Part IB, 2021, Paper 2

# Part IB, 2021, Paper 2

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Paper 2, Section I, $2 F$

commentLet $K:[0,1] \times[0,1] \rightarrow \mathbb{R}$ be a continuous function and let $C([0,1])$ denote the set of continuous real-valued functions on $[0,1]$. Given $f \in C([0,1])$, define the function $T f$ by the expression

$T f(x)=\int_{0}^{1} K(x, y) f(y) d y$

(a) Prove that $T$ is a continuous map $C([0,1]) \rightarrow C([0,1])$ with the uniform metric on $C([0,1])$.

(b) Let $d_{1}$ be the metric on $C([0,1])$ given by

$d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x$

Is $T$ continuous with respect to $d_{1} ?$

Paper 2, Section II, F

commentLet $k_{n}: \mathbb{R} \rightarrow \mathbb{R}$ be a sequence of functions satisfying the following properties:

$k_{n}(x) \geqslant 0$ for all $n$ and $x \in \mathbb{R}$ and there is $R>0$ such that $k_{n}$ vanishes outside $[-R, R]$ for all $n$

each $k_{n}$ is continuous and

$\int_{-\infty}^{\infty} k_{n}(t) d t=1$

- given $\varepsilon>0$ and $\delta>0$, there exists a positive integer $N$ such that if $n \geqslant N$, then

$\int_{-\infty}^{-\delta} k_{n}(t) d t+\int_{\delta}^{\infty} k_{n}(t) d t<\varepsilon$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a bounded continuous function and set

$f_{n}(x):=\int_{-\infty}^{\infty} k_{n}(t) f(x-t) d t$

Show that $f_{n}$ converges uniformly to $f$ on any compact subset of $\mathbb{R}$.

Let $g:[0,1] \rightarrow \mathbb{R}$ be a continuous function with $g(0)=g(1)=0$. Show that there is a sequence of polynomials $p_{n}$ such that $p_{n}$ converges uniformly to $g$ on $[0,1]$. $[$ Hint: consider the functions

$k_{n}(t)= \begin{cases}\left(1-t^{2}\right)^{n} / c_{n} & t \in[-1,1] \\ 0 & \text { otherwise }\end{cases}$

where $c_{n}$ is a suitably chosen constant.]

Paper 2, Section II, B

comment(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function and let $a>0, b>0$ be constants. Show that if

$|f(z)| \leqslant a|z|^{n / 2}+b$

for all $z \in \mathbb{C}$, where $n$ is a positive odd integer, then $f$ must be a polynomial with degree not exceeding $\lfloor n / 2\rfloor$ (closest integer part rounding down).

Does there exist a function $f$, analytic in $\mathbb{C} \backslash\{0\}$, such that $|f(z)| \geqslant 1 / \sqrt{|z|}$ for all nonzero $z ?$ Justify your answer.

(b) State Liouville's Theorem and use it to show the following.

(i) If $u$ is a positive harmonic function on $\mathbb{R}^{2}$, then $u$ is a constant function.

(ii) Let $L=\{z \mid z=a x+b, x \in \mathbb{R}\}$ be a line in $\mathbb{C}$ where $a, b \in \mathbb{C}, a \neq 0$. If $f: \mathbb{C} \rightarrow \mathbb{C}$ is an entire function such that $f(\mathbb{C}) \cap L=\emptyset$, then $f$ is a constant function.

Paper 2, Section I, $4 \mathrm{D}$

commentState Gauss's Law in the context of electrostatics.

A simple coaxial cable consists of an inner conductor in the form of a perfectly conducting, solid cylinder of radius $a$, surrounded by an outer conductor in the form of a perfectly conducting, cylindrical shell of inner radius $b>a$ and outer radius $c>b$. The cylinders are coaxial and the gap between them is filled with a perfectly insulating material. The cable may be assumed to be straight and arbitrarily long.

In a steady state, the inner conductor carries an electric charge $+Q$ per unit length, and the outer conductor carries an electric charge $-Q$ per unit length. The charges are distributed in a cylindrically symmetric way and no current flows through the cable.

Determine the electrostatic potential and the electric field as functions of the cylindrical radius $r$, for $0<r<\infty$. Calculate the capacitance $C$ of the cable per unit length and the electrostatic energy $U$ per unit length, and verify that these are related by

$U=\frac{Q^{2}}{2 C}$

Paper 2, Section II, $16 \mathrm{D}$

comment(a) Show that, for $|\mathbf{x}| \gg|\mathbf{y}|$,

$\frac{1}{|\mathbf{x}-\mathbf{y}|}=\frac{1}{|\mathbf{x}|}\left[1+\frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}|^{2}}+\frac{3(\mathbf{x} \cdot \mathbf{y})^{2}-|\mathbf{x}|^{2}|\mathbf{y}|^{2}}{2|\mathbf{x}|^{4}}+O\left(\frac{|\mathbf{y}|^{3}}{|\mathbf{x}|^{3}}\right)\right]$

(b) A particle with electric charge $q>0$ has position vector $(a, 0,0)$, where $a>0$. An earthed conductor (held at zero potential) occupies the plane $x=0$. Explain why the boundary conditions can be satisfied by introducing a fictitious 'image' particle of appropriate charge and position. Hence determine the electrostatic potential and the electric field in the region $x>0$. Find the leading-order approximation to the potential for $|\mathbf{x}| \gg a$ and compare with that of an electric dipole. Directly calculate the total flux of the electric field through the plane $x=0$ and comment on the result. Find the induced charge distribution on the surface of the conductor, and the total induced surface charge. Sketch the electric field lines in the plane $z=0$.

(c) Now consider instead a particle with charge $q$ at position $(a, b, 0)$, where $a>0$ and $b>0$, with earthed conductors occupying the planes $x=0$ and $y=0$. Find the leading-order approximation to the potential in the region $x, y>0$ for $|\mathbf{x}| \gg a, b$ and state what type of multipole potential this is.

Paper 2, Section I, A

commentConsider an axisymmetric container, initially filled with water to a depth $h_{I}$. A small circular hole of radius $r_{0}$ is opened in the base of the container at $z=0$.

(a) Determine how the radius $r$ of the container should vary with $z<h_{I}$ so that the depth of the water will decrease at a constant rate.

(b) For such a container, determine how the cross-sectional area $A$ of the free surface should decrease with time.

[You may assume that the flow rate through the opening is sufficiently small that Bernoulli's theorem for steady flows can be applied.]

Paper 2, Section II, E

commentDefine $\mathbb{H}$, the upper half plane model for the hyperbolic plane, and show that $\operatorname{PSL}_{2}(\mathbb{R})$ acts on $\mathbb{H}$ by isometries, and that these isometries preserve the orientation of $\mathbb{H}$.

Show that every orientation preserving isometry of $\mathbb{H}$ is in $P S L_{2}(\mathbb{R})$, and hence the full group of isometries of $\mathbb{H}$ is $G=P S L_{2}(\mathbb{R}) \cup P S L_{2}(\mathbb{R}) \tau$, where $\tau z=-\bar{z}$.

Let $\ell$ be a hyperbolic line. Define the reflection $\sigma_{\ell}$ in $\ell$. Now let $\ell, \ell^{\prime}$ be two hyperbolic lines which meet at a point $A \in \mathbb{H}$ at an angle $\theta$. What are the possibilities for the group $G$ generated by $\sigma_{\ell}$ and $\sigma_{\ell^{\prime}}$ ? Carefully justify your answer.

Paper 2, Section I, $1 G$

commentLet $M$ be a module over a Principal Ideal Domain $R$ and let $N$ be a submodule of $M$. Show that $M$ is finitely generated if and only if $N$ and $M / N$ are finitely generated.

Paper 2, Section II, G

commentLet $M$ be a module over a ring $R$ and let $S \subset M$. Define what it means that $S$ freely generates $M$. Show that this happens if and only if for every $R$-module $N$, every function $f: S \rightarrow N$ extends uniquely to a homomorphism $\phi: M \rightarrow N$.

Let $M$ be a free module over a (non-trivial) ring $R$ that is generated (not necessarily freely) by a subset $T \subset M$ of size $m$. Show that if $S$ is a basis of $M$, then $S$ is finite with $|S| \leqslant m$. Hence, or otherwise, deduce that any two bases of $M$ have the same number of elements. Denoting this number $\operatorname{rk} M$ and by quoting any result you need, show that if $R$ is a Euclidean Domain and $N$ is a submodule of $M$, then $N$ is free with $\operatorname{rk} N \leqslant \operatorname{rk} M$.

State the Primary Decomposition Theorem for a finitely generated module $M$ over a Euclidean Domain $R$. Deduce that any finite subgroup of the multiplicative group of a field is cyclic.

Paper 2, Section II, E

comment(a) Compute the characteristic polynomial and minimal polynomial of

$A=\left(\begin{array}{ccc} -2 & -6 & -9 \\ 3 & 7 & 9 \\ -1 & -2 & -2 \end{array}\right)$

Write down the Jordan normal form for $A$.

(b) Let $V$ be a finite-dimensional vector space over $\mathbb{C}, f: V \rightarrow V$ be a linear map, and for $\alpha \in \mathbb{C}, n \geqslant 1$, write

$W_{\alpha, n}:=\left\{v \in V \mid(f-\alpha I)^{n} v=0\right\}$

(i) Given $v \in W_{\alpha, n}, v \neq 0$, construct a non-zero eigenvector for $f$ in terms of $v$.

(ii) Show that if $w_{1}, \ldots, w_{d}$ are non-zero eigenvectors for $f$ with eigenvalues $\alpha_{1}, \ldots, \alpha_{d}$, and $\alpha_{i} \neq \alpha_{j}$ for all $i \neq j$, then $w_{1}, \ldots, w_{d}$ are linearly independent.

(iii) Show that if $v_{1} \in W_{\alpha_{1}, n}, \ldots, v_{d} \in W_{\alpha_{d}, n}$ are all non-zero, and $\alpha_{i} \neq \alpha_{j}$ for all $i \neq j$, then $v_{1}, \ldots, v_{d}$ are linearly independent.

Paper 2, Section II, 18H

commentLet $P$ be a transition matrix on state space $I$. What does it mean for a distribution $\pi$ to be an invariant distribution? What does it mean for $\pi$ and $P$ to be in detailed balance? Show that if $\pi$ and $P$ are in detailed balance, then $\pi$ is an invariant distribution.

(a) Assuming that an invariant distribution exists, state the relationship between this and

(i) the expected return time to a state $i$;

(ii) the expected time spent in a state $i$ between visits to a state $k$.

(b) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain with transition matrix $P=\left(p_{i j}\right)_{i, j \in I}$ where $I=\{0,1,2, \ldots\}$. The transition probabilities are given for $i \geqslant 1$ by

$p_{i j}= \begin{cases}q^{-(i+2)} & \text { if } j=i+1, \\ q^{-i} & \text { if } j=i-1 \\ 1-q^{-(i+2)}-q^{-i} & \text { if } j=i\end{cases}$

where $q \geqslant 2$. For $p \in(0,1]$ let $p_{01}=p=1-p_{00}$. Compute the following, justifying your answers:

(i) The expected time spent in states $\{2,4,6, \ldots\}$ between visits to state 1 ;

(ii) The expected time taken to return to state 1 , starting from 1 ;

(iii) The expected time taken to hit state 0 starting from $1 .$

Paper 2, Section I, C

commentConsider the differential operator

$\mathcal{L} y=\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}$

acting on real functions $y(x)$ with $0 \leqslant x \leqslant 1$.

(i) Recast the eigenvalue equation $\mathcal{L} y=-\lambda y$ in Sturm-Liouville form $\tilde{\mathcal{L}} y=-\lambda w y$, identifying $\tilde{\mathcal{L}}$ and $w$.

(ii) If boundary conditions $y(0)=y(1)=0$ are imposed, show that the eigenvalues form an infinite discrete set $\lambda_{1}<\lambda_{2}<\ldots$ and find the corresponding eigenfunctions $y_{n}(x)$ for $n=1,2, \ldots$. If $f(x)=x-x^{2}$ on $0 \leqslant x \leqslant 1$ is expanded in terms of your eigenfunctions i.e. $f(x)=\sum_{n=1}^{\infty} A_{n} y_{n}(x)$, give an expression for $A_{n}$. The expression can be given in terms of integrals that you need not evaluate.

Paper 2, Section II, A

commentThe Fourier transform $\tilde{f}(k)$ of a function $f(x)$ and its inverse are given by

$\tilde{f}(k)=\int_{-\infty}^{\infty} f(x) e^{-i k x} d x, \quad f(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(k) e^{i k x} d k$

(a) Calculate the Fourier transform of the function $f(x)$ defined by:

$f(x)= \begin{cases}1 & \text { for } 0<x<1 \\ -1 & \text { for }-1<x<0 \\ 0 & \text { otherwise }\end{cases}$

(b) Show that the inverse Fourier transform of $\tilde{g}(k)=e^{-\lambda|k|}$, for $\lambda$ a positive real constant, is given by

$g(x)=\frac{\lambda}{\pi\left(x^{2}+\lambda^{2}\right)}$

(c) Consider the problem in the quarter plane $0 \leqslant x, 0 \leqslant y$ :

$\begin{aligned} \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}} &=0 ; \\ u(x, 0) &= \begin{cases}1 & \text { for } 0<x<1, \\ 0 & \text { otherwise; }\end{cases} \\ u(0, y)=\lim _{x \rightarrow \infty} u(x, y)=\lim _{y \rightarrow \infty} u(x, y) &=0 . \end{aligned}$

Use the answers from parts (a) and (b) to show that

$u(x, y)=\frac{4 x y}{\pi} \int_{0}^{1} \frac{v d v}{\left[(x-v)^{2}+y^{2}\right]\left[(x+v)^{2}+y^{2}\right]}$

(d) Hence solve the problem in the quarter plane $0 \leqslant x, 0 \leqslant y$ :

$\begin{aligned} \frac{\partial^{2} w}{\partial x^{2}}+\frac{\partial^{2} w}{\partial y^{2}} &=0 ; \\ w(x, 0) &= \begin{cases}1 & \text { for } 0<x<1 \\ 0 & \text { otherwise }\end{cases} \\ w(0, y) &= \begin{cases}1 & \text { for } 0<y<1 \\ 0 & \text { otherwise }\end{cases} \\ \lim _{x \rightarrow \infty} w(x, y)=\lim _{y \rightarrow \infty} w(x, y) &=0 \end{aligned}$

[You may quote without proof any property of Fourier transforms.]

Paper 2, Section II, 17B

comment(a) Define Householder reflections and show that a real Householder reflection is symmetric and orthogonal. Moreover, show that if $H, A \in \mathbb{R}^{n \times n}$, where $H$ is a Householder reflection and $A$ is a full matrix, then the computational cost (number of arithmetic operations) of computing $H A H^{-1}$ can be $\mathcal{O}\left(n^{2}\right)$ operations, as opposed to $\mathcal{O}\left(n^{3}\right)$ for standard matrix products.

(b) Show that for any $A \in \mathbb{R}^{n \times n}$ there exists an orthogonal matrix $Q \in \mathbb{R}^{n \times n}$ such that

$Q A Q^{T}=T=\left[\begin{array}{ccccc} t_{1,1} & t_{1,2} & t_{1,3} & \cdots & t_{1, n} \\ t_{2,1} & t_{2,2} & t_{2,3} & \cdots & t_{2, n} \\ 0 & t_{3,2} & t_{3,3} & \cdots & t_{3, n} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & t_{n, n-1} & t_{n, n} \end{array}\right]$

In particular, $T$ has zero entries below the first subdiagonal. Show that one can compute such a $T$ and $Q$ (they may not be unique) using $\mathcal{O}\left(n^{3}\right)$ arithmetic operations.

[Hint: Multiply A from the left and right with Householder reflections.]

Paper 2, Section I, H

commentFind the solution to the following Optimization problem using the simplex algorithm:

Write down the dual problem and give its solution.

$\begin{aligned} & \text { maximise } 3 x_{1}+6 x_{2}+4 x_{3} \\ & \text { subject to } 2 x_{1}+3 x_{2}+x_{3} \leqslant 7 \text {, } \\ & 4 x_{1}+2 x_{2}+2 x_{3} \leqslant 5, \\ & x_{1}+x_{2}+2 x_{3} \leqslant 2, \quad x_{1}, x_{2}, x_{3} \geqslant 0 . \end{aligned}$

Paper 2, Section II, C

comment(a) Write down the expressions for the probability density $\rho$ and associated current density $j$ of a quantum particle in one dimension with wavefunction $\psi(x, t)$. Show that if $\psi$ is a stationary state then the function $j$ is constant.

For the non-normalisable free particle wavefunction $\psi(x, t)=A e^{i k x-i E t / \hbar}$ (where $E$ and $k$ are real constants and $A$ is a complex constant) compute the functions $\rho$ and $j$, and briefly give a physical interpretation of the functions $\psi, \rho$ and $j$ in this case.

(b) A quantum particle of mass $m$ and energy $E>0$ moving in one dimension is incident from the left in the potential $V(x)$ given by

$V(x)=\left\{\begin{array}{cl} -V_{0} & 0 \leqslant x \leqslant a \\ 0 & x<0 \text { or } x>a \end{array}\right.$

where $a$ and $V_{0}$ are positive constants. Write down the form of the wavefunction in the regions $x<0,0 \leqslant x \leqslant a$ and $x>a$.

Suppose now that $V_{0}=3 E$. Show that the probability $T$ of transmission of the particle into the region $x>a$ is given by

$T=\frac{16}{16+9 \sin ^{2}\left(\frac{a \sqrt{8 m E}}{\hbar}\right)}$

Paper 2, Section I, $\mathbf{6 H}$

commentThe efficacy of a new drug was tested as follows. Fifty patients were given the drug, and another fifty patients were given a placebo. A week later, the numbers of patients whose symptoms had gone entirely, improved, stayed the same and got worse were recorded, as summarised in the following table.

\begin{tabular}{|c|c|c|} \hline & Drug & Placebo \ \hline symptoms gone & 14 & 6 \ improved & 21 & 19 \ same & 10 & 10 \ worse & 5 & 15 \ \hline \end{tabular}

Conduct a $5 \%$ significance level test of the null hypothesis that the medicine and placebo have the same effect, against the alternative that their effects differ.

[Hint: You may find some of the following values relevant:

\begin{tabular}{|c|cccccc|} \hline Distribution & $\chi_{1}^{2}$ & $\chi_{2}^{2}$ & $\chi_{3}^{2}$ & $\chi_{4}^{2}$ & $\chi_{6}^{2}$ & $\chi_{8}^{2}$ \ \hline 95 th percentile & $3.84$ & $5.99$ & $7.81$ & $9.48$ & $12.59$ & $15.51$ \ \hline \end{tabular}

Paper 2, Section II, D

commentA particle of unit mass moves in a smooth one-dimensional potential $V(x)$. Its path $x(t)$ is such that the action integral

$S[x]=\int_{a}^{b} L(x, \dot{x}) d t$

has a stationary value, where $a$ and $b>a$ are constants, a dot denotes differentiation with respect to time $t$

$L(x, \dot{x})=\frac{1}{2} \dot{x}^{2}-V(x)$

is the Lagrangian function and the initial and final positions $x(a)$ and $x(b)$ are fixed.

By considering $S[x+\epsilon \xi]$ for suitably restricted functions $\xi(t)$, derive the differential equation governing the motion of the particle and obtain an integral expression for the second variation $\delta^{2} S$.

If $x(t)$ is a solution of the equation of motion and $x(t)+\epsilon u(t)+O\left(\epsilon^{2}\right)$ is also a solution of the equation of motion in the limit $\epsilon \rightarrow 0$, show that $u(t)$ satisfies the equation

$\ddot{u}+V^{\prime \prime}(x) u=0$

If $u(t)$ satisfies this equation and is non-vanishing for $a \leqslant t \leqslant b$, show that

$\delta^{2} S=\frac{1}{2} \int_{a}^{b}\left(\dot{\xi}-\frac{\dot{u} \xi}{u}\right)^{2} d t$

Consider the simple harmonic oscillator, for which

$V(x)=\frac{1}{2} \omega^{2} x^{2}$

where $2 \pi / \omega$ is the oscillation period. Show that the solution of the equation of motion is a local minimum of the action integral, provided that the time difference $b-a$ is less than half an oscillation period.