Part II, 2005, Paper 3

Algebraic Topology

• 3.II.20H

  Algebraic Topology | Part II, 2005

  Let $X$ be a space that is triangulable as a simplicial complex with no $n$-simplices. Show that any continuous map from $X$ to $S^{n}$ is homotopic to a constant map.

  [General theorems from the course may be used without proof, provided they are clearly stated.]

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Applications of Quantum Mechanics

• 3.II.33B

  Applications of Quantum Mechanics | Part II, 2005

  Let $\{\mathbf{l}\}$ be the set of lattice vectors of some lattice. Define the reciprocal lattice. What is meant by a Bravais lattice?

  Let $\mathbf{i}, \mathbf{j}, \mathbf{k}$ be mutually orthogonal unit vectors. A crystal has identical atoms at positions given by the vectors

  $$\begin{array}{ll} a\left[n_{1} \mathbf{i}+n_{2} \mathbf{j}+n_{3} \mathbf{k}\right], & a\left[\left(n_{1}+\frac{1}{2}\right) \mathbf{i}+\left(n_{2}+\frac{1}{2}\right) \mathbf{j}+n_{3} \mathbf{k}\right] \\ a\left[\left(n_{1}+\frac{1}{2}\right) \mathbf{i}+\mathbf{j}+\left(n_{3}+\frac{1}{2}\right) \mathbf{k}\right], & a\left[n_{1} \mathbf{i}+\left(n_{2}+\frac{1}{2}\right) \mathbf{j}+\left(n_{3}+\frac{1}{2}\right) \mathbf{k}\right] \end{array}$$

  where $\left(n_{1}, n_{2}, n_{3}\right)$ are arbitrary integers and $a$ is a constant. Show that these vectors define a Bravais lattice with basis vectors

  $$\mathbf{a}_{1}=a \frac{1}{2}(\mathbf{j}+\mathbf{k}), \quad \mathbf{a}_{2}=a \frac{1}{2}(\mathbf{i}+\mathbf{k}), \quad \mathbf{a}_{3}=a \frac{1}{2}(\mathbf{i}+\mathbf{j})$$

  Verify that a basis for the reciprocal lattice is

  $$\mathbf{b}_{1}=\frac{2 \pi}{a}(\mathbf{j}+\mathbf{k}-\mathbf{i}), \quad \mathbf{b}_{2}=\frac{2 \pi}{a}(\mathbf{i}+\mathbf{k}-\mathbf{j}), \quad \mathbf{b}_{3}=\frac{2 \pi}{a}(\mathbf{i}+\mathbf{j}-\mathbf{k})$$

  In Bragg scattering, an incoming plane wave of wave-vector $\mathbf{k}$ is scattered to an outgoing wave of wave-vector $\mathbf{k}^{\prime}$. Explain why $\mathbf{k}^{\prime}=\mathbf{k}+\mathbf{g}$ for some reciprocal lattice vector g. Given that $\theta$ is the scattering angle, show that

  $$\sin \frac{1}{2} \theta=\frac{|\mathbf{g}|}{2|\mathbf{k}|} .$$

  For the above lattice, explain why you would expect scattering through angles $\theta_{1}$ and $\theta_{2}$ such that

  $$\frac{\sin \frac{1}{2} \theta_{1}}{\sin \frac{1}{2} \theta_{2}}=\frac{\sqrt{3}}{2}$$

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Applied Probability

• 3.II.25I

  Applied Probability | Part II, 2005

  Consider an $\mathrm{M} / \mathrm{G} / r / 0$ loss system with arrival rate $\lambda$ and service-time distribution $F$. Thus, arrivals form a Poisson process of rate $\lambda$, service times are independent with common distribution $F$, there are $r$ servers and there is no space for waiting. Use Little's Lemma to obtain a relation between the long-run average occupancy $L$ and the stationary probability $\pi$ that the system is full.

  Cafe-Bar Duo has 23 serving tables. Each table can be occupied either by one person or two. Customers arrive either singly or in a pair; if a table is empty they are seated and served immediately, otherwise, they leave. The times between arrivals are independent exponential random variables of mean $20 / 3$. Each arrival is twice as likely to be a single person as a pair. A single customer stays for an exponential time of mean 20 , whereas a pair stays for an exponential time of mean 30 ; all these times are independent of each other and of the process of arrivals. The value of orders taken at each table is a constant multiple $2 / 5$ of the time that it is occupied.

  Express the long-run rate of revenue of the cafe as a function of the probability $\pi$ that an arriving customer or pair of customers finds the cafe full.

  By imagining a cafe with infinitely many tables, show that $\pi \leqslant \mathbb{P}(N \geqslant 23)$ where $N$ is a Poisson random variable of parameter $7 / 2$. Deduce that $\pi$ is very small. [Credit will be given for any useful numerical estimate, an upper bound of $10^{-3}$ being sufficient for full credit.]

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Asymptotic Methods

• 3.II $. 30$

  Asymptotic Methods | Part II, 2005

  Explain, without proof, how to obtain an asymptotic expansion, as $x \rightarrow \infty$, of

  $$I(x)=\int_{0}^{\infty} e^{-x t} f(t) d t$$

  if it is known that $f(t)$ possesses an asymptotic power series as $t \rightarrow 0$.

  Indicate the modification required to obtain an asymptotic expansion, under suitable conditions, of

  $$\int_{-\infty}^{\infty} e^{-x t^{2}} f(t) d t$$

  Find an asymptotic expansion as $z \rightarrow \infty$ of the function defined by

  $$I(z)=\int_{-\infty}^{\infty} \frac{e^{-t^{2}}}{(z-t)} d t \quad(\operatorname{Im}(z)<0)$$

  and its analytic continuation to $\operatorname{Im}(z) \geqslant 0$. Where are the Stokes lines, that is, the critical lines separating the Stokes regions?

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Classical Dynamics

• 3.I.9C

  Classical Dynamics | Part II, 2005

  Define the Poisson bracket $\{f, g\}$ between two functions $f\left(q_{a}, p_{a}\right)$ and $g\left(q_{a}, p_{a}\right)$ on phase space. If $f\left(q_{a}, p_{a}\right)$ has no explicit time dependence, and there is a Hamiltonian $H$, show that Hamilton's equations imply

  $$\frac{d f}{d t}=\{f, H\} .$$

  A particle with position vector $\mathbf{x}$ and momentum $\mathbf{p}$ has angular momentum $\mathbf{L}=\mathbf{x} \times \mathbf{p}$. Compute $\left\{p_{a}, L_{b}\right\}$ and $\left\{L_{a}, L_{b}\right\}$.

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• 3.II.15C

  Classical Dynamics | Part II, 2005

  (i) A point mass $m$ with position $q$ and momentum $p$ undergoes one-dimensional periodic motion. Define the action variable $I$ in terms of $q$ and $p$. Prove that an orbit of energy $E$ has period

  $$T=2 \pi \frac{d I}{d E} .$$

  (ii) Such a system has Hamiltonian

  $$H(q, p)=\frac{p^{2}+q^{2}}{\mu^{2}-q^{2}}$$

  where $\mu$ is a positive constant and $|q|<\mu$ during the motion. Sketch the orbits in phase space both for energies $E \gg 1$ and $E \ll 1$. Show that the action variable $I$ is given in terms of the energy $E$ by

  $$I=\frac{\mu^{2}}{2} \frac{E}{\sqrt{E+1}} .$$

  Hence show that for $E \gg 1$ the period of the orbit is $T \approx \frac{1}{2} \pi \mu^{3} / p_{0}$, where $p_{0}$ is the greatest value of the momentum during the orbit.

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Coding and Cryptography

• 3.I.4J

  Coding and Cryptography | Part II, 2005

  Briefly explain how and why a signature scheme is used. Describe the el Gamal scheme.

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• 3.II.12J

  Coding and Cryptography | Part II, 2005

  Define a cyclic code. Define the generator and check polynomials of a cyclic code and show that they exist.

  Show that Hamming's original code is a cyclic code with check polynomial $X^{4}+X^{2}+X+1$. What is its generator polynomial? Does Hamming's original code contain a subcode equivalent to its dual?

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Cosmology

• 3.I.10D

  Cosmology | Part II, 2005

  (a) Define and discuss the concept of the cosmological horizon and the Hubble radius for a homogeneous isotropic universe. Illustrate your discussion with the specific examples of the Einstein-de Sitter universe $\left(a \propto t^{2 / 3}\right.$ for $\left.t>0\right)$ and a de Sitter universe $\left(a \propto e^{H t}\right.$ with $H$ constant, $t>-\infty)$.

  (b) Explain the horizon problem for a decelerating universe in which $a(t) \propto t^{\alpha}$ with $\alpha<1$. How can inflation cure the horizon problem?

  (c) Consider a Tolman (radiation-filled) universe $\left(a(t) \propto t^{1 / 2}\right.$ ) beginning at $t_{\mathrm{r}} \sim$ $10^{-35} \mathrm{~s}$ and lasting until today at $t_{0} \approx 10^{17} \mathrm{~s}$. Estimate the horizon size today $d_{H}\left(t_{0}\right)$ and project this lengthscale backwards in time to show that it had a physical size of about 1 metre at $t \approx t_{\mathrm{r}}$.

  Prior to $t \approx t_{\mathrm{r}}$, assume an inflationary (de Sitter) epoch with constant Hubble parameter $H$ given by its value at $t \approx t_{\mathrm{r}}$ for the Tolman universe. How much expansion during inflation is required for the observable universe today to have begun inside one Hubble radius?

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Differential Geometry

• 3.II.23H

  Differential Geometry | Part II, 2005

  (i) Define geodesic curvature and state the Gauss-Bonnet theorem.

  (ii) Let $\alpha: I \rightarrow \mathbb{R}^{3}$ be a closed regular curve parametrized by arc-length, and assume that $\alpha$ has non-zero curvature everywhere. Let $n: I \rightarrow S^{2} \subset \mathbb{R}^{3}$ be the curve given by the normal vector $n(s)$ to $\alpha(s)$. Let $\bar{s}$ be the arc-length of the curve $n$ on $S^{2}$. Show that the geodesic curvature $k_{g}$ of $n$ is given by

  $$k_{g}=-\frac{d}{d s} \tan ^{-1}(\tau / k) \frac{d s}{d \bar{s}},$$

  where $k$ and $\tau$ are the curvature and torsion of $\alpha$.

  (iii) Suppose now that $n(s)$ is a simple curve (i.e. it has no self-intersections). Show that $n(I)$ divides $S^{2}$ into two regions of equal area.

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Dynamical Systems

• 3.I.7B

  Dynamical Systems | Part II, 2005

  Define the stable and unstable invariant subspaces of the linearisation of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ at a saddle point located at the origin in $\mathbb{R}^{n}$. How, according to the Stable Manifold Theorem, are the stable and unstable manifolds related to the invariant subspaces?

  Calculate the stable and unstable manifolds, correct to cubic order, for the system

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

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Electrodynamics

• 3.II $. 35 \mathrm{~B} \quad$

  Electrodynamics | Part II, 2005

  A non-relativistic particle of rest mass $m$ and charge $q$ is moving slowly with velocity $\mathbf{v}(t)$. The power $d P / d \Omega$ radiated per unit solid angle in the direction of a unit vector $\mathbf{n}$ is

  $$\frac{d P}{d \Omega}=\frac{\mu_{0}}{16 \pi^{2}}|\mathbf{n} \times q \dot{\mathbf{v}}|^{2} .$$

  Obtain Larmor's formula

  $$P=\frac{\mu_{0} q^{2}}{6 \pi}|\dot{\mathbf{v}}|^{2} .$$

  The particle has energy $\mathcal{E}$ and, starting from afar, makes a head-on collision with a fixed central force described by a potential $V(r)$, where $V(r)>\mathcal{E}$ for $r<r_{0}$ and $V(r)<\mathcal{E}$ for $r>r_{0}$. Let $W$ be the total energy radiated by the particle. Given that $W \ll \mathcal{E}$, show that

  $$W \approx \frac{\mu_{0} q^{2}}{3 \pi m^{2}} \sqrt{\frac{m}{2}} \int_{r_{0}}^{\infty}\left(\frac{d V}{d r}\right)^{2} \frac{d r}{\sqrt{V\left(r_{0}\right)-V(r)}}$$

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Further Complex Methods

• 3.I.8A

  Further Complex Methods | Part II, 2005

  The functions $f$ and $g$ have Laplace transforms $\widehat{f}$ and $\widehat{g}$, and satisfy $f(t)=0=g(t)$ for $t<0$. The convolution $h$ of $f$ and $g$ is defined by

  $$h(u)=\int_{0}^{u} f(u-v) g(v) d v$$

  and has Laplace transform $\widehat{h}$. Prove (the convolution theorem) that $\widehat{h}(p)=\widehat{f}(p) \widehat{g}(p)$.

  Given that $\int_{0}^{t}(t-s)^{-1 / 2} s^{-1 / 2} d s=\pi \quad(t>0)$, deduce the Laplace transform of the function $f(t)$, where

  $$f(t)=\left\{\begin{array}{l} t^{-1 / 2}, \quad t>0 \\ 0, \quad t \leqslant 0 \end{array}\right.$$

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• 3.II.14A

  Further Complex Methods | Part II, 2005

  Show that the equation

  $$z w^{\prime \prime}+2 k w^{\prime}+z w=0,$$

  where $k$ is constant, has solutions of the form

  $$w(z)=\int_{\gamma}\left(t^{2}+1\right)^{k-1} e^{z t} d t$$

  provided that the path $\gamma$ is chosen so that $\left[\left(t^{2}+1\right)^{k} e^{z t}\right]_{\gamma}=0$.

  (i) In the case Re $k>0$, show that there is a choice of $\gamma$ for which $w(0)=i B\left(k, \frac{1}{2}\right)$.

  (ii) In the case $k=n / 2$, where $n$ is any integer, show that $\gamma$ can be a finite contour and that the corresponding solution satisfies $w(0)=0$ if $n \leqslant-1$.

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Galois Theory

• 3.II.18G

  Galois Theory | Part II, 2005

  Find the Galois group of the polynomial

  $$x^{4}+x+1$$

  over $\mathbb{F}_{2}$ and $\mathbb{F}_{3}$. Hence or otherwise determine the Galois group over $\mathbb{Q}$.

  [Standard general results from Galois theory may be assumed.]

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Geometry of Group Actions

• 3.I.3G

  Geometry of Group Actions | Part II, 2005

  By considering fixed points in $\mathbb{C} \cup\{\infty\}$, prove that any complex Möbius transformation is conjugate either to a map of the form $z \mapsto k z$ for some $k \in \mathbb{C}$ or to $z \mapsto z+1$. Deduce that two Möbius transformations $g, h$ (neither the identity) are conjugate if and only if $\operatorname{tr}^{2}(g)=\operatorname{tr}^{2}(h)$.

  Does every Möbius transformation $g$ also have a fixed point in $\mathbb{H}^{3}$ ? Briefly justify your answer.

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Graph Theory

• 3.II.17F

  Graph Theory | Part II, 2005

  Let $X$ and $Y$ be disjoint sets of $n \geqslant 6$ vertices each. Let $G$ be a bipartite graph formed by adding edges between $X$ and $Y$ randomly and independently with probability $p=1 / 100$. Let $e(U, V)$ be the number of edges of $G$ between the subsets $U \subset X$ and $V \subset Y$. Let $k=\left\lceil n^{1 / 2}\right\rceil$. Consider three events $\mathcal{A}, \mathcal{B}$ and $\mathcal{C}$, as follows.

  $$\begin{array}{ll} \mathcal{A}: & \text { there exist } U \subset X, V \subset Y \text { with }|U|=|V|=k \text { and } e(U, V)=0 \\ \mathcal{B}: & \text { there exist } x \in X, W \subset Y \text { with }|W|=n-k \text { and } e(\{x\}, W)=0 \\ \mathcal{C}: & \text { there exist } Z \subset X, y \in Y \text { with }|Z|=n-k \text { and } e(Z,\{y\})=0 \end{array}$$

  Show that $\operatorname{Pr}(\mathcal{A}) \leqslant n^{2 k}(1-p)^{k^{2}}$ and $\operatorname{Pr}(\mathcal{B} \cup \mathcal{C}) \leqslant 2 n^{k+1}(1-p)^{n-k}$. Hence show that $\operatorname{Pr}(\mathcal{A} \cup \mathcal{B} \cup \mathcal{C})<3 n^{2 k}(1-p)^{n / 2}$ and so show that, almost surely, none of $\mathcal{A}, \mathcal{B}$ or $\mathcal{C}$ occur. Deduce that, almost surely, $G$ has a matching from $X$ to $Y$.

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Integrable Systems

• 3.II.31A

  Integrable Systems | Part II, 2005

  Let $Q(x, t)$ be an off-diagonal $2 \times 2$ matrix. The matrix NLS equation

  $$i Q_{t}-Q_{x x} \sigma_{3}+2 Q^{3} \sigma_{3}=0, \quad \sigma_{3}=\operatorname{diag}(1,-1),$$

  admits the Lax pair

  $$\begin{aligned} &\mu_{x}+i k\left[\sigma_{3}, \mu\right]=Q \mu \\ &\mu_{t}+2 i k^{2}\left[\sigma_{3}, \mu\right]=\left(2 k Q-i Q^{2} \sigma_{3}-i Q_{x} \sigma_{3}\right) \mu \end{aligned}$$

  where $k \in \mathbb{C}, \mu(x, t, k)$ is a $2 \times 2$ matrix and $\left[\sigma_{3}, \mu\right]$ denotes the matrix commutator.

  Let $S(k)$ be a $2 \times 2$ matrix-valued function decaying as $|k| \rightarrow \infty$. Let $\mu(x, t, k)$ satisfy the $2 \times 2$-matrix Riemann-Hilbert problem

  $$\begin{gathered} \mu^{+}(x, t, k)=\mu^{-}(x, t, k) e^{-i\left(k x+2 k^{2} t\right) \sigma_{3}} S(k) e^{i\left(k x+2 k^{2} t\right) \sigma_{3}}, \quad k \in \mathbb{R} \\ \mu=\operatorname{diag}(1,1)+\mathrm{O}\left(\frac{1}{k}\right), \quad k \rightarrow \infty \end{gathered}$$

  (a) Find expressions for $Q(x, t), A(x, t)$ and $B(x, t)$, in terms of the coefficients in the large $k$ expansion of $\mu$, so that $\mu$ solves

  $$\mu_{x}+i k\left[\sigma_{3}, \mu\right]-Q \mu=0$$

  and

  $$\mu_{t}+2 i k^{2}\left[\sigma_{3}, \mu\right]-(k A+B) \mu=0$$

  (b) Use the result of (a) to establish that

  $$A=2 Q, \quad B=-i\left(Q^{2}+Q_{x}\right) \sigma_{3}$$

  (c) Show that the above results provide a linearization of the matrix NLS equation. What is the disadvantage of this approach in comparison with the inverse scattering method?

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Linear Analysis

• 3.II.21F

  Linear Analysis | Part II, 2005

  Let $X$ be a normed vector space. Define the dual $X^{*}$ of $X$. Define the normed vector spaces $l^{s}=l^{s}(\mathbb{C})$ for all $1 \leqslant s \leqslant \infty$. [You are not required to prove that the norms you have given are indeed norms.]

  Now let $1<p, q<\infty$ be such that $p^{-1}+q^{-1}=1$. Show that $\left(l^{q}\right)^{*}$ is isometrically isomorphic to $l^{p}$ as a normed vector space. [You may assume any standard inequalities.]

  Show by a similar argument that $\left(l^{1}\right)^{*}$ is isomorphic to $l^{\infty}$. Does your argument also show that $\left(l^{\infty}\right)^{*}$ is isomorphic to $l^{1}$ ? If not, where does it fail?

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Logic and Set Theory

• 3.II.16F

  Logic and Set Theory | Part II, 2005

  State the Axiom of Foundation and the Principle of $\in$-Induction, and show that they are equivalent (in the presence of the other axioms of ZF). [You may assume the existence of transitive closures.]

  Explain briefly how the Principle of $\in$-Induction implies that every set is a member of some $V_{\alpha}$.

  For each natural number $n$, find the cardinality of $V_{n}$. For which ordinals $\alpha$ is the cardinality of $V_{\alpha}$ equal to that of the reals?

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Mathematical Biology

• 3.II.13E

  Mathematical Biology | Part II, 2005

  Protein synthesis by RNA can be represented by the stochastic system

  $$\begin{array}{lll} x_{1} \stackrel{\lambda_{1}}{\longrightarrow} x_{1}+1 & \text { and } & x_{1} \stackrel{\beta_{1} x_{1}}{\longrightarrow} x_{1}-1 \\ x_{2} \stackrel{\lambda_{2} x_{1}}{\longrightarrow} x_{2}+1 & \text { and } & x_{2} \stackrel{\beta_{2} x_{2}}{\longrightarrow} x_{2}-1 \end{array}$$

  in which $x_{1}$ is an environmental variable corresponding to the number of RNA molecules per cell and $x_{2}$ is a system variable, with birth rate proportional to $x_{1}$, corresponding to the number of protein molecules.

  (a) Use the normalized stationary Fluctuation-Dissipation Theorem (FDT) to calculate the (exact) normalized stationary variances $\eta_{11}=\sigma_{1}^{2} /<x_{1}>^{2}$ and $\eta_{22}=$ $\sigma_{2}^{2} /<x_{2}>^{2}$ in terms of the averages $<x_{1}>$ and $<x_{2}>$.

  (b) Separate $\eta_{22}$ into an intrinsic and an extrinsic term by considering the limits when $x_{1}$ does not fluctuate (intrinsic), and when $x_{2}$ responds deterministically to changes in $x_{1}$ (extrinsic). Explain how the extrinsic term represents the magnitude of environmental fluctuations and time-averaging.

  (c) Assume now that the birth rate of $x_{2}$ is changed from the "constitutive" mechanism $\lambda_{2} x_{1}$ in (1) to a "negative feedback" mechanism $\lambda_{2} x_{1} f\left(x_{2}\right)$, where $f$ is a monotonically decreasing function of $x_{2}$. Use the stationary FDT to approximate $\eta_{22}$ in terms of $h=\left|\partial \ln f / \partial \ln x_{2}\right|$. Apply your answer to the case $f\left(x_{2}\right)=k / x_{2}$.

  [Hint: To reduce the algebra introduce the elasticity $H_{22}=\partial \ln \left(R_{2}^{-} / R_{2}^{+}\right) / \partial \ln x_{2}$, where $R_{2}^{-}$and $R_{2}^{+}$are the death and birth rates of $x_{2}$ respectively.]

  (d) Explain the extrinsic term for the negative feedback system in terms of environmental fluctuations, time-averaging, and static susceptibility.

  (e) Explain why the FDT is exact for the constitutive system but approximate for the feedback system. When, generally speaking, does the FDT approximation work well?

  (f) Consider the following three experimental observations: (i) Large changes in $\lambda_{2}$ have no effect on $\eta_{22}$; (ii) When $x_{2}$ is perturbed by $1 \%$ from its stationary average, perturbations are corrected more rapidly in the feedback system than in the constitutive system; (iii) The feedback system displays lower values $\eta_{22}$ than the constitutive system.

  What does (i) imply about the relative importance of the noise terms? Can (ii) be directly explained by (iii), i.e., does rapid adjustment reduce noise? Justify your answers.

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Number Theory

• 3.I.1H

  Number Theory | Part II, 2005

  Let $\pi(x)$ be the number of primes $p \leqslant x$. State the Legendre formula, and prove that

  $$\lim _{x \rightarrow \infty} \frac{\pi(x)}{x}=0 .$$

  [You may use the formula

  $$\prod_{p \leqslant x}(1-1 / p)^{-1} \geqslant \log x$$

  without proof.]

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• 3.II.11H

  Number Theory | Part II, 2005

  Show that there are exactly two reduced positive definite integer binary quadratic forms with discriminant $-20$; write these forms down.

  State a criterion for an odd integer $n$ to be properly represented by a positive definite integer binary quadratic form of given discriminant $d$.

  Describe, in terms of congruences modulo 20, which primes other than 2,5 are properly represented by the form $x^{2}+5 y^{2}$, and justify your answer.

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Numerical Analysis

• 3.II.38A

  Numerical Analysis | Part II, 2005

  Consider the Runge-Kutta method

  $$\begin{aligned} k_{1} &=f\left(y_{n}\right) \\ k_{2} &=f\left(y_{n}+(1-a) h k_{1}+a h k_{2}\right), \\ y_{n+1} &=y_{n}+\frac{h}{2}\left(k_{1}+k_{2}\right) \end{aligned}$$

  for the solution of the scalar ordinary differential equation $y^{\prime}=f(y)$. Here $a$ is a real parameter.

  (a) Determine the order of the method.

  (b) Find the range of values of $a$ for which the method is A-stable.

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Optimization and Control

• 3.II.28I

  Optimization and Control | Part II, 2005

  Consider the problem

  $$\operatorname{minimize} E\left[x(T)^{2}+\int_{0}^{T} u(t)^{2} d t\right]$$

  where for $0 \leqslant t \leqslant T$,

  $$\dot{x}(t)=y(t) \text { and } \quad \dot{y}(t)=u(t)+\epsilon(t),$$

  $u(t)$ is the control variable, and $\epsilon(t)$ is Gaussian white noise. Show that the problem can be rewritten as one of controlling the scalar variable $z(t)$, where

  $$z(t)=x(t)+(T-t) y(t) .$$

  By guessing the form of the optimal value function and ensuring it satisfies an appropriate optimality equation, show that the optimal control is

  $$u(t)=-\frac{(T-t) z(t)}{1+\frac{1}{3}(T-t)^{3}} .$$

  Is this certainty equivalence control?

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Partial Differential Equations

• 3.II.29C

  Partial Differential Equations | Part II, 2005

  Write down a formula for the solution $u=u(t, x)$ of the $n$-dimensional heat equation

  $$w_{t}(t, x)-\Delta w=0, \quad w(0, x)=g(x),$$

  for $g: \mathbb{R}^{n} \rightarrow \mathbb{C}$ a given Schwartz function; here $w_{t}=\partial_{t} w$ and $\Delta$ is taken in the variables $x \in \mathbb{R}^{n}$. Show that

  $$w(t, x) \leqslant \frac{\int|g(x)| d x}{(4 \pi t)^{n / 2}}$$

  Consider the equation

  $$u_{t}-\Delta u=e^{i t} f(x),$$

  where $f: \mathbb{R}^{n} \rightarrow \mathbb{C}$ is a given Schwartz function. Show that $(*)$ has a solution of the form

  $$u(t, x)=e^{i t} v(x),$$

  where $v$ is a Schwartz function.

  Prove that the solution $u(t, x)$ of the initial value problem for $(*)$ with initial data $u(0, x)=g(x)$ satisfies

  $$\lim _{t \rightarrow+\infty}\left|u(t, x)-e^{i t} v(x)\right|=0 .$$

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Principles of Quantum Mechanics

• 3.II.32D

  Principles of Quantum Mechanics | Part II, 2005

  The angular momentum operators $\mathbf{J}^{(1)}$ and $\mathbf{J}^{(2)}$ refer to independent systems, each with total angular momentum one. The combination of these systems has a basis of states which are of product form $\left|m_{1} ; m_{2}\right\rangle=\left|1 m_{1}\right\rangle\left|1 m_{2}\right\rangle$ where $m_{1}$ and $m_{2}$ are the eigenvalues of $J_{3}^{(1)}$ and $J_{3}^{(2)}$ respectively. Let $|J M\rangle$ denote the alternative basis states which are simultaneous eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, where $\mathbf{J}=\mathbf{J}^{(1)}+\mathbf{J}^{(2)}$ is the combined angular momentum. What are the possible values of $J$ and $M$ ? Find expressions for all states with $J=1$ in terms of product states. How do these states behave when the constituent systems are interchanged?

  Two spin-one particles $A$ and $B$ have no mutual interaction but they each move in a potential $V(\mathbf{r})$ which is independent of spin. The single-particle energy levels $E_{i}$ and the corresponding wavefunctions $\psi_{i}(\mathbf{r})(i=1,2, \ldots)$ are the same for either $A$ or $B$. Given that $E_{1}<E_{2}<\ldots$, explain how to construct the two-particle states of lowest energy and combined total spin $J=1$ for the cases that (i) $A$ and $B$ are identical, and (ii) $A$ and $B$ are not identical.

  [You may assume $\hbar=1$ and use the result $\left.J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle .\right]$

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Principles of Statistics

• 3.II.26I

  Principles of Statistics | Part II, 2005

  In the context of decision theory, explain the meaning of the following italicized terms: loss function, decision rule, the risk of a decision rule, a Bayes rule with respect to prior $\pi$, and an admissible rule. Explain how a Bayes rule with respect to a prior $\pi$ can be constructed.

  Suppose that $X_{1}, \ldots, X_{n}$ are independent with common $N(0, v)$ distribution, where $v>0$ is supposed to have a prior density $f_{0}$. In a decision-theoretic approach to estimating $v$, we take a quadratic loss: $L(v, a)=(v-a)^{2}$. Write $X=\left(X_{1}, \ldots, X_{n}\right)$ and $|X|=\left(X_{1}^{2}+\ldots+X_{n}^{2}\right)^{1 / 2}$.

  By considering decision rules (estimators) of the form $\hat{v}(X)=\alpha|X|^{2}$, prove that if $\alpha \neq 1 /(n+2)$ then the estimator $\hat{v}(X)=\alpha|X|^{2}$ is not Bayes, for any choice of prior $f_{0}$.

  By considering decision rules of the form $\hat{v}(X)=\alpha|X|^{2}+\beta$, prove that if $\alpha \neq 1 / n$ then the estimator $\hat{v}(X)=\alpha|X|^{2}$ is not Bayes, for any choice of prior $f_{0}$.

  [You may use without proof the fact that, if $Z$ has a $N(0,1)$ distribution, then $E Z^{4}=3$.]

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Representation Theory

• 3.II.19G

  Representation Theory | Part II, 2005

  Let $G$ be the group with 21 elements generated by $a$ and $b$, subject to the relations $a^{7}=b^{3}=1$ and $b a=a^{2} b .$

  (i) Find the conjugacy classes of $G$.

  (ii) Find three non-isomorphic one-dimensional representations of $G$.

  (iii) For a subgroup $H$ of a finite group $K$, write down (without proof) the formula for the character of the $K$-representation induced from a representation of $H$.

  (iv) By applying Part (iii) to the case when $H$ is the subgroup $\langle a\rangle$ of $K=G$, find the remaining irreducible characters of $G$.

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Riemann Surfaces

• 3.II.22H

  Riemann Surfaces | Part II, 2005

  Explain what is meant by a meromorphic differential on a compact connected Riemann surface $S$. Show that if $f$ is a meromorphic function on $S$ then $d f$ defines a meromorphic differential on $S$. Show also that if $\eta$ and $\omega$ are two meromorphic differentials on $S$ which are not identically zero then $\eta=h \omega$ for some meromorphic function $h$. Show that zeros and poles of a meromorphic differential are well-defined and explain, without proof, how to obtain the genus of $S$ by counting zeros and poles of $\omega$.

  Let $V_{0} \subset \mathbb{C}^{2}$ be the affine curve with equation $u^{2}=v^{2}+1$ and let $V \subset \mathbb{P}^{2}$ be the corresponding projective curve. Show that $V$ is non-singular with two points at infinity, and that $d v$ extends to a meromorphic differential on $V$.

  [You may assume without proof that that the map

  $$(u, v)=\left(\frac{t^{2}+1}{t^{2}-1}, \frac{2 t}{t^{2}-1}\right), \quad t \in \mathbb{C} \backslash\{-1,1\},$$

  is onto $V_{0} \backslash\{(1,0)\}$ and extends to a biholomorphic map from $\mathbb{P}^{1}$ onto $V$.]

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Statistical Modelling

• 3.I.5I

  Statistical Modelling | Part II, 2005

  Consider the model $Y=X \beta+\epsilon$, where $Y$ is an $n$-dimensional observation vector, $X$ is an $n \times p$ matrix of rank $p, \epsilon$ is an $n$-dimensional vector with components $\epsilon_{1}, \ldots, \epsilon_{n}$, and $\epsilon_{1}, \ldots, \epsilon_{n}$ are independently and normally distributed, each with mean 0 and variance $\sigma^{2}$

  (a) Let $\hat{\beta}$ be the least-squares estimator of $\beta$. Show that

  $$\left(X^{T} X\right) \hat{\beta}=X^{T} Y$$

  and find the distribution of $\hat{\beta}$.

  (b) Define $\hat{Y}=X \hat{\beta}$. Show that $\hat{Y}$ has distribution $N\left(X \beta, \sigma^{2} H\right)$, where $H$ is a matrix that you should define.

  [You may quote without proof any results you require about the multivariate normal distribution.]

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Statistical Physics

• 3.II.34D

  Statistical Physics | Part II, 2005

  A free spinless particle moving in two dimensions is confined to a square box of side $L$. By imposing periodic boundary conditions show that the number of states in the energy range $\epsilon \rightarrow \epsilon+d \epsilon$ is $g(\epsilon) d \epsilon$, where

  $$g(\epsilon)=\frac{m L^{2}}{2 \pi \hbar^{2}}$$

  If, instead, the particle is an electron with magnetic moment $\mu$ moving in a constant external magnetic field $H$, show that

  $$g(\epsilon)= \begin{cases}\frac{m L^{2}}{2 \pi \hbar^{2}}, & -\mu H<\epsilon<\mu H \\ \frac{m L^{2}}{\pi \hbar^{2}}, & \mu H<\epsilon\end{cases}$$

  Let there be $N$ electrons in the box. Explain briefly how to construct the ground state of the system. Let $\epsilon$ be the Fermi energy. Show that when $\epsilon>\mu H$

  $$N=\frac{m L^{2}}{\pi \hbar^{2}} \epsilon .$$

  Show also that the magnetic moment $M$ of the system in its ground state is given by

  $$M=\frac{\mu^{2} m L^{2}}{\pi \hbar^{2}} H$$

  and that the ground state energy is

  $$\frac{1}{2} \frac{\pi \hbar^{2}}{m L^{2}} N^{2}-\frac{1}{2} M H$$

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Stochastic Financial Models

• 3.II.27J

  Stochastic Financial Models | Part II, 2005

  Suppose that over two periods a stock price moves on a binomial tree:

  

  (a) Find an arbitrage opportunity when the riskless rate equals $1 / 10$. Give precise details of when and how much you buy, borrow and sell.

  (b) From here on, assume instead that the riskless rate equals $1 / 4$. Determine the equivalent martingale measure. [No proof is required.]

  (c) Determine the time-zero price of an American put with strike 15 and expiry 2 . Assume you sell it at this price. Which hedge do you put on at time zero? Consider the scenario of two bad periods. How does your hedge work?

  (d) The buyer of the American put turns out to be an unsophisticated investor who fails to use his early exercise right when he should. Assume the first period was bad. How much profit can you make out of this? You should detail your exact strategy.

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Waves

• 3.II.37E

  Waves | Part II, 2005

  The real function $\phi(x, t)$ satisfies the equation

  $$\frac{\partial \phi}{\partial t}+U \frac{\partial \phi}{\partial x}=\frac{\partial^{3} \phi}{\partial x^{3}},$$

  where $U>0$ is a constant. Find the dispersion relation for waves of wavenumber $k$ and deduce whether wave crests move faster or slower than a wave packet.

  Suppose that $\phi(x, 0)$ is given by a Fourier transform as

  $$\phi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$$

  Use the method of stationary phase to find $\phi(V t, t)$ as $t \rightarrow \infty$ for fixed $V>U$.

  [You may use the result that $\int_{-\infty}^{\infty} e^{-a \xi^{2}} d \xi=(\pi / a)^{1 / 2}$ if $\left.\operatorname{Re}(a) \geqslant 0 .\right]$

  What can be said if $V<U$ ? [Detailed calculation is not required in this case.]

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