# Part II, 2002

### Jump to course

B2.10

commentFor $N \geq 1$, let $V_{N}$ be the (irreducible) projective plane curve $V_{N}: X^{N}+Y^{N}+Z^{N}=0$ over an algebraically closed field of characteristic zero.

Show that $V_{N}$ is smooth (non-singular). For $m, n \geq 1$, let $\alpha_{m, n}: V_{m n} \rightarrow V_{m}$ be the morphism $\alpha_{m, n}(X: Y: Z)=\left(X^{n}: Y^{n}: Z^{n}\right)$. Determine the degree of $\alpha_{m, n}$, its points of ramification and the corresponding ramification indices.

Applying the Riemann-Hurwitz formula to $\alpha_{1, n}$, determine the genus of $V_{n}$.

B3.10

commentLet $f=f(x, y)$ be an irreducible polynomial of degree $n \geq 2$ (over an algebraically closed field of characteristic zero) and $V_{0}=\{f=0\} \subset \mathbb{A}^{2}$ the corresponding affine plane curve. Assume that $V_{0}$ is smooth (non-singular) and that the projectivization $V \subset \mathbb{P}^{2}$ of $V_{0}$ intersects the line at infinity $\mathbb{P}^{2}-\mathbb{A}^{2}$ in $n$ distinct points. Show that $V$ is smooth and determine the divisor of the rational differential $\omega=\frac{d x}{f_{y}^{\prime}}$ on $V$. Deduce a formula for the genus of $V$.

B4.9

commentWrite an essay on the Riemann-Roch theorem and some of its applications.

B2.8

commentShow that the fundamental group $G$ of the Klein bottle is infinite. Show that $G$ contains an abelian subgroup of finite index. Show that $G$ is not abelian.

B3.7

commentFor a finite simplicial complex $X$, let $b_{i}(X)$ denote the rank of the finitely generated abelian group $H_{i} X$. Define the Euler characteristic $\chi(X)$ by the formula

$\chi(X)=\sum_{i}(-1)^{i} b_{i}(X) .$

Let $a_{i}$ denote the number of $i$-simplices in $X$, for each $i \geqslant 0$. Show that

$\chi(X)=\sum_{i}(-1)^{i} a_{i}$

B4.5

commentState the Mayer-Vietoris theorem for a finite simplicial complex $X$ which is the union of closed subcomplexes $A$ and $B$. Define all the maps in the long exact sequence. Prove that the sequence is exact at the term $H_{i} X$, for every $i \geqslant 0$.

A2.10

comment(i) Let $G$ be a directed network with nodes $N$, arcs $A$ and capacities specified on each of the arcs. Define the terms feasible flow, divergence, cut, upper and lower cut capacities. Given two disjoint sets of nodes $N^{+}$and $N^{-}$, what does it mean to say that a cut $Q$ separates $N^{+}$from $N^{-}$? Prove that the flux of a feasible flow $x$ from $N^{+}$to $N^{-}$is bounded above by the upper capacity of $Q$, for any cut $Q$ separating $N^{+}$from $N^{-}$.

(ii) Define the maximum-flow and minimum-cut problems. State the max-flow min-cut theorem and outline the main steps of the maximum-flow algorithm. Use the algorithm to find the maximum flow between the nodes 1 and 5 in a network whose node set is $\{1,2, \ldots, 5\}$, where the lower capacity of each arc is 0 and the upper capacity $c_{i j}$ of the directed arc joining node $i$ to node $j$ is given by the $(i, j)$-entry in the matrix

$\left(\begin{array}{ccccc} 0 & 7 & 9 & 8 & 0 \\ 0 & 0 & 6 & 8 & 4 \\ 0 & 9 & 0 & 2 & 10 \\ 0 & 3 & 7 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right)$

[The painted-network theorem can be used without proof but should be stated clearly. You may assume in your description of the maximum-flow algorithm that you are given an initial feasible flow.]

A3.10

comment(i) Consider the unconstrained geometric programme GP

$\text { minimise } g(t)=\sum_{i=1}^{n} c_{i} \prod_{j=1}^{m} t_{j}^{a_{i j}}$

subject to $t_{j}>0 \quad j=1, \ldots, m$.

State the dual problem to GP. Give a careful statement of the AM-GM inequality, and use it to prove the primal-dual inequality for GP.

(ii) Define min-path and max-tension problems. State and outline the proof of the max-tension min-path theorem.

A company has branches in five cities $A, B, C, D$ and $E$. The fares for direct flights between these cities are as follows:

\begin{tabular}{l|lllll} & $\mathrm{A}$ & $\mathrm{B}$ & $\mathrm{C}$ & $\mathrm{D}$ & $\mathrm{E}$ \ \hline $\mathrm{A}$ & $-$ & 50 & 40 & 25 & 10 \ $\mathrm{~B}$ & 50 & $-$ & 20 & 90 & 25 \ $\mathrm{C}$ & 40 & 20 & $-$ & 10 & 25 \ $\mathrm{D}$ & 25 & 90 & 10 & $-$ & 55 \ $\mathrm{E}$ & 10 & 25 & 25 & 55 & $-$ \end{tabular}

Formulate this as a min-path problem. Illustrate the max-tension min-path algorithm by finding the cost of travelling by the cheapest routes between $D$ and each of the other cities.

A4.11

commentWrite an essay on Strong Lagrangian problems. You should give an account of duality and how it relates to the Strong Lagrangian property. In particular, establish carefully the relationship between the Strong Lagrangian property and supporting hyperplanes.

Also, give an example of a class of problems that are Strong Lagrangian. [You should explain carefully why your example has the Strong Lagrangian property.]

B1.23

commentA quantum system, with Hamiltonian $H_{0}$, has continuous energy eigenstates $|E\rangle$ for all $E \geq 0$, and also a discrete eigenstate $|0\rangle$, with $H_{0}|0\rangle=E_{0}|0\rangle,\langle 0 \mid 0\rangle=1, E_{0}>0$. A time-independent perturbation $H_{1}$, such that $\left\langle E\left|H_{1}\right| 0\right\rangle \neq 0$, is added to $H_{0}$. If the system is initially in the state $|0\rangle$ obtain the formula for the decay rate

$w=\frac{2 \pi}{\hbar} \rho\left(E_{0}\right)\left|\left\langle E_{0}\left|H_{1}\right| 0\right\rangle\right|^{2},$

where $\rho$ is the density of states.

[You may assume that $\frac{1}{t}\left(\frac{\sin \frac{1}{2} \omega t}{\frac{1}{2} \omega}\right)^{2}$ behaves like $2 \pi \delta(\omega)$ for large $t$.]

Assume that, for a particle moving in one dimension,

$H_{0}=E_{0}|0\rangle\left\langle 0\left|+\int_{-\infty}^{\infty} p^{2}\right| p\right\rangle\langle p| d p, \quad H_{1}=f \int_{-\infty}^{\infty}(|p\rangle\langle 0|+| 0\rangle\langle p|) d p$

where $\left\langle p^{\prime} \mid p\right\rangle=\delta\left(p^{\prime}-p\right)$, and $f$ is constant. Obtain $w$ in this case.

B2.22

commentDefine the reciprocal lattice for a lattice $L$ with lattice vectors $\ell$.

A beam of electrons, with wave vector $\mathbf{k}$, is incident on a Bravais lattice $L$ with a large number of atoms, $N$. If the scattering amplitude for scattering on an individual atom in the direction $\hat{\mathbf{k}}^{\prime}$ is $f\left(\hat{\mathbf{k}}^{\prime}\right)$, show that the scattering amplitude for the whole lattice

$\sum_{\ell \in L} e^{i \mathbf{q} \cdot \ell} f\left(\hat{\mathbf{k}}^{\prime}\right), \quad \mathbf{q}=\mathbf{k}-|\mathbf{k}| \hat{\mathbf{k}}^{\prime}$

Derive the formula for the differential cross section

$\frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}=N\left|f\left(\hat{\mathbf{k}}^{\prime}\right)\right|^{2} \Delta(\mathbf{q}),$

obtaining an explicit form for $\Delta(\mathbf{q})$. Show that $\Delta(\mathbf{q})$ is strongly peaked when $\mathbf{q}=\mathbf{g}$, a reciprocal lattice vector. Show that this leads to the Bragg formula $2 d \sin \frac{\theta}{2}=\lambda$, where $\theta$ is the scattering angle, $\lambda$ the electron wavelength and $d$ the separation between planes of atoms in the lattice.

B3.23

commentA periodic potential is expressed as $V(\mathbf{x})=\sum_{\mathbf{g}} a_{\mathbf{g}} e^{i \mathbf{g} \cdot \mathbf{x}}$, where $\{\mathbf{g}\}$ are reciprocal lattice vectors and $a_{\mathbf{g}}{ }^{*}=a_{-\mathbf{g}}, a_{\mathbf{0}}=0$. In the nearly free electron model explain why it is appropriate, near the boundaries of energy bands, to consider a Bloch wave state

$\left|\psi_{\mathbf{k}}\right\rangle=\sum_{r} \alpha_{r}\left|\mathbf{k}_{r}\right\rangle, \quad \mathbf{k}_{r}=\mathbf{k}+\mathbf{g}_{r},$

where $|\mathbf{k}\rangle$ is a free electron state for wave vector $\mathbf{k},\left\langle\mathbf{k}^{\prime} \mid \mathbf{k}\right\rangle=\delta_{\mathbf{k}^{\prime} \mathbf{k}}$, and the sum is restricted to reciprocal lattice vectors $\mathbf{g}_{r}$ such that $\left|\mathbf{k}_{r}\right| \approx|\mathbf{k}|$. Obtain a determinantal formula for the possible energies $E(\mathbf{k})$ corresponding to Bloch wave states of this form.

[You may take $\mathbf{g}_{1}=\mathbf{0}$ and assume $e^{i \mathbf{b} \cdot \mathbf{x}}|\mathbf{k}\rangle=|\mathbf{k}+\mathbf{b}\rangle$ for any $\mathbf{b}$.]

Suppose the sum is restricted to just $\mathbf{k}$ and $\mathbf{k}+\mathbf{g}$. Show that there is a gap $2\left|a_{\mathbf{g}}\right|$ between energy bands. Setting $\mathbf{k}=-\frac{1}{2} \mathbf{g}+\mathbf{q}$, show that there are two Bloch wave states with energies near the boundaries of the energy bands

$E_{\pm}(\mathbf{k}) \approx \frac{\hbar^{2}|\mathbf{g}|^{2}}{8 m} \pm\left|a_{\mathbf{g}}\right|+\frac{\hbar^{2}|\mathbf{q}|^{2}}{2 m} \pm \frac{\hbar^{4}}{8 m^{2}\left|a_{\mathbf{g}}\right|}(\mathbf{q} \cdot \mathbf{g})^{2}$

What is meant by effective mass? Determine the value of the effective mass at the top and the bottom of the adjacent energy bands if $\mathbf{q}$ is parallel to $\mathrm{g}$.

B4.24

commentExplain the variational method for computing the ground state energy for a quantum Hamiltonian.

For the one-dimensional Hamiltonian

$H=\frac{1}{2} p^{2}+\lambda x^{4},$

obtain an approximate form for the ground state energy by considering as a trial state the state $|w\rangle$ defined by $a|w\rangle=0$, where $\langle w \mid w\rangle=1$ and $a=(w / 2 \hbar)^{\frac{1}{2}}(x+i p / w)$.

[It is useful to note that $\left\langle w\left|\left(a+a^{\dagger}\right)^{4}\right| w\right\rangle=\left\langle w\left|\left(a^{2} a^{\dagger 2}+a a^{\dagger} a a^{\dagger}\right)\right| w\right\rangle$.]

Explain why the states $a^{\dagger}|w\rangle$ may be used as trial states for calculating the first excited energy level.

B2.13

commentTwo enthusiastic probability students, Ros and Guil, sit an examination which starts at time 0 and ends at time $T$; they both decide to use the time to attempt a proof of a difficult theorem which carries a lot of extra marks.

Ros' strategy is to write the proof continuously at a constant speed $\lambda$ lines per unit time. In a time interval of length $\delta t$ he has a probability $\mu \delta t+o(\delta t)$ of realising he has made a mistake. If that happens he instantly panics, erases everything he has written and starts all over again.

Guil, on the other hand, keeps cool and thinks carefully about what he is doing. In a time interval of length $\delta t$, he has a probability $\lambda \delta t+o(\delta t)$ of writing the next line of proof and for each line he has written a probability $\mu \delta t+o(\delta t)$ of finding a mistake in that line, independently of all other lines he has written. When a mistake is found, he erases that line and carries on as usual, hoping for the best.

Both Ros and Guil realise that, even if they manage to finish the proof, they will not recognise that they have done so and will carry on writing as much as they can.

(a) Calculate $p_{l}(t)$, the probability that, for Ros, the length of his completed proof at time $t \geqslant l / \lambda$ is at least $l$.

(b) Let $q_{n}(t)$ be the probability that Guil has $n$ lines of proof at time $t>0$. Show that

$\frac{\partial Q}{\partial t}=(s-1)\left(\lambda Q-\mu \frac{\partial Q}{\partial s}\right)$

where $Q(s, t)=\sum_{n=0}^{\infty} s^{n} q_{n}(t)$.

(c) Suppose now that every time Ros starts all over again, the time until the next mistake has distribution $F$, independently of the past history. Write down a renewal-type integral equation satisfied by $l(t)$, the expected length of Ros' proof at time $t$. What is the expected length of proof produced by him at the end of the examination if $F$ is the exponential distribution with mean $1 / \mu$ ?

(d) What is the expected length of proof produced by Guil at the end of the examination if each line that he writes survives for a length of time with distribution $F$, independently of all other lines?

B3.13

comment(a) Define a renewal process and a discrete renewal process.

(b) State and prove the Discrete Renewal Theorem.

(c) The sequence $\mathbf{u}=\left\{u_{n}: n \geqslant 0\right\}$ satisfies

$u_{0}=1, \quad u_{n}=\sum_{i=1}^{n} f_{i} u_{n-i}, \quad \text { for } n \geqslant 1$

for some collection of non-negative numbers $\left(f_{i}: i \in \mathbb{N}\right)$ summing to 1 . Let $U(s)=$ $\sum_{n=1}^{\infty} u_{n} s^{n}, F(s)=\sum_{n=1}^{\infty} f_{n} s^{n}$. Show that

$F(s)=\frac{U(s)}{1+U(s)} .$

Give a probabilistic interpretation of the numbers $u_{n}, f_{n}$ and $m_{n}=\sum_{i=1}^{n} u_{i}$.

(d) Let the sequence $u_{n}$ be given by

$u_{2 n}=\left(\begin{array}{c} 2 n \\ n \end{array}\right)\left(\frac{1}{2}\right)^{2 n}, \quad u_{2 n+1}=0, \quad n \geqslant 1 .$

How is this related to the simple symmetric random walk on the integers $\mathbb{Z}$ starting from the origin, and its subsequent returns to the origin? Determine $F(s)$ in this case, either by calculating $U(s)$ or by showing that $F$ satisfies the quadratic equation

$F^{2}-2 F+s^{2}=0, \quad \text { for } \quad 0 \leqslant s<1 .$

B4.12

commentDefine a Poisson random measure. State and prove the Product Theorem for the jump times $J_{n}$ of a Poisson process with constant rate $\lambda$ and independent random variables $Y_{n}$ with law $\mu$. Write down the corresponding result for a Poisson process $\Pi$ in a space $E=\mathbb{R}^{d}$ with rate $\lambda(x)(x \in E)$ when we associate with each $X \in \Pi$ an independent random variable $m_{X}$ with density $\rho(X, d m)$.

Prove Campbell's Theorem, i.e. show that if $M$ is a Poisson random measure on the space $E$ with intensity measure $\nu$ and $a: E \rightarrow \mathbb{R}$ is a bounded measurable function then

$\mathbf{E}\left[e^{\theta \Sigma}\right]=\exp \left(\int_{E}\left(e^{\theta a(y)}-1\right) \nu(d y)\right)$

where

$\Sigma=\int_{E} a(y) M(d y)=\sum_{X \in \Pi} a(X)$

Stars are scattered over three-dimensional space $\mathbb{R}^{3}$ in a Poisson process $\Pi$ with density $\nu(X)\left(X \in \mathbb{R}^{3}\right)$. Masses of the stars are independent random variables; the mass $m_{X}$ of a star at $X$ has the density $\rho(X, d m)$. The gravitational potential at the origin is given by

$F=\sum_{X \in \Pi} \frac{G m_{X}}{|X|}$

where $G$ is a constant. Find the moment generating function $\mathbf{E}\left[e^{\theta F}\right]$.

A galaxy occupies a sphere of radius $R$ centred at the origin. The density of stars is $\nu(\mathbf{x})=1 /|\mathbf{x}|$ for points $\mathbf{x}$ inside the sphere; the mass of each star has the exponential distribution with mean $M$. Calculate the expected potential due to the galaxy at the origin. Let $C$ be a positive constant. Find the distribution of the distance from the origin to the nearest star whose contribution to the potential $F$ is at least $C$.

A1.10

comment(i) Describe the original Hamming code of length 7 . Show how to encode a message word, and how to decode a received word involving at most one error. Explain why the procedure works.

(ii) What is a linear binary code? What is its dual code? What is a cyclic binary code? Explain how cyclic binary codes of length $n$ correspond to polynomials in $\mathbb{F}_{2}[X]$ dividing $X^{n}+1$. Show that the dual of a cyclic code of length $n$ is cyclic of length $n$.

Using the factorization

$X^{7}+1=(X+1)\left(X^{3}+X+1\right)\left(X^{3}+X^{2}+1\right)$

in $\mathbb{F}_{2}[X]$, find all cyclic binary codes of length 7 . Identify those which are Hamming codes and their duals. Justify your answer.

A2.9

comment(i) Explain the idea of public key cryptography. Give an example of a public key system, explaining how it works.

(ii) What is a general feedback register of length $d$ with initial fill $\left(X_{0}, \ldots, X_{d-1}\right)$ ? What is the maximal period of such a register, and why? What does it mean for such a register to be linear?

Describe and justify the Berlekamp-Massey algorithm for breaking a cypher stream arising from a general linear feedback register of unknown length.

Use the Berlekamp-Massey algorithm to find a linear recurrence in $\mathbb{F}_{2}$ with first eight terms $1,1,0,0,1,0,1,1$.

B1.5

commentProve that every graph $G$ on $n \geqslant 3$ vertices with minimal degree $\delta(G) \geqslant \frac{n}{2}$ is Hamiltonian. For each $n \geqslant 3$, give an example to show that this result does not remain true if we weaken the condition to $\delta(G) \geqslant \frac{n}{2}-1$ ( $n$ even) or $\delta(G) \geqslant \frac{n-1}{2}$ ( $n$ odd).

Now let $G$ be a connected graph (with at least 2 vertices) without a cutvertex. Does $G$ Hamiltonian imply $G$ Eulerian? Does $G$ Eulerian imply $G$ Hamiltonian? Justify your answers.

B2.5

commentState and prove the local $L Y M$ inequality. Explain carefully when equality holds.

Define the colex order and state the Kruskal-Katona theorem. Deduce that, if $n$ and $r$ are fixed positive integers with $1 \leqslant r \leqslant n-1$, then for every $1 \leqslant m \leqslant\left(\begin{array}{l}n \\ r\end{array}\right)$ we have

$\min \left\{|\partial \mathcal{A}|: \mathcal{A} \subset[n]^{(r)},|\mathcal{A}|=m\right\}=\min \left\{|\partial \mathcal{A}|: \mathcal{A} \subset[n+1]^{(r)},|\mathcal{A}|=m\right\}$

By a suitable choice of $n, r$ and $m$, show that this result does not remain true if we replace the lower shadow $\partial A$ with the upper shadow $\partial^{+} \mathcal{A}$.

B4.1

commentWrite an essay on Ramsey theory. You should include the finite and infinite versions of Ramsey's theorem, together with a discussion of upper and lower bounds in the finite case.

[You may restrict your attention to colourings by just 2 colours.]

A1.13

comment(i) Suppose $Y_{1}, \ldots, Y_{n}$ are independent Poisson variables, and

$\mathbb{E}\left(Y_{i}\right)=\mu_{i}, \log \mu_{i}=\alpha+\beta^{T} x_{i}, 1 \leqslant i \leqslant n$

where $\alpha, \beta$ are unknown parameters, and $x_{1}, \ldots, x_{n}$ are given covariates, each of dimension $p$. Obtain the maximum-likelihood equations for $\alpha, \beta$, and explain briefly how you would check the validity of this model.

(ii) The data below show $y_{1}, \ldots, y_{33}$, which are the monthly accident counts on a major US highway for each of the 12 months of 1970 , then for each of the 12 months of 1971 , and finally for the first 9 months of 1972 . The data-set is followed by the (slightly edited) $R$ output. You may assume that the factors 'Year' and 'month' have been set up in the appropriate fashion. Give a careful interpretation of this $R$ output, and explain (a) how you would derive the corresponding standardised residuals, and (b) how you would predict the number of accidents in October 1972 .

$\begin{array}{llllllllllll}52 & 37 & 49 & 29 & 31 & 32 & 28 & 34 & 32 & 39 & 50 & 63 \\ 35 & 22 & 27 & 27 & 34 & 23 & 42 & 30 & 36 & 56 & 48 & 40 \\ 33 & 26 & 31 & 25 & 23 & 20 & 25 & 20 & 36 & & & \end{array}$

$>$ first.glm $-\operatorname{glm}(\mathrm{y} \sim$ Year $+$ month, poisson $) ;$ summary(first.glm $)$

Call:

$\operatorname{glm}($ formula $=\mathrm{y} \sim$ Year $+$ month, family $=$ poisson $)$

\begin{tabular}{lrlll} Coefficients: & & & & \ (Intercept) & Estimate & Std. Error & \multicolumn{1}{l}{ z value } & $\operatorname{Pr}(>|z|)$ \ Year1971 & $-0.81969$ & $0.09896$ & $38.600$ & $<2 e-16$ \ Year1972 & $-0.28794$ & $0.08267$ & $-3.483$ & $0.000496$ \ month2 & $-0.34484$ & $0.14176$ & $-2.433$ & $0.014994$ \ month3 & $-0.11466$ & $0.13296$ & $-0.862$ & $0.388459$ \ month4 & $-0.39304$ & $0.14380$ & $-2.733$ & $0.006271$ \ month5 & $-0.31015$ & $0.14034$ & $-2.210$ & $0.027108$ \ month6 & $-0.47000$ & $0.14719$ & $-3.193$ & $0.001408$ \ month7 & $-0.23361$ & $0.13732$ & $-1.701$ & $0.088889$ \ month8 & $-0.35667$ & $0.14226$ & $-2.507$ & $0.012168$ \ month9 & $-0.14310$ & $0.13397$ & $-1.068$ & $0.285444$ \ month10 & $0.10167$ & $0.13903$ & $0.731$ & $0.464628$ \ month11 & $0.13276$ & $0.13788$ & $0.963$ & $0.335639$ \ month12 & $0.18252$ & $0.13607$ & $1.341$ & $0.179812$ \end{tabular}

Signif. codes: 0 (

, $0.05$ '.**, $0.001$ (**, $0.01$ ((Dispersion parameter for poisson family taken to be 1 )

$\begin{array}{rlll}\text { Null deviance: } & 101.143 & \text { on } 32 \text { degrees of freedom } \\ \text { Residual deviance: } & 27.273 & \text { on } 19 \text { degrees of freedom }\end{array}$

Number of Fisher Scoring iterations: 3

A2.12

comment(i) Suppose that the random variable $Y$ has density function of the form

$f(y \mid \theta, \phi)=\exp \left[\frac{y \theta-b(\theta)}{\phi}+c(y, \phi)\right]$

where $\phi>0$. Show that $Y$ has expectation $b^{\prime}(\theta)$ and variance $\phi b^{\prime \prime}(\theta)$.

(ii) Suppose now that $Y_{1}, \ldots, Y_{n}$ are independent negative exponential variables, with $Y_{i}$ having density function $f\left(y_{i} \mid \mu_{i}\right)=\frac{1}{\mu_{i}} e^{-y_{i} / \mu_{i}}$ for $y_{i}>0$. Suppose further that $g\left(\mu_{i}\right)=\beta^{T} x_{i}$ for $1 \leqslant i \leqslant n$, where $g(\cdot)$ is a known 'link' function, and $x_{1}, \ldots, x_{n}$ are given covariate vectors, each of dimension $p$. Discuss carefully the problem of finding $\hat{\beta}$, the maximum-likelihood estimator of $\beta$, firstly for the case $g\left(\mu_{i}\right)=1 / \mu_{i}$, and secondly for the case $g(\mu)=\log \mu_{i}$; in both cases you should state the large-sample distribution of $\hat{\beta}$.

[Any standard theorems used need not be proved.]

A4.14

commentAssume that the $n$-dimensional observation vector $Y$ may be written as $Y=X \beta+\epsilon$, where $X$ is a given $n \times p$ matrix of rank $p, \beta$ is an unknown vector, with $\beta^{T}=\left(\beta_{1}, \ldots, \beta_{p}\right)$, and

$\epsilon \sim N_{n}\left(0, \sigma^{2} I\right)$

where $\sigma^{2}$ is unknown. Find $\hat{\beta}$, the least-squares estimator of $\beta$, and describe (without proof) how you would test

$H_{0}: \beta_{\nu}=0$

for a given $\nu$.

Indicate briefly two plots that you could use as a check of the assumption $(*)$.

Continued opposite Sulphur dioxide is one of the major air pollutants. A data-set presented by Sokal and Rohlf (1981) was collected on 41 US cities in 1969-71, corresponding to the following variables:

$Y=$ sulphur dioxide content of air in micrograms per cubic metre

$X 1=$ average annual temperature in degrees Fahrenheit

$X 2$ = number of manufacturing enterprises employing 20 or more workers

$X 3=$ population size (1970 census) in thousands

$X 4=$ average annual wind speed in miles per hour

$X 5=$ average annual precipitation in inches

$X 6=$ average annual of days with precipitation per year $.$

Interpret the $R$ output that follows below, quoting any standard theorems that you need to use.

$\begin{aligned} &>\text { next. } \operatorname{lm}-\operatorname{lm}(\log (\mathrm{Y}) \sim \mathrm{X} 1+\mathrm{X} 2+\mathrm{X} 3+\mathrm{X} 4+\mathrm{X} 5+\mathrm{X} 6) \\ &>\text { summary }(\text { next.lm }) \\ &\text { Call: } \operatorname{lm}(\text { formula }=\log (\mathrm{Y}) \sim \mathrm{X} 1+\mathrm{X} 2+\mathrm{X} 3+\mathrm{X} 4+\mathrm{X} 5+\mathrm{X} 6) \end{aligned}$

$\begin{aligned} & \text { Call: } \operatorname{lm}(\text { formula }=\log (\mathrm{Y}) \sim \mathrm{X} 1+\mathrm{X} 2+\mathrm{X} 3+\mathrm{X} 4+\mathrm{X} 5+\mathrm{X} 6) \end{aligned}$

Residuals :

$\begin{array}{rrrrr} \text { Min } & 1 Q & \text { Median } & 3 Q & \text { Max } \\ \hline .79548 & -0.25538 & -0.01968 & 0.28328 & 0.98029 \end{array}$

$\begin{array}{lllll}-0.79548 & -0.25538 & -0.01968 & 0.28328 & 0.98029\end{array}$

$\begin{array}{lrlcll}\text { Coefficients: } & & & & & \\ & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|t|) & \\ \text { (Intercept) } & 7.2532456 & 1.4483686 & 5.008 & 1.68 \mathrm{e}-05 & * * * \\ \text { X1 } & -0.0599017 & 0.0190138 & -3.150 & 0.00339 & * * \\ \text { X2 } & 0.0012639 & 0.0004820 & 2.622 & 0.01298 & * \\ \text { X3 } & -0.0007077 & 0.0004632 & -1.528 & 0.13580 & \\ \text { X4 } & -0.1697171 & 0.0555563 & -3.055 & 0.00436 & * * \\ \text { X5 } & 0.0173723 & 0.0111036 & 1.565 & 0.12695 & \\ \text { X6 } & 0.0004347 & 0.0049591 & 0.088 & 0.93066\end{array}$

Signif. codes: 0 '

, $0.05$ ':**, $0.001$ '**, $0.01$ 'Residual standard error: $0.448$ on 34 degrees of freedom

Multiple R-Squared: $0.6541$

F-statistic: $10.72$ on 6 and 34 degrees of freedom, p-value: $1.126 \mathrm{e}-06$

B1.8

commentWhat is meant by a "bump function" on $\mathbb{R}^{n}$ ? If $U$ is an open subset of a manifold $M$, prove that there is a bump function on $M$ with support contained in $U$.

Prove the following.

(i) Given an open covering $\mathcal{U}$ of a compact manifold $M$, there is a partition of unity on $M$ subordinate to $\mathcal{U}$.

(ii) Every compact manifold may be embedded in some Euclidean space.

B2.7

commentState, giving your reasons, whether the following are true or false.

(a) Diffeomorphic connected manifolds must have the same dimension.

(b) Every non-zero vector bundle has a nowhere-zero section.

(c) Every projective space admits a volume form.

(d) If a manifold $M$ has Euler characteristic zero, then $M$ is orientable.

B4.4

commentState and prove Stokes' Theorem for compact oriented manifolds-with-boundary.

[You may assume results relating local forms on the manifold with those on its boundary provided you state them clearly.]

Deduce that every differentiable map of the unit ball in $\mathbb{R}^{n}$ to itself has a fixed point.

B1.17

commentLet $f_{c}$ be the map of the closed interval $[0,1]$ to itself given by

$f_{c}(x)=c x(1-x), \text { where } 0 \leqslant c \leqslant 4 .$

Sketch the graphs of $f_{c}$ and (without proof) of $f_{c}^{2}$, find their fixed points, and determine which of the fixed points of $f_{c}$ are attractors. Does your argument work for $c=3 ?$

B3.17

commentLet $\mathcal{A}$ be a finite alphabet of letters and $\Sigma$ either the semi-infinite space or the doubly infinite space of sequences whose elements are drawn from $\mathcal{A}$. Define the natural topology on $\Sigma$. If $W$ is a set of words, denote by $\Sigma_{W}$ the subspace of $\Sigma$ consisting of those sequences none of whose subsequences is in $W$. Prove that $\Sigma_{W}$ is a closed subspace of $\Sigma$; and state and prove a necessary and sufficient condition for a closed subspace of $\Sigma$ to have the form $\Sigma_{W}$ for some $W$.

$\begin{aligned} &\text { If } \mathcal{A}=\{0,1\} \text { and } \quad W=\{000,111,010,101\} \end{aligned}$

what is the space $\Sigma_{W}$ ?

B4.17

commentLet $\mathcal{S}$ be a metric space, $F$ a map of $\mathcal{S}$ to itself and $P$ a point of $\mathcal{S}$. Define an attractor for $F$ and an omega point of the orbit of $P$ under $F$.

Let $f$ be the map of $\mathbb{R}$ to itself given by

$f(x)=x+\frac{1}{2}+c \sin ^{2} 2 \pi x$

where $c>0$ is so small that $f^{\prime}(x)>0$ for all $x$, and let $F$ be the map of $\mathbb{R} / \mathbb{Z}$ to itself induced by $f$. What points if any are

(a) attractors for $F^{2}$,

(b) omega points of the orbit of some point $P$ under $F$ ?

Is the cycle $\left\{0, \frac{1}{2}\right\}$ an attractor?

In the notation of the first two sentences, let $\mathcal{C}$ be a cycle of order $M$ and assume that $F$ is continuous. Prove that $\mathcal{C}$ is an attractor for $F$ if and only if each point of $\mathcal{C}$ is an attractor for $F^{M}$.

A1.6

comment(i) A system in $\mathbb{R}^{2}$ obeys the equations:

$\begin{aligned} &\dot{x}=x-x^{5}-2 x y^{4}-2 y^{3}\left(a-x^{2}\right) \\ &\dot{y}=y-x^{4} y-2 y^{5}+x^{3}\left(a-x^{2}\right) \end{aligned}$

where $a$ is a positive constant.

By considering the quantity $V=\alpha x^{4}+\beta y^{4}$, where $\alpha$ and $\beta$ are appropriately chosen, show that if $a>1$ then there is a unique fixed point and a unique limit cycle. How many fixed points are there when $a<1$ ?

(ii) Consider the second order system

$\ddot{x}-\left(a-b x^{2}\right) \dot{x}+x-x^{3}=0,$

where $a, b$ are constants.

(a) Find the fixed points and determine their stability.

(b) Show that if the fixed point at the origin is unstable and $3 a>b$ then there are no limit cycles.

[You may find it helpful to use the Liénard coordinate $z=\dot{x}-a x+\frac{1}{3} b x^{3}$.]

A2.6 B2.4

comment(i) Define the terms stable manifold and unstable manifold of a hyperbolic fixed point $\mathbf{x}_{0}$ of a dynamical system. State carefully the stable manifold theorem.

Give an approximation, correct to fourth order in $|\mathbf{x}|$, for the stable and unstable manifolds of the origin for the system

$\left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right)=\left(\begin{array}{c} x+x^{2}-y^{2} \\ -y+x^{2} \end{array}\right) .$

(ii) State, without proof, the centre manifold theorem. Show that the fixed point at the origin of the system

$\begin{aligned} &\dot{x}=y-x+a x^{3}, \\ &\dot{y}=r x-y-z y \\ &\dot{z}=-z+x y \end{aligned}$

where $a$ is a constant, is non-hyperbolic at $r=1$.

Using new coordinates $v=x+y, w=x-y$, find the centre manifold in the form

$w=\alpha v^{3}+\ldots, \quad z=\beta v^{2}+\gamma v^{4}+\ldots$

for constants $\alpha, \beta, \gamma$ to be determined. Hence find the evolution equation on the centre manifold in the form

$\dot{v}=\frac{1}{8}(a-1) v^{3}+\left(\frac{(3 a+1)(a+1)}{128}+\frac{(a-1)}{32}\right) v^{5}+\ldots$

Ignoring higher order terms, give conditions on $a$ that guarantee that the origin is asymptotically stable.

A3.6 B3.4

comment(i) Define the Floquet multiplier and Liapunov exponent for a periodic orbit $\hat{\mathbf{x}}(t)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$. Show that one multiplier is always unity, and that the other is given by

$\exp \left(\int_{0}^{T} \nabla \cdot \mathbf{f}(\hat{\mathbf{x}}(t)) d t\right)$

where $T$ is the period of the orbit.

The Van der Pol oscillator $\ddot{x}+\epsilon \dot{x}\left(x^{2}-1\right)+x=0,0<\epsilon \ll 1$ has a limit cycle $\hat{x}(t) \approx 2 \sin t$. Show using $(*)$ that this orbit is stable.

(ii) Show, by considering the normal form for a Hopf bifurcation from a fixed point $\mathbf{x}_{0}(\mu)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$, that in some neighbourhood of the bifurcation the periodic orbit is stable when it exists in the range of $\mu$ for which $\mathbf{x}_{0}$ is unstable, and unstable in the opposite case.

Now consider the system

$\left.\begin{array}{l} \dot{x}=x(1-y)+\mu x \\ \dot{y}=y(x-1)-\mu x \end{array}\right\} \quad x>0$

Show that the fixed point $(1+\mu, 1+\mu)$ has a Hopf bifurcation when $\mu=0$, and is unstable (stable) when $\mu>0(\mu<0)$.

Suppose that a periodic orbit exists in $\mu>0$. Show without solving for the orbit that the result of part (i) shows that such an orbit is unstable. Define a similar result for $\mu<0$.

What do you conclude about the existence of periodic orbits when $\mu \neq 0$ ? Check your answer by applying Dulac's criterion to the system, using the weighting $\rho=e^{-(x+y)}$.

A4.6

commentDefine the terms homoclinic orbit, heteroclinic orbit and heteroclinic loop. In the case of a dynamical system that possesses a homoclinic orbit, explain, without detailed calculation, how to calculate its stability.

A second order dynamical system depends on two parameters $\mu_{1}$ and $\mu_{2}$. When $\mu_{1}=\mu_{2}=0$ there is a heteroclinic loop between the points $P_{1}, P_{2}$ as in the diagram.

When $\mu_{1}, \mu_{2}$ are small there are trajectories that pass close to the fixed points $P_{1}, P_{2}$ :

By adapting the method used above for trajectories near homoclinic orbits, show that the distances $y_{n}, y_{n+1}$ to the stable manifold at $P_{1}$ on successive returns are related to $z_{n}$, $z_{n+1}$, the corresponding distances near $P_{2}$, by coupled equations of the form

$\left.\begin{array}{rl} z_{n} & =\left(y_{n}\right)^{\gamma_{1}}+\mu_{1}, \\ y_{n+1} & =\left(z_{n}\right)^{\gamma_{2}}+\mu_{2}, \end{array}\right\}$

where any arbitrary constants have been removed by rescaling, and $\gamma_{1}, \gamma_{2}$ depend on conditions near $P_{1}, P_{2}$. Show from these equations that there is a stable heteroclinic orbit $\left(\mu_{1}=\mu_{2}=0\right)$ if $\gamma_{1} \gamma_{2}>1$. Show also that in the marginal situation $\gamma_{1}=2, \gamma_{2}=\frac{1}{2}$ there can be a stable fixed point for small positive $y, z$ if $\mu_{2}<0, \mu_{2}^{2}<\mu_{1}$. Explain carefully the form of the orbit of the original dynamical system represented by the solution of the above map when $\mu_{2}^{2}=\mu_{1}$.

B1.21

commentExplain how one can write Maxwell's equations in relativistic form by introducing an antisymmetric field strength tensor $F_{a b}$.

In an inertial frame $S$, the electric and magnetic fields are $\mathbf{E}$ and $\mathbf{B}$. Suppose that there is a second inertial frame $S^{\prime}$ moving with velocity $v$ along the $x$-axis relative to $S$. Derive the rules for finding the electric and magnetic fields $\mathbf{E}^{\prime}$ and $\mathbf{B}^{\prime}$ in the frame $S^{\prime}$. Show that $|\mathbf{E}|^{2}-|\mathbf{B}|^{2}$ and $\mathbf{E} \cdot \mathbf{B}$ are invariant under Lorentz transformations.

Suppose that $\mathbf{E}=E_{0}(0,1,0)$ and $\mathbf{B}=E_{0}(0, \cos \theta, \sin \theta)$, where $0 \leq \theta<\pi / 2$. At what velocity must an observer be moving in the frame $S$ for the electric and magnetic fields to appear to be parallel?

Comment on the case $\theta=\pi / 2$.

B2.20

commentA particle of rest mass $m$ and charge $q$ moves in an electromagnetic field given by a potential $A_{a}$ along a trajectory $x^{a}(\tau)$, where $\tau$ is the proper time along the particle's worldline. The action for such a particle is

$I=\int\left(m \sqrt{-\eta_{a b} \dot{x}^{a} \dot{x}^{b}}-q A_{a} \dot{x}^{a}\right) d \tau .$

Show that the Euler-Lagrange equations resulting from this action reproduce the relativistic equation of motion for the particle.

Suppose that the particle is moving in the electrostatic field of a fixed point charge $Q$ with radial electric field $E_{r}$ given by

$E_{r}=\frac{Q}{4 \pi \epsilon_{0} r^{2}} .$

Show that one can choose a gauge such that $A_{i}=0$ and only $A_{0} \neq 0$. Find $A_{0}$.

Assume that the particle executes planar motion, which in spherical polar coordinates $(r, \theta, \phi)$ can be taken to be in the plane $\theta=\pi / 2$. Derive the equations of motion for $t$ and $\phi$.

By using the fact that $\eta_{a b} \dot{x}^{a} \dot{x}^{b}=-1$, find the equation of motion for $r$, and hence show that the shape of the orbit is described by

$\frac{d r}{d \phi}=\pm \frac{r^{2}}{\ell} \sqrt{\left(E+\frac{\gamma}{r}\right)^{2}-1-\frac{\ell^{2}}{r^{2}}}$

where $E(>1)$ and $\ell$ are constants of integration and $\gamma$ is to be determined.

By putting $u=1 / r$ or otherwise, show that if $\gamma^{2}<\ell^{2}$ then the orbits are bounded and generally not closed, and show that the angle between successive minimal values of $r$ is $2 \pi\left(1-\gamma^{2} / \ell^{2}\right)^{-1 / 2}$.

B4.21

commentDerive Larmor's formula for the rate at which radiation is produced by a particle of charge $q$ moving along a trajectory $\mathbf{x}(t)$.

A non-relativistic particle of mass $m$, charge $q$ and energy $E$ is incident along a radial line in a central potential $V(r)$. The potential is vanishingly small for $r$ very large, but increases without bound as $r \rightarrow 0$. Show that the total amount of energy $\mathcal{E}$ radiated by the particle is

$\mathcal{E}=\frac{\mu_{0} q^{2}}{3 \pi m^{2}} \sqrt{\frac{m}{2}} \int_{r_{0}}^{\infty} \frac{1}{\sqrt{E-V(r)}}\left(\frac{d V}{d r}\right)^{2} d r$

where $V\left(r_{0}\right)=E$.

Suppose that $V$ is the Coulomb potential $V(r)=A / r$. Evaluate $\mathcal{E}$.

A $1 . 5 \quad$ B $1 . 4 \quad$

comment(i) Show that, in a region where there is no magnetic field and the charge density vanishes, the electric field can be expressed either as minus the gradient of a scalar potential $\phi$ or as the curl of a vector potential A. Verify that the electric field derived from

$\mathbf{A}=\frac{1}{4 \pi \epsilon_{0}} \frac{\mathbf{p} \wedge \mathbf{r}}{r^{3}}$

is that of an electrostatic dipole with dipole moment $\mathbf{p}$.

[You may assume the following identities:

$\begin{gathered} \nabla(\mathbf{a} \cdot \mathbf{b})=\mathbf{a} \wedge(\nabla \wedge \mathbf{b})+\mathbf{b} \wedge(\nabla \wedge \mathbf{a})+(\mathbf{a} \cdot \nabla) \mathbf{b}+(\mathbf{b} \cdot \nabla) \mathbf{a} \\ \nabla \wedge(\mathbf{a} \wedge \mathbf{b})=(\mathbf{b} \cdot \nabla) \mathbf{a}-(\mathbf{a} \cdot \nabla) \mathbf{b}+\mathbf{a} \nabla \cdot \mathbf{b}-\mathbf{b} \nabla \cdot \mathbf{a} .] \end{gathered}$

(ii) An infinite conducting cylinder of radius $a$ is held at zero potential in the presence of a line charge parallel to the axis of the cylinder at distance $s_{0}>a$, with charge density $q$ per unit length. Show that the electric field outside the cylinder is equivalent to that produced by replacing the cylinder with suitably chosen image charges.

A2.5

comment(i) Show that the Lorentz force corresponds to a curvature force and the gradient of a magnetic pressure, and that it can be written as the divergence of a second rank tensor, the Maxwell stress tensor.

Consider the potential field $\mathbf{B}$ given by $\mathbf{B}=-\nabla \Phi$, where

$\Phi(x, y)=\left(\frac{B_{0}}{k}\right) \cos k x e^{-k y}$

referred to cartesian coordinates $(x, y, z)$. Obtain the Maxwell stress tensor and verify that its divergence vanishes.

(ii) The magnetic field in a stellar atmosphere is maintained by steady currents and the Lorentz force vanishes. Show that there is a scalar field $\alpha$ such that $\nabla \wedge \mathbf{B}=\alpha \mathbf{B}$ and $\mathbf{B} \cdot \nabla \alpha=0$. Show further that if $\alpha$ is constant, then $\nabla^{2} \mathbf{B}+\alpha^{2} \mathbf{B}=0$. Obtain a solution in the form $\mathbf{B}=\left(B_{1}(z), B_{2}(z), 0\right)$; describe the structure of this field and sketch its variation in the $z$-direction.

A3.5 B3.3

comment(i) A plane electromagnetic wave in a vacuum has an electric field

$\mathbf{E}=\left(E_{1}, E_{2}, 0\right) \cos (k z-\omega t),$

referred to cartesian axes $(x, y, z)$. Show that this wave is plane polarized and find the orientation of the plane of polarization. Obtain the corresponding plane polarized magnetic field and calculate the rate at which energy is transported by the wave.

(ii) Suppose instead that

$\mathbf{E}=\left(E_{1} \cos (k z-\omega t), E_{2} \cos (k z-\omega t+\phi), 0\right),$

with $\phi$ a constant, $0<\phi<\pi$. Show that, if the axes are now rotated through an angle $\psi$ so as to obtain an elliptically polarized wave with an electric field

$\mathbf{E}^{\prime}=\left(F_{1} \cos (k z-\omega t+\chi), F_{2} \sin (k z-\omega t+\chi), 0\right),$

then

$\tan 2 \psi=\frac{2 E_{1} E_{2} \cos \phi}{E_{1}^{2}-E_{2}^{2}} .$

Show also that if $E_{1}=E_{2}=E$ there is an elliptically polarized wave with

$\mathbf{E}^{\prime}=\sqrt{2} E\left(\cos \left(k z-\omega t+\frac{1}{2} \phi\right) \cos \frac{1}{2} \phi, \sin \left(k z-\omega t+\frac{1}{2} \phi\right) \sin \frac{1}{2} \phi, 0\right) .$

A4.5

commentState the four integral relationships between the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ and explain their physical significance. Derive Maxwell's equations from these relationships and show that $\mathbf{E}$ and $\mathbf{B}$ can be described by a scalar potential $\phi$ and a vector potential A which satisfy the inhomogeneous wave equations

$\begin{gathered} \nabla^{2} \phi-\epsilon_{0} \mu_{0} \frac{\partial^{2} \phi}{\partial t^{2}}=-\frac{\rho}{\epsilon_{0}} \\ \nabla^{2} \mathbf{A}-\epsilon_{0} \mu_{0} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=-\mu_{0} \mathbf{j} \end{gathered}$

If the current $\mathbf{j}$ satisfies Ohm's law and the charge density $\rho=0$, show that plane waves of the form

$\mathbf{A}=A(z, t) e^{i \omega t} \hat{\mathbf{x}},$

where $\hat{\mathbf{x}}$ is a unit vector in the $x$-direction of cartesian axes $(x, y, z)$, are damped. Find an approximate expression for $A(z, t)$ when $\omega \ll \sigma / \epsilon_{0}$, where $\sigma$ is the electrical conductivity.

B1.25

commentState the minimum dissipation theorem for Stokes flow in a bounded domain.

Fluid of density $\rho$ and viscosity $\mu$ fills an infinite cylindrical annulus $a \leq r \leq b$ between a fixed cylinder $r=a$ and a cylinder $r=b$ which rotates about its axis with constant angular velocity $\Omega$. In cylindrical polar coordinates $(r, \theta, z)$, the fluid velocity is $\mathbf{u}=(0, v(r), 0)$. The Reynolds number $\rho \Omega b^{2} / \mu$ is not necessarily small. Show that $v(r)=A r+B / r$, where $A$ and $B$ are constants to be determined.

[You may assume that $\nabla^{2} \mathbf{u}=\left(0, \nabla^{2} v-v / r^{2}, 0\right)$ and $(\mathbf{u} \cdot \nabla) \mathbf{u}=\left(-v^{2} / r, 0,0\right) .$ ]

Show that the outer cylinder exerts a couple $G_{0}$ per unit length on the fluid, where

$G_{0}=\frac{4 \pi \mu \Omega a^{2} b^{2}}{b^{2}-a^{2}} .$

[You may assume that, in standard notation, $e_{r \theta}=\frac{r}{2} \frac{d}{d r}\left(\frac{v}{r}\right)$.]

Suppose now that $b \geq \sqrt{2} a$ and that the cylinder $r=a$ is replaced by a fixed cylinder whose cross-section is a square of side $2 a$ centred on $r=0$, all other conditions being unchanged. The flow may still be assumed steady. Explaining your argument carefully, show that the couple $G$ now required to maintain the motion of the outer cylinder is greater than $G_{0}$.

B2.24

commentA thin layer of liquid of kinematic viscosity $\nu$ flows under the influence of gravity down a plane inclined at an angle $\alpha$ to the horizontal $(0 \leq \alpha \leq \pi / 2)$. With origin $O$ on the plane, and axes $O x$ down the line of steepest slope and $O y$ normal to the plane, the free surface is given by $y=h(x, t)$, where $|\partial h / \partial x| \ll 1$. The pressure distribution in the liquid may be assumed to be hydrostatic. Using the approximations of lubrication theory, show that

$\frac{\partial h}{\partial t}=\frac{g}{3 \nu} \frac{\partial}{\partial x}\left\{h^{3}\left(\cos \alpha \frac{\partial h}{\partial x}-\sin \alpha\right)\right\} .$

Now suppose that

$h=h_{0}+\eta(x, t)$

where

$\eta(x, 0)=\eta_{0} e^{-x^{2} / a^{2}}$

and $h_{0}, \eta_{0}$ and $a$ are constants with $\eta_{0} \ll a, h_{0}$. Show that, to leading order,

$\eta(x, t)=\frac{a \eta_{0}}{\left(a^{2}+4 D t\right)^{1 / 2}} \exp \left\{-\frac{(x-U t)^{2}}{a^{2}+4 D t}\right\}$

where $U$ and $D$ are constants to be determined.

Explain in physical terms the meaning of this solution.

B3.24

comment(i) Suppose that, with spherical polar coordinates, the Stokes streamfunction

$\Psi_{\lambda}(r, \theta)=r^{\lambda} \sin ^{2} \theta \cos \theta$

represents a Stokes flow and thus satisfies the equation $D^{2}\left(D^{2} \Psi_{\lambda}\right)=0$, where

$D^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{\sin \theta}{r^{2}} \frac{\partial}{\partial \theta} \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} .$

Show that the possible values of $\lambda$ are $5,3,0$ and $-2$. For which of these values is the corresponding flow irrotational? Sketch the streamlines of the flow for the case $\lambda=3$.

(ii) A spherical drop of liquid of viscosity $\mu_{1}$, radius $a$ and centre at $r=0$, is suspended in another liquid of viscosity $\mu_{2}$ which flows with streamfunction

$\Psi \sim \Psi_{\infty}(r, \theta)=\alpha r^{3} \sin ^{2} \theta \cos \theta$

far from the drop. The two liquids are of equal densities, surface tension is sufficiently strong to keep the drop spherical, and inertia is negligible. Show that

$\Psi= \begin{cases}\left(A r^{5}+B r^{3}\right) \sin ^{2} \theta \cos \theta & (r<a), \\ \left(\alpha r^{3}+C+D / r^{2}\right) \sin ^{2} \theta \cos \theta & (r>a)\end{cases}$

and obtain four equations determining the constants $A, B, C$ and $D$. (You need not solve these equations.)

[You may assume, with standard notation, that

$\left.u_{r}=\frac{1}{r^{2} \sin \theta} \frac{\partial \Psi}{\partial \theta} \quad, \quad u_{\theta}=-\frac{1}{r \sin \theta} \frac{\partial \Psi}{\partial r} \quad, \quad e_{r \theta}=\frac{1}{2}\left\{r \frac{\partial}{\partial r}\left(\frac{u_{\theta}}{r}\right)+\frac{1}{r} \frac{\partial u_{r}}{\partial \theta}\right\} .\right]$

B4.26

commentWrite an essay on boundary-layer theory and its application to the generation of lift in aerodynamics.

You should include discussion of the derivation of the boundary-layer equation, the similarity transformation leading to the Falkner-Skan equation, the influence of an adverse pressure gradient, and the mechanism(s) by which circulation is generated in flow past bodies with a sharp trailing edge.

A2.13 B2.21

(i) A Hamiltonian $H_{0}$ has energy eigenvalues $E_{r}$