Part IB, 2008, Paper 4

Analysis II

• 4.I.3F

  Analysis II | Part IB, 2008

  Let $X$ be the vector space of all continuous real-valued functions on the unit interval $[0,1]$. Show that the functions

  $$\|f\|_{1}=\int_{0}^{1}|f(t)| d t \quad \text { and } \quad\|f\|_{\infty}=\sup \{|f(t)|: 0 \leqslant t \leqslant 1\}$$

  both define norms on $X$.

  Consider the sequence $\left(f_{n}\right)$ defined by $f_{n}(t)=n t^{n}(1-t)$. Does $\left(f_{n}\right)$ converge in the norm $\|-\|_{1}$ ? Does it converge in the norm $\|-\|_{\infty}$ ? Justify your answers.

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• 4.II.13F

  Analysis II | Part IB, 2008

  Explain what it means for two norms on a real vector space to be Lipschitz equivalent. Show that if two norms are Lipschitz equivalent, then one is complete if and only if the other is.

  Let $\|-\|$ be an arbitrary norm on the finite-dimensional space $\mathbb{R}^{n}$, and let $\|-\|_{2}$ denote the standard (Euclidean) norm. Show that for every $\mathbf{x} \in \mathbb{R}^{n}$ with $\|\mathbf{x}\|_{2}=1$, we have

  $$\|\mathbf{x}\| \leqslant\left\|\mathbf{e}_{1}\right\|+\left\|\mathbf{e}_{2}\right\|+\cdots+\left\|\mathbf{e}_{n}\right\|$$

  where $\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}\right)$ is the standard basis for $\mathbb{R}^{n}$, and deduce that the function $\|-\|$ is continuous with respect to $\|-\|_{2}$. Hence show that there exists a constant $m>0$ such that $\|\mathbf{x}\| \geqslant m$ for all $\mathbf{x}$ with $\|\mathbf{x}\|_{2}=1$, and deduce that $\|-\|$ and $\|-\|_{2}$ are Lipschitz equivalent.

  [You may assume the Bolzano-Weierstrass Theorem.]

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Complex Analysis

• 4.I.4E

  Complex Analysis | Part IB, 2008

  Suppose that $f$ and $g$ are two functions which are analytic on the whole complex plane $\mathbb{C}$. Suppose that there is a sequence of distinct points $z_{1}, z_{2}, \ldots$ with $\left|z_{i}\right| \leqslant 1$ such that $f\left(z_{i}\right)=g\left(z_{i}\right)$. Show that $f(z)=g(z)$ for all $z \in \mathbb{C}$. [You may assume any results on Taylor expansions you need, provided they are clearly stated.]

  What happens if the assumption that $\left|z_{i}\right| \leqslant 1$ is dropped?

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Complex Methods

• 4.II.15C

  Complex Methods | Part IB, 2008

  Let $H$ be the domain $\mathbb{C}-\{x+i y: x \leq 0, y=0\}$ (i.e., $\mathbb{C}$ cut along the negative $x$-axis). Show, by a suitable choice of branch, that the mapping

  $$z \mapsto w=-i \log z$$

  maps $H$ onto the strip $S=\{z=x+i y,-\pi<x<\pi\}$.

  How would a different choice of branch change the result?

  Let $G$ be the domain $\{z \in \mathbb{C}:|z|<1,|z+i|>\sqrt{2}\}$. Find an analytic transformation that maps $G$ to $S$, where $S$ is the strip defined above.

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Electromagnetism

• 4.I.7B

  Electromagnetism | Part IB, 2008

  The energy stored in a static electric field $\mathbf{E}$ is

  $$U=\frac{1}{2} \int \rho \phi d V,$$

  where $\phi$ is the associated electric potential, $\mathbf{E}=-\nabla \phi$, and $\rho$ is the volume charge density.

  (i) Assuming that the energy is calculated over all space and that $\mathbf{E}$ vanishes at infinity, show that the energy can be written as

  $$U=\frac{\epsilon_{0}}{2} \int|\mathbf{E}|^{2} d V$$

  (ii) Find the electric field produced by a spherical shell with total charge $Q$ and radius $R$, assuming it to vanish inside the shell. Find the energy stored in the electric field.

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Fluid Dynamics

• 4.II.18B

  Fluid Dynamics | Part IB, 2008

  (i) Starting from Euler's equation for an incompressible fluid show that for potential flow with $\mathbf{u}=\nabla \phi$,

  $$\frac{\partial \phi}{\partial t}+\frac{1}{2} u^{2}+\chi=f(t)$$

  where $u=|\mathbf{u}|, \chi=p / \rho+V$, the body force per unit mass is $-\nabla V$ and $f(t)$ is an arbitrary function of time.

  (ii) Hence show that, for the steady flow of a liquid of density $\rho$ through a pipe of varying cross-section that is subject to a pressure difference $\Delta p=p_{1}-p_{2}$ between its two ends, the mass flow through the pipe per unit time is given by

  $$m \equiv \frac{d M}{d t}=S_{1} S_{2} \sqrt{\frac{2 \rho \Delta p}{S_{1}^{2}-S_{2}^{2}}}$$

  where $S_{1}$ and $S_{2}$ are the cross-sectional areas of the two ends.

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Geometry

• 4.II.12G

  Geometry | Part IB, 2008

  Let $\gamma:[a, b] \rightarrow S$ be a curve on a smoothly embedded surface $S \subset \mathbf{R}^{3}$. Define the energy of $\gamma$. Show that if $\gamma$ is a stationary point for the energy for proper variations of $\gamma$, then $\gamma$ satisfies the geodesic equations

  $$\begin{aligned} \frac{d}{d t}\left(E \dot{\gamma}_{1}+F \dot{\gamma}_{2}\right) &=\frac{1}{2}\left(E_{u} \dot{\gamma}_{1}^{2}+2 F_{u} \dot{\gamma}_{1} \dot{\gamma}_{2}+G_{u} \dot{\gamma}_{2}^{2}\right) \\ \frac{d}{d t}\left(F \dot{\gamma}_{1}+G \dot{\gamma}_{2}\right) &=\frac{1}{2}\left(E_{v} \dot{\gamma}_{1}^{2}+2 F_{v} \dot{\gamma}_{1} \dot{\gamma}_{2}+G_{v} \dot{\gamma}_{2}^{2}\right) \end{aligned}$$

  where $\gamma=\left(\gamma_{1}, \gamma_{2}\right)$ in terms of a smooth parametrization $(u, v)$ for $S$, with first fundamental form $E d u^{2}+2 F d u d v+G d v^{2}$.

  Now suppose that for every $c, d$ the curves $u=c, v=d$ are geodesics.

  (i) Show that $(F / \sqrt{G})_{v}=(\sqrt{G})_{u}$ and $(F / \sqrt{E})_{u}=(\sqrt{E})_{v}$.

  (ii) Suppose moreover that the angle between the curves $u=c, v=d$ is independent of $c$ and $d$. Show that $E_{v}=0=G_{u}$.

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Groups, Rings and Modules

• 4.I.2G

  Groups, Rings and Modules | Part IB, 2008

  Let $n \geq 2$ be an integer. Show that the polynomial $\left(X^{n}-1\right) /(X-1)$ is irreducible over $\mathbb{Z}$ if and only if $n$ is prime.

  [You may use Eisenstein's criterion without proof.]

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• 4.II.11G

  Groups, Rings and Modules | Part IB, 2008

  Let $R$ be a ring and $M$ an $R$-module. What does it mean to say that $M$ is a free $R$-module? Show that $M$ is free if there exists a submodule $N \subseteq M$ such that both $N$ and $M / N$ are free.

  Let $M$ and $M^{\prime}$ be $R$-modules, and $N \subseteq M, N^{\prime} \subseteq M^{\prime}$ submodules. Suppose that $N \cong N^{\prime}$ and $M / N \cong M^{\prime} / N^{\prime}$. Determine (by proof or counterexample) which of the following statements holds:

  (1) If $N$ is free then $M \cong M^{\prime}$.

  (2) If $M / N$ is free then $M \cong M^{\prime}$.

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Linear Algebra

• 4.I.1E

  Linear Algebra | Part IB, 2008

  Describe (without proof) what it means to put an $n \times n$ matrix of complex numbers into Jordan normal form. Explain (without proof) the sense in which the Jordan normal form is unique.

  Put the following matrix in Jordan normal form:

  $$\left(\begin{array}{ccc} -7 & 3 & -5 \\ 7 & -1 & 5 \\ 17 & -6 & 12 \end{array}\right)$$

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• 4.II.10E

  Linear Algebra | Part IB, 2008

  What is meant by a Hermitian matrix? Show that if $A$ is Hermitian then all its eigenvalues are real and that there is an orthonormal basis for $\mathbb{C}^{n}$ consisting of eigenvectors of $A$.

  A Hermitian matrix is said to be positive definite if $\langle A x, x\rangle>0$ for all $x \neq 0$. We write $A>0$ in this case. Show that $A$ is positive definite if, and only if, all of its eigenvalues are positive. Show that if $A>0$ then $A$ has a unique positive definite square root $\sqrt{A}$.

  Let $A, B$ be two positive definite Hermitian matrices with $A-B>0$. Writing $C=\sqrt{A}$ and $X=\sqrt{A}-\sqrt{B}$, show that $C X+X C>0$. By considering eigenvalues of $X$, or otherwise, show that $X>0$.

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Markov Chains

• 4.I.9H

  Markov Chains | Part IB, 2008

  A Markov chain on the state-space $I=\{1,2,3,4,5,6,7\}$ has transition matrix

  $$P=\left(\begin{array}{ccccccc} 0 & 1 / 2 & 1 / 4 & 0 & 1 / 4 & 0 & 0 \\ 1 / 3 & 0 & 1 / 2 & 0 & 0 & 1 / 6 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 / 2 & 0 & 1 / 2 \end{array}\right)$$

  Classify the chain into its communicating classes, deciding for each what the period is, and whether the class is recurrent.

  For each $i, j \in I$ say whether the $\operatorname{limit}^{-1} \lim _{n \rightarrow \infty} p_{i j}^{(n)}$ exists, and evaluate the limit when it does exist.

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Methods

• 4.I.5A

  Methods | Part IB, 2008

  Find the half-range Fourier cosine series for $f(x)=x^{2}, 0<x<1$. Hence show that

  $$\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6} .$$

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• 4.II.16A

  Methods | Part IB, 2008

  Assume $F(x)$ satisfies

  $$\int_{-\infty}^{\infty}|F(x)| d x<\infty$$

  and that the series

  $$g(\tau)=\sum_{n=-\infty}^{\infty} F(2 n \pi+\tau)$$

  converges uniformly in $[0 \leqslant \tau \leqslant 2 \pi]$.

  If $\tilde{F}$ is the Fourier transform of $F$, prove that

  $$g(\tau)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} \tilde{F}(n) e^{i n \tau}$$

  [Hint: prove that $g$ is periodic and express its Fourier expansion coefficients in terms of $\tilde{F}]$.

  In the case that $F(x)=e^{-|x|}$, evaluate the sum

  $$\sum_{n=-\infty}^{\infty} \frac{1}{1+n^{2}} .$$

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Metric and Topological Spaces

• 4.II.14F

  Metric and Topological Spaces | Part IB, 2008

  Explain what is meant by a base for a topology. Illustrate your definition by describing bases for the topology induced by a metric on a set, and for the product topology on the cartesian product of two topological spaces.

  A topological space $(X, \mathcal{T})$ is said to be separable if there is a countable subset $C \subseteq X$ which is dense, i.e. such that $C \cap U \neq \emptyset$ for every nonempty $U \in \mathcal{T}$. Show that a product of two separable spaces is separable. Show also that a metric space is separable if and only if its topology has a countable base, and deduce that every subspace of a separable metric space is separable.

  Now let $X=\mathbb{R}$ with the topology $\mathcal{T}$ having as a base the set of all half-open intervals

  $$[a, b)=\{x \in \mathbb{R}: a \leqslant x<b\}$$

  with $a<b$. Show that $X$ is separable, but that the subspace $Y=\{(x,-x): x \in \mathbb{R}\}$ of $X \times X$ is not separable.

  [You may assume standard results on countability.]

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Numerical Analysis

• 4.I.8D

  Numerical Analysis | Part IB, 2008

  Show that the Chebyshev polynomials, $T_{n}(x)=\cos \left(n \cos ^{-1} x\right), n=0,1,2, \ldots$ obey the orthogonality relation

  $$\int_{-1}^{1} \frac{T_{n}(x) T_{m}(x)}{\sqrt{1-x^{2}}} d x=\frac{\pi}{2} \delta_{n, m}\left(1+\delta_{n, 0}\right)$$

  State briefly how an optimal choice of the parameters $a_{k}, x_{k}, k=1,2 \ldots n$ is made in the Gaussian quadrature formula

  $$\int_{-1}^{1} \frac{f(x)}{\sqrt{1-x^{2}}} d x \sim \sum_{k=1}^{n} a_{k} f\left(x_{k}\right)$$

  Find these parameters for the case $n=3$.

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Optimization

• 4.II.20H

  Optimization | Part IB, 2008

  (i) Suppose that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$, and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ are continuously differentiable. Suppose that the problem

  $\max f(x)$ subject to $g(x)=b$

  is solved by a unique $\bar{x}=\bar{x}(b)$ for each $b \in \mathbb{R}^{m}$, and that there exists a unique $\lambda(b) \in \mathbb{R}^{m}$ such that

  $$\varphi(b) \equiv f(\bar{x}(b))=\sup _{x}\left\{f(x)+\lambda(b)^{T}(b-g(x))\right\}$$

  Assuming that $\bar{x}$ and $\lambda$ are continuously differentiable, prove that

  $$\frac{\partial \varphi}{\partial b_{i}}(b)=\lambda_{i}(b)$$

  (ii) The output of a firm is a function of the capital $K$ deployed, and the amount $L$ of labour employed, given by

  $$f(K, L)=K^{\alpha} L^{\beta}$$

  where $\alpha, \beta \in(0,1)$. The firm's manager has to optimize the output subject to the budget constraint

  $$K+w L=b,$$

  where $w>0$ is the wage rate and $b>0$ is the available budget. By casting the problem in Lagrangian form, find the optimal solution and verify the relation $(*)$.

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Quantum Mechanics

• 4.I.6A

  Quantum Mechanics | Part IB, 2008

  What is meant by a stationary state? What form does the wavefunction take in such a state? A particle has wavefunction $\psi(x, t)$, such that

  $$\psi(x, 0)=\sqrt{\frac{1}{2}}\left(\chi_{1}(x)+\chi_{2}(x)\right)$$

  where $\chi_{1}$ and $\chi_{2}$ are normalised eigenstates of the Hamiltonian with energies $E_{1}$ and $E_{2}$. Write down $\psi(x, t)$ at time $t$. Show that the expectation value of $A$ at time $t$ is

  $$\langle A\rangle_{\psi}=\frac{1}{2} \int_{-\infty}^{\infty}\left(\chi_{1}^{*} \hat{A} \chi_{1}+\chi_{2}^{*} \hat{A} \chi_{2}\right) d x+\operatorname{Re}\left(e^{i\left(E_{1}-E_{2}\right) t / \hbar} \int_{-\infty}^{\infty} \chi_{1}^{*} \hat{A} \chi_{2} d x\right)$$

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Special Relativity

• 4.II.17C

  Special Relativity | Part IB, 2008

  Write down the formulae for the one-dimensional Lorentz transformation $(x, t) \rightarrow$ $\left(x^{\prime}, t^{\prime}\right)$ for frames moving with relative velocity $v$ along the $x$-direction. Derive the relativistic formula for the addition of velocities $v$ and $u$.

  A train, of proper length $L$, travels past a station at velocity $v>0$. The origin of the inertial frame $S$, with coordinates $(x, t)$, in which the train is stationary, is located at the mid-point of the train. The origin of the inertial frame $S^{\prime}$, with coordinates $\left(x^{\prime}, t^{\prime}\right)$, in which the station is stationary, is located at the mid-point of the platform. Coordinates are chosen such that when the origins coincide then $t=t^{\prime}=0$.

  Observers A and B, stationary in $S$, are located, respectively, at the front and rear of the train. Observer C, stationary in $S^{\prime}$, is located at the origin of $S^{\prime}$. At $t^{\prime}=0$, C sends two signals, which both travel at speed $u$, where $v<u \leq c$, one directed towards $\mathrm{A}$ and the other towards $\mathrm{B}$, who receive the signals at respective times $t_{A}$ and $t_{B}$. $\mathrm{C}$ observes these events to occur, respectively, at times $t_{A}^{\prime}$ and $t_{B}^{\prime}$. At $t^{\prime}=0, \mathrm{C}$ also observes that the two ends of the platform coincide with the positions of $A$ and $B$.

  (a) Draw two space-time diagrams, one for $S$ and the other for $S^{\prime}$, showing the trajectories of the observers and the events that take place.

  (b) What is the length of the platform in terms of $L$ ? Briefly illustrate your answer by reference to the space-time diagrams.

  (c) Calculate the time differences $t_{B}-t_{A}$ and $t_{B}^{\prime}-t_{A}^{\prime}$.

  (d) Setting $u=c$, use this example to discuss briefly the fact that two events observed to be simultaneous in one frame need not be observed to be simultaneous in another.

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Statistics

• 4.II.19H

  Statistics | Part IB, 2008

  (i) Consider the linear model

  $$Y_{i}=\alpha+\beta x_{i}+\varepsilon_{i}$$

  where observations $Y_{i}, i=1, \ldots, n$, depend on known explanatory variables $x_{i}$, $i=1, \ldots, n$, and independent $N\left(0, \sigma^{2}\right)$ random variables $\varepsilon_{i}, i=1, \ldots, n$.

  Derive the maximum-likelihood estimators of $\alpha, \beta$ and $\sigma^{2}$.

  Stating clearly any results you require about the distribution of the maximum-likelihood estimators of $\alpha, \beta$ and $\sigma^{2}$, explain how to construct a test of the hypothesis that $\alpha=0$ against an unrestricted alternative.

  (ii) A simple ballistic theory predicts that the range of a gun fired at angle of elevation $\theta$ should be given by the formula

  $$Y=\frac{V^{2}}{g} \sin 2 \theta$$

  where $V$ is the muzzle velocity, and $g$ is the gravitational acceleration. Shells are fired at 9 different elevations, and the ranges observed are as follows:

  $\begin{array}{cccccccccc}\theta \text { (degrees) } & 5 & 15 & 25 & 35 & 45 & 55 & 65 & 75 & 85 \\ \sin 2 \theta & 0.1736 & 0.5 & 0.7660 & 0.9397 & 1 & 0.9397 & 0.7660 & 0.5 & 0.1736 \\ Y(\mathrm{~m}) & 4322 & 11898 & 17485 & 20664 & 21296 & 19491 & 15572 & 10027 & 3458\end{array}$

  The model

  $$Y_{i}=\alpha+\beta \sin 2 \theta_{i}+\varepsilon_{i}$$

  is proposed. Using the theory of part (i) above, find expressions for the maximumlikelihood estimators of $\alpha$ and $\beta$.

  The $t$-test of the null hypothesis that $\alpha=0$ against an unrestricted alternative does not reject the null hypothesis. Would you be willing to accept the model $(*)$ ? Briefly explain your answer.

  [You may need the following summary statistics of the data. If $x_{i}=\sin 2 \theta_{i}$, then $\bar{x} \equiv n^{-1} \sum x_{i}=0.63986, \bar{Y}=13802, S_{x x} \equiv \sum\left(x_{i}-\bar{x}\right)^{2}=0.81517, S_{x y}=\sum Y_{i}\left(x_{i}-\bar{x}\right)=$ 17186. ]

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