Part IB, 2012, Paper 4

# Part IB, 2012, Paper 4

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Paper 4, Section I, E

commentLet $f: \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}$ be a bilinear function. Show that $f$ is differentiable at any point in $\mathbb{R}^{n} \times \mathbb{R}^{m}$ and find its derivative.

Paper 4, Section II, E

commentState and prove the Bolzano-Weierstrass theorem in $\mathbb{R}^{n}$. [You may assume the Bolzano-Weierstrass theorem in $\mathbb{R}$.]

Let $X \subset \mathbb{R}^{n}$ be a subset and let $f: X \rightarrow X$ be a mapping such that $d(f(x), f(y))=d(x, y)$ for all $x, y \in X$, where $d$ is the Euclidean distance in $\mathbb{R}^{n}$. Prove that if $X$ is closed and bounded, then $f$ is a bijection. Is this result still true if we drop the boundedness assumption on $X$ ? Justify your answer.

Paper 4, Section I, $4 \mathrm{E}$

commentLet $h: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function with $h(i) \neq h(-i)$. Does there exist a holomorphic function $f$ defined in $|z|<1$ for which $f^{\prime}(z)=\frac{h(z)}{1+z^{2}}$ ? Does there exist a holomorphic function $f$ defined in $|z|>1$ for which $f^{\prime}(z)=\frac{h(z)}{1+z^{2}}$ ? Justify your answers.

Paper 4, Section II, A

commentState the convolution theorem for Fourier transforms.

The function $\phi(x, y)$ satisfies

$\nabla^{2} \phi=0$

on the half-plane $y \geqslant 0$, subject to the boundary conditions

$\begin{gathered} \phi \rightarrow 0 \text { as } y \rightarrow \infty \text { for all } x \\ \phi(x, 0)= \begin{cases}1, & |x| \leqslant 1 \\ 0, & |x|>1\end{cases} \end{gathered}$

Using Fourier transforms, show that

$\phi(x, y)=\frac{y}{\pi} \int_{-1}^{1} \frac{1}{y^{2}+(x-t)^{2}} \mathrm{~d} t$

and hence that

$\phi(x, y)=\frac{1}{\pi}\left[\tan ^{-1}\left(\frac{1-x}{y}\right)+\tan ^{-1}\left(\frac{1+x}{y}\right)\right]$

Paper 4, Section I, B

commentDefine the notions of magnetic flux, electromotive force and resistance, in the context of a single closed loop of wire. Use the Maxwell equation

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

to derive Faraday's law of induction for the loop, assuming the loop is at rest.

Suppose now that the magnetic field is $\mathbf{B}=(0,0, B \tanh t)$ where $B$ is a constant, and that the loop of wire, with resistance $R$, is a circle of radius a lying in the $(x, y)$ plane. Calculate the current in the wire as a function of time.

Explain briefly why, even in a time-independent magnetic field, an electromotive force may be produced in a loop of wire that moves through the field, and state the law of induction in this situation.

Paper 4, Section II, A

commentThe equations governing the flow of a shallow layer of inviscid liquid of uniform depth $H$ rotating with angular velocity $\frac{1}{2} f$ about the vertical $z$-axis are

$\begin{aligned} \frac{\partial u}{\partial t}-f v &=-g \frac{\partial \eta}{\partial x} \\ \frac{\partial v}{\partial t}+f u &=-g \frac{\partial \eta}{\partial y} \\ \frac{\partial \eta}{t}+H\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) &=0 \end{aligned}$

where $u, v$ are the $x$ - and $y$-components of velocity, respectively, and $\eta$ is the elevation of the free surface. Show that these equations imply that

$\frac{\partial q}{\partial t}=0, \quad \text { where } \quad q=\omega-\frac{f \eta}{H} \text { and } \omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$

Consider an initial state where there is flow in the $y$-direction given by

$\begin{aligned} u &=\eta=0, \quad-\infty<x<\infty \\ v &= \begin{cases}\frac{g}{2 f} e^{2 x}, & x<0 \\ -\frac{g}{2 f} e^{-2 x}, & x>0\end{cases} \end{aligned}$

Find the initial potential vorticity.

Show that when this initial state adjusts, there is a final steady state independent of $y$ in which $\eta$ satisfies

$\frac{\partial^{2} \eta}{\partial x^{2}}-\frac{\eta}{a^{2}}= \begin{cases}e^{2 x}, & x<0 \\ e^{-2 x}, & x>0\end{cases}$

where $a^{2}=g H / f^{2}$.

In the case $a=1$, find the final free surface elevation that is finite at large $|x|$ and which is continuous and has continuous slope at $x=0$, and show that it is negative for all $x$.

Paper 4, Section II, G

commentLet $\Sigma \subset \mathbb{R}^{3}$ be a smooth closed surface. Define the principal curvatures $\kappa_{\max }$ and $\kappa_{\min }$ at a point $p \in \Sigma$. Prove that the Gauss curvature at $p$ is the product of the two principal curvatures.

A point $p \in \Sigma$ is called a parabolic point if at least one of the two principal curvatures vanishes. Suppose $\Pi \subset \mathbb{R}^{3}$ is a plane and $\Sigma$ is tangent to $\Pi$ along a smooth closed curve $C=\Pi \cap \Sigma \subset \Sigma$. Show that $C$ is composed of parabolic points.

Can both principal curvatures vanish at a point of $C$ ? Briefly justify your answer.

Paper 4, Section I, $2 G$

commentAn idempotent element of a ring $R$ is an element $e$ satisfying $e^{2}=e$. A nilpotent element is an element e satisfying $e^{N}=0$ for some $N \geqslant 0$.

Let $r \in R$ be non-zero. In the ring $R[X]$, can the polynomial $1+r X$ be (i) an idempotent, (ii) a nilpotent? Can $1+r X$ satisfy the equation $(1+r X)^{3}=(1+r X)$ ? Justify your answers.

Paper 4, Section II, G

commentLet $R$ be a commutative ring with unit 1. Prove that an $R$-module is finitely generated if and only if it is a quotient of a free module $R^{n}$, for some $n>0$.

Let $M$ be a finitely generated $R$-module. Suppose now $I$ is an ideal of $R$, and $\phi$ is an $R$-homomorphism from $M$ to $M$ with the property that

$\phi(M) \subset I \cdot M=\left\{m \in M \mid m=r m^{\prime} \quad \text { with } \quad r \in I, m^{\prime} \in M\right\}$

Prove that $\phi$ satisfies an equation

$\phi^{n}+a_{n-1} \phi^{n-1}+\cdots+a_{1} \phi+a_{0}=0$

where each $a_{j} \in I$. [You may assume that if $T$ is a matrix over $R$, then $\operatorname{adj}(T) T=$ $\operatorname{det} T$ (id), with id the identity matrix.]

Deduce that if $M$ satisfies $I \cdot M=M$, then there is some $a \in R$ satisfying

$a-1 \in I \quad \text { and } \quad a M=0 .$

Give an example of a finitely generated $\mathbb{Z}$-module $M$ and a proper ideal $I$ of $\mathbb{Z}$ satisfying the hypothesis $I \cdot M=M$, and for your example, give an explicit such element $a$.

Paper 4, Section I, F

commentLet $V$ be a complex vector space with basis $\left\{e_{1}, \ldots, e_{n}\right\}$. Define $T: V \rightarrow V$ by $T\left(e_{i}\right)=e_{i}-e_{i+1}$ for $i<n$ and $T\left(e_{n}\right)=e_{n}-e_{1}$. Show that $T$ is diagonalizable and find its eigenvalues. [You may use any theorems you wish, as long as you state them clearly.]

Paper 4, Section II, F

commentLet $V$ be a finite-dimensional real vector space of dimension $n$. A bilinear form $B: V \times V \rightarrow \mathbb{R}$ is nondegenerate if for all $\mathbf{v} \neq 0$ in $V$, there is some $\mathbf{w} \in V$ with $B(\mathbf{v}, \mathbf{w}) \neq 0$. For $\mathbf{v} \in V$, define $\langle\mathbf{v}\rangle^{\perp}=\{\mathbf{w} \in V \mid B(\mathbf{v}, \mathbf{w})=0\}$. Assuming $B$ is nondegenerate, show that $V=\langle\mathbf{v}\rangle \oplus\langle\mathbf{v}\rangle^{\perp}$ whenever $B(\mathbf{v}, \mathbf{v}) \neq 0$.

Suppose that $B$ is a nondegenerate, symmetric bilinear form on $V$. Prove that there is a basis $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\}$ of $V$ with $B\left(\mathbf{v}_{i}, \mathbf{v}_{j}\right)=0$ for $i \neq j$. [If you use the fact that symmetric matrices are diagonalizable, you must prove it.]

Define the signature of a quadratic form. Explain how to determine the signature of the quadratic form associated to $B$ from the basis you constructed above.

A linear subspace $V^{\prime} \subset V$ is said to be isotropic if $B(\mathbf{v}, \mathbf{w})=0$ for all $\mathbf{v}, \mathbf{w} \in V^{\prime}$. Show that if $B$ is nondegenerate, the maximal dimension of an isotropic subspace of $V$ is $(n-|\sigma|) / 2$, where $\sigma$ is the signature of the quadratic form associated to $B$.

Paper 4, Section I, H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be an irreducible Markov chain with $p_{i j}^{(n)}=P\left(X_{n}=j \mid X_{0}=i\right)$. Define the meaning of the statements:

(i) state $i$ is transient,

(ii) state $i$ is aperiodic.

Give a criterion for transience that can be expressed in terms of the probabilities $\left(p_{i i}^{(n)}, n=0,1, \ldots\right)$.

Prove that if a state $i$ is transient then all states are transient.

Prove that if a state $i$ is aperiodic then all states are aperiodic.

Suppose that $p_{i i}^{(n)}=0$ unless $n$ is divisible by 3 . Given any other state $j$, prove that $p_{j j}^{(n)}=0$ unless $n$ is divisible by 3 .

Paper 4, Section I, D

commentShow that the general solution of the wave equation

$\frac{1}{c^{2}} \frac{\partial^{2} y}{\partial t^{2}}-\frac{\partial^{2} y}{\partial x^{2}}=0$

can be written in the form

$y(x, t)=f(x-c t)+g(x+c t)$

Hence derive the solution $y(x, t)$ subject to the initial conditions

$y(x, 0)=0, \quad \frac{\partial y}{\partial t}(x, 0)=\psi(x)$

Paper 4, Section II, D

commentLet $D \subset \mathbb{R}^{2}$ be a two-dimensional domain with boundary $S=\partial D$, and let

$G_{2}=G_{2}\left(\mathbf{r}, \mathbf{r}_{0}\right)=\frac{1}{2 \pi} \log \left|\mathbf{r}-\mathbf{r}_{0}\right|$

where $\mathbf{r}_{0}$ is a point in the interior of $D$. From Green's second identity,

$\int_{S}\left(\phi \frac{\partial \psi}{\partial n}-\psi \frac{\partial \phi}{\partial n}\right) d \ell=\int_{D}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d a$

derive Green's third identity

$u\left(\mathbf{r}_{0}\right)=\int_{D} G_{2} \nabla^{2} u d a+\int_{S}\left(u \frac{\partial G_{2}}{\partial n}-G_{2} \frac{\partial u}{\partial n}\right) d \ell$

[Here $\frac{\partial}{\partial n}$ denotes the normal derivative on $S$.]

Consider the Dirichlet problem on the unit $\operatorname{disc} D_{1}=\left\{\mathbf{r} \in \mathbb{R}^{2}:|\mathbf{r}| \leqslant 1\right\}$ :

$\begin{aligned} \nabla^{2} u=0, & \mathbf{r} \in D_{1} \\ u(\mathbf{r})=f(\mathbf{r}), & \mathbf{r} \in S_{1}=\partial D_{1} \end{aligned}$

Show that, with an appropriate function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$, the solution can be obtained by the formula

$u\left(\mathbf{r}_{0}\right)=\int_{S_{1}} f(\mathbf{r}) \frac{\partial}{\partial n} G\left(\mathbf{r}, \mathbf{r}_{0}\right) d \ell$

State the boundary conditions on $G$ and explain how $G$ is related to $G_{2}$.

For $\mathbf{r}, \mathbf{r}_{0} \in \mathbb{R}^{2}$, prove the identity

$\left|\frac{\mathbf{r}}{|\mathbf{r}|}-\mathbf{r}_{0}\right| \mathbf{r}||=\left|\frac{\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|}-\mathbf{r}\right| \mathbf{r}_{0}|| \text {, }$

and deduce that if the point $\mathbf{r}$ lies on the unit circle, then

$\left|\mathbf{r}-\mathbf{r}_{0}\right|=\left|\mathbf{r}_{0}\right|\left|\mathbf{r}-\mathbf{r}_{0}^{*}\right|, \text { where } \mathbf{r}_{0}^{*}=\frac{\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|^{2}}$

Hence, using the method of images, or otherwise, find an expression for the function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$. [An expression for $\frac{\partial}{\partial n} G$ is not required.]

Paper 4, Section II, F

commentSuppose $A_{1}$ and $A_{2}$ are topological spaces. Define the product topology on $A_{1} \times A_{2}$. Let $\pi_{i}: A_{1} \times A_{2} \rightarrow A_{i}$ be the projection. Show that a map $F: X \rightarrow A_{1} \times A_{2}$ is continuous if and only if $\pi_{1} \circ F$ and $\pi_{2} \circ F$ are continuous.

Prove that if $A_{1}$ and $A_{2}$ are connected, then $A_{1} \times A_{2}$ is connected.

Let $X$ be the topological space whose underlying set is $\mathbb{R}$, and whose open sets are of the form $(a, \infty)$ for $a \in \mathbb{R}$, along with the empty set and the whole space. Describe the open sets in $X \times X$. Are the maps $f, g: X \times X \rightarrow X$ defined by $f(x, y)=x+y$ and $g(x, y)=x y$ continuous? Justify your answers.

Paper 4, Section I, D

commentState the Dahlquist equivalence theorem regarding convergence of a multistep method.

The multistep method, with a real parameter $a$,

$y_{n+3}+(2 a-3)\left(y_{n+2}-y_{n+1}\right)-y_{n}=h a\left(f_{n+2}-f_{n+1}\right)$

is of order 2 for any $a$, and also of order 3 for $a=6$. Determine all values of $a$ for which the method is convergent, and find the order of convergence.

Paper 4, Section II, 20H

commentDescribe the Ford-Fulkerson algorithm.

State conditions under which the algorithm is guaranteed to terminate in a finite number of steps. Explain why it does so, and show that it finds a maximum flow. [You may assume that the value of a flow never exceeds the value of any cut.]

In a football league of $n$ teams the season is partly finished. Team $i$ has already won $w_{i}$ matches. Teams $i$ and $j$ are to meet in $m_{i j}$ further matches. Thus the total number of remaining matches is $M=\sum_{i<j} m_{i j}$. Assume there will be no drawn matches. We wish to determine whether it is possible for the outcomes of the remaining matches to occur in such a way that at the end of the season the numbers of wins by the teams are $\left(x_{1}, \ldots, x_{n}\right)$.

Invent a network flow problem in which the maximum flow from source to sink equals $M$ if and only if $\left(x_{1}, \ldots, x_{n}\right)$ is a feasible vector of final wins.

Illustrate your idea by answering the question of whether or not $x=(7,5,6,6)$ is a possible profile of total end-of-season wins when $n=4, w=(1,2,3,4)$, and $M=14$ with

$\left(m_{i j}\right)=\left(\begin{array}{cccc} - & 2 & 2 & 2 \\ 2 & - & 1 & 1 \\ 2 & 1 & - & 6 \\ 2 & 1 & 6 & - \end{array}\right)$

Paper 4, Section I, $\mathbf{6 C}$

commentIn terms of quantum states, what is meant by energy degeneracy?

A particle of mass $m$ is confined within the box $0<x<a, 0<y<a$ and $0<z<c$. The potential vanishes inside the box and is infinite outside. Find the allowed energies by considering a stationary state wavefunction of the form

$\chi(x, y, z)=X(x) Y(y) Z(z)$

Write down the normalised ground state wavefunction. Assuming that $c<a<\sqrt{2} c$, give the energies of the first three excited states.

Paper 4, Section II, H

commentFrom each of 3 populations, $n$ data points are sampled and these are believed to obey

$y_{i j}=\alpha_{i}+\beta_{i}\left(x_{i j}-\bar{x}_{i}\right)+\epsilon_{i j}, \quad j \in\{1, \ldots, n\}, i \in\{1,2,3\},$

where $\bar{x}_{i}=(1 / n) \sum_{j} x_{i j}$, the $\epsilon_{i j}$ are independent and identically distributed as $N\left(0, \sigma^{2}\right)$, and $\sigma^{2}$ is unknown. Let $\bar{y}_{i}=(1 / n) \sum_{j} y_{i j}$.

(i) Find expressions for $\hat{\alpha}_{i}$ and $\hat{\beta}_{i}$, the least squares estimates of $\alpha_{i}$ and $\beta_{i}$.

(ii) What are the distributions of $\hat{\alpha}_{i}$ and $\hat{\beta}_{i}$ ?

(iii) Show that the residual sum of squares, $R_{1}$, is given by

$R_{1}=\sum_{i=1}^{3}\left[\sum_{j=1}^{n}\left(y_{i j}-\bar{y}_{i}\right)^{2}-\hat{\beta}_{i}^{2} \sum_{j=1}^{n}\left(x_{i j}-\bar{x}_{i}\right)^{2}\right]$

Calculate $R_{1}$ when $n=9,\left\{\hat{\alpha}_{i}\right\}_{i=1}^{3}=\{1.6,0.6,0.8\},\left\{\hat{\beta}_{i}\right\}_{i=1}^{3}=\{2,1,1\}$,

$\left\{\sum_{j=1}^{9}\left(y_{i j}-\bar{y}_{i}\right)^{2}\right\}_{i=1}^{3}=\{138,82,63\}, \quad\left\{\sum_{j=1}^{9}\left(x_{i j}-\bar{x}_{i}\right)^{2}\right\}_{i=1}^{3}=\{30,60,40\}$

(iv) $H_{0}$ is the hypothesis that $\alpha_{1}=\alpha_{2}=\alpha_{3}$. Find an expression for the maximum likelihood estimator of $\alpha_{1}$ under the assumption that $H_{0}$ is true. Calculate its value for the above data.

(v) Explain (stating without proof any relevant theory) the rationale for a statistic which can be referred to an $F$ distribution to test $H_{0}$ against the alternative that it is not true. What should be the degrees of freedom of this $F$ distribution? What would be the outcome of a size $0.05$ test of $H_{0}$ with the above data?

Paper 4, Section II, B

commentConsider a functional

$I=\int_{a}^{b} F\left(x, y, y^{\prime}\right) d x$

where $F$ is smooth in all its arguments, $y(x)$ is a $C^{1}$ function and $y^{\prime}=\frac{d y}{d x}$. Consider the function $y(x)+h(x)$ where $h(x)$ is a small $C^{1}$ function which vanishes at $a$ and $b$. Obtain formulae for the first and second variations of $I$ about the function $y(x)$. Derive the Euler-Lagrange equation from the first variation, and state its variational interpretation.

Suppose now that

$I=\int_{0}^{1}\left(y^{\prime 2}-1\right)^{2} d x$

where $y(0)=0$ and $y(1)=\beta$. Find the Euler-Lagrange equation and the formula for the second variation of $I$. Show that the function $y(x)=\beta x$ makes $I$ stationary, and that it is a (local) minimizer if $\beta>\frac{1}{\sqrt{3}}$.

Show that when $\beta=0$, the function $y(x)=0$ is not a minimizer of $I$.