Part II, 2006, Paper 3

Algebraic Topology

• 3.II.20H

  Algebraic Topology | Part II, 2006

  Let $X$ be the union of two circles identified at a point: the "figure eight". Classify all the connected double covering spaces of $X$. If we view these double coverings just as topological spaces, determine which of them are homeomorphic to each other and which are not.

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Applications of Quantum Mechanics

• 3.II.33A

  Applications of Quantum Mechanics | Part II, 2006

  Consider a one-dimensional crystal of lattice space $b$, with atoms having positions $x_{s}$ and momenta $p_{s}, s=0,1,2, \ldots, N-1$, such that the classical Hamiltonian is

  $$H=\sum_{s=0}^{N-1}\left(\frac{p_{s}^{2}}{2 m}+\frac{1}{2} m \lambda^{2}\left(x_{s+1}-x_{s}-b\right)^{2}\right)$$

  where we identify $x_{N}=x_{0}$. Show how this may be quantized to give the energy eigenstates consisting of a ground state $|0\rangle$ together with free phonons with energy $\hbar \omega\left(k_{r}\right)$ where $k_{r}=2 \pi r / N b$ for suitable integers $r$. Obtain the following expression for the quantum operator $x_{s}$

  $$x_{s}=s b+\left(\frac{\hbar}{2 m N}\right)^{\frac{1}{2}} \sum_{r} \frac{1}{\sqrt{\omega\left(k_{r}\right)}}\left(a_{r} e^{i k_{r} s b}+a_{r}^{\dagger} e^{-i k_{r} s b}\right)$$

  where $a_{r}, a_{r}^{\dagger}$ are annihilation and creation operators, respectively.

  An interaction involves the matrix element

  $$M=\sum_{s=0}^{N-1}\left\langle 0\left|e^{i q x_{s}}\right| 0\right\rangle .$$

  Calculate this and show that $|M|^{2}$ has its largest value when $q=2 \pi n / b$ for integer $n$.

  Disregard the case $\omega\left(k_{r}\right)=0$.

  [You may use the relations

  $$\sum_{s=0}^{N-1} e^{i k_{r} s b}= \begin{cases}N, & r=N b \\ 0 & \text { otherwise }\end{cases}$$

  and $e^{A+B}=e^{A} e^{B} e^{-\frac{1}{2}[A, B]}$ if $[A, B]$ commutes with $A$ and with $\left.B .\right]$

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Applied Probability

• 3.II.25J

  Applied Probability | Part II, 2006

  A passenger plane with $N$ numbered seats is about to take off; $N-1$ seats have already been taken, and now the last passenger enters the cabin. The first $N-1$ passengers were advised by the crew, rather imprudently, to take their seats completely at random, but the last passenger is determined to sit in the place indicated on his ticket. If his place is free, he takes it, and the plane is ready to fly. However, if his seat is taken, he insists that the occupier vacates it. In this case the occupier decides to follow the same rule: if the free seat is his, he takes it, otherwise he insists on his place being vacated. The same policy is then adopted by the next unfortunate passenger, and so on. Each move takes a random time which is exponentially distributed with mean $\mu^{-1}$. What is the expected duration of the plane delay caused by these displacements?

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Asymptotic Methods

• 3.II $. 30$ B

  Asymptotic Methods | Part II, 2006

  The Airy function $\operatorname{Ai}(z)$ is defined by

  $$\operatorname{Ai}(z)=\frac{1}{2 \pi i} \int_{C} \exp \left(-\frac{1}{3} t^{3}+z t\right) d t$$

  where the contour $C$ begins at infinity along the ray $\arg (t)=4 \pi / 3$ and ends at infinity along the ray $\arg (t)=2 \pi / 3$. Restricting attention to the case where $z$ is real and positive, use the method of steepest descent to obtain the leading term in the asymptotic expansion for $\operatorname{Ai}(z)$ as $z \rightarrow \infty$ :

  $$\operatorname{Ai}(z) \sim \frac{\exp \left(-\frac{2}{3} z^{3 / 2}\right)}{2 \pi^{1 / 2} z^{1 / 4}}$$

  $\left[\right.$ Hint: put $\left.t=z^{1 / 2} \tau .\right]$

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Classical Dynamics

• 3.I.9C

  Classical Dynamics | Part II, 2006

  A pendulum of length $\ell$ oscillates in the $x y$ plane, making an angle $\theta(t)$ with the vertical $y$ axis. The pivot is attached to a moving lift that descends with constant acceleration $a$, so that the position of the bob is

  $$x=\ell \sin \theta, \quad y=\frac{1}{2} a t^{2}+\ell \cos \theta .$$

  Given that the Lagrangian for an unconstrained particle is

  $$L=\frac{1}{2} m\left(\dot{x}^{2}+\dot{y}^{2}\right)+m g y,$$

  determine the Lagrangian for the pendulum in terms of the generalized coordinate $\theta$. Derive the equation of motion in terms of $\theta$. What is the motion when $a=g$ ?

  Find the equilibrium configurations for arbitrary $a$. Determine which configuration is stable when

  $$\text { (i) } a<g$$

  and when

  $$\text { (ii) } a>g \text {. }$$

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• 3.II.15C

  Classical Dynamics | Part II, 2006

  A particle of mass $m$ is constrained to move on the surface of a sphere of radius $\ell$.

  The Lagrangian is given in spherical polar coordinates by

  $$L=\frac{1}{2} m \ell^{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+m g \ell \cos \theta,$$

  where gravity $g$ is constant. Find the two constants of the motion.

  The particle is projected horizontally with velocity $v$ from a point whose depth below the centre is $\ell \cos \theta=D$. Find $v$ such that the particle trajectory

  (i) just grazes the horizontal equatorial plane $\theta=\pi / 2$;

  (ii) remains at depth $D$ for all time $t$.

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Coding and Cryptography

• 3.I.4G

  Coding and Cryptography | Part II, 2006

  What does it mean to say that a binary code $C$ has length $n$, size $m$ and minimum distance $d$ ? Let $A(n, d)$ be the largest value of $m$ for which there exists an $[n, m, d]$-code. Prove that

  $$\frac{2^{n}}{V(n, d-1)} \leqslant A(n, d) \leqslant \frac{2^{n}}{V\left(n,\left\lfloor\frac{1}{2}(d-1)\right\rfloor\right)}$$

  where $V(n, r)=\sum_{j=0}^{r}\left(\begin{array}{l}n \\ j\end{array}\right)$.

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• 3.II.12G

  Coding and Cryptography | Part II, 2006

  Describe the RSA system with public key $(N, e)$ and private key $(N, d)$. Briefly discuss the possible advantages or disadvantages of taking (i) $e=2^{16}+1$ or (ii) $d=2^{16}+1$.

  Explain how to factor $N$ when both the private key and public key are known.

  Describe the bit commitment problem, and briefly indicate how RSA can be used to solve it.

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Cosmology

• 3.I.10D

  Cosmology | Part II, 2006

  (a) Consider a spherically symmetric star with outer radius $R$, density $\rho(r)$ and pressure $P(r)$. By balancing the gravitational force on a shell at radius $r$ against the force due to the pressure gradient, derive the pressure support equation

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

  where $m(r)=\int_{0}^{r} \rho\left(r^{\prime}\right) 4 \pi r^{\prime 2} d r^{\prime}$. Show that this implies

  $$\frac{d}{d r}\left(\frac{r^{2}}{\rho} \frac{d P}{d r}\right)=-4 \pi G r^{2} \rho$$

  Suggest appropriate boundary conditions at $r=0$ and $r=R$, together with a brief justification.

  (b) Describe qualitatively the endpoint of stellar evolution for our sun when all its nuclear fuel is spent. Your discussion should briefly cover electron degeneracy pressure and the relevance of stability against inverse beta-decay.

  [Note that $m_{n}-m_{p} \approx 2.6 m_{e}$, where $m_{n}, m_{p}, m_{e}$ are the masses of the neutron, proton and electron respectively.]

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Differential Geometry

• 3.II.23H

  Differential Geometry | Part II, 2006

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

  (a) Define the Gauss map $N: S \rightarrow S^{2}$ of $S$. Given $p \in S$, show that the derivative of $N$,

  $$d N_{p}: T_{p} S \rightarrow T_{N(p)} S^{2}=T_{p} S$$

  is self-adjoint.

  (b) Show that if $N$ is a diffeomorphism, then the Gaussian curvature is positive everywhere. Is the converse true?

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Dynamical Systems

• 3.I.7E

  Dynamical Systems | Part II, 2006

  State the normal-form equations for (a) a saddle-node bifurcation, (b) a transcritical bifurcation, and (c) a pitchfork bifurcation, for a dynamical system $\dot{x}=f(x, \mu)$.

  Consider the system

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

  Compute the extended centre manifold near $x=y=\mu=0$, and the evolution equation on the centre manifold, both correct to second order in $x$ and $\mu$. Deduce the type of bifurcation and show that the equation can be put in normal form, to the same order, by a change of variables of the form $T=\alpha t, X=x-\beta \mu, \tilde{\mu}=\gamma(\mu)$ for suitably chosen $\alpha, \beta$ and $\gamma(\mu)$.

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Fluid Dynamics II

• 3.II.36B

  Fluid Dynamics II | Part II, 2006

  Define the rate of strain tensor $e_{i j}$ in terms of the velocity components $u_{i}$.

  Write down the relation between $e_{i j}$, the pressure $p$ and the stress tensor $\sigma_{i j}$ in an incompressible Newtonian fluid of viscosity $\mu$.

  Prove that $2 \mu e_{i j} e_{i j}$ is the local rate of dissipation per unit volume in the fluid.

  Incompressible fluid of density $\rho$ and viscosity $\mu$ occupies the semi-infinite domain $y>0$ above a rigid plane boundary $y=0$ that oscillates with velocity $(V \cos \omega t, 0,0)$, where $V$ and $\omega$ are constants. The fluid is at rest at $y=\infty$. Determine the velocity field produced by the boundary motion after any transients have decayed.

  Evaluate the time-averaged rate of dissipation in the fluid, per unit area of boundary.

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Further Complex Methods

• 3.I.8E

  Further Complex Methods | Part II, 2006

  Show that, for $b \neq 0$,

  $$\mathcal{P} \int_{0}^{\infty} \frac{\cos u}{u^{2}-b^{2}} d u=-\frac{\pi}{2 b} \sin b$$

  where $\mathcal{P}$ denotes the Cauchy principal value.

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• 3.II.14E

  Further Complex Methods | Part II, 2006

  It is given that the hypergeometric function $F(a, b ; c ; z)$ is the solution of the hypergeometric equation determined by the Papperitz symbol

  $$P\left\{\begin{array}{ccc} 0 & \infty & 1 \\ 0 & a & 0 \\ 1-c & b & c-a-b \end{array}\right\}$$

  that is analytic at $z=0$ and satisfies $F(a, b ; c ; 0)=1$, and that for $\operatorname{Re}(c-a-b)>0$

  $$F(a, b ; c ; 1)=\frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a) \Gamma(c-b)} .$$

  [You may assume that $a, b, c$ are such that $F(a, b ; c ; 1)$ exists.]

  (a) Show, by manipulating Papperitz symbols, that

  $$F(a, b ; c ; z)=(1-z)^{-a} F\left(a, c-b ; c ; \frac{z}{z-1}\right) \quad(|\arg (1-z)|<\pi) .$$

  (b) Let $w_{1}(z)=(-z)^{-a} F\left(a, 1+a-c ; 1+a-b ; \frac{1}{z}\right)$, where $|\arg (-z)|<\pi$. Show that $w_{1}(z)$ satisfies the hypergeometric equation determined by $(*)$.

  (c) By considering the limit $z \rightarrow \infty$ in parts (a) and (b) above, deduce that, for $|\arg (-z)|<\pi$,

  $$F(a, b ; c ; z)=\frac{\Gamma(c) \Gamma(b-a)}{\Gamma(b) \Gamma(c-a)} w_{1}(z)+(\text { a similar term with } a \text { and } b \text { interchanged })$$

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Galois Theory

• 3.II.18H

  Galois Theory | Part II, 2006

  Let $K$ be a field and $m$ a positive integer, not divisible by the characteristic of $K$. Let $L$ be the splitting field of the polynomial $X^{m}-1$ over $K$. Show that $\operatorname{Gal}(L / K)$ is isomorphic to a subgroup of $(\mathbb{Z} / m \mathbb{Z})^{*}$.

  Now assume that $K$ is a finite field with $q$ elements. Show that $[L: K]$ is equal to the order of the residue class of $q$ in the group $(\mathbb{Z} / m \mathbb{Z})^{*}$. Hence or otherwise show that the splitting field of $X^{11}-1$ over $\mathbb{F}_{4}$ has degree 5 .

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Geometry and Groups

• 3.I.3F

  Geometry and Groups | Part II, 2006

  Let $G$ be a discrete subgroup of the Möbius group. Define the limit set of $G$ in $S^{2}$. If $G$ contains two loxodromic elements whose fixed point sets in $S^{2}$ are different, show that the limit set of $G$ contains no isolated points.

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Graph Theory

• 3.II.17F

  Graph Theory | Part II, 2006

  Let $R(s)$ be the least integer $n$ such that every colouring of the edges of $K_{n}$ with two colours contains a monochromatic $K_{s}$. Prove that $R(s)$ exists.

  Prove that a connected graph of maximum degree $d \geqslant 2$ and order $d^{k}$ contains two vertices distance at least $k$ apart.

  Let $C(s)$ be the least integer $n$ such that every connected graph of order $n$ contains, as an induced subgraph, either a complete graph $K_{s}$, a star $K_{1, s}$ or a path $P_{s}$ of length $s$. Show that $C(s) \leqslant R(s)^{s}$.

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Integrable Systems

• 3.II.31E

  Integrable Systems | Part II, 2006

  The solution of the initial value problem of the $\mathrm{KdV}$ equation is given by

  $$q(x, t)=-2 i \lim _{k \rightarrow \infty} k \frac{\partial N}{\partial x}(x, t, k),$$

  where the scalar function $N(x, t, k)$ can be obtained by solving the following RiemannHilbert problem:

  $$\frac{M(x, t, k)}{a(k)}=N(x, t,-k)+\frac{b(k)}{a(k)} \exp \left(2 i k x+8 i k^{3} t\right) N(x, t, k), \quad k \in \mathbb{R},$$

  $M, N$ and $a$ are the boundary values of functions of $k$ that are analytic for $\operatorname{Im} k>0$ and tend to unity as $k \rightarrow \infty$. The functions $a(k)$ and $b(k)$ can be determined from the initial condition $q(x, 0)$.

  Assume that $M$ can be written in the form

  $$\frac{M}{a}=\mathcal{M}(x, t, k)+\frac{c \exp \left(-2 p x+8 p^{3} t\right) N(x, t, i p)}{k-i p}, \quad \operatorname{Im} k \geqslant 0,$$

  where $\mathcal{M}$ as a function of $k$ is analytic for $\operatorname{Im} k>0$ and tends to unity as $k \rightarrow \infty ; c$ and $p$ are constants and $p>0$.

  (a) By solving the above Riemann-Hilbert problem find a linear equation relating $N(x, t, k)$ and $N(x, t, i p)$.

  (b) By solving this equation explicitly in the case that $b=0$ and letting $c=2 i p e^{-2 x_{0}}$, compute the one-soliton solution.

  (c) Assume that $q(x, 0)$ is such that $a(k)$ has a simple zero at $k=i p$. Discuss the dominant form of the solution as $t \rightarrow \infty$ and $x / t=O(1)$.

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Linear Analysis

• 3.II.21G

  Linear Analysis | Part II, 2006

  Let $X$ be a complex Banach space. We say a sequence $x^{i} \in X$ converges to $x \in X$ weakly if $\phi\left(x^{i}\right) \rightarrow \phi(x)$ for all $\phi \in X^{*}$. Let $T: X \rightarrow Y$ be bounded and linear. Show that if $x^{i}$ converges to $x$ weakly, then $T x^{i}$ converges to $T x$ weakly.

  Now let $X=\ell_{2}$. Show that for a sequence $x^{i} \in X, i=1,2, \ldots$, with $\left\|x^{i}\right\| \leqslant 1$, there exists a subsequence $x^{i_{k}}$ such that $x^{i_{k}}$ converges weakly to some $x \in X$ with $\|x\| \leqslant 1$.

  Now let $Y=\ell_{1}$, and show that $y^{i} \in Y$ converges to $y \in Y$ weakly if and only if $y^{i} \rightarrow y$ in the usual sense.

  Define what it means for a linear operator $T: X \rightarrow Y$ to be compact, and deduce from the above that any bounded linear $T: \ell_{2} \rightarrow \ell_{1}$ is compact.

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Logic and Set Theory

• 3.II.16H

  Logic and Set Theory | Part II, 2006

  Explain what is meant by a structure for a first-order signature $\Sigma$, and describe how first-order formulae over $\Sigma$ are interpreted in a given structure. Show that if $B$ is a substructure of $A$, and $\phi$ is a quantifier-free formula (with $n$ free variables), then

  $$\llbracket \phi \rrbracket_{B}=\llbracket \phi \rrbracket_{A} \cap B^{n}$$

  A first-order theory is said to be inductive if its axioms all have the form

  $$\left(\forall x_{1}, \ldots, x_{n}\right)\left(\exists y_{1}, \ldots, y_{m}\right) \phi$$

  where $\phi$ is quantifier-free (and either of the strings $x_{1}, \ldots, x_{n}$ or $y_{1}, \ldots, y_{m}$ may be empty). If $T$ is an inductive theory, and $A$ is a structure for the appropriate signature, show that the poset of those substructures of $A$ which are $T$-models is chain-complete.

  Which of the following can be expressed as inductive theories over the signature with one binary predicate symbol $\leqslant$ ? Justify your answers.

  (a) The theory of totally ordered sets without greatest or least elements.

  (b) The theory of totally ordered sets with greatest and least elements.

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Mathematical Biology

• 3.I.6B

  Mathematical Biology | Part II, 2006

  The SIR epidemic model for an infectious disease divides the population $N$ into three categories of susceptible $S(t)$, infected $I(t)$ and recovered (non-infectious) $R(t)$. It is supposed that the disease is non-lethal, so that the population does not change in time.

  Explain the reasons for the terms in the following model equations:

  $$\frac{d S}{d t}=p R-r I S, \quad \frac{d I}{d t}=r I S-a I, \quad \frac{d R}{d t}=a I-p R$$

  At time $t=0, S \approx N$ while $I, R \ll 1$.

  (a) Show that if $r N<a$ no epidemic occurs.

  (b) Now suppose that $p>0$ and there is an epidemic. Show that the system has a nontrivial fixed point, and that it is stable to small disturbances. Show also that for both small and large $p$ both the trace and the determinant of the Jacobian matrix are $O(p)$, and deduce that the matrix has complex eigenvalues for sufficiently small $p$, and real eigenvalues for sufficiently large $p$.

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• 3.II.13B

  Mathematical Biology | Part II, 2006

  A chemical system with concentrations $u(x, t), v(x, t)$ obeys the coupled reactiondiffusion equations

  $$\begin{aligned} &\frac{d u}{d t}=r u+u^{2}-u v+\kappa_{1} \frac{d^{2} u}{d x^{2}} \\ &\frac{d v}{d t}=s\left(u^{2}-v\right)+\kappa_{2} \frac{d^{2} v}{d x^{2}} \end{aligned}$$

  where $r, s, \kappa_{1}, \kappa_{2}$ are constants with $s, \kappa_{1}, \kappa_{2}$ positive.

  (a) Find conditions on $r, s$ such that there is a steady homogeneous solution $u=u_{0}$, $v=u_{0}^{2}$ which is stable to spatially homogeneous perturbations.

  (b) Investigate the stability of this homogeneous solution to disturbances proportional to $\exp (i k x)$. Assuming that a solution satisfying the conditions of part (a) exists, find the region of parameter space in which the solution is stable to space-dependent disturbances, and show in particular that one boundary of this region for fixed $s$ is given by

  $$d \equiv \sqrt{\frac{\kappa_{2}}{\kappa_{1}}}=\sqrt{2 s}+\frac{1}{u_{0}} \sqrt{s\left(2 u_{0}^{2}-u_{0}\right)}$$

  Sketch the various regions of existence and stability of steady, spatially homogeneous solutions in the $\left(d, u_{0}\right)$ plane for the case $s=2$.

  (c) Show that the critical wavenumber $k=k_{c}$ for the onset of the instability satisfies the relation

  $$k_{c}^{2}=\frac{1}{\sqrt{\kappa_{1} \kappa_{2}}}\left[\frac{s(d-\sqrt{2 s})}{d(2 \sqrt{2 s}-d)}\right] .$$

  Explain carefully what happens when $d<\sqrt{2 s}$ and when $d>2 \sqrt{2 s}$.

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Number Theory

• 3.I.1H

  Number Theory | Part II, 2006

  Let $N=p_{1} p_{2} \ldots p_{r}$ be a product of distinct primes, and let $\lambda(N)$ be the least common multiple of $p_{1}-1, p_{2}-1, \ldots, p_{r}-1$. Prove that

  $$a^{\lambda(N)} \equiv 1 \bmod N \quad \text { when } \quad(a, N)=1 .$$

  Now take $N=7 \times 13 \times 19$, and prove that

  $$a^{N-1} \equiv 1 \bmod N \quad \text { when } \quad(a, N)=1 .$$

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• 3.II.11H

  Number Theory | Part II, 2006

  State the prime number theorem, and Dirichlet's theorem on primes in arithmetic progression.

  If $p$ is an odd prime number, prove that $-1$ is a quadratic residue modulo $p$ if and only if $p \equiv 1 \bmod 4$.

  Let $p_{1}, \ldots, p_{m}$ be distinct prime numbers, and define

  $$N_{1}=4 p_{1} \ldots p_{m}-1, \quad N_{2}=4\left(p_{1} \ldots p_{m}\right)^{2}+1$$

  Prove that $N_{1}$ has at least one prime factor which is congruent to $3 \bmod 4$, and that every prime factor of $N_{2}$ must be congruent to $1 \bmod 4$.

  Deduce that there are infinitely many primes which are congruent to $1 \bmod 4$, and infinitely many primes which are congruent to $3 \bmod 4$.

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Numerical Analysis

• 3.II.38C

  Numerical Analysis | Part II, 2006

  (a) For the equation $y^{\prime}=f(t, y)$, consider the following multistep method with $s$ steps,

  $$\sum_{i=0}^{s} \rho_{i} y_{n+i}=h \sum_{i=0}^{s} \sigma_{i} f\left(t_{n+i}, y_{n+i}\right)$$

  where $h$ is the step size and $\rho_{i}, \sigma_{i}$ are specified constants with $\rho_{s}=1$. Prove that this method is of order $p$ if and only if

  $$\sum_{i=0}^{s} \rho_{i} Q\left(t_{n+i}\right)=h \sum_{i=0}^{s} \sigma_{i} Q^{\prime}\left(t_{n+i}\right)$$

  for any polynomial $Q$ of degree $\leqslant p$. Deduce that there is no $s$-step method of order $2 s+1$.

  [You may use the fact that, for any $a_{i}, b_{i}$, the Hermite interpolation problem

  $$Q\left(x_{i}\right)=a_{i}, \quad Q^{\prime}\left(x_{i}\right)=b_{i}, \quad i=0, \ldots, s$$

  is uniquely solvable in the space of polynomials of degree $2 s+1 .]$

  (b) State the Dahlquist equivalence theorem regarding the convergence of a multistep method. Determine all the values of the real parameter $a \neq 0$ for which the multistep method

  $$y_{n+3}+(2 a-3)\left[y_{n+2}-y_{n+1}\right]-y_{n}=h a\left[f_{n+2}+f_{n+1}\right]$$

  is convergent, and determine the order of convergence.

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Optimization and Control

• 3.II.28I

  Optimization and Control | Part II, 2006

  A discrete-time controlled Markov process evolves according to

  $$X_{t+1}=\lambda X_{t}+u_{t}+\varepsilon_{t}, \quad t=0,1, \ldots,$$

  where the $\varepsilon$ are independent zero-mean random variables with common variance $\sigma^{2}$, and $\lambda$ is a known constant.

  Consider the problem of minimizing

  $$F_{t, T}(x)=\mathbb{E}\left[\sum_{j=t}^{T-1} \beta^{j-t} C\left(X_{j}, u_{j}\right)+\beta^{T-t} R\left(X_{T}\right)\right],$$

  where $C(x, u)=\frac{1}{2}\left(u^{2}+a x^{2}\right), \beta \in(0,1)$ and $R(x)=\frac{1}{2} a_{0} x^{2}+b_{0}$. Show that the optimal control at time $j$ takes the form $u_{j}=k_{T-j} X_{j}$ for certain constants $k_{i}$. Show also that the minimized value for $F_{t, T}(x)$ is of the form

  $$\frac{1}{2} a_{T-t} x^{2}+b_{T-t}$$

  for certain constants $a_{j}, b_{j}$. Explain how these constants are to be calculated. Prove that the equation

  $$f(z) \equiv a+\frac{\lambda^{2} \beta z}{1+\beta z}=z$$

  has a unique positive solution $z=a_{*}$, and that the sequence $\left(a_{j}\right)_{j} \geqslant 0$ converges monotonically to $a_{*}$.

  Prove that the sequence $\left(b_{j}\right)_{j \geqslant 0}$ converges, to the limit

  $$b_{*} \equiv \frac{\beta \sigma^{2} a_{*}}{2(1-\beta)} .$$

  Finally, prove that $k_{j} \rightarrow k_{*} \equiv-\beta a_{*} \lambda /\left(1+\beta a_{*}\right)$.

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Partial Differential Equations

• 3.II.29A

  Partial Differential Equations | Part II, 2006

  Write down the formula for the solution $u=u(t, x)$ for $t>0$ of the initial value problem for the $n$-dimensional heat equation

  $$\begin{gathered} \frac{\partial u}{\partial t}-\Delta u=0 \\ u(0, x)=g(x) \end{gathered}$$

  for $g: \mathbb{R}^{n} \rightarrow \mathbb{C}$ a given smooth bounded function.

  State and prove the Duhamel principle giving the solution $v(t, x)$ for $t>0$ to the inhomogeneous initial value problem

  $$\begin{aligned} &\frac{\partial v}{\partial t}-\Delta v=f \\ &v(0, x)=g(x) \end{aligned}$$

  for $f=f(t, x)$ a given smooth bounded function.

  For the case $n=4$ and when $f=f(x)$ is a fixed Schwartz function (independent of $t)$, find $v(t, x)$ and show that $w(x)=\lim _{t \rightarrow+\infty} v(t, x)$ is a solution of

  $$-\Delta w=f \text {. }$$

  [Hint: you may use without proof the fact that the fundamental solution of the Laplacian on $\mathbb{R}^{4}$ is $\left.-1 /\left(4 \pi^{2}|x|^{2}\right) .\right]$

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Principles of Quantum Mechanics

• 3.II.32D

  Principles of Quantum Mechanics | Part II, 2006

  Consider a Hamiltonian $H$ with known eigenstates and eigenvalues (possibly degenerate). Derive a general method for calculating the energies of a new Hamiltonian $H+\lambda V$ to first order in the parameter $\lambda$. Apply this method to find approximate expressions for the new energies close to an eigenvalue $E$ of $H$, given that there are just two orthonormal eigenstates $|1\rangle$ and $|2\rangle$ corresponding to $E$ and that

  $$\langle 1|V| 1\rangle=\langle 2|V| 2\rangle=\alpha, \quad\langle 1|V| 2\rangle=\langle 2|V| 1\rangle=\beta$$

  A charged particle of mass $m$ moves in two-dimensional space but is confined to a square box $0 \leqslant x, y \leqslant a$. In the absence of any potential within this region the allowed wavefunctions are

  $$\psi_{p q}(x, y)=\frac{2}{a} \sin \frac{p \pi x}{a} \sin \frac{q \pi y}{a}, \quad p, q=1,2, \ldots$$

  inside the box, and zero outside. A weak electric field is now applied, modifying the Hamiltonian by a term $\lambda x y / a^{2}$, where $\lambda m a^{2} / \hbar^{2}$ is small. Show that the three lowest new energy levels for the particle are approximately

  $$\frac{\hbar^{2} \pi^{2}}{m a^{2}}+\frac{\lambda}{4}, \quad \frac{5 \hbar^{2} \pi^{2}}{2 m a^{2}}+\lambda\left(\frac{1}{4} \pm\left(\frac{4}{3 \pi}\right)^{4}\right)$$

  [It may help to recall that $2 \sin \theta \sin \varphi=\cos (\theta-\varphi)-\cos (\theta+\varphi)$.]

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Principles of Statistics

• 3.II.26J

  Principles of Statistics | Part II, 2006

  Write an essay on the rôle of the Metropolis-Hastings algorithm in computational Bayesian inference on a parametric model. You may for simplicity assume that the parameter space is finite. Your essay should:

  (a) explain what problem in Bayesian inference the Metropolis-Hastings algorithm is used to tackle;

  (b) fully justify that the algorithm does indeed deliver the required information about the model;

  (c) discuss any implementational issues that need care.

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Representation Theory

• 3.II.19F

  Representation Theory | Part II, 2006

  (a) Let $G=\mathrm{SU}_{2}$, and let $V_{n}$ be the space of homogeneous polynomials of degree $n$ in the variables $x$ and $y$. Thus $\operatorname{dim} V_{n}=n+1$. Define the action of $G$ on $V_{n}$ and show that $V_{n}$ is an irreducible representation of $G$.

  (b) Decompose $V_{3} \otimes V_{3}$ into irreducible representations. Decompose $\wedge^{2} V_{3}$ and $\mathrm{S}^{2} V_{3}$ into irreducible representations.

  (c) Given any representation $V$ of a group $G$, define the dual representation $V^{*}$. Show that $V_{n}^{*}$ is isomorphic to $V_{n}$ as a representation of $\mathrm{SU}_{2}$.

  [You may use any results from the lectures provided that you state them clearly.]

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Statistical Modelling

• 3.I.5I

  Statistical Modelling | Part II, 2006

  Consider a generalized linear model for independent observations $Y_{1}, \ldots, Y_{n}$, with $\mathbb{E}\left(Y_{i}\right)=\mu_{i}$ for $i=1, \ldots, n$. What is a linear predictor? What is meant by the link function? If $Y_{i}$ has model function (or density) of the form

  $$f\left(y_{i} ; \mu_{i}, \sigma^{2}\right)=\exp \left[\frac{1}{\sigma^{2}}\left\{\theta\left(\mu_{i}\right) y_{i}-K\left(\theta\left(\mu_{i}\right)\right)\right\}\right] a\left(\sigma^{2}, y_{i}\right)$$

  for $y_{i} \in \mathcal{Y} \subseteq \mathbb{R}, \mu_{i} \in \mathcal{M} \subseteq \mathbb{R}, \sigma^{2} \in \Phi \subseteq(0, \infty)$, where $a\left(\sigma^{2}, y_{i}\right)$ is a known positive function, define the canonical link function.

  Now suppose that $Y_{1}, \ldots, Y_{n}$ are independent with $Y_{i} \sim \operatorname{Bin}\left(1, \mu_{i}\right)$ for $i=1, \ldots, n$. Derive the canonical link function.

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Statistical Physics

• 3.II.34D

  Statistical Physics | Part II, 2006

  What is meant by the chemical potential of a thermodynamic system? Derive the Gibbs distribution with variable particle number $N$, for a system at temperature $T$ and chemical potential $\mu$. (You may assume that the volume does not vary.)

  Consider a non-interacting gas of fermions in a box of fixed volume, at temperature $T$ and chemical potential $\mu$. Use the Gibbs distribution to find the mean occupation number of a one-particle quantum state of energy $\varepsilon$. Assuming that the density of states is $C \varepsilon^{1 / 2}$, for some constant $C$, deduce that the mean number of particles with energies between $\varepsilon$ and $\varepsilon+d \varepsilon$ is

  $$\frac{C \varepsilon^{\frac{1}{2}} d \varepsilon}{e^{(\varepsilon-\mu) / T}+1} .$$

  Why can $\mu$ be identified with the Fermi energy $\varepsilon_{F}$ when $T=0$ ? Estimate the number of particles with energies greater than $\varepsilon_{F}$ when $T$ is small but non-zero.

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Stochastic Financial Models

• 3.II.27I

  Stochastic Financial Models | Part II, 2006

  Let $r$ denote the riskless rate and let $\sigma>0$ be a fixed volatility parameter.

  (a) Let $\left(S_{t}\right)_{t \geqslant 0}$ be a Black-Scholes asset with zero dividends:

  $$S_{t}=S_{0} \exp \left(\sigma B_{t}+\left(r-\sigma^{2} / 2\right) t\right),$$

  where $B$ is standard Brownian motion. Derive the Black-Scholes partial differential equation for the price of a European option on $S$ with bounded payoff $\varphi\left(S_{T}\right)$ at expiry $T$ :

  $$\partial_{t} V+\frac{1}{2} \sigma^{2} S^{2} \partial_{S S} V+r S \partial_{S} V-r V=0, \quad V(T, \cdot)=\varphi(\cdot)$$

  [You may use the fact that for $C^{2}$ functions $f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ satisfying exponential growth conditions, and standard Brownian motion $B$, the process

  $$C_{t}^{f}=f\left(t, B_{t}\right)-\int_{0}^{t}\left(\partial_{t} f+\frac{1}{2} \partial_{B B} f\right)\left(s, B_{s}\right) d s$$

  is a martingale.]

  (b) Indicate the changes in your argument when the asset pays dividends continuously at rate $D>0$. Find the corresponding Black-Scholes partial differential equation.

  (c) Assume $D=0$. Find a closed form solution for the time-0 price of a European power option with payoff $S_{T}^{n}$.

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Topics in Analysis

• 3.I.2G

  Topics in Analysis | Part II, 2006

  Let $a_{0}, a_{1}, a_{2}, \ldots$ be positive integers and, for each $n$, let

  $$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+} \cdot \ddots+\frac{1}{a_{n}}}$$

  with $\left(p_{n}, q_{n}\right)=1$.

  Obtain an expression for the matrix $\left(\begin{array}{cc}p_{n} & p_{n-1} \\ q_{n} & q_{n-1}\end{array}\right)$ and use it to show that $p_{n} q_{n-1}-q_{n} p_{n-1}=(-1)^{n+1} .$

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