Part II, 2001

Algebraic Curves

• B2.10

  Algebraic Curves | Part II, 2001

  Let $f: \mathbb{P}^{2} \rightarrow \mathbb{P}^{2}$ be the rational map given by $f\left(X_{0}: X_{1}: X_{2}\right)=\left(X_{1} X_{2}: X_{0} X_{2}\right.$ : $\left.X_{0} X_{1}\right)$. Determine whether $f$ is defined at the following points: $(1: 1: 1),(0: 1: 1),(0:$ $0: 1)$.

  Let $C \subset \mathbb{P}^{2}$ be the curve defined by $X_{1}^{2} X_{2}-X_{0}^{3}=0$. Define a bijective morphism $\alpha: \mathbb{P}^{1} \rightarrow C$. Prove that $\alpha$ is not an isomorphism.

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• B3.10

  Algebraic Curves | Part II, 2001

  Let $C$ be the projective curve (over an algebraically closed field $k$ of characteristic zero) defined by the affine equation

  $$x^{5}+y^{5}=1$$

  Determine the points at infinity of $C$ and show that $C$ is smooth.

  Determine the divisors of the rational functions $x, y \in k(C)$.

  Show that $\omega=d x / y^{4}$ is a regular differential on $C$.

  Compute the divisor of $\omega$. What is the genus of $C$ ?

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• B4.9

  Algebraic Curves | Part II, 2001

  Write an essay on curves of genus one (over an algebraically closed field $k$ of characteristic zero). Legendre's normal form should not be discussed.

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Algebraic Topology

• B2.8

  Algebraic Topology | Part II, 2001

  Show that the fundamental group of the 2-torus $S^{1} \times S^{1}$ is isomorphic to $\mathbf{Z} \times \mathbf{Z}$.

  Show that an injective continuous map from the circle $S^{1}$ to itself induces multiplication by $\pm 1$ on the fundamental group.

  Show that there is no retraction from the solid torus $S^{1} \times D^{2}$ to its boundary.

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• B3.7

  Algebraic Topology | Part II, 2001

  Write down the Mayer-Vietoris sequence and describe all the maps involved.

  Use the Mayer-Vietoris sequence to compute the homology of the $n$-sphere $S^{n}$ for all $n$.

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• B4.5

  Algebraic Topology | Part II, 2001

  Write an essay on the definition of simplicial homology groups. The essay should include a discussion of orientations, of the action of a simplicial map and a proof of $\partial^{2}=0$.

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Algorithms and Networks

• A2.10

  Algorithms and Networks | Part II, 2001

  (i) Let $G$ be a directed network with nodes $N$ and $\operatorname{arcs} A$. Let $S \subset N$ be a subset of the nodes, $x$ be a flow on $G$, and $y$ be the divergence of $x$. Describe carefully what is meant by a cut $Q=[S, N \backslash S]$. Define the arc-cut incidence $e_{Q}$, and the flux of $x$ across $Q$. Define also the divergence $y(S)$ of $S$. Show that $y(S)=x \cdot e_{Q}$.

  Now suppose that capacity constraints are specified on each of the arcs. Define the upper cut capacity $c^{+}(Q)$ of $Q$. State the feasible distribution problem for a specified divergence $b$, and show that the problem only has a solution if $b(N)=0$ and $b(S) \leqslant c^{+}(Q)$ for all cuts $Q=[S, N \backslash S]$.

  (ii) Describe an algorithm to find a feasible distribution given a specified divergence $b$ and capacity constraints on each arc. Explain what happens when no feasible distribution exists.

  Illustrate the algorithm by either finding a feasible circulation, or demonstrating that one does not exist, in the network given below. Arcs are labelled with capacity constraint intervals.

  

  Part II

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• A3.10

  Algorithms and Networks | Part II, 2001

  (i) Let $P$ be the problem

  $$\text { minimize } f(x) \quad \text { subject to } h(x)=b, \quad x \in X \text {. }$$

  Explain carefully what it means for the problem $P$ to be Strong Lagrangian.

  Outline the main steps in a proof that a quadratic programming problem

  $$\operatorname{minimize} \frac{1}{2} x^{T} Q x+c^{T} x \quad \text { subject to } A x \geqslant b$$

  where $Q$ is a symmetric positive semi-definite matrix, is Strong Lagrangian.

  [You should carefully state the results you need, but should not prove them.]

  (ii) Consider the quadratic programming problem:

  $$\begin{aligned} \operatorname{minimize} & x_{1}^{2}+2 x_{1} x_{2}+2 x_{2}^{2}+x_{1}-4 x_{2} \\ \text { subject to } 3 x_{1}+2 x_{2} \leqslant 6, \quad x_{1}+x_{2} \geqslant 1 . \end{aligned}$$

  State necessary and sufficient conditions for $\left(x_{1}, x_{2}\right)$ to be optimal, and use the activeset algorithm (explaining your steps briefly) to solve the problem starting with initial condition $(2,0)$. Demonstrate that the solution you have found is optimal by showing that it satisfies the necessary and sufficient conditions stated previously.

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• A4.11

  Algorithms and Networks | Part II, 2001

  State the optimal distribution problem. Carefully describe the simplex-on-a-graph algorithm for solving optimal distribution problems when the flow in each arc in the network is constrained to lie in the interval $[0, \infty)$. Explain how the algorithm can be initialised if there is no obvious feasible solution with which to begin. Describe the adjustments that are needed for the algorithm to cope with more general capacity constraints $x(j) \in\left[c^{-}(j), c^{+}(j)\right]$ for each arc $j$ (where $c^{\pm}(j)$ may be finite or infinite).

  Part II

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Applications of Quantum Mechanics

• B1.23

  Applications of Quantum Mechanics | Part II, 2001

  A steady beam of particles, having wavenumber $k$ and moving in the $z$ direction, scatters on a spherically-symmetric potential. Write down the asymptotic form of the wave function at large $r$.

  The incoming wave is written as a partial-wave series

  $$\sum_{\ell=0}^{\infty} \chi_{\ell}(k r) P_{\ell}(\cos \theta)$$

  Show that for large $r$

  $$\chi_{\ell}(k r) \sim \frac{\ell+\frac{1}{2}}{i k r}\left(e^{i k r}-(-1)^{\ell} e^{-i k r}\right)$$

  and calculate $\chi_{0}(k r)$ and $\chi_{1}(k r)$ for all $r$.

  Write down the second-order differential equation satisfied by the $\chi_{\ell}(k r)$. Construct a second linearly-independent solution for each $\ell$ that is singular at $r=0$ and, when it is suitably normalised, has large- $r$ behaviour

  $$\frac{\ell+\frac{1}{2}}{i k r}\left(e^{i k r}+(-1)^{\ell} e^{-i k r}\right)$$

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• B2.22

  Applications of Quantum Mechanics | Part II, 2001

  A particle of charge $e$ moves freely within a cubical box of side $a$. Its initial wavefunction is

  $$(2 / a)^{-\frac{3}{2}} \sin (\pi x / a) \sin (\pi y / a) \sin (\pi z / a) .$$

  A uniform electric field $\mathcal{E}$ in the $x$ direction is switched on for a time $T$. Derive from first principles the probability, correct to order $\mathcal{E}^{2}$, that after the field has been switched off the wave function will be found to be

  $$(2 / a)^{-\frac{3}{2}} \sin (2 \pi x / a) \sin (\pi y / a) \sin (\pi z / a) .$$

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• B3.23

  Applications of Quantum Mechanics | Part II, 2001

  Write down the commutation relations satisfied by the cartesian components of the total angular momentum operator $\mathbf{J}$.

  In quantum mechanics an operator $\mathbf{V}$ is said to be a vector operator if, under rotations, its components transform in the same way as those of the coordinate operator r. Show from first principles that this implies that its cartesian components satisfy the commutation relations

  $$\left[J_{j}, V_{k}\right]=i \epsilon_{j k l} V_{l}$$

  Hence calculate the total angular momentum of the nonvanishing states $V_{j}|0\rangle$, where $|0\rangle$ is the vacuum state.

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• B4.24

  Applications of Quantum Mechanics | Part II, 2001

  Derive the Bloch form of the wave function $\psi(x)$ of an electron moving in a onedimensional crystal lattice.

  The potential in such an $N$-atom lattice is modelled by

  $$V(x)=\sum_{n}\left(-\frac{\hbar^{2} U}{2 m} \delta(x-n L)\right)$$

  Assuming that $\psi(x)$ is continuous across each lattice site, and applying periodic boundary conditions, derive an equation for the allowed electron energy levels. Show that for suitable values of $U L$ they have a band structure, and calculate the number of levels in each band when $U L>2$. Verify that when $U L \gg 1$ the levels are very close to those corresponding to a solitary atom.

  Describe briefly how the band structure in a real 3-dimensional crystal differs from that of this simple model.

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Applied Probability

• B2.13

  Applied Probability | Part II, 2001

  Let $M$ be a Poisson random measure on $E=\mathbb{R} \times[0, \pi)$ with constant intensity $\lambda$. For $(x, \theta) \in E$, denote by $l(x, \theta)$ the line in $\mathbb{R}^{2}$ obtained by rotating the line $\{(x, y): y \in \mathbb{R}\}$ through an angle $\theta$ about the origin.

  Consider the line process $L=M \circ l^{-1}$.

  (i) What is the distribution of the number of lines intersecting the disc $\left\{z \in \mathbb{R}^{2}:|z| \leqslant a\right\}$ ?

  (ii) What is the distribution of the distance from the origin to the nearest line?

  (iii) What is the distribution of the distance from the origin to the $k$ th nearest line?

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• B3.13

  Applied Probability | Part II, 2001

  Consider an $M / G / 1$ queue with arrival rate $\lambda$ and traffic intensity less

  than 1. Prove that the moment-generating function of a typical busy period, $M_{B}(\theta)$, satisfies

  $$M_{B}(\theta)=M_{S}\left(\theta-\lambda+\lambda M_{B}(\theta)\right),$$

  where $M_{S}(\theta)$ is the moment-generating function of a typical service time.

  If service times are exponentially distributed with parameter $\mu>\lambda$, show that

  $$M_{B}(\theta)=\frac{\lambda+\mu-\theta-\left\{(\lambda+\mu-\theta)^{2}-4 \lambda \mu\right\}^{1 / 2}}{2 \lambda}$$

  for all sufficiently small but positive values of $\theta$.

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• B4.12

  Applied Probability | Part II, 2001

  Define a renewal process and a renewal reward process.

  State and prove the strong law of large numbers for these processes.

  [You may assume the strong law of large numbers for independent, identically-distributed random variables.

  State and prove Little's formula.

  Customers arrive according to a Poisson process with rate $\nu$ at a single server, but a restricted waiting room causes those who arrive when $n$ customers are already present to be lost. Accepted customers have service times which are independent and identicallydistributed with mean $\alpha$ and independent of the arrival process. Let $P_{j}$ be the equilibrium probability that an arriving customer finds $j$ customers already present.

  Using Little's formula, or otherwise, determine a relationship between $P_{0}, P_{n}, \nu$ and

  Part II

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Coding and Cryptography

• A1.10

  Coding and Cryptography | Part II, 2001

  (i) Explain briefly how and why a signature scheme is used. Describe the el Gamal scheme,

  (ii) Define a cyclic code. Define the generator of a cyclic code and show that it exists. Prove a necessary and sufficient condition for a polynomial to be the generator of a cyclic code of length $n$.

  What is the $\mathrm{BCH}$ code? Show that the $\mathrm{BCH}$ code associated with $\left\{\beta, \beta^{2}\right\}$, where $\beta$ is a root of $X^{3}+X+1$ in an appropriate field, is Hamming's original code.

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• A2.9

  Coding and Cryptography | Part II, 2001

  (i) Give brief answers to the following questions.

  (a) What is a stream cypher?

  (b) Explain briefly why a one-time pad is safe if used only once but becomes unsafe if used many times.

  (c) What is a feedback register of length $d$ ? What is a linear feedback register of length $d ?$

  (d) A cypher stream is given by a linear feedback register of known length $d$. Show that, given plain text and cyphered text of length $2 d$, we can find the complete cypher stream.

  (e) State and prove a similar result for a general feedback register.

  (ii) Describe the construction of a Reed-Muller code. Establish its information rate and its weight

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Combinatorics

• B1.5

  Combinatorics | Part II, 2001

  Let $\mathcal{A} \subset[n]^{(r)}$ where $r \leqslant n / 2$. Prove that, if $\mathcal{A}$ is 1-intersecting, then $|\mathcal{A}| \leqslant\left(\begin{array}{l}n-1 \\ r-1\end{array}\right)$. State an upper bound on $|\mathcal{A}|$ that is valid if $\mathcal{A}$ is $t$-intersecting and $n$ is large compared to $r$ and $t$.

  Let $\mathcal{B} \subset \mathcal{P}([n])$ be maximal 1-intersecting; that is, $\mathcal{B}$ is 1-intersecting but if $\mathcal{B} \subset \mathcal{C} \subset \mathcal{P}([n])$ and $\mathcal{B} \neq \mathcal{C}$ then $\mathcal{C}$ is not 1-intersecting. Show that $|\mathcal{B}|=2^{n-1}$.

  Let $\mathcal{B} \subset \mathcal{P}([n])$ be 2 -intersecting. Show that $|\mathcal{B}| \geqslant 2^{n-2}$ is possible. Can the inequality be strict?

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• B2.5

  Combinatorics | Part II, 2001

  As usual, $R_{k}^{(r)}(m)$ denotes the smallest integer $n$ such that every $k$-colouring of $[n]^{(r)}$ yields a monochromatic $m$-subset $M \in[n]^{(m)}$. Prove that $R_{2}^{(2)}(m)>2^{m / 2}$ for $m \geqslant 3$.

  Let $\mathcal{P}([n])$ have the colex order, and for $a, b \in \mathcal{P}([n])$ let $\delta(a, b)=\max a \triangle b$; thus $a<b$ means $\delta(a, b) \in b$. Show that if $a<b<c$ then $\delta(a, b) \neq \delta(b, c)$, and that $\delta(a, c)=\max \{\delta(a, b), \delta(b, c)\} .$

  Given a red-blue colouring of $[n]^{(2)}$, the 4 -colouring

  $$\chi: \mathcal{P}([n])^{(3)} \rightarrow\{\text { red, blue }\} \times\{0,1\}$$

  is defined as follows:

  $$\chi(\{a, b, c\})= \begin{cases}(\text { red, } 0) & \text { if }\{\delta(a, b), \delta(b, c)\} \text { is red and } \delta(a, b)<\delta(b, c) \\ (\text { red }, 1) & \text { if }\{\delta(a, b), \delta(b, c)\} \text { is red and } \delta(a, b)>\delta(b, c) \\ (\text { blue, } 0) & \text { if }\{\delta(a, b), \delta(b, c)\} \text { is blue and } \delta(a, b)<\delta(b, c) \\ \text { (blue, } 1) & \text { if }\{\delta(a, b), \delta(b, c)\} \text { is blue and } \delta(a, b)>\delta(b, c)\end{cases}$$

  where $a<b<c$. Show that if $M=\left\{a_{0}, a_{1}, \ldots, a_{m}\right\} \in \mathcal{P}([n])^{(m+1)}$ is monochromatic then $\left\{\delta_{1}, \ldots, \delta_{m}\right\} \in[n]^{(m)}$ is monochromatic, where $\delta_{i}=\delta\left(a_{i-1}, a_{i}\right)$ and $a_{0}<a_{1}<\cdots<a_{m}$.

  Deduce that $R_{4}^{(3)}(m+1)>2^{2^{m / 2}}$ for $m \geqslant 3$.

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• B4.1

  Combinatorics | Part II, 2001

  Write an essay on extremal graph theory. You should give proofs of at least two major theorems and you should also include a description of alternative proofs or further results.

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Computational Statistics and Statistical Modelling

• A1.13

  Computational Statistics and Statistical Modelling | Part II, 2001

  (i) Assume 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 $\operatorname{rank} p, \beta$ is an unknown vector, and

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

  Let $Q(\beta)=(Y-X \beta)^{T}(Y-X \beta)$. Find $\widehat{\beta}$, the least-squares estimator of $\beta$, and show that

  $$Q(\widehat{\beta})=Y^{T}(I-H) Y$$

  where $H$ is a matrix that you should define.

  (ii) Show that $\sum_{i} H_{i i}=p$. Show further for the special case of

  $$Y_{i}=\beta_{1}+\beta_{2} x_{i}+\beta_{3} z_{i}+\epsilon_{i}, \quad 1 \leqslant i \leqslant n$$

  where $\Sigma x_{i}=0, \Sigma z_{i}=0$, that

  $$H=\frac{1}{n} \mathbf{1 1}{ }^{T}+a x x^{T}+b\left(x z^{T}+z x^{T}\right)+c z z^{T} ;$$

  here, $\mathbf{1}$ is a vector of which every element is one, and $a, b, c$, are constants that you should derive.

  Hence show that, if $\widehat{Y}=X \widehat{\beta}$ is the vector of fitted values, then

  $$\frac{1}{\sigma^{2}} \operatorname{var}\left(\widehat{Y}_{i}\right)=\frac{1}{n}+a x_{i}^{2}+2 b x_{i} z_{i}+c z_{i}^{2}, \quad 1 \leqslant i \leqslant n .$$

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• A2.12

  Computational Statistics and Statistical Modelling | Part II, 2001

  (i) Suppose that $Y_{1}, \ldots, Y_{n}$ are independent random variables, and that $Y_{i}$ has probability density function

  $$f\left(y_{i} \mid \theta_{i}, \phi\right)=\exp \left[\left(y_{i} \theta_{i}-b\left(\theta_{i}\right)\right) / \phi+c\left(y_{i}, \phi\right)\right]$$

  Assume that $E\left(Y_{i}\right)=\mu_{i}$, and that $g\left(\mu_{i}\right)=\beta^{T} x_{i}$, where $g(\cdot)$ is a known 'link' function, $x_{1}, \ldots, x_{n}$ are known covariates, and $\beta$ is an unknown vector. Show that

  $$\mathbb{E}\left(Y_{i}\right)=b^{\prime}\left(\theta_{i}\right), \operatorname{var}\left(Y_{i}\right)=\phi b^{\prime \prime}\left(\theta_{i}\right)=V_{i} \text {, say, }$$

  and hence

  $$\frac{\partial l}{\partial \beta}=\sum_{i=1}^{n} \frac{\left(y_{i}-\mu_{i}\right) x_{i}}{g^{\prime}\left(\mu_{i}\right) V_{i}}, \text { where } l=l(\beta, \phi) \text { is the log-likelihood. }$$

  (ii) The table below shows the number of train miles (in millions) and the number of collisions involving British Rail passenger trains between 1970 and 1984 . Give a detailed interpretation of the $R$ output that is shown under this table:

  $\begin{array}{llll} & \text { year } & \text { collisions } & \text { miles } \\ 1 & 1970 & 3 & 281 \\ 2 & 1971 & 6 & 276 \\ 3 & 1972 & 4 & 268 \\ 4 & 1973 & 7 & 269 \\ 5 & 1974 & 6 & 281 \\ 6 & 1975 & 2 & 271 \\ 7 & 1976 & 2 & 265 \\ 8 & 1977 & 4 & 264 \\ 9 & 1978 & 1 & 267 \\ 10 & 1979 & 7 & 265 \\ 11 & 1980 & 3 & 267 \\ 12 & 1981 & 5 & 260 \\ 13 & 1982 & 6 & 231 \\ 14 & 1983 & 1 & 249\end{array}$

  Call:

  glm(formula $=$ collisions $\sim$ year $+\log ($ miles $)$, family $=$ poisson)

  Coefficients:

  $\begin{array}{lrrrr} & \text { Estimate } & \text { Std. Error } & \text { z value } & \operatorname{Pr}(>|z|) \\ \text { (Intercept) } & 127.14453 & 121.37796 & 1.048 & 0.295 \\ \text { year } & -0.05398 & 0.05175 & -1.043 & 0.297 \\ \log \text { (miles) } & -3.41654 & 4.18616 & -0.816 & 0.414\end{array}$

  (Dispersion parameter for poisson family taken to be 1)

  Null deviance: $15.937$ on 13 degrees of freedom

  Residual deviance: $14.843$ on 11 degrees of freedom

  Number of Fisher Scoring iterations: 4

  Part II

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• A4.14

  Computational Statistics and Statistical Modelling | Part II, 2001

  (i) Assume that independent observations $Y_{1}, \ldots, Y_{n}$ are such that

  $$Y_{i} \sim \operatorname{Binomial}\left(t_{i}, \pi_{i}\right), \log \frac{\pi_{i}}{1-\pi_{i}}=\beta^{T} x_{i} \quad \text { for } 1 \leqslant i \leqslant n$$

  where $x_{1}, \ldots, x_{n}$ are given covariates. Discuss carefully how to estimate $\beta$, and how to test that the model fits.

  (ii) Carmichael et al. (1989) collected data on the numbers of 5 -year old children with "dmft", i.e. with 5 or more decayed, missing or filled teeth, classified by social class, and by whether or not their tap water was fluoridated or non-fluoridated. The numbers of such children with dmft, and the total numbers, are given in the table below:

  \begin{tabular}{l|ll} Social Class & Fluoridated & Non-fluoridated \ \hline I & $12 / 117$ & $12 / 56$ \ II & $26 / 170$ & $48 / 146$ \ III & $11 / 52$ & $29 / 64$ \ Unclassified & $24 / 118$ & $49 / 104$ \end{tabular}

  A (slightly edited) version of the $R$ output is given below. Explain carefully what model is being fitted, whether it does actually fit, and what the parameter estimates and Std. Errors are telling you. (You may assume that the factors SClass (social class) and Fl (with/without) have been correctly set up.)

  

  $\begin{array}{lrrrr} & \text { Estimate } & \text { Std. } & \text { Error } & \text { z value } \\ \text { (Intercept) } & -2.2716 & 0.2396 & -9.480 \\ \text { SClassII } & 0.5099 & 0.2628 & 1.940 \\ \text { SClassIII } & 0.9857 & 0.3021 & 3.262 \\ \text { SClassUnc } & 1.0020 & 0.2684 & 3.734 \\ \text { Flwithout } & 1.0813 & 0.1694 & 6.383\end{array}$

  

  Here 'Yes' is the vector of numbers with dmft, taking values $12,12, \ldots, 24,49$, 'Total' is the vector of Total in each category, taking values $117,56, \ldots, 118,104$, and SClass, Fl are the factors corresponding to Social class and Fluoride status, defined in the obvious way.

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Differentiable Manifolds

• B1.8

  Differentiable Manifolds | Part II, 2001

  Define an immersion and an embedding of one manifold in another. State a necessary and sufficient condition for an immersion to be an embedding and prove its necessity.

  Assuming the existence of "bump functions" on Euclidean spaces, state and prove a version of Whitney's embedding theorem.

  Deduce that $\mathbb{R}^{n}$ embeds in $\mathbb{R}^{(n+1)^{2}}$.

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• B2.7

  Differentiable Manifolds | Part II, 2001

  State Stokes' Theorem.

  Prove that, if $M^{m}$ is a compact connected manifold and $\Phi: U \rightarrow \mathbb{R}^{m}$ is a surjective chart on $M$, then for any $\omega \in \Omega^{m}(M)$ there is $\eta \in \Omega^{m-1}(M)$ such that $\operatorname{supp}(\omega+d \eta) \subseteq \Phi^{-1}\left(\mathbf{B}^{m}\right)$, where $\mathbf{B}^{m}$ is the unit ball in $\mathbb{R}^{m}$.

  [You may assume that, if $\omega \in \Omega^{m}\left(\mathbb{R}^{m}\right)$ with $\operatorname{supp}(\omega) \subseteq \mathbf{B}^{m}$ and $\int_{\mathbb{R}^{m}} \omega=0$, then $\exists \eta \in \Omega^{m-1}\left(\mathbb{R}^{m}\right)$ with $\operatorname{supp}(\eta) \subseteq \mathbf{B}^{m}$ such that $\left.d \eta=\omega .\right]$

  By considering the $m$-form

  $$\omega=x_{1} d x_{2} \wedge \ldots \wedge d x_{m+1}+\cdots+x_{m+1} d x_{1} \wedge \ldots \wedge d x_{m}$$

  on $\mathbb{R}^{m+1}$, or otherwise, deduce that $H^{m}\left(S^{m}\right) \cong \mathbb{R}$.

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• B4.4

  Differentiable Manifolds | Part II, 2001

  Describe the Mayer-Vietoris exact sequence for forms on a manifold $M$ and show how to derive from it the Mayer-Vietoris exact sequence for the de Rham cohomology.

  Calculate $H^{*}\left(\mathbb{R} \mathbb{P}^{n}\right)$.

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Dynamical Systems

• B1.17

  Dynamical Systems | Part II, 2001

  Define topological conjugacy and $C^{1}$-conjugacy.

  Let $a, b$ be real numbers with $a>b>0$ and let $F_{a}, F_{b}$ be the maps of $(0, \infty)$ to itself given by $F_{a}(x)=a x, F_{b}(x)=b x$. For which pairs $a, b$ are $F_{a}$ and $F_{b}$ topologically conjugate? Would the answer be the same for $C^{1}$-conjugacy? Justify your statements.

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• B3.17

  Dynamical Systems | Part II, 2001

  If $A=\left(\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right)$ show that $A^{n+2}=A^{n+1}+A^{n}$ for all $n \geqslant 0$. Show that $A^{5}$ has trace 11 and deduce that the subshift map defined by $A$ has just two cycles of exact period 5. What are they?

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• B4.17

  Dynamical Systems | Part II, 2001

  Define the rotation number $\rho(f)$ of an orientation-preserving circle map $f$ and the rotation number $\rho(F)$ of a lift $F$ of $f$. Prove that $\rho(f)$ and $\rho(F)$ are well-defined. Prove also that $\rho(F)$ is a continuous function of $F$.

  State without proof the main consequence of $\rho(f)$ being rational.

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Dynamics of Differential Equations

• A1.6

  Dynamics of Differential Equations | Part II, 2001

  (i) Given a differential equation $\dot{x}=f(x)$ for $x \in \mathbb{R}^{n}$, explain what it means to say that the solution is given by a flow $\phi: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$. Define the orbit, $o(x)$, through a point $x$ and the $\omega$-limit set, $\omega(x)$, of $x$. Define also a homoclinic orbit to a fixed point $x_{0}$. Sketch a flow in $\mathbb{R}^{2}$ with a homoclinic orbit, and identify (without detailed justification) the $\omega$-limit sets $\omega(x)$ for each point $x$ in your diagram.

  (ii) Consider the differential equations

  $$\dot{x}=z y, \quad \dot{y}=-z x, \quad \dot{z}=-z^{2} .$$

  Transform the equations to polar coordinates $(r, \theta)$ in the $(x, y)$ plane. Solve the equation for $z$ to find $z(t)$, and hence find $\theta(t)$. Hence, or otherwise, determine (with justification) the $\omega$-limit set for all points $\left(x_{0}, y_{0}, z_{0}\right) \in \mathbb{R}^{3}$.

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• A2.6 B2.4

  Dynamics of Differential Equations | Part II, 2001

  (i) Define a Liapounov function for a flow $\phi$ on $\mathbb{R}^{n}$. Explain what it means for a fixed point of the flow to be Liapounov stable. State and prove Liapounov's first stability theorem.

  (ii) Consider the damped pendulum

  $$\ddot{\theta}+k \dot{\theta}+\sin \theta=0,$$

  where $k>0$. Show that there are just two fixed points (considering the phase space as an infinite cylinder), and that one of these is the origin and is Liapounov stable. Show further that the origin is asymptotically stable, and that the the $\omega$-limit set of each point in the phase space is one or other of the two fixed points, justifying your answer carefully.

  [You should state carefully any theorems you use in your answer, but you need not prove them.]

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• A3.6 B3.4

  Dynamics of Differential Equations | Part II, 2001

  (i) Define a hyperbolic fixed point $x_{0}$ of a flow determined by a differential equation $\dot{x}=f(x)$ where $x \in R^{n}$ and $f$ is $C^{1}$ (i.e. differentiable). State the Hartman-Grobman Theorem for flow near a hyperbolic fixed point. For nonlinear flows in $R^{2}$ with a hyperbolic fixed point $x_{0}$, does the theorem necessarily allow us to distinguish, on the basis of the linearized flow near $x_{0}$ between (a) a stable focus and a stable node; and (b) a saddle and a stable node? Justify your answers briefly.

  (ii) Show that the system:

  $$\begin{aligned} &\dot{x}=-(\mu+1)+(\mu-3) x-y+6 x^{2}+12 x y+5 y^{2}-2 x^{3}-6 x^{2} y-5 x y^{2} \\ &\dot{y}=2-2 x+(\mu-5) y+4 x y+6 y^{2}-2 x^{2} y-6 x y^{2}-5 y^{3} \end{aligned}$$

  has a fixed point $\left(x_{0}, 0\right)$ on the $x$-axis. Show that there is a bifurcation at $\mu=0$ and determine the stability of the fixed point for $\mu>0$ and for $\mu<0$.

  Make a linear change of variables of the form $u=x-x_{0}+\alpha y, v=x-x_{0}+\beta y$, where $\alpha$ and $\beta$ are constants to be determined, to bring the system into the form:

  $$\begin{aligned} &\dot{u}=v+u\left[\mu-\left(u^{2}+v^{2}\right)\right] \\ &\dot{v}=-u+v\left[\mu-\left(u^{2}+v^{2}\right)\right] \end{aligned}$$

  and hence determine whether the periodic orbit produced in the bifurcation is stable or unstable, and whether it exists in $\mu<0$ or $\mu>0$.

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• A4.6

  Dynamics of Differential Equations | Part II, 2001

  Write a short essay about periodic orbits in flows in two dimensions. Your essay should include criteria for the existence and non-existence of periodic orbits, and should mention (with sketches) at least two bifurcations that create or destroy periodic orbits in flows as a parameter is altered (though a detailed analysis of any bifurcation is not required).

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Electrodynamics

• B1.21

  Electrodynamics | Part II, 2001

  Explain the multipole expansion in electrostatics, and devise formulae for the total charge, dipole moments and quadrupole moments given by a static charge distribution $\rho(\mathbf{r})$.

  A nucleus is modelled as a uniform distribution of charge inside the ellipsoid

  $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1$$

  The total charge of the nucleus is $Q$. What are the dipole moments and quadrupole moments of this distribution?

  Describe qualitatively what happens if the nucleus starts to oscillate.

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• B2.20

  Electrodynamics | Part II, 2001

  In a superconductor, there are superconducting charge carriers with number density $n$, mass $m$ and charge $q$. Starting from the quantum mechanical wavefunction $\Psi=R e^{i \Phi}$ (with real $R$ and $\Phi$ ), construct a formula for the electric current and explain carefully why your result is gauge invariant.

  Now show that inside a superconductor a static magnetic field obeys the equation

  $$\nabla^{2} \mathbf{B}=\frac{\mu_{0} n q^{2}}{m} \mathbf{B}$$

  A superconductor occupies the region $z>0$, while for $z<0$ there is a vacuum with a constant magnetic field in the $x$ direction. Show that the magnetic field cannot penetrate deep into the superconductor.

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• B4.21

  Electrodynamics | Part II, 2001

  The Liénard-Wiechert potential for a particle of charge $q$, assumed to be moving non-relativistically along the trajectory $y^{\mu}(\tau), \tau$ being the proper time along the trajectory,

  $$A^{\mu}(x, t)=\left.\frac{\mu_{0} q}{4 \pi} \frac{d y^{\mu} / d \tau}{(x-y(\tau))_{\nu} d y^{\nu} / d \tau}\right|_{\tau=\tau_{0}} .$$

  Explain how to calculate $\tau_{0}$ given $x^{\mu}=(x, t)$ and $y^{\mu}=\left(y, t^{\prime}\right)$.

  Derive Larmor's formula for the rate at which electromagnetic energy is radiated from a particle of charge $q$ undergoing an acceleration $a$.

  Suppose that one considers the classical non-relativistic hydrogen atom with an electron of mass $m$ and charge $-e$ orbiting a fixed proton of charge $+e$, in a circular orbit of radius $r_{0}$. What is the total energy of the electron? As the electron is accelerated towards the proton it will radiate, thereby losing energy and causing the orbit to decay. Derive a formula for the lifetime of the orbit.

  Part II

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Electromagnetism

• A $1 . 5 \quad$ B $1 . 4 \quad$

  Electromagnetism | Part II, 2001

  (i) Write down the two Maxwell equations that govern steady magnetic fields. Show that the boundary conditions satisfied by the magnetic field on either side of a current sheet, $\mathbf{J}$, with unit normal to the sheet $\mathbf{n}$, are

  $$\mathbf{n} \wedge \mathbf{B}_{2}-\mathbf{n} \wedge \mathbf{B}_{1}=\mu_{0} \mathbf{J}$$

  State without proof the force per unit area on $\mathbf{J}$.

  (ii) Conducting gas occupies the infinite slab $0 \leqslant x \leqslant a$. It carries a steady current $\mathbf{j}=(0,0, j)$ and a magnetic field $\mathbf{B}=(0, B, 0)$ where $\mathbf{j}$, $\mathbf{B}$ depend only on $x$. The pressure is $p(x)$. The equation of hydrostatic equilibrium is $\nabla p=\mathbf{j} \wedge \mathbf{B}$. Write down the equations to be solved in this case. Show that $p+\left(1 / 2 \mu_{0}\right) B^{2}$ is independent of $x$. Using the suffixes 1,2 to denote values at $x=0, a$, respectively, verify that your results are in agreement with those of Part (i) in the case of $a \rightarrow 0$.

  Suppose that

  $$j(x)=\frac{\pi j_{0}}{2 a} \sin \left(\frac{\pi x}{a}\right), \quad B_{1}=0, \quad p_{2}=0$$

  Find $B(x)$ everywhere in the slab.

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• A2.5

  Electromagnetism | Part II, 2001

  (i) Write down the expression for the electrostatic potential $\phi(\mathbf{r})$ due to a distribution of charge $\rho(\mathbf{r})$ contained in a volume $V$. Perform the multipole expansion of $\phi(\mathbf{r})$ taken only as far as the dipole term.

  (ii) If the volume $V$ is the sphere $|\mathbf{r}| \leqslant a$ and the charge distribution is given by

  $$\rho(\mathbf{r})= \begin{cases}r^{2} \cos \theta & r \leqslant a \\ 0 & r>a\end{cases}$$

  where $r, \theta$ are spherical polar coordinates, calculate the charge and dipole moment. Hence deduce $\phi$ as far as the dipole term.

  Obtain an exact solution for $r>a$ by solving the boundary value problem using trial solutions of the forms

  $$\phi=\frac{A \cos \theta}{r^{2}} \text { for } r>a,$$

  and

  $$\phi=B r \cos \theta+C r^{4} \cos \theta \text { for } r<a .$$

  Show that the solution obtained from the multipole expansion is in fact exact for $r>a$.

  [You may use without proof the result

  $$\left.\nabla^{2}\left(r^{k} \cos \theta\right)=(k+2)(k-1) r^{k-2} \cos \theta, \quad k \in \mathbb{N} .\right]$$

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• A3.5 B3.3

  Electromagnetism | Part II, 2001

  (i) Develop the theory of electromagnetic waves starting from Maxwell equations in vacuum. You should relate the wave-speed $c$ to $\epsilon_{0}$ and $\mu_{0}$ and establish the existence of plane, plane-polarized waves in which $\mathbf{E}$ takes the form

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

  You should give the form of the magnetic field $\mathbf{B}$ in this case.

  (ii) Starting from Maxwell's equation, establish Poynting's theorem.

  $$-\mathbf{j} \cdot \mathbf{E}=\frac{\partial W}{\partial t}+\nabla \cdot \mathbf{S},$$

  where $W=\frac{\epsilon_{0}}{2} \mathbf{E}^{2}+\frac{1}{2 \mu_{0}} \mathbf{B}^{2}$ and $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \wedge \mathbf{B}$. Give physical interpretations of $W, S$ and the theorem.

  Compute the averages over space and time of $W$ and $\mathbf{S}$ for the plane wave described in (i) and relate them. Comment on the result.

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• A4.5 $\quad$

  Electromagnetism | Part II, 2001

  Write down the form of Ohm's Law that applies to a conductor if at a point $\mathbf{r}$ it is moving with velocity $\mathbf{v}(\mathbf{r})$.

  Use two of Maxwell's equations to prove that

  $$\int_{C}(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot d \mathbf{r}=-\frac{d}{d t} \int_{S} \mathbf{B} \cdot d \mathbf{S}$$

  where $C(t)$ is a moving closed loop, $\mathbf{v}$ is the velocity at the point $\mathbf{r}$ on $C$, and $S$ is a surface spanning $C$. The time derivative on the right hand side accounts for changes in both $C$ and B. Explain briefly the physical importance of this result.

  Find and sketch the magnetic field $\mathbf{B}$ described in the vector potential

  $$\mathbf{A}(r, \theta, z)=\left(0, \frac{1}{2} b r z, 0\right)$$

  in cylindrical polar coordinates $(r, \theta, z)$, where $b>0$ is constant.

  A conducting circular loop of radius $a$ and resistance $R$ lies in the plane $z=h(t)$ with its centre on the $z$-axis.

  Find the magnitude and direction of the current induced in the loop as $h(t)$ changes with time, neglecting self-inductance.

  At time $t=0$ the loop is at rest at $z=0$. For time $t>0$ the loop moves with constant velocity $d h / d t=v>0$. Ignoring the inertia of the loop, use energy considerations to find the force $F(t)$ necessary to maintain this motion.

  [ In cylindrical polar coordinates

  $$\left.\operatorname{curl} \mathbf{A}=\left(\frac{1}{r} \frac{\partial A_{z}}{\partial \theta}-\frac{\partial A_{\theta}}{\partial z}, \frac{\partial A_{r}}{\partial z}-\frac{\partial A_{z}}{\partial r}, \frac{1}{r} \frac{\partial}{\partial r}\left(r A_{\theta}\right)-\frac{1}{r} \frac{\partial A_{r}}{\partial \theta}\right)\right]$$

  Part II

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Fluid Dynamics II

• B1.25

  Fluid Dynamics II | Part II, 2001

  The energy equation for the motion of a viscous, incompressible fluid states that

  $$\frac{d}{d t} \int_{V(t)} \frac{1}{2} \rho u^{2} d V=\int_{S(t)} u_{i} \sigma_{i j} n_{j} d S-2 \mu \int_{V(t)} e_{i j} e_{i j} d V$$

  Interpret each term in this equation and explain the meaning of the symbols used.

  For steady rectilinear flow in a (not necessarily circular) pipe having rigid stationary walls, deduce a relation between the viscous dissipation per unit length of the pipe, the pressure gradient $G$, and the volume flux $Q$.

  Starting from the Navier-Stokes equations, calculate the velocity field for steady rectilinear flow in a circular pipe of radius $a$. Using the relationship derived above, or otherwise, find in terms of $G$ the viscous dissipation per unit length for this flow.

  [In cylindrical polar coordinates,

  $$\left.\nabla^{2} w(r)=\frac{1}{r} \frac{d}{d r}\left(r \frac{d w}{d r}\right) .\right]$$

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• B2.24

  Fluid Dynamics II | Part II, 2001

  Explain what is meant by a Stokes flow and show that, in such a flow, in the absence of body forces, $\partial \sigma_{i j} / \partial x_{j}=0$, where $\sigma_{i j}$ is the stress tensor.

  State and prove the reciprocal theorem for Stokes flow.

  When a rigid sphere of radius $a$ translates with velocity $\mathbf{U}$ through unbounded fluid at rest at infinity, it may be shown that the traction per unit area, $\boldsymbol{\sigma} \cdot \mathbf{n}$, exerted by the sphere on the fluid, has the uniform value $3 \mu \mathbf{U} / 2 a$ over the sphere surface. Find the drag on the sphere.

  Suppose that the same sphere is free of external forces and is placed with its centre at the origin in an unbounded Stokes flow given in the absence of the sphere as $\mathbf{u}_{s}(\mathbf{x})$. By applying the reciprocal theorem to the perturbation to the flow generated by the presence of the sphere, and assuming this to tend to zero sufficiently rapidly at infinity, show that the instantaneous velocity of the centre of the sphere is

  $$\mathbf{V}=\frac{1}{4 \pi a^{2}} \int_{r=a} \mathbf{u}_{s}(\mathbf{x}) d S$$

  Part II

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• B3.24

  Fluid Dynamics II | Part II, 2001

  A planar flow of an inviscid, incompressible fluid is everywhere in the $x$-direction and has velocity profile

  $$u=\left\{\begin{array}{cc} U & y>0, \\ 0 & y<0 . \end{array}\right.$$

  By examining linear perturbations to the vortex sheet at $y=0$ that have the form $e^{i k x-i \omega t}$, show that

  $$\omega=\frac{1}{2} k U(1 \pm i)$$

  and deduce the temporal stability of the sheet to disturbances of wave number $k$.

  Use this result to determine also the spatial growth rate and propagation speed of disturbances of frequency $\omega$ introduced at a fixed spatial position.

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• B4.26

  Fluid Dynamics II | Part II, 2001

  Starting from the steady planar vorticity equation

  $$\mathbf{u} \cdot \nabla \omega=\nu \nabla^{2} \omega,$$

  outline briefly the derivation of the boundary layer equation

  $$u u_{x}+v u_{y}=U d U / d x+\nu u_{y y},$$

  explaining the significance of the symbols used.

  Viscous fluid occupies the region $y>0$ with rigid stationary walls along $y=0$ for $x>0$ and $x<0$. There is a line sink at the origin of strength $\pi Q, Q>0$, with $Q / \nu \gg 1$. Assuming that vorticity is confined to boundary layers along the rigid walls:

  (a) Find the flow outside the boundary layers.

  (b) Explain why the boundary layer thickness $\delta$ along the wall $x>0$ is proportional to $x$, and deduce that

  $$\delta=\left(\frac{\nu}{Q}\right)^{\frac{1}{2}} x$$

  (c) Show that the boundary layer equation admits a solution having stream function

  $$\psi=(\nu Q)^{1 / 2} f(\eta) \quad \text { with } \quad \eta=y / \delta$$

  Find the equation and boundary conditions satisfied by $f$.

  (d) Verify that a solution is

  $$f^{\prime}=\frac{6}{1+\cosh (\eta \sqrt{2}+c)}-1$$

  provided that $c$ has one of two values to be determined. Should the positive or negative value be chosen?

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Foundations of Quantum Mechanics

• A2.13 B2.21

  Foundations of Quantum Mechanics | Part II, 2001

  (i) Hermitian operators $\hat{x}, \hat{p}$, satisfy $[\hat{x}, \hat{p}]=i \hbar$. The eigenvectors $|p\rangle$, satisfy $\hat{p}|p\rangle=p|p\rangle$ and $\left\langle p^{\prime} \mid p\right\rangle=\delta\left(p^{\prime}-p\right)$. By differentiating with respect to $b$ verify that

  $$e^{-i b \hat{x} / \hbar} \hat{p} e^{i b \hat{x} / \hbar}=\hat{p}+b$$

  and hence show that

  $$e^{i b \hat{x} / \hbar}|p\rangle=|p+b\rangle$$

  Show that

  $$\langle p|\hat{x}| \psi\rangle=i \hbar \frac{\partial}{\partial p}\langle p \mid \psi\rangle$$

  and

  $$\langle p|\hat{p}| \psi\rangle=p\langle p \mid \psi\rangle .$$

  (ii) A quantum system has Hamiltonian $H=H_{0}+H_{1}$, where $H_{1}$ is a small perturbation. The eigenvalues of $H_{0}$ are $\epsilon_{n}$. Give (without derivation) the formulae for the first order and second order perturbations in the energy level of a non-degenerate state. Suppose that the $r$ th energy level of $H_{0}$ has $j$ degenerate states. Explain how to determine the eigenvalues of $H$ corresponding to these states to first order in $H_{1}$.

  In a particular quantum system an orthonormal basis of states is given by $\left|n_{1}, n_{2}\right\rangle$, where $n_{i}$ are integers. The Hamiltonian is given by

  $$H=\sum_{n_{1}, n_{2}}\left(n_{1}^{2}+n_{2}^{2}\right)\left|n_{1}, n_{2}\right\rangle\left\langle n_{1}, n_{2}\left|+\sum_{n_{1}, n_{2}, n_{1}^{\prime}, n_{2}^{\prime}} \lambda_{\left|n_{1}-n_{1}^{\prime}\right|,\left|n_{2}-n_{2}^{\prime}\right|}\right| n_{1}, n_{2}\right\rangle\left\langle n_{1}^{\prime}, n_{2}^{\prime}\right|,$$

  where $\lambda_{r, s}=\lambda_{s, r}, \lambda_{0,0}=0$ and $\lambda_{r, s}=0$ unless $r$ and $s$ are both even.

  Obtain an expression for the ground state energy to second order in the perturbation, $\lambda_{r, s}$. Find the energy eigenvalues of the first excited state to first order in the perturbation. Determine a matrix (which depends on two independent parameters) whose eigenvalues give the first order energy shift of the second excited state.

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• A3.13 B3.21

  Foundations of Quantum Mechanics | Part II, 2001

  (i) Write the Hamiltonian for the harmonic oscillator,

  $$H=\frac{p^{2}}{2 m}+\frac{1}{2} m \omega^{2} x^{2}$$

  in terms of creation and annihilation operators, defined by

  $$a^{\dagger}=\left(\frac{m \omega}{2 \hbar}\right)^{\frac{1}{2}}\left(x-i \frac{p}{m \omega}\right), \quad a=\left(\frac{m \omega}{2 \hbar}\right)^{\frac{1}{2}}\left(x+i \frac{p}{m \omega}\right)$$

  Obtain an expression for $\left[a^{\dagger}, a\right]$ by using the usual commutation relation between $p$ and $x$. Deduce the quantized energy levels for this system.

  (ii) Define the number operator, $N$, in terms of creation and annihilation operators, $a^{\dagger}$ and $a$. The normalized eigenvector of $N$ with eigenvalue $n$ is $|n\rangle$. Show that $n \geq 0$.

  Determine $a|n\rangle$ and $a^{\dagger}|n\rangle$ in the basis defined by $\{|n\rangle\}$.

  Show that

  $$a^{\dagger m} a^{m}|n\rangle=\left\{\begin{aligned} \frac{n !}{(n-m) !}|n\rangle, & m \leq n \\ 0, & m>n \end{aligned}\right.$$

  Verify the relation

  $$|0\rangle\langle 0|=\sum_{m=0} \frac{1}{m !}(-1)^{m} a^{\dagger m} a^{m}$$

  by considering the action of both sides of the equation on an arbitrary basis vector.

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• A4.15 B4.22

  Foundations of Quantum Mechanics | Part II, 2001

  (i) The two states of a spin- $\frac{1}{2}$ particle corresponding to spin pointing along the $z$ axis are denoted by $|\uparrow\rangle$ and $|\downarrow\rangle$. Explain why the states

  $$|\uparrow, \theta\rangle=\cos \frac{\theta}{2}|\uparrow\rangle+\sin \frac{\theta}{2}|\downarrow\rangle, \quad \quad|\downarrow, \theta\rangle=-\sin \frac{\theta}{2}|\uparrow\rangle+\cos \frac{\theta}{2}|\downarrow\rangle$$

  correspond to the spins being aligned along a direction at an angle $\theta$ to the $z$ direction.

  The spin- 0 state of two spin- $\frac{1}{2}$ particles is

  $$|0\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle_{1}|\downarrow\rangle_{2}-|\downarrow\rangle_{1}|\uparrow\rangle_{2}\right)$$

  Show that this is independent of the direction chosen to define $|\uparrow\rangle_{1,2},|\downarrow\rangle_{1,2}$. If the spin of particle 1 along some direction is measured to be $\frac{1}{2} \hbar$ show that the spin of particle 2 along the same direction is determined, giving its value.

  [The Pauli matrices are given by

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

  (ii) Starting from the commutation relation for angular momentum in the form

  $$\left[J_{3}, J_{\pm}\right]=\pm \hbar J_{\pm}, \quad\left[J_{+}, J_{-}\right]=2 \hbar J_{3},$$

  obtain the possible values of $j, m$, where $m \hbar$ are the eigenvalues of $J_{3}$ and $j(j+1) \hbar^{2}$ are the eigenvalues of $\mathbf{J}^{2}=\frac{1}{2}\left(J_{+} J_{-}+J_{-} J_{+}\right)+J_{3}^{2}$. Show that the corresponding normalized eigenvectors, $|j, m\rangle$, satisfy

  $$J_{\pm}|j, m\rangle=\hbar((j \mp m)(j \pm m+1))^{1 / 2}|j, m \pm 1\rangle,$$

  and that

  $$\frac{1}{n !} J_{-}^{n}|j, j\rangle=\hbar^{n}\left(\frac{(2 j) !}{n !(2 j-n) !}\right)^{1 / 2}|j, j-n\rangle, \quad n \leq 2 j$$

  The state $|w\rangle$ is defined by

  $$|w\rangle=e^{w J_{-} / \hbar}|j, j\rangle$$

  for any complex $w$. By expanding the exponential show that $\langle w \mid w\rangle=\left(1+|w|^{2}\right)^{2 j}$. Verify that

  $$e^{-w J_{-} / \hbar} J_{3} e^{w J_{-} / \hbar}=J_{3}-w J_{-}$$

  and hence show that

  $$J_{3}|w\rangle=\hbar\left(j-w \frac{\partial}{\partial w}\right)|w\rangle$$

  If $H=\alpha J_{3}$ verify that $\left|e^{i \alpha t}\right\rangle e^{-i j \alpha t}$ is a solution of the time-dependent Schrödinger equation.

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Functional Analysis

• A1 $. 3 \quad$

  Functional Analysis | Part II, 2001

  (i) Define the adjoint of a bounded, linear map $u: H \rightarrow H$ on the Hilbert space $H$. Find the adjoint of the map

  $$u: H \rightarrow H ; \quad x \mapsto \phi(x) a$$

  where $a, b \in H$ and $\phi \in H^{*}$ is the linear map $x \mapsto\langle b, x\rangle$.

  Now let $J$ be an incomplete inner product space and $u: J \rightarrow J$ a bounded, linear map. Is it always true that there is an adjoint $u^{*}: J \rightarrow J$ ?

  (ii) Let $\mathcal{H}$ be the space of analytic functions $f: \mathbb{D} \rightarrow \mathbb{C}$ on the unit disc $\mathbb{D}$ for which

  $$\iint_{\mathbb{D}}|f(z)|^{2} d x d y<\infty \quad(z=x+i y)$$

  You may assume that this is a Hilbert space for the inner product:

  $$\langle f, g\rangle=\iint_{\mathbb{D}} \overline{f(z)} g(z) d x d y .$$

  Show that the functions $u_{k}: z \mapsto \alpha_{k} z^{k}(k=0,1,2, \ldots)$ form an orthonormal sequence in $\mathcal{H}$ when the constants $\alpha_{k}$ are chosen appropriately.

  Prove carefully that every function $f \in \mathcal{H}$ can be written as the sum of a convergent series $\sum_{k=0}^{\infty} f_{k} u_{k}$ in $\mathcal{H}$ with $f_{k} \in \mathbb{C}$.

  For each smooth curve $\gamma$ in the disc $\mathbb{D}$ starting from 0 , prove that

  $$\phi: \mathcal{H} \rightarrow \mathbb{C} ; \quad f \mapsto \int_{\gamma} f(z) d z$$

  is a continuous, linear map. Show that the norm of $\phi$ satisfies

  $$\|\phi\|^{2}=\frac{1}{\pi} \log \left(\frac{1}{1-|w|^{2}}\right)$$

  where $w$ is the endpoint of $\gamma$.

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• A2.3 B2.2

  Functional Analysis | Part II, 2001

  (i) State the Stone-Weierstrass theorem for complex-valued functions. Use it to show that the trigonometric polynomials are dense in the space $C(\mathbb{T})$ of continuous, complexvalued functions on the unit circle $\mathbb{T}$ with the uniform norm.

  Show further that, for $f \in C(\mathbb{T})$, the $n$th Fourier coefficient

  $$\widehat{f}(n)=\frac{1}{2 \pi} \int_{0}^{2 \pi} f\left(e^{i \theta}\right) e^{-i n \theta} d \theta$$

  tends to 0 as $|n|$ tends to infinity.

  (ii) (a) Let $X$ be a normed space with the property that the series $\sum_{n=1}^{\infty} x_{n}$ converges whenever $\left(x_{n}\right)$ is a sequence in $X$ with $\sum_{n=1}^{\infty}\left\|x_{n}\right\|$ convergent. Show that $X$ is a Banach space.

  (b) Let $K$ be a compact metric space and $L$ a closed subset of $K$. Let $R: C(K) \rightarrow$ $C(L)$ be the map sending $f \in C(K)$ to its restriction $R(f)=f \mid L$ to $L$. Show that $R$ is a bounded, linear map and that its image is a subalgebra of $C(L)$ separating the points of

  Show further that, for each function $g$ in the image of $R$, there is a function $f \in C(K)$ with $R(f)=g$ and $\|f\|_{\infty}=\|g\|_{\infty}$. Deduce that every continuous, complexvalued function on $L$ can be extended to a continuous function on all of $K$.

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• A3.3 B3.2

  Functional Analysis | Part II, 2001

  (i) Define the notion of a measurable function between measurable spaces. Show that a continuous function $\mathbb{R}^{2} \rightarrow \mathbb{R}$ is measurable with respect to the Borel $\sigma$-fields on $\mathbb{R}^{2}$ and $\mathbb{R}$.

  By using this, or otherwise, show that, when $f, g: X \rightarrow \mathbb{R}$ are measurable with respect to some $\sigma$-field $\mathcal{F}$ on $X$ and the Borel $\sigma$-field on $\mathbb{R}$, then $f+g$