Part II, 2007

Algebraic Topology

• $3 . \mathrm{II} . 20 \mathrm{H}$

  Algebraic Topology | Part II, 2007

  Define what it means for a group $G$ to act on a topological space $X$. Prove that, if $G$ acts freely, in a sense that you should specify, then the quotient map $X \rightarrow X / G$ is a covering map and there is a surjective group homomorphism from the fundamental group of $X / G$ to $G$.

  comment

• 1.II.21H

  Algebraic Topology | Part II, 2007

  (i) Compute the fundamental group of the Klein bottle. Show that this group is not abelian, for example by defining a suitable homomorphism to the symmetric group $S_{3}$.

  (ii) Let $X$ be the closed orientable surface of genus 2 . How many (connected) double coverings does $X$ have? Show that the fundamental group of $X$ admits a homomorphism onto the free group on 2 generators.

  comment

• 2.II.21H

  Algebraic Topology | Part II, 2007

  State the Mayer-Vietoris sequence for a simplicial complex $X$ which is a union of two subcomplexes $A$ and $B$. Define the homomorphisms in the sequence (but do not check that they are well-defined). Prove exactness of the sequence at the term $H_{i}(A \cap B)$.

  comment

• 4.II $. 21 \mathrm{H}$

  Algebraic Topology | Part II, 2007

  Compute the homology of the space obtained from the torus $S^{1} \times S^{1}$ by identifying $S^{1} \times\{p\}$ to a point and $S^{1} \times\{q\}$ to a point, for two distinct points $p$ and $q$ in $S^{1} .$

  comment

Applications of Quantum Mechanics

• 1.II.33A

  Applications of Quantum Mechanics | Part II, 2007

  In a certain spherically symmetric potential, the radial wavefunction for particle scattering in the $l=0$ sector ( $S$-wave), for wavenumber $k$ and $r \gg 0$, is

  $$R(r, k)=\frac{A}{k r}\left(g(-k) e^{-i k r}-g(k) e^{i k r}\right)$$

  where

  $$g(k)=\frac{k+i \kappa}{k-i \alpha}$$

  with $\kappa$ and $\alpha$ real, positive constants. Scattering in sectors with $l \neq 0$ can be neglected. Deduce the formula for the $S$-matrix in this case and show that it satisfies the expected symmetry and reality properties. Show that the phase shift is

  $$\delta(k)=\tan ^{-1} \frac{k(\kappa+\alpha)}{k^{2}-\kappa \alpha}$$

  What is the scattering length for this potential?

  From the form of the radial wavefunction, deduce the energies of the bound states, if any, in this system. If you were given only the $S$-matrix as a function of $k$, and no other information, would you reach the same conclusion? Are there any resonances here?

  [Hint: Recall that $S(k)=e^{2 i \delta(k)}$ for real $k$, where $\delta(k)$ is the phase shift.]

  comment

• 2.II.33A

  Applications of Quantum Mechanics | Part II, 2007

  Describe the variational method for estimating the ground state energy of a quantum system. Prove that an error of order $\epsilon$ in the wavefunction leads to an error of order $\epsilon^{2}$ in the energy.

  Explain how the variational method can be generalized to give an estimate of the energy of the first excited state of a quantum system.

  Using the variational method, estimate the energy of the first excited state of the anharmonic oscillator with Hamiltonian

  $$H=-\frac{d^{2}}{d x^{2}}+x^{2}+x^{4}$$

  How might you improve your estimate?

  [Hint: If $I_{2 n}=\int_{-\infty}^{\infty} x^{2 n} e^{-a x^{2}} d x$ then

  $$\left.I_{0}=\sqrt{\frac{\pi}{a}}, \quad I_{2}=\sqrt{\frac{\pi}{a}} \frac{1}{2 a}, \quad I_{4}=\sqrt{\frac{\pi}{a}} \frac{3}{4 a^{2}}, \quad I_{6}=\sqrt{\frac{\pi}{a}} \frac{15}{8 a^{3}}\right]$$

  comment

• 3.II.33A

  Applications of Quantum Mechanics | Part II, 2007

  Consider the Hamiltonian

  $$H=\mathbf{B}(t) \cdot \mathbf{S}$$

  for a particle of spin $\frac{1}{2}$ fixed in space, in a rotating magnetic field, where

  $$S_{1}=\frac{\hbar}{2}\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad S_{2}=\frac{\hbar}{2}\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad S_{3}=\frac{\hbar}{2}\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

  and

  $$\mathbf{B}(t)=B(\sin \alpha \cos \omega t, \sin \alpha \sin \omega t, \cos \alpha)$$

  with $B, \alpha$ and $\omega$ constant, and $B>0, \omega>0$.

  There is an exact solution of the time-dependent Schrödinger equation for this Hamiltonian,

  $$\chi(t)=\left(\cos \left(\frac{1}{2} \lambda t\right)-i \frac{B-\omega \cos \alpha}{\lambda} \sin \left(\frac{1}{2} \lambda t\right)\right) e^{-i \omega t / 2} \chi_{+}+i\left(\frac{\omega}{\lambda} \sin \alpha \sin \left(\frac{1}{2} \lambda t\right)\right) e^{i \omega t / 2} \chi_{-}$$

  where $\lambda \equiv\left(\omega^{2}-2 \omega B \cos \alpha+B^{2}\right)^{1 / 2}$ and

  $$\chi_{+}=\left(\begin{array}{c} \cos \frac{\alpha}{2} \\ e^{i \omega t} \sin \frac{\alpha}{2} \end{array}\right), \quad \chi_{-}=\left(\begin{array}{c} e^{-i \omega t} \sin \frac{\alpha}{2} \\ -\cos \frac{\alpha}{2} \end{array}\right)$$

  Show that, for $\omega \ll B$, this exact solution simplifies to a form consistent with the adiabatic approximation. Find the dynamic phase and the geometric phase in the adiabatic regime. What is the Berry phase for one complete cycle of $\mathbf{B}$ ?

  The Berry phase can be calculated as an integral of the form

  $$\Gamma=i \oint\left\langle\psi \mid \nabla_{\mathbf{R}} \psi\right\rangle \cdot d \mathbf{R}$$

  Evaluate $\Gamma$ for the adiabatic evolution described above.

  comment

• 4.II.33A

  Applications of Quantum Mechanics | Part II, 2007

  Consider a 1-dimensional chain of $2 N$ atoms of mass $m$ (with $N$ large and with periodic boundary conditions). The interactions between neighbouring atoms are modelled by springs with alternating spring constants $K$ and $G$, with $K>G$.

  

  In equilibrium, the separation of the atoms is $a$, the natural length of the springs.

  Find the frequencies of the longitudinal modes of vibration for this system, and show that they are labelled by a wavenumber $q$ that is restricted to a Brillouin zone. Identify the acoustic and optical bands of the vibration spectrum, and determine approximations for the frequencies near the centre of the Brillouin zone. What is the frequency gap between the acoustic and optical bands at the zone boundary?

  Describe briefly the properties of the phonons in this system.

  comment

Applied Probability

• 1.II.26J

  Applied Probability | Part II, 2007

  An open air rock concert is taking place in beautiful Pine Valley, and enthusiastic fans from the entire state of Alifornia are heading there long before the much anticipated event. The arriving cars have to be directed to one of three large (practically unlimited) parking lots, $a, b$ and $c$ situated near the valley entrance. The traffic cop at the entrance to the valley decides to direct every third car (in the order of their arrival) to a particular lot. Thus, cars $1,4,7,10$ and so on are directed to lot $a$, cars $2,5,8,11$ to lot $b$ and cars $3,6,9,12$ to lot $c$.

  Suppose that the total arrival process $N(t), t \geqslant 0$, at the valley entrance is Poisson, of rate $\lambda>0$ (the initial time $t=0$ is taken to be considerably ahead of the actual event). Consider the processes $X^{a}(t), X^{b}(t)$ and $X^{c}(t)$ where $X^{i}(t)$ is the number of cars arrived in lot $i$ by time $t, i=a, b, c$. Assume for simplicity that the time to reach a parking lot from the entrance is negligible so that the car enters its specified lot at the time it crosses the valley entrance.

  (a) Give the probability density function of the time of the first arrival in each of the processes $X^{a}(t), X^{b}(t), X^{c}(t)$.

  (b) Describe the distribution of the time between two subsequent arrivals in each of these processes. Are these times independent? Justify your answer.

  (c) Which of these processes are delayed renewal processes (where the distribution of the first arrival time differs from that of the inter-arrival time)?

  (d) What are the corresponding equilibrium renewal processes?

  (e) Describe how the direction rule should be changed for $X^{a}(t), X^{b}(t)$ and $X^{c}(t)$ to become Poisson processes, of rate $\lambda / 3$. Will these Poisson processes be independent? Justify your answer.

  comment

• 2.II.26J

  Applied Probability | Part II, 2007

  In this question we work with a continuous-time Markov chain where the rate of jump $i \rightarrow j$ may depend on $j$ but not on $i$. A virus can be in one of $s$ strains $1, \ldots, s$, and it mutates to strain $j$ with rate $r_{j} \geqslant 0$ from each strain $i \neq j$. (Mutations are caused by the chemical environment.) Set $R=r_{1}+\ldots+r_{s}$.

  (a) Write down the Q-matrix (the generator) of the chain $\left(X_{t}\right)$ in terms of $r_{j}$ and $R$.

  (b) If $R=0$, that is, $r_{1}=\ldots=r_{s}=0$, what are the communicating classes of the chain $\left(X_{t}\right)$ ?

  (c) From now on assume that $R>0$. State and prove a necessary and sufficient condition, in terms of the numbers $r_{j}$, for the chain $\left(X_{t}\right)$ to have a single communicating class (which therefore should be closed).

  (d) In general, what is the number of closed communicating classes in the chain $\left(X_{t}\right)$ ? Describe all open communicating classes of $\left(X_{t}\right)$.

  (e) Find the equilibrium distribution of $\left(X_{t}\right)$. Is the chain $\left(X_{t}\right)$ reversible? Justify your answer.

  (f) Write down the transition matrix $\widehat{P}=\left(\widehat{p}_{i j}\right)$ of the discrete-time jump chain for $\left(X_{t}\right)$ and identify its equilibrium distribution. Is the jump chain reversible? Justify your answer.

  comment

• 3.II.25J

  Applied Probability | Part II, 2007

  For a discrete-time Markov chain, if the probability of transition $i \rightarrow j$ does not depend on $i$ then the chain is reduced to a sequence of independent random variables (states). In this case, the chain forgets about its initial state and enters equilibrium after a single transition. In the continuous-time case, a Markov chain whose rates $q_{i j}$ of transition $i \rightarrow j$ depend on $j$ but not on $i \neq j$ still 'remembers' its initial state and reaches equilibrium only in the limit as the time grows indefinitely. This question is an illustration of this property.

  A protean sea sponge may change its colour among $s$ varieties $1, \ldots, s$, under the influence of the environment. The rate of transition from colour $i$ to $j$ equals $r_{j} \geqslant 0$ and does not depend on $i, i \neq j$. Consider a Q-matrix $Q=\left(q_{i j}\right)$ with entries

  $$q_{i j}= \begin{cases}r_{j}, & i \neq j \\ -R+r_{i}, & i=j\end{cases}$$

  where $R=r_{1}+\ldots+r_{s}$. Assume that $R>0$ and let $\left(X_{t}\right)$ be the continuous-time Markov chain with generator $Q$. Given $t \geqslant 0$, let $P(t)=\left(p_{i j}(t)\right)$ be the matrix of transition probabilities in time $t$ in chain $\left(X_{t}\right)$.

  (a) State the exponential relation between the matrices $Q$ and $P(t)$.

  (b) Set $\pi_{j}=r_{j} / R, j=1, \ldots, s$. Check that $\pi_{1}, \ldots, \pi_{s}$ are equilibrium probabilities for the chain $\left(X_{t}\right)$. Is this a unique equilibrium distribution? What property of the vector with entries $\pi_{j}$ relative to the matrix $Q$ is involved here?

  (c) Let $\mathbf{x}$ be a vector with components $x_{1}, \ldots, x_{s}$ such that $x_{1}+\ldots+x_{s}=0$. Show that $\mathbf{x}^{\mathrm{T}} Q=-R \mathbf{x}^{\mathrm{T}}$. Compute $\mathbf{x}^{\mathrm{T}} P(t)$

  (d) Now let $\delta_{i}$ denote the (column) vector whose entries are 0 except for the $i$ th one which equals 1. Observe that the $i$ th row of $P(t)$ is $\delta_{i}^{\mathrm{T}} P(t)$. Prove that $\delta_{i}^{\mathrm{T}} P(t)=\pi^{\mathrm{T}}+e^{-t R}\left(\delta_{i}^{\mathrm{T}}-\pi^{\mathrm{T}}\right) .$

  (e) Deduce the expression for transition probabilities $p_{i j}(t)$ in terms of rates $r_{j}$ and their sum $R$.

  comment

• 4.II.26J

  Applied Probability | Part II, 2007

  A population of rare Monarch butterflies functions as follows. At the times of a Poisson process of rate $\lambda$ a caterpillar is produced from an egg. After an exponential time, the caterpillar is transformed into a pupa which, after an exponential time, becomes a butterfly. The butterfly lives for another exponential time and then dies. (The Poissonian assumption reflects the fact that butterflies lay a huge number of eggs most of which do not develop.) Suppose that all lifetimes are independent (of the arrival process and of each other) and let their rate be $\mu$. Assume that the population is in an equilibrium and let $C$ be the number of caterpillars, $R$ the number of pupae and $B$ the number of butterflies (so that the total number of insects, in any metamorphic form, equals $N=C+R+B)$. Let $\pi_{(c, r, b)}$ be the equilibrium probability $\mathbb{P}(C=c, R=r, B=b)$ where $c, r, b=0,1, \ldots$

  (a) Specify the rates of transitions $(c, r, b) \rightarrow\left(c^{\prime}, r^{\prime}, b^{\prime}\right)$ for the resulting continuous-time Markov chain $\left(X_{t}\right)$ with states $(c, r, b)$. (The rates are non-zero only when $c^{\prime}=c$ or $c^{\prime}=c \pm 1$ and similarly for other co-ordinates.) Check that the holding rate for state $(c, r, b)$ is $\lambda+\mu n$ where $n=c+r+b$.

  (b) Let $Q$ be the Q-matrix from (a). Consider the invariance equation $\pi Q=0$. Verify that the only solution is

  $$\pi_{(c, r, b)}=\frac{(3 \lambda / \mu)^{n}}{3^{n} c ! r ! b !} \exp \left(-\frac{3 \lambda}{\mu}\right), \quad n=c+r+b$$

  (c) Derive the marginal equilibrium probabilities $\mathbb{P}(N=n)$ and the conditional equilibrium probabilities $\mathbb{P}(C=c, R=r, B=b \mid N=n)$.

  (d) Determine whether the chain $\left(X_{t}\right)$ is positive recurrent, null-recurrent or transient.

  (e) Verify that the equilibrium probabilities $\mathbb{P}(N=n)$ are the same as in the corresponding $M / G I / \infty$ system (with the correct specification of the arrival rate and the service-time distribution).

  comment

Asymptotic Methods

• 1.II.30B

  Asymptotic Methods | Part II, 2007

  State Watson's lemma, describing the asymptotic behaviour of the integral

  $$I(\lambda)=\int_{0}^{A} e^{-\lambda t} f(t) d t, \quad A>0$$

  as $\lambda \rightarrow \infty$, given that $f(t)$ has the asymptotic expansion

  $$f(t) \sim t^{\alpha} \sum_{n=0}^{\infty} a_{n} t^{n \beta}$$

  as $t \rightarrow 0_{+}$, where $\beta>0$ and $\alpha>-1$.

  Give an account of Laplace's method for finding asymptotic expansions of integrals of the form

  $$J(z)=\int_{-\infty}^{\infty} e^{-z p(t)} q(t) d t$$

  for large real $z$, where $p(t)$ is real for real $t$.

  Deduce the following asymptotic expansion of the contour integral

  $$\int_{-\infty-i \pi}^{\infty+i \pi} \exp (z \cosh t) d t=2^{1 / 2} i e^{z} \Gamma\left(\frac{1}{2}\right)\left[z^{-1 / 2}+\frac{1}{8} z^{-3 / 2}+O\left(z^{-5 / 2}\right)\right]$$

  as $z \rightarrow \infty$.

  comment

• 3.II.30B

  Asymptotic Methods | Part II, 2007

  Explain the method of stationary phase for determining the behaviour of the integral

  $$I(x)=\int_{a}^{b} d u e^{i x f(u)}$$

  for large $x$. Here, the function $f(u)$ is real and differentiable, and $a, b$ and $x$ are all real.

  Apply this method to show that the first term in the asymptotic behaviour of the function

  $$\Gamma(m+1)=\int_{0}^{\infty} d u u^{m} e^{-u}$$

  where $m=i n$ with $n>0$ and real, is

  $$\Gamma(i n+1) \sim \sqrt{2 \pi} e^{-i n} \exp \left[\left(i n+\frac{1}{2}\right)\left(\frac{i \pi}{2}+\log n\right)\right]$$

  as $n \rightarrow \infty$

  comment

• 4.II.31B

  Asymptotic Methods | Part II, 2007

  Consider the time-independent Schrödinger equation

  $$\frac{d^{2} \psi}{d x^{2}}+\lambda^{2} q(x) \psi(x)=0$$

  where $\lambda \gg 1$ denotes $\hbar^{-1}$ and $q(x)$ denotes $2 m[E-V(x)]$. Suppose that

  and consider a bound state $\psi(x)$. Write down the possible Liouville-Green approximate solutions for $\psi(x)$ in each region, given that $\psi \rightarrow 0$ as $|x| \rightarrow \infty$.

  Assume that $q(x)$ may be approximated by $q^{\prime}(a)(x-a)$ near $x=a$, where $q^{\prime}(a)>0$, and by $q^{\prime}(b)(x-b)$ near $x=b$, where $q^{\prime}(b)<0$. The Airy function $\operatorname{Ai}(z)$ satisfies

  $$\frac{d^{2}(\mathrm{Ai})}{d z^{2}}-z(\mathrm{Ai})=0$$

  and has the asymptotic expansions

  $$\operatorname{Ai}(z) \sim \frac{1}{2} \pi^{-1 / 2} z^{-1 / 4} \exp \left(-\frac{2}{3} z^{3 / 2}\right) \quad \text { as } \quad z \rightarrow+\infty$$

  and

  $$\operatorname{Ai}(z) \sim \pi^{-1 / 2}|z|^{-1 / 4} \cos \left[\left(\frac{2}{3}|z|^{3 / 2}\right)-\frac{\pi}{4}\right] \quad \text { as } \quad z \rightarrow-\infty .$$

  Deduce that the energies $E$ of bound states are given approximately by the WKB condition:

  $$\lambda \int_{a}^{b} q^{1 / 2}(x) d x=\left(n+\frac{1}{2}\right) \pi \quad(n=0,1,2, \ldots)$$

  $$\begin{aligned} & q(x)>0 \quad \text { for } \quad a<x<b \\ & \text { and } q(x)<0 \text { for }-\infty<x<a \text { and } b<x<\infty \end{aligned}$$

  comment

Classical Dynamics

• 1.I.9C

  Classical Dynamics | Part II, 2007

  The action for a system with generalized coordinates, $q_{i}(t)$, for a time interval $\left[t_{1}, t_{2}\right]$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L\left(q_{i}, \dot{q}_{i}\right) d t$$

  where $L$ is the Lagrangian, and where the end point values $q_{i}\left(t_{1}\right)$ and $q_{i}\left(t_{2}\right)$ are fixed at specified values. Derive Lagrange's equations from the principle of least action by considering the variation of $S$ for all possible paths.

  What is meant by the statement that a particular coordinate $q_{j}$ is ignorable? Show that there is an associated constant of the motion, to be specified in terms of $L$.

  A particle of mass $m$ is constrained to move on the surface of a sphere of radius $a$ under a potential, $V(\theta)$, for which the Lagrangian is given by

  $$L=\frac{m}{2} a^{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)-V(\theta)$$

  Identify an ignorable coordinate and find the associated constant of the motion, expressing it as a function of the generalized coordinates. Evaluate the quantity

  $$H=\dot{q}_{i} \frac{\partial L}{\partial \dot{q}_{i}}-L$$

  in terms of the same generalized coordinates, for this case. Is $H$ also a constant of the motion? If so, why?

  comment

• 2.I.9C

  Classical Dynamics | Part II, 2007

  The Lagrangian for a particle of mass $m$ and charge $e$ moving in a magnetic field with position vector $\mathbf{r}=(x, y, z)$ is given by

  $$L=\frac{1}{2} m \dot{\mathbf{r}}^{2}+e \frac{\dot{\mathbf{r}} \cdot \mathbf{A}}{c}$$

  where the vector potential $\mathbf{A}(\mathbf{r})$, which does not depend on time explicitly, is related to the magnetic field $\mathbf{B}$ through

  $$\mathbf{B}=\nabla \times \mathbf{A}$$

  Write down Lagrange's equations and use them to show that the equation of motion of the particle can be written in the form

  $$m \ddot{\mathbf{r}}=e \frac{\dot{\mathbf{r}} \times \mathbf{B}}{c}$$

  Deduce that the kinetic energy, $T$, is constant.

  When the magnetic field is of the form $\mathbf{B}=(0,0, d F / d x)$ for some specified function $F(x)$, show further that

  $$\dot{x}^{2}=\frac{2 T}{m}-\frac{(e F(x)+C)^{2}}{m^{2} c^{2}}+D$$

  where $C$ and $D$ are constants.

  comment

• 2.II.15C

  Classical Dynamics | Part II, 2007

  (a) A Hamiltonian system with $n$ degrees of freedom is described by the phase space coordinates $\left(q_{1}, q_{2}, \ldots, q_{n}\right)$ and momenta $\left(p_{1}, p_{2}, \ldots, p_{n}\right)$. Show that the phase-space volume element

  $$d \tau=d q_{1} d q_{2} \ldots . d q_{n} d p_{1} d p_{2} \ldots . d p_{n}$$

  is conserved under time evolution.

  (b) The Hamiltonian, $H$, for the system in part (a) is independent of time. Show that if $F\left(q_{1}, \ldots, q_{n}, p_{1}, \ldots, p_{n}\right)$ is a constant of the motion, then the Poisson bracket $[F, H]$ vanishes. Evaluate $[F, H]$ when

  $$F=\sum_{k=1}^{n} p_{k}$$

  and

  $$H=\sum_{k=1}^{n} p_{k}^{2}+V\left(q_{1}, q_{2}, \ldots, q_{n}\right)$$

  where the potential $V$ depends on the $q_{k}(k=1,2, \ldots, n)$ only through quantities of the form $q_{i}-q_{j}$ for $i \neq j$.

  (c) For a system with one degree of freedom, state what is meant by the transformation

  $$(q, p) \rightarrow(Q(q, p), P(q, p))$$

  being canonical. Show that the transformation is canonical if and only if the Poisson bracket $[Q, P]=1$.

  comment

• 3.I.9C

  Classical Dynamics | Part II, 2007

  A particle of mass $m_{1}$ is constrained to move in the horizontal $(x, y)$ plane, around a circle of fixed radius $r_{1}$ whose centre is at the origin of a Cartesian coordinate system $(x, y, z)$. A second particle of mass $m_{2}$ is constrained to move around a circle of fixed radius $r_{2}$ that also lies in a horizontal plane, but whose centre is at $(0,0, a)$. It is given that the Lagrangian $L$ of the system can be written as

  $$L=\frac{m_{1}}{2} r_{1}^{2} \dot{\phi}_{1}^{2}+\frac{m_{2}}{2} r_{2}^{2} \dot{\phi}_{2}^{2}+\omega^{2} r_{1} r_{2} \cos \left(\phi_{2}-\phi_{1}\right)$$

  using the particles' cylindrical polar angles $\phi_{1}$ and $\phi_{2}$ as generalized coordinates. Deduce the equations of motion and use them to show that $m_{1} r_{1}^{2} \dot{\phi}_{1}+m_{2} r_{2}^{2} \dot{\phi}_{2}$ is constant, and that $\psi=\phi_{2}-\phi_{1}$ obeys an equation of the form

  $$\ddot{\psi}=-k^{2} \sin \psi$$

  where $k$ is a constant to be determined.

  Find two values of $\psi$ corresponding to equilibria, and show that one of the two equilibria is stable. Find the period of small oscillations about the stable equilibrium.

  comment

• 4.I $9 \mathrm{C} \quad$

  Classical Dynamics | Part II, 2007

  (a) Show that the principal moments of inertia for the oblate spheroid of mass $M$ defined by

  $$\frac{\left(x_{1}^{2}+x_{2}^{2}\right)}{a^{2}}+\frac{x_{3}^{2}}{a^{2}\left(1-e^{2}\right)} \leqslant 1$$

  are given by $\left(I_{1}, I_{2}, I_{3}\right)=\frac{2}{5} M a^{2}\left(1-\frac{1}{2} e^{2}, 1-\frac{1}{2} e^{2}, 1\right)$. Here $a$ is the semi-major axis and $e$ is the eccentricity.

  [You may assume that a sphere of radius a has principal moments of inertia $\frac{2}{5} M a^{2}$.]

  (b) The spheroid in part (a) rotates about an axis that is not a principal axis. Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ as viewed in the body frame are

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

  and

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

  Show that $\omega_{3}$ is constant. Show further that the angular momentum vector precesses around the $x_{3}$ axis with period

  $$P=\frac{2 \pi\left(2-e^{2}\right)}{e^{2} \omega_{3}}$$

  comment

• 4.II.15C

  Classical Dynamics | Part II, 2007

  The Hamiltonian for an oscillating particle with one degree of freedom is

  $$H=\frac{p^{2}}{2 m}+V(q, \lambda)$$

  The mass $m$ is a constant, and $\lambda$ is a function of time $t$ alone. Write down Hamilton's equations and use them to show that

  $$\frac{d H}{d t}=\frac{\partial H}{\partial \lambda} \frac{d \lambda}{d t}$$

  Now consider a case in which $\lambda$ is constant and the oscillation is exactly periodic. Denote the constant value of $H$ in that case by $E$. Consider the quantity $I=$ $(2 \pi)^{-1} \oint p d q$, where the integral is taken over a single oscillation cycle. For any given function $V(q, \lambda)$ show that $I$ can be expressed as a function of $E$ and $\lambda$ alone, namely

  $$I=I(E, \lambda)=\frac{(2 m)^{1 / 2}}{2 \pi} \oint(E-V(q, \lambda))^{1 / 2} d q$$

  where the sign of the integrand alternates between the two halves of the oscillation cycle. Let $\tau$ be the period of oscillation. Show that the function $I(E, \lambda)$ has partial derivatives

  $$\frac{\partial I}{\partial E}=\frac{\tau}{2 \pi} \quad \text { and } \quad \frac{\partial I}{\partial \lambda}=-\frac{1}{2 \pi} \oint \frac{\partial V}{\partial \lambda} d t$$

  You may assume without proof that $\partial / \partial E$ and $\partial / \partial \lambda$ may be taken inside the integral.

  Now let $\lambda$ change very slowly with time $t$, by a negligible amount during an oscillation cycle. Assuming that, to sufficient approximation,

  $$\frac{d\langle H\rangle}{d t}=\frac{\partial\langle H\rangle}{\partial \lambda} \frac{d \lambda}{d t}$$

  where $\langle H\rangle$ is the average value of $H$ over an oscillation cycle, and that

  $$\frac{d I}{d t}=\frac{\partial I}{\partial E} \frac{d\langle H\rangle}{d t}+\frac{\partial I}{\partial \lambda} \frac{d \lambda}{d t}$$

  deduce that $d I / d t=0$, carefully explaining your reasoning.

  When

  $$V(q, \lambda)=\lambda q^{2 n}$$

  with $n$ a positive integer and $\lambda$ positive, deduce that

  $$\langle H\rangle=C \lambda^{1 /(n+1)}$$

  for slowly-varying $\lambda$, where $C$ is a constant.

  [Do not try to solve Hamilton's equations. Rather, consider the form taken by $I$. ]

  comment

Coding and Cryptography

• $3 . \mathrm{I} . 4 \mathrm{G} \quad$

  Coding and Cryptography | Part II, 2007

  Compute the rank and minimum distance of the cyclic code with generator polynomial $g(X)=X^{3}+X+1$ and parity-check polynomial $h(X)=X^{4}+X^{2}+X+1$. Now let $\alpha$ be a root of $g(X)$ in the field with 8 elements. We receive the word $r(X)=X^{5}+X^{3}+X \quad\left(\bmod X^{7}-1\right)$. Verify that $r(\alpha)=\alpha^{4}$, and hence decode $r(X)$ using minimum-distance decoding.

  comment

• 1.I.4G

  Coding and Cryptography | Part II, 2007

  Let $\Sigma_{1}$ and $\Sigma_{2}$ be alphabets of sizes $m$ and $a$. What does it mean to say that $f: \Sigma_{1} \rightarrow \Sigma_{2}^{*}$ is a decipherable code? State the inequalities of Kraft and Gibbs, and deduce that if letters are drawn from $\Sigma_{1}$ with probabilities $p_{1}, \ldots, p_{m}$ then the expected word length is at least $H\left(p_{1}, \ldots, p_{m}\right) / \log a$.

  comment

• 1.II.11G

  Coding and Cryptography | Part II, 2007

  Define the bar product $C_{1} \mid C_{2}$ of 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 $C_{2}$. 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 $R M(d, r)$ for $0 \leqslant r \leqslant d$. Show that if $0 \leqslant r \leqslant d-1$, then the dual of $R M(d, r)$ is again a Reed-Muller code.

  comment

• 2.I.4G

  Coding and Cryptography | Part II, 2007

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

  comment

• 2.II.11G

  Coding and Cryptography | Part II, 2007

  Define the capacity of a discrete memoryless channel. State Shannon's second coding theorem and use it to show that the discrete memoryless channel with channel matrix

  $$\left(\begin{array}{ll} 1 & 0 \\ \frac{1}{2} & \frac{1}{2} \end{array}\right)$$

  has capacity $\log 5-2$.

  comment

• 4.I.4G

  Coding and Cryptography | Part II, 2007

  What is a linear feedback shift register? Explain the Berlekamp-Massey method for recovering the feedback polynomial of a linear feedback shift register from its output. Illustrate in the case when we observe output

  $101011001000 \ldots \ldots$

  comment

Cosmology

• 1.I.10A

  Cosmology | Part II, 2007

  Describe the motion of light rays in an expanding universe with scale factor $a(t)$, and derive the redshift formula

  $$1+z=\frac{a\left(t_{0}\right)}{a\left(t_{\mathrm{e}}\right)},$$

  where the light is emitted at time $t_{\mathrm{e}}$ and observed at time $t_{0}$.

  A galaxy at comoving position $\mathbf{x}$ is observed to have a redshift $z$. Given that the galaxy emits an amount of energy $L$ per unit time, show that the total energy per unit time crossing a sphere centred on the galaxy and intercepting the earth is $L /(1+z)^{2}$. Hence, show that the energy per unit time per unit area passing the earth is

  $$\frac{L}{(1+z)^{2}} \frac{1}{4 \pi|\mathbf{x}|^{2} a^{2}\left(t_{0}\right)}$$

  comment

• 1.II.15A

  Cosmology | Part II, 2007

  In a homogeneous and isotropic universe, the scale factor $a(t)$ obeys the Friedmann equation

  $$\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho,$$

  where $\rho$ is the matter density, which, together with the pressure $P$, satisfies

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

  Here, $k$ is a constant curvature parameter. Use these equations to show that the rate of change of the Hubble parameter $H=\dot{a} / a$ satisfies

  $$\dot{H}+H^{2}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)$$

  Suppose that an expanding Friedmann universe is filled with radiation (density $\rho_{R}$ and pressure $\left.P_{R}=\rho_{R} c^{2} / 3\right)$ as well as a "dark energy" component (density $\rho_{\Lambda}$ and pressure $\left.P_{\Lambda}=-\rho_{\Lambda} c^{2}\right)$. Given that the energy densities of these two components are measured today $\left(t=t_{0}\right)$ to be

  $$\rho_{R 0}=\beta \frac{3 H_{0}^{2}}{8 \pi G} \quad \text { and } \quad \rho_{\Lambda 0}=\frac{3 H_{0}^{2}}{8 \pi G} \quad \text { with constant } \beta>0 \quad \text { and } \quad a\left(t_{0}\right)=1,$$

  show that the curvature parameter must satisfy $k c^{2}=\beta H_{0}^{2}$. Hence derive the following relations for the Hubble parameter and its time derivative:

  $$\begin{aligned} H^{2} &=\frac{H_{0}^{2}}{a^{4}}\left(\beta-\beta a^{2}+a^{4}\right) \\ \dot{H} &=-\beta \frac{H_{0}^{2}}{a^{4}}\left(2-a^{2}\right) \end{aligned}$$

  Show qualitatively that universes with $\beta>4$ will recollapse to a Big Crunch in the future. [Hint: Sketch $a^{4} H^{2}$ and $a^{4} \dot{H}$ versus $a^{2}$ for representative values of $\beta$.]

  For $\beta=4$, find an explicit solution for the scale factor $a(t)$ satisfying $a(0)=0$. Find the limiting behaviours of this solution for large and small $t$. Comment briefly on their significance.

  comment

• 2.I.10A

  Cosmology | Part II, 2007

  The number density of photons in thermal equilibrium at temperature $T$ takes the form

  $$n=\frac{8 \pi}{c^{3}} \int \frac{\nu^{2} d \nu}{\exp (h \nu / k T)-1}$$

  At time $t=t_{\mathrm{dec}}$ and temperature $T=T_{\mathrm{dec}}$, photons decouple from thermal equilibrium. By considering how the photon frequency redshifts as the universe expands, show that the form of the equilibrium frequency distribution is preserved, with the temperature for $t>t_{\mathrm{dec}}$ defined by

  $$T \equiv \frac{a\left(t_{\mathrm{dec}}\right)}{a(t)} T_{\mathrm{dec}}$$

  Show that the photon number density $n$ and energy density $\epsilon$ can be expressed in the form

  $$n=\alpha T^{3}, \quad \epsilon=\xi T^{4},$$

  where the constants $\alpha$ and $\xi$ need not be evaluated explicitly.

  comment

• 3.I.10A

  Cosmology | Part II, 2007

  The number density of a non-relativistic species in thermal equilibrium is given by

  $$n=g_{s}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} \exp \left[\left(\mu-m c^{2}\right) / k T\right]$$

  Suppose that thermal and chemical equilibrium is maintained between protons p (mass $m_{\mathrm{p}}$, degeneracy $g_{s}=2$ ), neutrons $\mathrm{n}$ (mass $m_{\mathrm{n}} \approx m_{\mathrm{p}}$, degeneracy $g_{s}=2$ ) and helium-4 nuclei ${ }^{4} \mathrm{He}\left(\right.$ mass $m_{\mathrm{He}} \approx 4 m_{\mathrm{p}}$, degeneracy $g_{s}=1$ ) via the interaction

  $$2 \mathrm{p}+2 \mathrm{n} \leftrightarrow{ }^{4} \mathrm{He}+\gamma$$

  where you may assume the photons $\gamma$ have zero chemical potential $\mu_{\gamma}=0$. Given that the binding energy of helium-4 obeys $B_{\mathrm{He}} / c^{2} \equiv 2 m_{\mathrm{p}}+2 n_{\mathrm{n}}-m_{\mathrm{He}} \ll m_{\mathrm{He}}$, show that the ratio of the number densities can be written as

  $$\frac{n_{\mathrm{p}}^{2} n_{\mathrm{n}}^{2}}{n_{\mathrm{He}}}=2\left(\frac{2 \pi m_{\mathrm{p}} k T}{h^{2}}\right)^{9 / 2} \exp \left(-B_{\mathrm{He}} / k T\right)$$

  Explain briefly why the baryon-to-photon ratio $\eta \equiv n_{B} / n_{\gamma}$ remains constant during the expansion of the universe, where $n_{B} \approx n_{\mathrm{p}}+n_{\mathrm{n}}+4 n_{\mathrm{He}}$ and $n_{\gamma} \approx\left(16 \pi /(h c)^{3}\right)(k T)^{3}$.

  By considering the fractional densities $X_{i} \equiv n_{i} / n_{B}$ of the species $i$, re-express the ratio ( $\uparrow$ ) in the form

  $$\frac{X_{\mathrm{p}}^{2} X_{\mathrm{n}}^{2}}{X_{\mathrm{He}}}=\eta^{-3} \frac{1}{32}\left(\frac{\pi}{2}\right)^{3 / 2}\left(\frac{m_{\mathrm{p}} c^{2}}{k T}\right)^{9 / 2} \exp \left(-B_{\mathrm{He}} / k T\right)$$

  Given that $B_{\mathrm{He}} \approx 30 \mathrm{MeV}$, verify (very approximately) that this ratio approaches unity when $k T \approx 0.3 \mathrm{MeV}$. In reality, helium-4 is not formed until after deuterium production at a considerably lower temperature. Explain briefly the reason for this delay.

  comment

• 3.II.15A

  Cosmology | Part II, 2007

  A spherically symmetric star with outer radius $R$ has mass density $\rho(r)$ and pressure $P(r)$, where $r$ is the distance from the centre of the star. Show that hydrostatic equilibrium implies the pressure support equation,

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

  where $m(r)$ is the mass inside radius $r$. State without proof any results you may need.

  Write down an integral expression for the total gravitational potential energy $E_{\text {grav }}$ of the star. Hence use $(†))$ to deduce the virial theorem

  $$\tag{*} E_{\mathrm{grav}}=-3\langle P\rangle V,$$

  where $\langle P\rangle$ is the average pressure and $V$ is the volume of the star.

  Given that a non-relativistic ideal gas obeys $P=2 E_{\mathrm{kin}} / 3 \mathrm{~V}$ and that an ultrarelativistic gas obeys $P=E_{\text {kin }} / 3 V$, where $E_{\text {kin }}$ is the kinetic energy, discuss briefly the gravitational stability of a star in these two limits.

  At zero temperature, the number density of particles obeying the Pauli exclusion principle is given by

  $$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{p_{\mathrm{F}}} p^{2} d p=\frac{4 \pi g_{s}}{3}\left(\frac{p_{\mathrm{F}}}{h}\right)^{3}$$

  where $p_{\mathrm{F}}$ is the Fermi momentum, $g_{s}$ is the degeneracy and $h$ is Planck's constant. Deduce that the non-relativistic internal energy $E_{\text {kin }}$ of these particles is

  $$E_{\mathrm{kin}}=\frac{4 \pi g_{s} V h^{2}}{10 m_{p}}\left(\frac{p_{\mathrm{F}}}{h}\right)^{5}$$

  where $m_{p}$ is the mass of a particle. Hence show that the non-relativistic Fermi degeneracy pressure satisfies

  $$P \sim \frac{h^{2}}{m_{p}} n^{5 / 3} .$$

  Use the virial theorem $(*)$ to estimate that the radius $R$ of a star supported by Fermi degeneracy pressure is approximately

  $$R \sim \frac{h^{2} M^{-1 / 3}}{G m_{p}^{8 / 3}},$$

  where $M$ is the total mass of the star.

  [Hint: Assume $\rho(r)=m_{p} n(r) \sim m_{p}\langle n\rangle$ and note that $\left.M \approx\left(4 \pi R^{3} / 3\right) m_{p}\langle n\rangle .\right]$

  comment

• 4.I.10A

  Cosmology | Part II, 2007

  The equation governing density perturbation modes $\delta_{\mathbf{k}}(t)$ in a matter-dominated universe (with $a(t)=\left(t / t_{0}\right)^{2 / 3}$ ) is

  $$\ddot{\delta}_{\mathbf{k}}+2 \frac{\dot{a}}{a} \dot{\delta}_{\mathbf{k}}-\frac{3}{2}\left(\frac{\dot{a}}{a}\right)^{2} \delta_{\mathbf{k}}=0$$

  where $\mathbf{k}$ is the comoving wavevector. Find the general solution for the perturbation, showing that there is a growing mode such that

  $$\delta_{\mathbf{k}}(t) \approx \frac{a(t)}{a\left(t_{i}\right)} \delta_{\mathbf{k}}\left(t_{i}\right) \quad\left(t \gg t_{i}\right)$$

  Show that the physical wavelength corresponding to the comoving wavenumber $k=|\mathbf{k}|$ crosses the Hubble radius $c H^{-1}$ at a time $t_{k}$ given by

  $$\frac{t_{k}}{t_{0}}=\left(\frac{k_{0}}{k}\right)^{3} \quad, \quad \text { where } \quad k_{0}=\frac{2 \pi}{c H_{0}^{-1}}$$

  According to inflationary theory, the amplitude of the variance at horizon-crossing is constant, that is, $\left\langle\left|\delta_{\mathbf{k}}\left(t_{k}\right)\right|^{2}\right\rangle=A V^{-1} / k^{3}$ where $A$ and $V$ (the volume) are constants. Given this amplitude and the results obtained above, deduce that the power spectrum today takes the form

  $$P(k) \equiv V\left\langle\left|\delta_{\mathbf{k}}\left(t_{0}\right)\right|^{2}\right\rangle=\frac{A}{k_{0}^{4}} k$$

  comment

Differential Geometry

• 1.II.24H

  Differential Geometry | Part II, 2007

  Let $f: X \rightarrow Y$ be a smooth map between manifolds without boundary. Recall that $f$ is a submersion if $d f_{x}: T_{x} X \rightarrow T_{f(x)} Y$ is surjective for all $x \in X$. The canonical submersion is the standard projection of $\mathbb{R}^{k}$ onto $\mathbb{R}^{l}$ for $k \geqslant l$, given by

  $$\left(x_{1}, \ldots, x_{k}\right) \mapsto\left(x_{1}, \ldots, x_{l}\right)$$

  (i) Let $f$ be a submersion, $x \in X$ and $y=f(x)$. Show that there exist local coordinates around $x$ and $y$ such that $f$, in these coordinates, is the canonical submersion. [You may assume the inverse function theorem.]

  (ii) Show that submersions map open sets to open sets.

  (iii) If $X$ is compact and $Y$ connected, show that every submersion is surjective. Are there submersions of compact manifolds into Euclidean spaces $\mathbb{R}^{k}$ with $k \geqslant 1$ ?

  comment

• 2.II.24H

  Differential Geometry | Part II, 2007

  (i) What is a minimal surface? Explain why minimal surfaces always have non-positive Gaussian curvature.

  (ii) A smooth map $f: S_{1} \rightarrow S_{2}$ between two surfaces in 3-space is said to be conformal if

  $$\left\langle d f_{p}\left(v_{1}\right), d f_{p}\left(v_{2}\right)\right\rangle=\lambda(p)\left\langle v_{1}, v_{2}\right\rangle$$

  for all $p \in S_{1}$ and all $v_{1}, v_{2} \in T_{p} S_{1}$, where $\lambda(p) \neq 0$ is a number which depends only on $p$.

  Let $S$ be a surface without umbilical points. Prove that $S$ is a minimal surface if and only if the Gauss map $N: S \rightarrow S^{2}$ is conformal.

  (iii) Show that isothermal coordinates exist around a non-planar point in a minimal surface.

  comment

• 3.II.23H

  Differential Geometry | Part II, 2007

  (i) Let $f: X \rightarrow Y$ be a smooth map between manifolds without boundary. Define critical point, critical value and regular value. State Sard's theorem.

  (ii) Explain how to define the degree modulo 2 of a smooth map $f$, indicating clearly the hypotheses on $X$ and $Y$. Show that a smooth map with non-zero degree modulo 2 must be surjective.

  (iii) Let $S$ be the torus of revolution obtained by rotating the circle $(y-2)^{2}+z^{2}=1$ in the $y z$-plane around the $z$-axis. Describe the critical points and the critical values of the Gauss map $N$ of $S$. Find the degree modulo 2 of $N$. Justify your answer by means of a sketch or otherwise.

  comment

• 4.II.24H

  Differential Geometry | Part II, 2007

  (i) What is a geodesic? Show that geodesics are critical points of the energy functional.

  (ii) Let $S$ be a surface which admits a parametrization $\phi(u, v)$ defined on an open subset $W$ of $\mathbb{R}^{2}$ such that $E=G=U+V$ and $F=0$, where $U=U(u)$ is a function of $u$ alone and $V=V(v)$ is a function of $v$ alone. Let $\gamma: I \rightarrow \phi(W)$ be a geodesic and write $\gamma(t)=\phi(u(t), v(t))$. Show that

  $$[U(u(t))+V(v(t))]\left[V(v(t)) \dot{u}^{2}-U(u(t)) \dot{v}^{2}\right]$$

  is independent of $t$.

  comment

Dynamical Systems

• 1.I.7E

  Dynamical Systems | Part II, 2007

  Given a non-autonomous $k$ th-order differential equation

  $$\frac{d^{k} y}{d t^{k}}=g\left(t, y, \frac{d y}{d t}, \frac{d^{2} y}{d t^{2}}, \ldots, \frac{d^{k-1} y}{d t^{k-1}}\right)$$

  with $y \in \mathbb{R}$, explain how it may be written in the autonomous first-order form $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ for suitably chosen vectors $\mathbf{x}$ and $\mathbf{f}$.

  Given an autonomous system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$, define the corresponding flow $\boldsymbol{\phi}_{t}(\mathbf{x})$. What is $\phi_{s}\left(\phi_{t}(\mathbf{x})\right)$ equal to? Define the orbit $\mathcal{O}(\mathbf{x})$ through $\mathbf{x}$ and the limit set $\omega(\mathbf{x})$ of $\mathbf{x}$. Define a homoclinic orbit.

  comment

• 2.I.7E

  Dynamical Systems | Part II, 2007

  Find and classify the fixed points of the system

  $$\begin{aligned} &\dot{x}=\left(1-x^{2}\right) y \\ &\dot{y}=x\left(1-y^{2}\right) \end{aligned}$$

  What are the values of their Poincaré indices? Prove that there are no periodic orbits. Sketch the phase plane.

  comment

• 3.I.7E

  Dynamical Systems | Part II, 2007

  State the Poincaré-Bendixson Theorem for a system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$.

  Prove that if $k^{2}<4$ then the system

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

  has a periodic orbit in the region $2 /\left(2+k^{2}\right) \leqslant x^{2}+y^{2} \leqslant 1$.

  comment

• 3.II.14E

  Dynamical Systems | Part II, 2007

  The Lorenz equations are

  $$\begin{aligned} &\dot{x}=\sigma(y-x) \\ &\dot{y}=r x-y-x z \\ &\dot{z}=x y-b z \end{aligned}$$

  where $r, \sigma$ and $b$ are positive constants and $(x, y, z) \in \mathbb{R}^{3}$.

  (i) Show that the origin is globally asymptotically stable for $0<r<1$ by considering a function $V(x, y, z)=\frac{1}{2}\left(x^{2}+A y^{2}+B z^{2}\right)$ with a suitable choice of constants $A$ and $B$

  (ii) State, without proof, the Centre Manifold Theorem.

  Show that the fixed point at the origin is nonhyperbolic at $r=1$. What are the dimensions of the linear stable and (non-extended) centre subspaces at this point?

  (iii) Let $\sigma=1$ from now on. Make the substitutions $u=x+y, v=x-y$ and $\mu=r-1$ and derive the resulting equations for $\dot{u}, \dot{v}$ and $\dot{z}$.

  The extended centre manifold is given by

  $$v=V(u, \mu), \quad z=Z(u, \mu)$$

  where $V$ and $Z$ can be expanded as power series about <