^{2}\right)$ show that the origin is an asymptotically stable fixed point of

  $$\begin{aligned} &\dot{x}=-2 x+y+x^{3}-x y^{2} \\ &\dot{y}=-x-2 y+6 x^{2} y+4 y^{3} \end{aligned}$$

  Show also that its domain of stability includes $x^{2}+y^{2}<\frac{1}{2}$ and is contained in $x^{2}+y^{2} \leqslant 2$.

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• 2.II.14E

  Dynamical Systems | Part II, 2006

  Let $F: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Explain what is meant by saying (a) that $F$ has a horseshoe, (b) that $F$ is chaotic (Glendinning's definition).

  Consider the tent map defined on the interval $[0,1]$ by

  $$F(x)= \begin{cases}\mu x & 0 \leqslant x<\frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}$$

  with $1<\mu \leqslant 2$.

  Find the non-zero fixed point $x_{0}$ and the points $x_{-1}<\frac{1}{2}<x_{-2}$ that satisfy

  $$F^{2}\left(x_{-2}\right)=F\left(x_{-1}\right)=x_{0} .$$

  Sketch a graph of $F$ and $F^{2}$ showing the points corresponding to $x_{-2}, x_{-1}$ and $x_{0}$. Hence show that $F^{2}$ has a horseshoe if $\mu \geqslant 2^{1 / 2}$.

  Explain briefly why $F^{4}$ has a horseshoe when $2^{1 / 4} \leqslant \mu<2^{1 / 2}$ and why there are periodic points arbitrarily close to $x_{0}$ for $\mu \geqslant 2^{1 / 2}$, but no such points for $2^{1 / 4} \leqslant \mu<2^{1 / 2}$.

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Fluid Dynamics II

• 2.II.36B

  Fluid Dynamics II | Part II, 2006

  A very long cylinder of radius a translates steadily at speed $V$ in a direction perpendicular to its axis and parallel to a plane boundary. The centre of the cylinder remains a distance $a+b$ above the plane, where $b \ll a$, and the motion takes place through an incompressible fluid of viscosity $\mu$.

  Consider the force $F$ per unit length parallel to the plane that must be applied to the cylinder to maintain the motion. Explain why $F$ scales according to $F \propto \mu V(a / b)^{1 / 2}$.

  Approximating the lower cylindrical surface by a parabola, or otherwise, determine the velocity and pressure gradient fields in the space between the cylinder and the plane. Hence, by considering the shear stress on the plane, or otherwise, calculate $F$ explicitly.

  [You may use

  $$\int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-1} d x=\pi, \quad \int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-2} d x=\frac{1}{2} \pi \quad \text { and } \quad \int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-3} d x=\frac{3}{8} \pi$$

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Further Complex Methods

• 2.I.8E

  Further Complex Methods | Part II, 2006

  The function $F(t)$ is defined, for $\operatorname{Re} t>-1$, by

  $$F(t)=\int_{0}^{\infty} \frac{u^{t} e^{-u}}{1+u} d u$$

  and by analytic continuation elsewhere in the complex $t$-plane. By considering the integral of a suitable function round a Hankel contour, obtain the analytic continuation of $F(t)$ and hence show that singularities of $F(t)$ can occur only at $z=-1,-2,-3, \ldots$.

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Galois Theory

• 2.II.18H

  Galois Theory | Part II, 2006

  Write an essay on ruler and compass construction.

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General Relativity

• 2.II.35A

  General Relativity | Part II, 2006

  The Schwarzschild metric is

  $$d s^{2}=\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)-\left(1-\frac{2 M}{r}\right) d t^{2}$$

  Writing $u=1 / r$, obtain the equation

  $$\frac{d^{2} u}{d \phi^{2}}+u=3 M u^{2}$$

  determining the spatial orbit of a null (massless) particle moving in the equatorial plane $\theta=\pi / 2$.

  Verify that two solutions of $(*)$ are

  $$\begin{aligned} \text { (i) } u &=\frac{1}{3 M}, \quad \text { and } \\ \text { (ii) } \quad u &=\frac{1}{3 M}-\frac{1}{M} \frac{1}{\cosh \phi+1} . \end{aligned}$$

  What is the significance of solution (i)? Sketch solution (ii) and describe its relation to solution (i).

  Show that, near $\phi=\cosh ^{-1} 2$, one may approximate the solution (ii) by

  $$r \sin \left(\phi-\cosh ^{-1} 2\right) \approx \sqrt{27} M$$

  and hence obtain the impact parameter.

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Geometry and Groups

• 2.I.3F

  Geometry and Groups | Part II, 2006

  Determine whether the following elements of $\operatorname{PSL}_{2}(\mathbb{R})$ are elliptic, parabolic, or hyperbolic. Justify your answers.

  $$\left(\begin{array}{cc} 5 & 8 \\ -2 & -3 \end{array}\right), \quad\left(\begin{array}{cc} -3 & 1 \\ 2 & -1 \end{array}\right)$$

  In the case of the first of these transformations find the fixed points.

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Graph Theory

• 2.II.17F

  Graph Theory | Part II, 2006

  Let $G$ be a bipartite graph with vertex classes $X$ and $Y$. State Hall's necessary condition for $G$ to have a matching from $X$ to $Y$, and prove that it is sufficient.

  Deduce a necessary and sufficient condition for $G$ to have $|X|-d$ independent edges, where $d$ is a natural number.

  Show that the maximum size of a set of independent edges in $G$ is equal to the minimum size of a subset $S \subset V(G)$ such that every edge of $G$ has an end vertex in $S$.

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Integrable Systems

• 2.II.31E

  Integrable Systems | Part II, 2006

  Let $\phi(t)$ satisfy the singular integral equation

  $$\left(t^{4}+t^{3}-t^{2}\right) \frac{\phi(t)}{2}+\frac{\left(t^{4}-t^{3}-t^{2}\right)}{2 \pi i} \oint_{C} \frac{\phi(\tau)}{\tau-t} d \tau=(A-1) t^{3}+t^{2}$$

  where $C$ denotes the circle of radius 2 centred on the origin, $\oint$ denotes the principal value integral and $A$ is a constant. Derive the associated Riemann-Hilbert problem, and compute the canonical solution of the corresponding homogeneous problem.

  Find the value of $A$ such that $\phi(t)$ exists, and compute the unique solution $\phi(t)$ if $A$ takes this value.

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Linear Analysis

• 2.II.22G

  Linear Analysis | Part II, 2006

  Let $X$ be a metric space. Define what it means for a subset $E \subset X$ to be of first or second category. State and prove a version of the Baire category theorem. For $1 \leqslant p \leqslant \infty$, show that the set $\ell_{p}$ is of first category in the normed space $\ell_{r}$ when $r>p$ and $\ell_{r}$ is given its standard norm. What about $r=p$ ?

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Logic and Set Theory

• 2.II.16H

  Logic and Set Theory | Part II, 2006

  Which of the following statements are true, and which false? Justify your answers.

  (a) For any ordinals $\alpha$ and $\beta$ with $\beta \neq 0$, there exist ordinals $\gamma$ and $\delta$ with $\delta<\beta$ such that $\alpha=\beta . \gamma+\delta$.

  (b) For any ordinals $\alpha$ and $\beta$ with $\beta \neq 0$, there exist ordinals $\gamma$ and $\delta$ with $\delta<\beta$ such that $\alpha=\gamma \cdot \beta+\delta$.

  (c) $\alpha \cdot(\beta+\gamma)=\alpha \cdot \beta+\alpha \cdot \gamma$ for all $\alpha, \beta, \gamma$.

  (d) $(\alpha+\beta) \cdot \gamma=\alpha \cdot \gamma+\beta \cdot \gamma$ for all $\alpha, \beta, \gamma$.

  (e) Any ordinal of the form $\omega . \alpha$ is a limit ordinal.

  (f) Any limit ordinal is of the form $\omega . \alpha$.

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Mathematical Biology

• 2.I.6B

  Mathematical Biology | Part II, 2006

  Two interacting populations of prey and predators, with populations $u(t), v(t)$ respectively, obey the evolution equations (with all parameters positive)

  $$\begin{aligned} &\frac{d u}{d t}=u\left(\mu_{1}-\alpha_{1} v-\delta u\right) \\ &\frac{d v}{d t}=v\left(-\mu_{2}+\alpha_{2} u\right)-\epsilon \end{aligned}$$

  Give an explanation in terms of population dynamics of each of the terms in these equations.

  Show that if $\alpha_{2} \mu_{1}>\delta \mu_{2}$ there are two non-trivial fixed points with $u, v \neq 0$, provided $\epsilon$ is sufficiently small. Find the trace and determinant of the Jacobian in terms of $u, v$ and show that, when $\delta$ and $\epsilon$ are very small, the fixed point with $u \approx \mu_{1} / \delta$, $v \approx \epsilon \delta / \mu_{1} \alpha_{2}$ is always unstable.

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• 2.II.13B

  Mathematical Biology | Part II, 2006

  Consider the discrete predator-prey model for two populations $N_{t}, P_{t}$ of prey and predators, respectively:

  $$\left.\begin{array}{rl} N_{t+T} & =r N_{t} \exp \left(-a P_{t}\right) \\ P_{t+T} & =s N_{t}\left(1-b \exp \left(-a P_{t}\right)\right) \end{array}\right\},$$

  where $r, s, a, b$ are constants, all assumed to be positive.

  (a) Give plausible explanations of the meanings of $T, r, s, a, b$.

  (b) Nondimensionalize equations $(*)$ to show that with appropriate rescaling they may be reduced to the form

  $$\left.\begin{array}{l} n_{t+1}=r n_{t} \exp \left(-p_{t}\right) \\ p_{t+1}=n_{t}\left(1-b \exp \left(-p_{t}\right)\right) \end{array}\right\}$$

  (c) Now assume that $b<1, r>1$. Show that the origin is unstable, and that there is a nontrivial fixed point $(n, p)=\left(n_{c}(b, r), p_{c}(b, r)\right)$. Investigate the stability of this point by writing $\left(n_{t}, p_{t}\right)=\left(n_{c}+n_{t}^{\prime}, p_{c}+p_{t}^{\prime}\right)$ and linearizing. Express the linearized equations as a second order recurrence relation for $n_{t}^{\prime}$, and hence show that $n_{t}^{\prime}$ satisfies an equation of the form

  $$n_{t}^{\prime}=A \lambda_{1}^{t}+B \lambda_{2}^{t}$$

  where the quantities $\lambda_{1,2}$ satisfy $\lambda_{1}+\lambda_{2}=1+b n_{c} / r, \lambda_{1} \lambda_{2}=n_{c}$ and $A, B$ are constants. Give a similar expression for $p_{t}^{\prime}$ for the same values of $A, B$.

  Show that when $r$ is just greater than unity the $\lambda_{i}(i=1,2)$ are real and both less than unity, while if $n_{c}$ is just greater than unity then the $\lambda_{i}$ are complex with modulus greater than one. Show also that $n_{c}$ increases monotonically with $r$ and that if the roots are real neither of them can be unity.

  Deduce that the fixed point is stable for sufficiently small $r$ but loses stability for a value of $r$ that depends on $b$ but is certainly less than $e=\exp (1)$. Give an equation that determines the value of $r$ where stability is lost, and an equation that gives the argument of the eigenvalue at this point. Sketch the behaviour of the moduli of the eigenvalues as functions of $r$.

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Number Fields

• 2.II.20G

  Number Fields | Part II, 2006

  Let $K=\mathbb{Q}(\sqrt{26})$ and let $\varepsilon=5+\sqrt{26}$. By Dedekind's theorem, or otherwise, show that the ideal equations

  $$2=[2, \varepsilon+1]^{2}, \quad 5=[5, \varepsilon+1][5, \varepsilon-1], \quad[\varepsilon+1]=[2, \varepsilon+1][5, \varepsilon+1]$$

  hold in $K$. Deduce that $K$ has class number 2 .

  Show that $\varepsilon$ is the fundamental unit in $K$. Hence verify that all solutions in integers $x, y$ of the equation $x^{2}-26 y^{2}=\pm 10$ are given by

  $$x+\sqrt{26} y=\pm \varepsilon^{n}(\varepsilon \pm 1) \quad(n=0, \pm 1, \pm 2, \ldots) .$$

  [It may be assumed that the Minkowski constant for $K$ is $\frac{1}{2}$.]

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Number Theory

• 2.I.1H

  Number Theory | Part II, 2006

  Prove that all binary quadratic forms of discriminant $-7$ are equivalent to $x^{2}+x y+2 y^{2} .$

  Determine which prime numbers $p$ are represented by $x^{2}+x y+2 y^{2}$.

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Numerical Analysis

• 2.II.38C

  Numerical Analysis | Part II, 2006

  In the unit square the Poisson equation $\nabla^{2} u=f$, with zero Dirichlet boundary conditions, is being solved by the five-point formula using a square grid of mesh size $h=1 /(M+1)$,

  $$u_{i, j-1}+u_{i, j+1}+u_{i-1, j}+u_{i+1, j}-4 u_{i, j}=h^{2} f_{i, j} .$$

  Let $u(x, y)$ be the exact solution, and let $e_{i, j}=u_{i, j}-u(i h, j h)$ be the error of the five-point formula at the $(i, j)$ th grid point. Justifying each step, prove that

  $$\left[\sum_{i, j=1}^{M}\left|e_{i, j}\right|^{2}\right]^{1 / 2} \leqslant c h, \quad h \rightarrow 0$$

  where $c$ is some constant.

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Optimization and Control

• 2.II.29I

  Optimization and Control | Part II, 2006

  A policy $\pi$ is to be chosen to maximize

  $$F(\pi, x)=\mathbb{E}_{\pi}\left[\sum_{t \geqslant 0} \beta^{t} r\left(x_{t}, u_{t}\right) \mid x_{0}=x\right]$$

  where $0<\beta \leqslant 1$. Assuming that $r \geqslant 0$, prove that $\pi$ is optimal if $F(\pi, x)$ satisfies the optimality equation.

  An investor receives at time $t$ an income of $x_{t}$ of which he spends $u_{t}$, subject to $0 \leqslant u_{t} \leqslant x_{t}$. The reward is $r\left(x_{t}, u_{t}\right)=u_{t}$, and his income evolves as

  $$x_{t+1}=x_{t}+\left(x_{t}-u_{t}\right) \varepsilon_{t},$$

  where $\left(\varepsilon_{t}\right)_{t \geqslant 0}$ is a sequence of independent random variables with common mean $\theta>0$. If $0<\beta \leqslant 1 /(1+\theta)$, show that the optimal policy is to take $u_{t}=x_{t}$ for all $t$.

  What can you say about the problem if $\beta>1 /(1+\theta) ?$

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Partial Differential Equations

• 2.II.30A

  Partial Differential Equations | Part II, 2006

  Define a fundamental solution of a constant-coefficient linear partial differential operator, and prove that the distribution defined by the function $N: \mathbb{R}^{3} \rightarrow \mathbb{R}$

  $$N(x)=(4 \pi|x|)^{-1}$$

  is a fundamental solution of the operator $-\Delta$ on $\mathbb{R}^{3}$.

  State and prove the mean value property for harmonic functions on $\mathbb{R}^{3}$ and deduce that any two smooth solutions of

  $$-\Delta u=f, \quad f \in C^{\infty}\left(\mathbb{R}^{3}\right)$$

  which satisfy the condition

  $$\lim _{|x| \rightarrow \infty} u(x)=0$$

  are in fact equal

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Principles of Quantum Mechanics

• 2.II.32A

  Principles of Quantum Mechanics | Part II, 2006

  Let $|\uparrow\rangle$ and $|\downarrow\rangle$ denote the eigenstates of $S_{z}$ for a particle of spin $\frac{1}{2}$. Show that

  $$|\uparrow \theta\rangle=\cos \frac{\theta}{2}|\uparrow\rangle+\sin \frac{\theta}{2}|\downarrow\rangle, \quad|\downarrow \theta\rangle=-\sin \frac{\theta}{2}|\uparrow\rangle+\cos \frac{\theta}{2}|\downarrow\rangle$$

  are eigenstates of $S_{z} \cos \theta+S_{x} \sin \theta$ for any $\theta$. Show also that the composite state

  $$|\chi\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle|\downarrow\rangle-|\downarrow\rangle|\uparrow\rangle),$$

  for two spin- $\frac{1}{2}$ particles, is unchanged under a transformation

  $$|\uparrow\rangle \mapsto|\uparrow \theta\rangle, \quad|\downarrow\rangle \mapsto|\downarrow \theta\rangle$$

  applied to all one-particle states. Hence, by considering the action of certain components of the spin operator for the composite system, show that $|\chi\rangle$ is a state of total spin zero.

  Two spin- $\frac{1}{2}$ particles A and B have combined spin zero (as in the state $|\chi\rangle$ above) but are widely separated in space. A magnetic field is applied to particle B in such a way that its spin states are transformed according to $(*)$, for a certain value of $\theta$, while the spin states of particle A are unaffected. Once this has been done, a measurement is made of $S_{z}$ for particle A, followed by a measurement of $S_{z}$ for particle B. List the possible results for this pair of measurements and find the total probability, in terms of $\theta$, for each pair of outcomes to occur. For which outcomes is the two-particle system left in an eigenstate of the combined total spin operator, $S^{2}$, and what is the eigenvalue for each such outcome?

  $$\left[\sigma_{x}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{y}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{z}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right) \cdot\right]$$

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Principles of Statistics

• 2.II.27J

  Principles of Statistics | Part II, 2006

  Let $\{f(\cdot \mid \theta): \theta \in \Theta\}$ be a parametric family of densities for observation $X$. What does it mean to say that the statistic $T \equiv T(X)$ is sufficient for $\theta$ ? What does it mean to say that $T$ is minimal sufficient?

  State the Rao-Blackwell theorem. State the Cramér-Rao lower bound for the variance of an unbiased estimator of a (scalar) parameter, taking care to specify any assumptions needed.

  Let $X_{1}, \ldots, X_{n}$ be a sample from a $U(0, \theta)$ distribution, where the positive parameter $\theta$ is unknown. Find a minimal sufficient statistic $T$ for $\theta$. If $h(T)$ is an unbiased estimator for $\theta$, find the form of $h$, and deduce that this estimator is minimum-variance unbiased. Would it be possible to reach this conclusion using the Cramér-Rao lower bound?

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Probability and Measure

• 2.II $. 25 \mathrm{~J} \quad$

  Probability and Measure | Part II, 2006

  (a) What is meant by saying that $(\Omega, \mathcal{A}, \mu)$ is a measure space? Your answer should include clear definitions of any terms used.

  (b) Consider the following sequence of Borel-measurable functions on the measure space $(\mathbb{R}, \mathcal{L}, \lambda)$, with the Lebesgue $\sigma$-algebra $\mathcal{L}$ and Lebesgue measure $\lambda$ :

  $$f_{n}(x)=\left\{\begin{array}{ll} 1 / n & \text { if } 0 \leqslant x \leqslant e^{n} ; \\ 0 & \text { otherwise } \end{array} \quad \text { for } n \in \mathbb{N}\right.$$

  For each $p \in[1, \infty]$, decide whether the sequence $\left(f_{n}\right)_{n \in \mathbb{N}}$ converges in $L^{p}$ as $n \rightarrow \infty$.

  Does $\left(f_{n}\right)_{n \in \mathbb{N}}$ converge almost everywhere?

  Does $\left(f_{n}\right)_{n \in \mathbb{N}}$ converge in measure?

  Justify your answers.

  For parts (c) and (d), let $\left(f_{n}\right)_{n \in \mathbb{N}}$ be a sequence of real-valued, Borel-measurable functions on a probability space $(\Omega, \mathcal{A}, \mu)$.

  (c) Prove that $\left\{x \in \Omega: f_{n}(x)\right.$ converges to a finite limit $\} \in \mathcal{A}$.

  (d) Show that $f_{n} \rightarrow 0$ almost surely if and only if $\sup _{m \geqslant n}\left|f_{m}\right| \rightarrow 0$ in probability.

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Representation Theory

• 2.II.19F

  Representation Theory | Part II, 2006

  (a) Let $G$ be $S_{4}$, the symmetric group on four letters. Determine the character table of $G$.

  [Begin by listing the conjugacy classes and their orders.]

  (b) For each irreducible representation $V$ of $G=S_{4}$, decompose $\operatorname{Res}_{A_{4}}^{S_{4}}(V)$ into irreducible representations. You must justify your answer.

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Riemann Surfaces

• 2.II.23F

  Riemann Surfaces | Part II, 2006

  Define the terms Riemann surface, holomorphic map between Riemann surfaces, and biholomorphic map.

  (a) Prove that if two holomorphic maps $f, g$ coincide on a non-empty open subset of a connected Riemann surface $R$ then $f=g$ everywhere on $R$.

  (b) Prove that if $f: R \rightarrow S$ is a non-constant holomorphic map between Riemann surfaces and $p \in R$ then there is a choice of co-ordinate charts $\phi$ near $p$ and $\psi$ near $f(p)$, such that $\left(\psi \circ f \circ \phi^{-1}\right)(z)=z^{n}$, for some non-negative integer $n$. Deduce that a holomorphic bijective map between Riemann surfaces is biholomorphic.

  [The inverse function theorem for holomorphic functions on open domains in $\mathbb{C}$ may be used without proof if accurately stated.]

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Statistical Modelling

• 2.I.5I

  Statistical Modelling | Part II, 2006

  Let $Y_{1}, \ldots, Y_{n}$ be independent Poisson random variables with means $\mu_{1}, \ldots, \mu_{n}$, for $i=1, \ldots, n$, where $\log \left(\mu_{i}\right)=\beta x_{i}$, for some known constants $x_{i}$ and an unknown parameter $\beta$. Find the log-likelihood for $\beta$.

  By first computing the first and second derivatives of the log-likelihood for $\beta$, explain the algorithm you would use to find the maximum likelihood estimator, $\hat{\beta}$.

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Statistical Physics

• 2.II.34D

  Statistical Physics | Part II, 2006

  What is meant by the heat capacity $C_{V}$ of a thermodynamic system? By establishing a suitable Maxwell identity, show that

  $$\left.\frac{\partial C_{V}}{\partial V}\right|_{T}=\left.T \frac{\partial^{2} P}{\partial T^{2}}\right|_{V} .$$

  In a certain model of $N$ interacting particles in a volume $V$ and at temperature $T$, the partition function is

  $$Z=\frac{1}{N !}(V-a N)^{N}(b T)^{3 N / 2}$$

  where $a$ and $b$ are constants. Find the equation of state and the entropy for this gas of particles. Find the energy and hence the heat capacity $C_{V}$ of the gas, and verify that the relation $(*)$ is satisfied.

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Stochastic Financial Models

• 2.II.28I

  Stochastic Financial Models | Part II, 2006

  (a) In the context of a single-period financial market with $n$ traded assets and a single riskless asset earning interest at rate $r$, what is an arbitrage? What is an equivalent martingale measure? Explain marginal utility pricing, and how it leads to an equivalent martingale measure.

  (b) Consider the following single-period market with two assets. The first is a riskless bond, worth 1 at time 0 , and 1 at time 1 . The second is a share, worth 1 at time 0 and worth $S_{1}$ at time 1 , where $S_{1}$ is uniformly distributed on the interval $[0, a]$, where $a>0$. Under what condition on $a$ is this model arbitrage free? When it is, characterise the set $\mathcal{E}$ of equivalent martingale measures.

  An agent with $C^{2}$ utility $U$ and with wealth $w$ at time 0 aims to pick the number $\theta$ of shares to hold so as to maximise his expected utility of wealth at time 1 . Show that he will choose $\theta$ to be positive if and only if $a>2$.

  An option pays $\left(S_{1}-1\right)^{+}$at time 1 . Assuming that $a=2$, deduce that the agent's price for this option will be $1 / 4$, and show that the range of possible prices for this option as the pricing measure varies in $\mathcal{E}$ is the interval $\left(0, \frac{1}{2}\right)$.

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Topics in Analysis

• 2.I.2G

  Topics in Analysis | Part II, 2006

  (a) State Chebyshev's equal ripple criterion.

  (b) Let $f:[-1,1] \rightarrow \mathbb{R}$ be defined by

  $$f(x)=\cos 4 \pi x$$

  and let $g$ be a polynomial of degree 7 . Prove that there exists an $x \in[-1,1]$ such that $|f(x)-g(x)| \geqslant 1$.

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• 2.II.11G

  Topics in Analysis | Part II, 2006

  (a) Let $K$ be a closed subset of the unit disc in $\mathbb{C}$. Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a rational function with all its poles of modulus strictly greater than 1 . Explain why $f$ can be approximated uniformly on $K$ by polynomials.

  [Standard results from complex analysis may be assumed.]

  (b) With $K$ as above, define $\Lambda$ to be the set of all $\lambda \in \mathbb{C} \backslash K$ such that the function $(z-\lambda)^{-1}$ can be uniformly approximated on $K$ by polynomials. If $\lambda \in \Lambda$, prove that there is some $\delta>0$ such that $\mu \in \Lambda$ whenever $|\lambda-\mu|<\delta$.

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