Part II, 2013

Algebraic Geometry

• Paper 1, Section II, H

  Algebraic Geometry | Part II, 2013

  Let $V \subset \mathbb{A}^{n}$ be an affine variety over an algebraically closed field $k$. What does it mean to say that $V$ is irreducible? Show that any non-empty affine variety $V \subset \mathbb{A}^{n}$ is the union of a finite number of irreducible affine varieties $V_{j} \subset \mathbb{A}^{n}$.

  Define the ideal $I(V)$ of $V$. Show that $I(V)$ is a prime ideal if and only if $V$ is irreducible.

  Assume that the base field $k$ has characteristic zero. Determine the irreducible components of

  $$V\left(X_{1} X_{2}, X_{1} X_{3}+X_{2}^{2}-1, X_{1}^{2}\left(X_{1}-X_{3}\right)\right) \subset \mathbb{A}^{3}$$

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• Paper 2, Section II, H

  Algebraic Geometry | Part II, 2013

  Let $V \subset \mathbb{P}^{3}$ be an irreducible quadric surface.

  (i) Show that if $V$ is singular, then every nonsingular point lies in exactly one line in $V$, and that all the lines meet in the singular point, which is unique.

  (ii) Show that if $V$ is nonsingular then each point of $V$ lies on exactly two lines of $V$.

  Let $V$ be nonsingular, $P_{0}$ a point of $V$, and $\Pi \subset \mathbb{P}^{3}$ a plane not containing $P_{0}$. Show that the projection from $P_{0}$ to $\Pi$ is a birational map $f: V \rightarrow \rightarrow \Pi$. At what points does $f$ fail to be regular? At what points does $f^{-1}$ fail to be regular? Justify your answers.

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• Paper 3, Section II, H

  Algebraic Geometry | Part II, 2013

  Let $C \subset \mathbb{P}^{2}$ be the plane curve given by the polynomial

  $$X_{0}^{n}-X_{1}^{n}-X_{2}^{n}$$

  over the field of complex numbers, where $n \geqslant 3$.

  (i) Show that $C$ is nonsingular.

  (ii) Compute the divisors of the rational functions

  $$x=\frac{X_{1}}{X_{0}}, \quad y=\frac{X_{2}}{X_{0}}$$

  on $C$.

  (iii) Consider the morphism $\phi=\left(X_{0}: X_{1}\right): C \rightarrow \mathbb{P}^{1}$. Compute its ramification points and degree.

  (iv) Show that a basis for the space of regular differentials on $C$ is

  $$\left\{x^{i} y^{j} \omega_{0} \mid i, j \geqslant 0, i+j \leqslant n-3\right\}$$

  where $\omega_{0}=d x / y^{n-1} .$

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• Paper 4, Section II, H

  Algebraic Geometry | Part II, 2013

  Let $C$ be a nonsingular projective curve, and $D$ a divisor on $C$ of degree $d$.

  (i) State the Riemann-Roch theorem for $D$, giving a brief explanation of each term. Deduce that if $d>2 g-2$ then $\ell(D)=1-g+d$.

  (ii) Show that, for every $P \in C$,

  $$\ell(D-P) \geqslant \ell(D)-1$$

  Deduce that $\ell(D) \leqslant 1+d$. Show also that if $\ell(D)>1$, then $\ell(D-P)=\ell(D)-1$ for all but finitely many $P \in C$.

  (iii) Deduce that for every $d \geqslant g-1$ there exists a divisor $D$ of degree $d$ with $\ell(D)=1-g+d$.

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Algebraic Topology

• Paper 1, Section II, 21G

  Algebraic Topology | Part II, 2013

  (i) Define the notion of the fundamental group $\pi_{1}\left(X, x_{0}\right)$ of a path-connected space $X$ with base point $x_{0}$.

  (ii) Prove that if a group $G$ acts freely and properly discontinuously on a simply connected space $Z$, then $\pi_{1}\left(G \backslash Z, x_{0}\right)$ is isomorphic to $G$. [You may assume the homotopy lifting property, provided that you state it clearly.]

  (iii) Suppose that $p, q$ are distinct points on the 2 -sphere $S^{2}$ and that $X=S^{2} /(p \sim q)$. Exhibit a simply connected space $Z$ with an action of a group $G$ as in (ii) such that $X=G \backslash Z$, and calculate $\pi_{1}\left(X, x_{0}\right)$.

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• Paper 2, Section II, G

  Algebraic Topology | Part II, 2013

  (i) State the Seifert-van Kampen theorem.

  (ii) Assuming any standard results about the fundamental group of a circle that you wish, calculate the fundamental group of the $n$-sphere, for every $n \geqslant 2$.

  (iii) Suppose that $n \geqslant 3$ and that $X$ is a path-connected topological $n$-manifold. Show that $\pi_{1}\left(X, x_{0}\right)$ is isomorphic to $\pi_{1}\left(X-\{P\}, x_{0}\right)$ for any $P \in X-\left\{x_{0}\right\}$.

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• Paper 3, Section II, G

  Algebraic Topology | Part II, 2013

  (i) State, but do not prove, the Mayer-Vietoris theorem for the homology groups of polyhedra.

  (ii) Calculate the homology groups of the $n$-sphere, for every $n \geqslant 0$.

  (iii) Suppose that $a \geqslant 1$ and $b \geqslant 0$. Calculate the homology groups of the subspace $X$ of $\mathbb{R}^{a+b}$ defined by $\sum_{i=1}^{a} x_{i}^{2}-\sum_{j=a+1}^{a+b} x_{j}^{2}=1$.

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• Paper 4, Section II, G

  Algebraic Topology | Part II, 2013

  (i) State, but do not prove, the Lefschetz fixed point theorem.

  (ii) Show that if $n$ is even, then for every map $f: S^{n} \rightarrow S^{n}$ there is a point $x \in S^{n}$ such that $f(x)=\pm x$. Is this true if $n$ is odd? [Standard results on the homology groups for the $n$-sphere may be assumed without proof, provided they are stated clearly.]

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Applications of Quantum Mechanics

• Paper 1, Section II, D

  Applications of Quantum Mechanics | Part II, 2013

  Consider a quantum system with Hamiltonian $\widehat{H}$and energy levels

  $$E_{0}<E_{1}<E_{2}<\ldots$$

  For any state $|\psi\rangle$ define the Rayleigh-Ritz quotient $R[\psi]$ and show the following:

  (i) the ground state energy $E_{0}$ is the minimum value of $R[\psi]$;

  (ii) all energy eigenstates are stationary points of $R[\psi]$ with respect to variations of $|\psi\rangle$.

  Under what conditions can the value of $R\left[\psi_{\alpha}\right]$ for a trial wavefunction $\psi_{\alpha}$ (depending on some parameter $\alpha$ ) be used as an estimate of the energy $E_{1}$ of the first excited state? Explain your answer.

  For a suitably chosen trial wavefunction which is the product of a polynomial and a Gaussian, use the Rayleigh-Ritz quotient to estimate $E_{1}$ for a particle of mass $m$ moving in a potential $V(x)=g|x|$, where $g$ is a constant.

  [You may use the integral formulae,

  $$\begin{aligned} \int_{0}^{\infty} x^{2 n} \exp \left(-p x^{2}\right) d x &=\frac{(2 n-1) ! !}{2(2 p)^{n}} \sqrt{\frac{\pi}{p}} \\ \int_{0}^{\infty} x^{2 n+1} \exp \left(-p x^{2}\right) d x &=\frac{n !}{2 p^{n+1}} \end{aligned}$$

  where $n$ is a non-negative integer and $p$ is a constant. ]

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• Paper 2, Section II, D

  Applications of Quantum Mechanics | Part II, 2013

  (i) A particle of momentum $\hbar k$ and energy $E=\hbar^{2} k^{2} / 2 m$ scatters off a sphericallysymmetric target in three dimensions. Define the corresponding scattering amplitude $f$ as a function of the scattering angle $\theta$. Expand the scattering amplitude in partial waves of definite angular momentum $l$, and determine the coefficients of this expansion in terms of the phase shifts $\delta_{l}(k)$ appearing in the following asymptotic form of the wavefunction, valid at large distance from the target,

  $$\psi(\mathbf{r}) \sim \sum_{l=0}^{\infty} \frac{2 l+1}{2 i k}\left[e^{2 i \delta_{l}} \frac{e^{i k r}}{r}-(-1)^{l} \frac{e^{-i k r}}{r}\right] P_{l}(\cos \theta) .$$

  Here, $r=|\mathbf{r}|$ is the distance from the target and $P_{l}$ are the Legendre polynomials.

  [You may use without derivation the following approximate relation between plane and spherical waves (valid asymptotically for large $r$ ):

  $$\exp (i k z) \sim \sum_{l=0}^{\infty}(2 l+1) i^{l} \frac{\sin \left(k r-\frac{1}{2} l \pi\right)}{k r} P_{l}(\cos \theta) .$$

  (ii) Suppose that the potential energy takes the form $V(r)=\lambda U(r)$ where $\lambda \ll 1$ is a dimensionless coupling. By expanding the wavefunction in a power series in $\lambda$, derive the Born Approximation to the scattering amplitude in the form

  $$f(\theta)=-\frac{2 m \lambda}{\hbar^{2}} \int_{0}^{\infty} U(r) \frac{\sin q r}{q} r d r$$

  up to corrections of order $\lambda^{2}$, where $q=2 k \sin (\theta / 2)$. [You may quote any results you need for the Green's function for the differential operator $\nabla^{2}+k^{2}$ provided they are stated clearly.]

  (iii) Derive the corresponding order $\lambda$ contribution to the phase shift $\delta_{l}(k)$ of angular momentum $l$.

  [You may use the orthogonality relations

  $$\int_{-1}^{+1} P_{l}(w) P_{m}(w) d w=\frac{2}{(2 l+1)} \delta_{l m}$$

  and the integral formula

  $$\int_{0}^{1} P_{l}\left(1-2 x^{2}\right) \sin (a x) d x=\frac{a}{2}\left[j_{l}\left(\frac{a}{2}\right)\right]^{2}$$

  where $j_{l}(z)$ is a spherical Bessel function.]

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• Paper 3, Section II, D

  Applications of Quantum Mechanics | Part II, 2013

  Write down the classical Hamiltonian for a particle of mass $m$, electric charge $-e$ and momentum p moving in the background of an electromagnetic field with vector and scalar potentials $\mathbf{A}(\mathbf{x}, t)$ and $\phi(\mathbf{x}, t)$.

  Consider the case of a constant uniform magnetic field, $\mathbf{B}=(0,0, B)$ and $\mathbf{E}=0$. Working in the gauge with $\mathbf{A}=(-B y, 0,0)$ and $\phi=0$, show that Hamilton's equations,

  $$\dot{\mathbf{x}}=\frac{\partial H}{\partial \mathbf{p}}, \quad \dot{\mathbf{p}}=-\frac{\partial H}{\partial \mathbf{x}},$$

  admit solutions corresponding to circular motion in the $x-y$ plane with angular frequency $\omega_{B}=e B / m$.

  Show that, in the same gauge, the coordinates $\left(x_{0}, y_{0}, 0\right)$ of the centre of the circle are related to the instantaneous position $\mathbf{x}=(x, y, z)$ and momentum $\mathbf{p}=\left(p_{x}, p_{y}, p_{z}\right)$ of the particle by

  $$x_{0}=x-\frac{p_{y}}{e B}, \quad y_{0}=\frac{p_{x}}{e B} .$$

  Write down the quantum Hamiltonian $\hat{H}$ for the system. In the case of a uniform constant magnetic field discussed above, find the allowed energy levels. Working in the gauge specified above, write down quantum operators corresponding to the classical quantities $x_{0}$ and $y_{0}$ defined in (1) above and show that they are conserved.

  [In this question you may use without derivation any facts relating to the energy spectrum of the quantum harmonic oscillator provided they are stated clearly.]

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• Paper 4, Section II, D

  Applications of Quantum Mechanics | Part II, 2013

  Define the Floquet matrix for a particle moving in a periodic potential in one dimension and explain how it determines the allowed energy bands of the system.

  A potential barrier in one dimension has the form

  $$V(x)= \begin{cases}V_{0}(x), & |x|<a / 4 \\ 0, & |x|>a / 4\end{cases}$$

  where $V_{0}(x)$ is a smooth, positive function of $x$. The reflection and transmission amplitudes for a particle of wavenumber $k>0$, incident from the left, are $r(k)$ and $t(k)$ respectively. For a particle of wavenumber $-k$, incident from the right, the corresponding amplitudes are $r^{\prime}(k)$ and $t^{\prime}(k)=t(k)$. In the following, for brevity, we will suppress the $k$-dependence of these quantities.

  Consider the periodic potential $\tilde{V}$, defined by $\tilde{V}(x)=V(x)$ for $|x|<a / 2$ and by $\tilde{V}(x+a)=\widetilde{V}(x)$ elsewhere. Write down two linearly independent solutions of the corresponding Schrödinger equation in the region $-3 a / 4<x<-a / 4$. Using the scattering data given above, extend these solutions to the region $a / 4<x<3 a / 4$. Hence find the Floquet matrix of the system in terms of the amplitudes $r, r^{\prime}$ and $t$ defined above.

  Show that the edges of the allowed energy bands for this potential lie at $E=\hbar^{2} k^{2} / 2 m$, where

  $$k a=i \log \left(t \pm \sqrt{r r^{\prime}}\right)$$

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Applied Probability

• Paper 1, Section II, J

  Applied Probability | Part II, 2013

  Let $\left(X_{t}, t \geqslant 0\right)$ be a Markov chain on $\{0,1, \ldots\}$ with $Q$-matrix given by

  $$\begin{aligned} q_{n, n+1} &=\lambda_{n} \\ q_{n, 0} &=\lambda_{n} \varepsilon_{n} \quad(n>0) \\ q_{n, m} &=0 \quad \text { if } m \notin\{0, n, n+1\} \end{aligned}$$

  where $\varepsilon_{n}, \lambda_{n}>0$.

  (i) Show that $X$ is transient if and only if $\sum_{n} \varepsilon_{n}<\infty$. [You may assume without proof that $x(1-\delta) \leqslant \log (1+x) \leqslant x$ for all $\delta>0$ and all sufficiently small positive $x$.]

  (ii) Assume that $\sum_{n} \varepsilon_{n}<\infty$. Find a necessary and sufficient condition for $X$ to be almost surely explosive. [You may assume without proof standard results about pure birth processes, provided that they are stated clearly.]

  (iii) Find a stationary measure for $X$. For the case $\lambda_{n}=\lambda$ and $\varepsilon_{n}=\alpha /(n+1)(\lambda, \alpha>0)$, show that $X$ is positive recurrent if and only if $\alpha>1$.

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• Paper 2, Section II, J

  Applied Probability | Part II, 2013

  (i) Define a Poisson process as a Markov chain on the non-negative integers and state three other characterisations.

  (ii) Let $\lambda(s)(s \geqslant 0)$ be a continuous positive function. Let $\left(X_{t}, t \geqslant 0\right)$ be a right-continuous process with independent increments, such that

  $$\begin{aligned} \mathbb{P}\left(X_{t+h}=X_{t}+1\right) &=\lambda(t) h+o(h) \\ \mathbb{P}\left(X_{t+h}=X_{t}\right) &=1-\lambda(t) h+o(h) \end{aligned}$$

  where the $o(h)$ terms are uniform in $t \in[0, \infty)$. Show that $X_{t}$ is a Poisson random variable with parameter $\Lambda(t)=\int_{0}^{t} \lambda(s) d s$.

  (iii) Let $X=\left(X_{n}: n=1,2, \ldots\right)$ be a sequence of independent and identically distributed positive random variables with continuous density function $f$. We define the sequence of successive records, $\left(K_{n}, n=0,1, \ldots\right)$, by $K_{0}:=0$ and, for $n \geqslant 0$,

  $$K_{n+1}:=\inf \left\{m>K_{n}: X_{m}>X_{K_{n}}\right\}$$

  The record process,$\left(R_{t}, t \geqslant 0\right)$, is then defined by

  $$R_{t}:=\#\left\{n \geqslant 1: X_{K_{n}} \leqslant t\right\}$$

  Explain why the increments of $R$ are independent. Show that $R_{t}$ is a Poisson random variable with parameter $-\log \{1-F(t)\}$ where $F(t)=\int_{0}^{t} f(s) d s$.

  [You may assume the following without proof: For fixed $t>0$, let $Y$ (respectively, $Z$ ) be the subsequence of $X$ obtained by retaining only those elements that are greater than (respectively, smaller than) $t$. Then $Y$ (respectively, $Z$ ) is a sequence of independent variables each having the distribution of $X_{1}$ conditioned on $X_{1}>t$ (respectively, $X_{1}<t$ ); and $Y$ and $Z$ are independent.]

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• Paper 3, Section II, J

  Applied Probability | Part II, 2013

  Define the Moran model. Describe briefly the infinite sites model of mutations.

  We henceforth consider a population with $N$ individuals evolving according to the rules of the Moran model. In addition we assume:

  • the allelic type of any individual at any time lies in a given countable state space $S$;

  • individuals are subject to mutations at constant rate $u=\theta / N$, independently of the population dynamics;

  • each time a mutation occurs, if the allelic type of the individual was $x \in S$, it changes to $y \in S$ with probability $P(x, y)$, where $P(x, y)$ is a given Markovian transition matrix on $S$ that is symmetric:

  $$P(x, y)=P(y, x) \quad(x, y \in S)$$

  (i) Show that, if two individuals are sampled at random from the population at some time $t$, then the time to their most recent common ancestor has an exponential distribution, with a parameter that you should specify.

  (ii) Let $\Delta+1$ be the total number of mutations that accumulate on the two branches separating these individuals from their most recent common ancestor. Show that $\Delta+1$ is a geometric random variable, and specify its probability parameter $p$.

  (iii) The first individual is observed to be of type $x \in S$. Explain why the probability that the second individual is also of type $x$ is

  $$\mathbb{P}\left(X_{\Delta}=x \mid X_{0}=x\right),$$

  where $\left(X_{n}, n \geqslant 0\right)$ is a Markov chain on $S$ with transition matrix $P$ and is independent of $\Delta$.

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• Paper 4, Section II, J

  Applied Probability | Part II, 2013

  (i) Define an $M / M / 1$ queue. Justifying briefly your answer, specify when this queue has a stationary distribution, and identify that distribution. State and prove Burke's theorem for this queue.

  (ii) Let $\left(L_{1}(t), \ldots, L_{N}(t), t \geqslant 0\right)$ denote a Jackson network of $N$ queues, where the entrance and service rates for queue $i$ are respectively $\lambda_{i}$ and $\mu_{i}$, and each customer leaving queue $i$ moves to queue $j$ with probability $p_{i j}$ after service. We assume $\sum_{j} p_{i j}<1$ for each $i=1, \ldots, N$; with probability $1-\sum_{j} p_{i j}$ a customer leaving queue $i$ departs from the system. State Jackson's theorem for this network. [You are not required to prove it.] Are the processes $\left(L_{1}(t), \ldots, L_{N}(t), t \geqslant 0\right)$ independent at equilibrium? Justify your answer.

  (iii) Let $D_{i}(t)$ be the process of final departures from queue $i$. Show that, at equilibrium, $\left(L_{1}(t), \ldots, L_{N}(t)\right)$ is independent of $\left(D_{i}(s), 1 \leqslant i \leqslant N, 0 \leqslant s \leqslant t\right)$. Show that, for each fixed $i=1, \ldots, N,\left(D_{i}(t), t \geqslant 0\right)$ is a Poisson process, and specify its rate.

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Asymptotic Methods

• Paper 1, Section II, B

  Asymptotic Methods | Part II, 2013

  Suppose $\alpha>0$. Define what it means to say that

  $$F(x) \sim \frac{1}{\alpha x} \sum_{n=0}^{\infty} n !\left(\frac{-1}{\alpha x}\right)^{n}$$

  is an asymptotic expansion of $F(x)$ as $x \rightarrow \infty$. Show that $F(x)$ has no other asymptotic expansion in inverse powers of $x$ as $x \rightarrow \infty$.

  To estimate the value of $F(x)$ for large $x$, one may use an optimal truncation of the asymptotic expansion. Explain what is meant by this, and show that the error is an exponentially small quantity in $x$.

  Derive an integral respresentation for a function $F(x)$ with the above asymptotic expansion.

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• Paper 3, Section II, B

  Asymptotic Methods | Part II, 2013

  Let

  $$I(x)=\int_{0}^{\pi} f(t) e^{i x \psi(t)} d t$$

  where $f(t)$ and $\psi(t)$ are smooth, and $\psi^{\prime}(t) \neq 0$ for $t>0 ;$ also $f(0) \neq 0$, $\psi(0)=a$, $\psi^{\prime}(0)=\psi^{\prime \prime}(0)=0$ and $\psi^{\prime \prime \prime}(0)=6 b>0$. Show that, as $x \rightarrow+\infty$,

  $$I(x) \sim f(0) e^{i(x a+\pi / 6)}\left(\frac{1}{27 b x}\right)^{1 / 3} \Gamma(1 / 3) .$$

  Consider the Bessel function

  $$J_{n}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (n t-x \sin t) d t$$

  Show that, as $n \rightarrow+\infty$,

  $$J_{n}(n) \sim \frac{\Gamma(1 / 3)}{\pi} \frac{1}{(48)^{1 / 6}} \frac{1}{n^{1 / 3}}$$

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• Paper 4, Section II, B

  Asymptotic Methods | Part II, 2013

  Show that the equation

  $$\frac{d^{2} y}{d x^{2}}+\frac{2}{x} \frac{d y}{d x}+\left(\frac{1}{x^{2}}-1\right) y=0$$

  has an irregular singular point at infinity. Using the Liouville-Green method, show that one solution has the asymptotic expansion

  $$y(x) \sim \frac{1}{x} e^{x}\left(1+\frac{1}{2 x}+\ldots\right)$$

  as $x \rightarrow \infty$

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Classical Dynamics

• Paper 1, Section I, B

  Classical Dynamics | Part II, 2013

  Consider an $n$-dimensional dynamical system with generalized coordinates and momenta $\left(q_{i}, p_{i}\right), i=1,2, \ldots, n$.

  (a) Define the Poisson bracket $\{f, g\}$ of two functions $f\left(q_{i}, p_{i}, t\right)$ and $g\left(q_{i}, p_{i}, t\right)$.

  (b) Assuming Hamilton's equations of motion, prove that if a function $G\left(q_{i}, p_{i}\right)$ Poisson commutes with the Hamiltonian, that is $\{G, H\}=0$, then $G$ is a constant of the motion.

  (c) Assume that $q_{j}$ is an ignorable coordinate, that is the Hamiltonian does not depend on it explicitly. Using the formalism of Poisson brackets prove that the conjugate momentum $p_{j}$ is conserved.

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• Paper 2, Section I, B

  Classical Dynamics | Part II, 2013

  (i) Consider a rigid body with principal moments of inertia $I_{1}, I_{2}, I_{3}$. Derive Euler's equations of torque-free motion,

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}, \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1}, \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}, \end{aligned}$$

  with components of the angular velocity $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ given in the body frame.

  (ii) Use Euler's equations to show that the energy $E$ and the square of the total angular momentum $\mathbf{L}^{2}$ of the body are conserved.

  (iii) Consider a torque-free motion of a symmetric top with $I_{1}=I_{2}=\frac{1}{2} I_{3}$. Show that in the body frame the vector of angular velocity $\boldsymbol{\omega}$ precesses about the body-fixed $\mathbf{e}_{3}$ axis with constant angular frequency equal to $\omega_{3}$.

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• Paper 2, Section II, B

  Classical Dynamics | Part II, 2013

  (i) The action for a system with a generalized coordinate $q$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  (a) State the Principle of Least Action and derive the Euler-Lagrange equation.

  (b) Consider an arbitrary function $f(q, t)$. Show that $L^{\prime}=L+d f / d t$ leads to the same equation of motion.

  (ii) A wire frame $A B C$ in a shape of an equilateral triangle with side $a$ rotates in a horizontal plane with constant angular frequency $\omega$ about a vertical axis through $A$. A bead of mass $m$ is threaded on $B C$ and moves without friction. The bead is connected to $B$ and $C$ by two identical light springs of force constant $k$ and equilibrium length $a / 2$.

  (a) Introducing the displacement $\eta$ of the particle from the mid point of $B C$, determine the Lagrangian $L(\eta, \dot{\eta})$.

  (b) Derive the equation of motion. Identify the integral of the motion.

  (c) Describe the motion of the bead. Find the condition for there to be a stable equilibrium and find the frequency of small oscillations about it when it exists.

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• Paper 3, Section I, B

  Classical Dynamics | Part II, 2013

  Two equal masses $m$ are connected to each other and to fixed points by three springs of force constant $5 k, k$ and $5 k$ as shown in the figure.

  

  (i) Write down the Lagrangian and derive the equations describing the motion of the system in the direction parallel to the springs.

  (ii) Find the normal modes and their frequencies. Comment on your results.

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• Paper 4, Section I, B

  Classical Dynamics | Part II, 2013

  The Lagrangian for a heavy symmetric top of mass $M$, pinned at point $O$ which is a distance $l$ from the centre of mass, is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  (i) Starting with the fixed space frame $\left(\tilde{\mathbf{e}}_{\mathbf{1}}, \tilde{\mathbf{e}}_{2}, \tilde{\mathbf{e}}_{3}\right)$ and choosing $O$ at its origin, sketch the top with embedded body frame axis $\mathbf{e}_{3}$ being the symmetry axis. Clearly identify the Euler angles $(\theta, \phi, \psi)$.

  (ii) Obtain the momenta $p_{\theta}, p_{\phi}$ and $p_{\psi}$ and the Hamiltonian $H\left(\theta, \phi, \psi, p_{\theta}, p_{\phi}, p_{\psi}\right)$. Derive Hamilton's equations. Identify the three conserved quantities.

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• Paper 4, Section II, B

  Classical Dynamics | Part II, 2013

  The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

  $$L=\frac{m \dot{\mathbf{r}}^{2}}{2}-q(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$$

  (i) Write down the Hamiltonian of the particle.

  (ii) Write down Hamilton's equations of motion for the particle.

  (iii) Show that Hamilton's equations are invariant under the gauge transformation

  $$\phi \rightarrow \phi-\frac{\partial \Lambda}{\partial t}, \quad \mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda,$$

  for an arbitrary function $\Lambda(\mathbf{r}, t)$.

  (iv) The particle moves in the presence of a field such that $\phi=0$ and $\mathbf{A}=\left(-\frac{1}{2} y B, \frac{1}{2} x B, 0\right)$, where $(x, y, z)$ are Cartesian coordinates and $B$ is a constant.

  (a) Find a gauge transformation such that only one component of $\mathbf{A}(x, y, z)$ remains non-zero.

  (b) Determine the motion of the particle.

  (v) Now assume that $B$ varies very slowly with time on a time-scale much longer than $(q B / m)^{-1}$. Find the quantity which remains approximately constant throughout the motion.

  [You may use the expression for the action variable $I=\frac{1}{2 \pi} \oint p_{i} d q_{i}$.]

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Coding and Cryptography

• Paper 1, Section I, H

  Coding and Cryptography | Part II, 2013

  A binary Huffman code is used for encoding symbols $1, \ldots, m$ occurring with respective probabilities $p_{1} \geqslant \cdots \geqslant p_{m}>0$ where $\sum_{1 \leqslant j \leqslant m} p_{j}=1$. Let $s_{1}$ be the length of a shortest codeword and $s_{m}$ the length of a longest codeword. Determine the maximal and minimal values of each of $s_{1}$ and $s_{m}$, and find binary trees for which they are attained.

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• Paper 1, Section II, H

  Coding and Cryptography | Part II, 2013

  Define the bar product $C_{1} \mid C_{2}$ of binary linear codes $C_{1}$ and $C_{2}$, where $C_{2}$ is a subcode of $C_{1}$. Relate the rank and minimum distance of $C_{1} \mid C_{2}$ to those of $C_{1}$ and $C_{2}$ and justify your answer. Show that if $C^{\perp}$ denotes the dual code of $C$, then

  $$\left(C_{1} \mid C_{2}\right)^{\perp}=C_{2}^{\perp} \mid C_{1}^{\perp}$$

  Using the bar product construction, or otherwise, define the Reed-Muller code $\mathrm{RM}(d, r)$ for $0 \leqslant r \leqslant d$. Show that if $0 \leqslant r \leqslant d-1$, then the dual of $\mathrm{RM}(d, r)$ is again a Reed-Muller code.

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• Paper 2, Section I, H

  Coding and Cryptography | Part II, 2013

  Let $A(n, d)$ denote the maximum size of a binary code of length $n$ with minimum distance $d$. For fixed $\delta$ with $0<\delta<1 / 2$, let $\alpha(\delta)=\limsup _{n} \frac{1}{n} \log _{2} A(n, n \delta)$. Show that

  $$1-H(\delta) \leqslant \alpha(\delta) \leqslant 1-H(\delta / 2)$$

  where $H(p)=-p \log _{2} p-(1-p) \log _{2}(1-p)$.

  [You may assume the GSV and Hamming bounds and any form of Stirling's theorem provided you state them clearly.]

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• Paper 2, Section II, H

  Coding and Cryptography | Part II, 2013

  Define a BCH code of length $n$, where $n$ is odd, over the field of 2 elements with design distance $\delta$. Show that the minimum weight of such a code is at least $\delta$. [Results about the van der Monde determinant may be quoted without proof, provided they are stated clearly.]

  Consider a BCH code of length 31 over the field of 2 elements with design distance 8 . Show that the minimum distance is at least 11. [Hint: Let $\alpha$ be a primitive element in the field of $2^{5}$ elements, and consider the minimal polynomial for certain powers of $\left.\alpha .\right]$

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• Paper 3, Section I, H

  Coding and Cryptography | Part II, 2013

  Describe briefly the Rabin cipher with modulus $N$, explaining how it can be deciphered by the intended recipient and why it is difficult for an eavesdropper to decipher it.

  The Cabinet decides to communicate using Rabin ciphers to maintain confidentiality. The Cabinet Secretary encrypts a message, represented as a positive integer $m$, using the Rabin cipher with modulus $N$ (with $0<m<N$ ) and publishes both the encrypted message and the modulus. The Defence Secretary deciphers this message to read it but then foolishly encrypts it again using a Rabin cipher with a different modulus $N^{\prime}$ (with $\left.m<N^{\prime}\right)$ and publishes the newly encrypted message and $N^{\prime}$. Mr Rime (the Leader of the Opposition) knows this has happened. Explain how Rime can work out what the original message was using the two different encrypted versions.

  Can Rime decipher other messages sent out by the Cabinet using the original modulus $N$ ?

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• Paper 4, Section I, $4 \mathrm{H}$

  Coding and Cryptography | Part II, 2013

  Describe how a stream cipher works. What is a one-time pad?

  A one-time pad is used to send the message $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} y_{7}$ which is encoded as 0101011. In error, it is reused to send the message $y_{0} x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$ which is encoded as 0100010 . Show that there are two possibilities for the substring $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$, and find them.

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Cosmology

• Paper 1, Section I, D

  Cosmology | Part II, 2013

  The Friedmann equation and the fluid conservation equation for a closed isotropic and homogeneous cosmology are given by

  $$\begin{aligned} &\frac{\dot{a}^{2}}{a^{2}}=\frac{8 \pi G \rho}{3}-\frac{1}{a^{2}} \\ &\dot{\rho}+3 \frac{\dot{a}}{a}(\rho+P)=0 \end{aligned}$$

  where the speed of light is set equal to unity, $G$ is the gravitational constant, $a(t)$ is the expansion scale factor, $\rho$ is the fluid mass density and $P$ is the fluid pressure, and overdots denote differentiation with respect to the time coordinate $t$.

  If the universe contains only blackbody radiation and $a=0$ defines the zero of time $t$, show that

  $$a^{2}(t)=t\left(t_{*}-t\right)$$

  where $t_{*}$ is a constant. What is the physical significance of the time $t_{*}$ ? What is the value of the ratio $a(t) / t$ at the time when the scale factor is largest? Sketch the curve of $a(t)$ and identify its geometric shape.

  Briefly comment on whether this cosmological model is a good description of the observed universe at any time in its history.

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• Paper 1, Section II, D

  Cosmology | Part II, 2013

  A spherically symmetric star of total mass $M_{s}$ has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is the radial distance from its centre. These quantities are related by the equations of hydrostatic equilibrium and mass conservation:

  $$\begin{aligned} \frac{d P}{d r} &=-\frac{G M(r) \rho}{r^{2}} \\ \frac{d M}{d r} &=4 \pi \rho r^{2} \end{aligned}$$

  where $M(r)$ is the mass inside radius $r$.

  By integrating from the centre of the star at $r=0$, where $P=P_{c}$, to the surface of the star at $r=R_{s}$, where $P=P_{s}$, show that

  $$4 \pi R_{s}^{3} P_{s}=\Omega+3 \int_{0}^{M_{s}} \frac{P}{\rho} d M,$$

  where $\Omega$ is the total gravitational potential energy. Show that

  $$-\Omega>\frac{G M_{s}^{2}}{2 R_{s}}$$

  If the surface pressure is negligible and the star is a perfect gas of particles of mass $m$ with number density $n$ and $P=n k_{B} T$ at temperature $T$, and radiation pressure can be ignored, then show that

  $$3 \int_{0}^{M_{s}} \frac{P}{\rho} d M=\frac{3 k_{B}}{m} \bar{T},$$

  where $\bar{T}$ is the mean temperature of the star, which you should define.

  Hence, show that the mean temperature of the star satisfies the inequality

  $$\bar{T}>\frac{G M_{s} m}{6 k_{B} R_{s}}$$

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• Paper 2, Section I, D

  Cosmology | Part II, 2013

  The linearised equation for the growth of small inhomogeneous density perturbations $\delta_{\mathbf{k}}$ with comoving wavevector $\mathbf{k}$ in an isotropic and homogeneous universe is

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

  where $\rho$ is the matter density, $c_{s}=(d P / d \rho)^{1 / 2}$ is the sound speed, $P$ is the pressure, $a(t)$ is the expansion scale factor of the unperturbed universe, and overdots denote differentiation with respect to time $t$.

  Define the Jeans wavenumber and explain its physical meaning.

  Assume the unperturbed Friedmann universe has zero curvature and cosmological constant and it contains only zero-pressure matter, so that $a(t)=a_{0} t^{2 / 3}$. Show that the solution for the growth of density perturbations is given by

  $$\delta_{\mathbf{k}}=A(\mathbf{k}) t^{2 / 3}+B(\mathbf{k}) t^{-1}$$

  Comment briefly on the cosmological significance of this result.

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• Paper 3, Section I, D

  Cosmology | Part II, 2013

  The number densities of protons of mass $m_{p}$ or neutrons of mass $m_{n}$ in kinetic equilibrium at temperature $T$, in the absence of any chemical potentials, are each given by (with $i=n$ or $p$ )

  $$n_{i}=g_{i}\left(\frac{m_{i} k_{B} T}{2 \pi \hbar^{2}}\right)^{3 / 2} \exp \left[-m_{i} c^{2} / k_{B} T\right]$$

  where $k_{B}$ is Boltzmann's constant and $g_{i}$ is the spin degeneracy.

  Use this to show, to a very good approximation, that the ratio of the number of neutrons to protons at a temperature $T \simeq 1 \mathrm{MeV} / k_{B}$ is given by

  $$\frac{n_{n}}{n_{p}}=\exp \left[-\left(m_{n}-m_{p}\right) c^{2} / k_{B} T\right]$$

  where $\left(m_{n}-m_{p}\right) c^{2}=1.3 \mathrm{MeV}$. Explain any approximations you have used.

  The reaction rate for weak interactions between protons and neutrons at energies $5 \mathrm{MeV} \geqslant k_{B} T \geqslant 0.8 \mathrm{MeV}$ is given by $\Gamma=\left(k_{B} T / 1 \mathrm{MeV}\right)^{5} \mathrm{~s}^{-1}$ and the expansion rate of the universe at these energies is given by $H=\left(k_{B} T / 1 M e V\right)^{2} s^{-1}$. Give an example of a weak interaction that can maintain equilibrium abundances of protons and neutrons at these energies. Show how the final abundance of neutrons relative to protons can be calculated and use it to estimate the mass fraction of the universe in helium- 4 after nucleosynthesis.

  What would have happened to the helium abundance if the proton and neutron masses had been exactly equal?

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• Paper 3, Section II, D

  Cosmology | Part II, 2013

  The contents of a spatially homogeneous and isotropic universe are modelled as a finite mass $M$ of pressureless material whose radius $r(t)$ evolves from some constant reference radius $r_{0}$ in proportion to the time-dependent scale factor $a(t)$, with

  $$r(t)=a(t) r_{0}$$

  (i) Show that this motion leads to expansion governed by Hubble's Law. If this universe is expanding, explain why there will be a shift in the frequency of radiation between its emission from a distant object and subsequent reception by an observer. Define the redshift $z$ of the observed object in terms of the values of the scale factor $a(t)$ at the times of emission and reception.

  (ii) The expanding universal mass $M$ is given a small rotational perturbation, with angular velocity $\omega$, and its angular momentum is subsequently conserved. If deviations from spherical expansion can be neglected, show that its linear rotational velocity will fall as $V \propto a^{-n}$, where you should determine the value of $n$. Show that this perturbation will become increasingly insignificant compared to the expansion velocity as the universe expands if $a \propto t^{2 / 3}$.

  (iii) A distant cloud of intermingled hydrogen (H) atoms and carbon monoxide (CO) molecules has its redshift determined simultaneously in two ways: by detecting $21 \mathrm{~cm}$ radiation from atomic hydrogen and by detecting radiation from rotational transitions in CO molecules. The ratio of the $21 \mathrm{~cm}$ atomic transition frequency to the CO rotational transition frequency is proportional to $\alpha^{2}$, where $\alpha$ is the fine structure constant. It is suggested that there may be a small difference in the value of the constant $\alpha$ between the times of emission and reception of the radiation from the cloud.

  Show that the difference in the redshift values for the cloud, $\Delta z=z_{C O}-z_{21}$, determined separately by observations of the $\mathrm{H}$ and $\mathrm{CO}$ transitions, is related to $\delta \alpha=$ $\alpha_{r}-\alpha_{e}$, the difference in $\alpha$ values at the times of reception and emission, by

  $$\Delta z=2\left(\frac{\delta \alpha}{\alpha_{r}}\right)\left(1+z_{C O}\right)$$

  (iv) The universe today contains $30 \%$ of its total density in the form of pressureless matter and $70 \%$ in the form of a dark energy with constant redshift-independent density. If these are the only two significant constituents of the universe, show that their densities were equal when the scale factor of the universe was approximately equal to $75 \%$ of its present value.

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• Paper 4, Section I, D

  Cosmology | Part II, 2013

  List the relativistic species of bosons and fermions from the standard model of particle physics that are present in the early universe when the temperature falls to $1 \mathrm{MeV} / \mathrm{k}_{B}$.

  Which of the particles above will be interacting when the temperature is above $1 \mathrm{MeV} / k_{B}$ and between $1 \mathrm{MeV} / k_{B} \geq T \geq 0.51 \mathrm{MeV} / k_{B}$, respectively?

  Explain what happens to the populations of particles present when the temperature falls to $0.51 \mathrm{MeV} / k_{B}$.

  The entropy density of fermion and boson species with temperature $T$ is $s \propto g_{s} T^{3}$, where $g_{s}$ is the number of relativistic spin degrees of freedom, that is,

  $$g_{s}=\sum_{\text {bosons }} g_{i}+\frac{7}{8} \sum_{\text {fermions }} g_{i}$$

  Show that when the temperature of the universe falls below $0.51 \mathrm{MeV} / k_{B}$ the ratio of the neutrino and photon temperatures will be given by

  $$\frac{T_{\nu}}{T_{\gamma}}=\left(\frac{4}{11}\right)^{1 / 3}$$

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Differential Geometry

• Paper 1, Section II, H

  Differential Geometry | Part II, 2013

  For $f: X \rightarrow Y$ a smooth map of manifolds, define the concepts of critical point, critical value and regular value.

  With the obvious identification of $\mathbf{C}$ with $\mathbf{R}^{2}$, and hence also of $\mathbf{C}^{3}$ with $\mathbf{R}^{6}$, show that the complex-valued polynomial $z_{1}^{3}+z_{2}^{2}+z_{3}^{2}$ determines a smooth map $f: \mathbf{R}^{6} \rightarrow \mathbf{R}^{2}$ whose only critical point is at the origin. Hence deduce that $V:=f^{-1}((0,0)) \backslash\{\mathbf{0}\} \subset \mathbf{R}^{6}$ is a 4-dimensional manifold, and find the equations of its tangent space at any given point $\left(z_{1}, z_{2}, z_{3}\right) \in V$.

  Now let $S^{5} \subset \mathbf{C}^{3}=\mathbf{R}^{6}$ be the unit 5 -sphere, defined by $\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}=1$. Given a point $P=\left(z_{1}, z_{2}, z_{3}\right) \in S^{5} \cap V$, by considering the vector $\left(2 z_{1}, 3 z_{2}, 3 z_{3}\right) \in \mathbf{C}^{3}=\mathbf{R}^{6}$ or otherwise, show that not all tangent vectors to $V$ at $P$ are tangent to $S^{5}$. Deduce that $S^{5} \cap V \subset \mathbf{R}^{6}$ is a compact three-dimensional manifold.

  [Standard results may be quoted without proof if stated carefully.]

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• Paper 2, Section II, H

  Differential Geometry | Part II, 2013

  Let $\alpha:[0, L] \rightarrow \mathbf{R}^{3}$ be a regular curve parametrized by arc length having nowherevanishing curvature. State the Frenet relations between the tangent, normal and binormal vectors at a point, and their derivatives.

  Let $S \subset \mathbf{R}^{3}$ be a smooth oriented surface. Define the Gauss map $N: S \rightarrow S^{2}$, and show that its derivative at $P \in S, d N_{P}: T_{P} S \rightarrow T_{P} S$, is self-adjoint. Define the Gaussian curvature of $S$ at $P$.

  Now suppose that $\alpha:[0, L] \rightarrow \mathbf{R}^{3}$ has image in $S$ and that its normal curvature is zero for all $s \in[0, L]$. Show that the Gaussian curvature of $S$ at a point $P=\alpha(s)$ of the curve is $K(P)=-\tau(s)^{2}$, where $\tau(s)$ denotes the torsion of the curve.

  If $S \subset \mathbf{R}^{3}$ is a standard embedded torus, show that there is a curve on $S$ for which the normal curvature vanishes and the Gaussian curvature of $S$ is zero at all points of the curve.

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• Paper 3, Section II, H

  Differential Geometry | Part II, 2013

  We say that a parametrization $\phi: U \rightarrow S \subset \mathbf{R}^{3}$ of a smooth surface $S$ is isothermal if the coefficients of the first fundamental form satisfy $F=0$ and $E=G=\lambda(u, v)^{2}$, for some smooth non-vanishing function $\lambda$ on $U$. For an isothermal parametrization, prove that

  $$\phi_{u u}+\phi_{v v}=2 \lambda^{2} H \mathbf{N}$$

  where $\mathbf{N}$ denotes the unit normal vector and $H$ the mean curvature, which you may assume is given by the formula

  $$H=\frac{g+e}{2 \lambda^{2}}$$

  where $g=-\left\langle\mathbf{N}_{u}, \phi_{u}\right\rangle$ and $e=-\left\langle\mathbf{N}_{v}, \phi_{v}\right\rangle$ are coefficients in the second fundamental form.

  Given a parametrization $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ of a surface $S \subset \mathbf{R}^{3}$, we consider the complex valued functions on $U$ :

  $$\theta_{1}=x_{u}-i x_{v}, \quad \theta_{2}=y_{u}-i y_{v}, \quad \theta_{3}=z_{u}-i z_{v}$$

  Show that $\phi$ is isothermal if and only if $\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}=0$. If $\phi$ is isothermal, show that $S$ is a minimal surface if and only if $\theta_{1}, \theta_{2}, \theta_{3}$ are holomorphic functions of the complex variable $\zeta=u+i v$

  Consider the holomorphic functions on $D:=\mathbf{C} \backslash \mathbf{R}_{\geqslant 0}$ (with complex coordinate $\zeta=u+i v$ on $\mathbf{C})$ given by

  $$\theta_{1}:=\frac{1}{2}\left(1-\zeta^{-2}\right), \quad \theta_{2}:=-\frac{i}{2}\left(1+\zeta^{-2}\right), \quad \theta_{3}:=-\zeta^{-1}$$

  Find a smooth map $\phi(u, v)=(x(u, v), y(u, v), z(u, v)): D \rightarrow \mathbf{R}^{3}$ for which $\phi(-1,0)=\mathbf{0}$ and the $\theta_{i}$ defined by (2) satisfy the equations (1). Show furthermore that $\phi$ extends to a smooth map $\tilde{\phi}: \mathbf{C}^{*} \rightarrow \mathbf{R}^{3}$. If $w=x+i y$ is the complex coordinate on $\mathbf{C}$, show that

  $$\widetilde{\phi}(\exp (i w))=(\cosh y \cos x+1, \cosh y \sin x, y)$$

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• Paper 4, Section II, H

  Differential Geometry | Part II, 2013

  Define what is meant by the geodesic curvature $k_{g}$ of a regular curve $\alpha: I \rightarrow S$ parametrized by arc length on a smooth oriented surface $S \subset \mathbf{R}^{3}$. If $S$ is the unit sphere in $\mathbf{R}^{3}$ and $\alpha: I \rightarrow S$ is a parametrized geodesic circle of radius $\phi$, with $0<\phi<\pi / 2$, justify the fact that $\left|k_{g}\right|=\cot \phi$.

  State the general form of the Gauss-Bonnet theorem with boundary on an oriented surface $S$, explaining briefly the terms which occur.

  Let $S \subset \mathbf{R}^{3}$ now denote the circular cone given by $z>0$ and $x^{2}+y^{2}=z^{2} \tan ^{2} \phi$, for a fixed choice of $\phi$ with $0<\phi<\pi / 2$, and with a fixed choice of orientation. Let $\alpha: I \rightarrow S$ be a simple closed piecewise regular curve on $S$, with (signed) exterior angles $\theta_{1}, \ldots, \theta_{N}$ at the vertices (that is, $\theta_{i}$ is the angle between limits of tangent directions, with sign determined by the orientation). Suppose furthermore that the smooth segments of $\alpha$ are geodesic curves. What possible values can $\theta_{1}+\cdots+\theta_{N}$ take? Justify your answer.

  [You may assume that a simple closed curve in $\mathbf{R}^{2}$ bounds a region which is homeomorphic to a disc. Given another simple closed curve in the interior of this region, you may assume that the two curves bound a region which is homeomorphic to an annulus.]

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Dynamical Systems

• Paper 1, Section I, C

  Dynamical Systems | Part II, 2013

  Consider the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$ which has a hyperbolic fixed point at the origin.

  Define the stable and unstable invariant subspaces of the system linearised about the origin. Give a constraint on the dimensions of these two subspaces.

  Define the local stable and unstable manifolds of the origin for the system. How are these related to the invariant subspaces of the linearised system?

  For the system

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

  calculate the stable and unstable manifolds of the origin, each correct up to and including cubic order.

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• Paper 2, Section I, $7 \mathrm{C}$

  Dynamical Systems | Part II, 2013

  Let $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ be a two-dimensional dynamical system with a fixed point at $\mathbf{x}=\mathbf{0}$. Define a Lyapunov function $V(\mathbf{x})$ and explain what it means for $\mathbf{x}=\mathbf{0}$ to be Lyapunov stable.

  For the system

  $$\begin{aligned} &\dot{x}=-x-2 y+x^{3} \\ &\dot{y}=-y+x+\frac{1}{2} y^{3}+x^{2} y \end{aligned}$$

  determine the values of $C$ for which $V=x^{2}+C y^{2}$ is a Lyapunov function in a sufficiently small neighbourhood of the origin.

  For the case $C=2$, find $V_{1}$ and $V_{2}$ such that $V(\mathbf{x})<V_{1}$ at $t=0$ implies that $V \rightarrow 0$ as $t \rightarrow \infty$ and $V(\mathbf{x})>V_{2}$ at $t=0$ implies that $V \rightarrow \infty$ as $t \rightarrow \infty$

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• Paper 3, Section I, C

  Dynamical Systems | Part II, 2013

  A one-dimensional map is defined by

  $$x_{n+1}=F\left(x_{n}, \mu\right)$$

  where $\mu$ is a parameter. What is the condition for a bifurcation of a fixed point $x_{*}$ of $F$ ?

  Let $F(x, \mu)=x\left(x^{2}-2 x+\mu\right)$. Find the fixed points and show that bifurcations occur when $\mu=-1, \mu=1$ and $\mu=2$. Sketch the bifurcation diagram, showing the locus and stability of the fixed points in the $(x, \mu)$ plane and indicating the type of each bifurcation.

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• Paper 3, Section II, C

  Dynamical Systems | Part II, 2013

  Let $f: I \rightarrow I$ be a continuous map of an interval $I \subset \mathbb{R}$. Explain what is meant by the statements (a) $f$ has a horseshoe and (b) $f$ is chaotic according to Glendinning's definition of chaos.

  Assume that $f$ has a 3-cycle $\left\{x_{0}, x_{1}, x_{2}\right\}$ with $x_{1}=f\left(x_{0}\right), x_{2}=f\left(x_{1}\right), x_{0}=f\left(x_{2}\right)$, $x_{0}<x_{1}<x_{2}$. Prove that $f^{2}$ has a horseshoe. [You may assume the Intermediate Value Theorem.]

  Represent the effect of $f$ on the intervals $I_{a}=\left[x_{0}, x_{1}\right]$ and $I_{b}=\left[x_{1}, x_{2}\right]$ by means of a directed graph. Explain how the existence of the 3 -cycle corresponds to this graph.

  The map $g: I \rightarrow I$ has a 4-cycle $\left\{x_{0}, x_{1}, x_{2}, x_{3}\right\}$ with $x_{1}=g\left(x_{0}\right), x_{2}=g\left(x_{1}\right)$, $x_{3}=g\left(x_{2}\right)$