Part IB, 2020, Paper 2

# Part IB, 2020, Paper 2

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Paper 2, Section I, $2 E$

commentLet $\tau$ be the collection of subsets of $\mathbb{C}$ of the form $\mathbb{C} \backslash f^{-1}(0)$, where $f$ is an arbitrary complex polynomial. Show that $\tau$ is a topology on $\mathbb{C}$.

Given topological spaces $X$ and $Y$, define the product topology on $X \times Y$. Equip $\mathbb{C}^{2}$ with the topology given by the product of $(\mathbb{C}, \tau)$ with itself. Let $g$ be an arbitrary two-variable complex polynomial. Is the subset $\mathbb{C}^{2} \backslash g^{-1}(0)$ always open in this topology? Justify your answer.

Paper 2, Section II, E

commentLet $C[0,1]$ be the space of continuous real-valued functions on $[0,1]$, and let $d_{1}, d_{\infty}$ be the metrics on it given by

$d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x \quad \text { and } \quad d_{\infty}(f, g)=\max _{x \in[0,1]}|f(x)-g(x)|$

Show that id : $\left(C[0,1], d_{\infty}\right) \rightarrow\left(C[0,1], d_{1}\right)$ is a continuous map. Do $d_{1}$ and $d_{\infty}$ induce the same topology on $C[0,1]$ ? Justify your answer.

Let $d$ denote for any $m \in \mathbb{N}$ the uniform metric on $\mathbb{R}^{m}: d\left(\left(x_{i}\right),\left(y_{i}\right)\right)=\max _{i}\left|x_{i}-y_{i}\right|$. Let $\mathcal{P}_{n} \subset C[0,1]$ be the subspace of real polynomials of degree at most $n$. Define a Lipschitz map between two metric spaces, and show that evaluation at a point gives a Lipschitz map $\left(C[0,1], d_{\infty}\right) \rightarrow(\mathbb{R}, d)$. Hence or otherwise find a bijection from $\left(\mathcal{P}_{n}, d_{\infty}\right)$ to $\left(\mathbb{R}^{n+1}, d\right)$ which is Lipschitz and has a Lipschitz inverse.

Let $\tilde{\mathcal{P}}_{n} \subset \mathcal{P}_{n}$ be the subset of polynomials with values in the range $[-1,1]$.

(i) Show that $\left(\tilde{\mathcal{P}}_{n}, d_{\infty}\right)$ is compact.

(ii) Show that $d_{1}$ and $d_{\infty}$ induce the same topology on $\tilde{\mathcal{P}}_{n}$.

Any theorems that you use should be clearly stated.

[You may use the fact that for distinct constants $a_{i}$, the following matrix is invertible:

$\left(\begin{array}{ccccc} 1 & a_{0} & a_{0}^{2} & \ldots & a_{0}^{n} \\ 1 & a_{1} & a_{1}^{2} & \ldots & a_{1}^{n} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & a_{n} & a_{n}^{2} & \ldots & a_{n}^{n} \end{array}\right)$

Paper 2, Section II, B

commentFor the function

$f(z)=\frac{1}{z(z-2)}$

find the Laurent expansions

(i) about $z=0$ in the annulus $0<|z|<2$,

(ii) about $z=0$ in the annulus $2<|z|<\infty$,

(iii) about $z=1$ in the annulus $0<|z-1|<1$.

What is the nature of the singularity of $f$, if any, at $z=0, z=\infty$ and $z=1$ ?

Using an integral of $f$, or otherwise, evaluate

$\int_{0}^{2 \pi} \frac{2-\cos \theta}{5-4 \cos \theta} d \theta$

Paper 2, Section I, D

commentTwo concentric spherical shells with radii $R$ and $2 R$ carry fixed, uniformly distributed charges $Q_{1}$ and $Q_{2}$ respectively. Find the electric field and electric potential at all points in space. Calculate the total energy of the electric field.

Paper 2, Section II, D

comment(a) A surface current $\mathbf{K}=K \mathbf{e}_{x}$, with $K$ a constant and $\mathbf{e}_{x}$ the unit vector in the $x$-direction, lies in the plane $z=0$. Use Ampère's law to determine the magnetic field above and below the plane. Confirm that the magnetic field is discontinuous across the surface, with the discontinuity given by

$\lim _{z \rightarrow 0^{+}} \mathbf{e}_{z} \times \mathbf{B}-\lim _{z \rightarrow 0^{-}} \mathbf{e}_{z} \times \mathbf{B}=\mu_{0} \mathbf{K}$

where $\mathbf{e}_{z}$ is the unit vector in the $z$-direction.

(b) A surface current $\mathbf{K}$ flows radially in the $z=0$ plane, resulting in a pile-up of charge $Q$ at the origin, with $d Q / d t=I$, where $I$ is a constant.

Write down the electric field $\mathbf{E}$ due to the charge at the origin, and hence the displacement current $\epsilon_{0} \partial \mathbf{E} / \partial t$.

Confirm that, away from the plane and for $\theta<\pi / 2$, the magnetic field due to the displacement current is given by

$\mathbf{B}(r, \theta)=\frac{\mu_{0} I}{4 \pi r} \tan \left(\frac{\theta}{2}\right) \mathbf{e}_{\phi}$

where $(r, \theta, \phi)$ are the usual spherical polar coordinates. [Hint: Use Stokes' theorem applied to a spherical cap that subtends an angle $\theta$.]

Paper 2, Section I, C

commentIncompressible fluid of constant viscosity $\mu$ is confined to the region $0<y<h$ between two parallel rigid plates. Consider two parallel viscous flows: flow A is driven by the motion of one plate in the $x$-direction with the other plate at rest; flow B is driven by a constant pressure gradient in the $x$-direction with both plates at rest. The velocity mid-way between the plates is the same for both flows.

The viscous friction in these flows is known to produce heat locally at a rate

$Q=\mu\left(\frac{\partial u}{\partial y}\right)^{2}$

per unit volume, where $u$ is the $x$-component of the velocity. Determine the ratio of the total rate of heat production in flow A to that in flow B.

Paper 2, Section II, C

commentA vertical cylindrical container of radius $R$ is partly filled with fluid of constant density to depth $h$. The free surface is perturbed so that the fluid occupies the region

$0<r<R, \quad-h<z<\zeta(r, \theta, t)$

where $(r, \theta, z)$ are cylindrical coordinates and $\zeta$ is the perturbed height of the free surface. For small perturbations, a linearised description of surface waves in the cylinder yields the following system of equations for $\zeta$ and the velocity potential $\phi$ :

$\begin{aligned} \nabla^{2} \phi &=0, \quad 0<r<R, \quad-h<z<0 \\ \frac{\partial \phi}{\partial t}+g \zeta &=0 \quad \text { on } \quad z=0 \\ \frac{\partial \zeta}{\partial t}-\frac{\partial \phi}{\partial z} &=0 \quad \text { on } \quad z=0 \\ \frac{\partial \phi}{\partial z} &=0 \quad \text { on } \quad z=-h \\ \frac{\partial \phi}{\partial r} &=0 \quad \text { on } \quad r=R \end{aligned}$

(a) Describe briefly the physical meaning of each equation.

(b) Consider axisymmetric normal modes of the form

$\phi=\operatorname{Re}\left(\hat{\phi}(r, z) e^{-i \sigma t}\right), \quad \zeta=\operatorname{Re}\left(\hat{\zeta}(r) e^{-i \sigma t}\right)$

Show that the system of equations $(1)-(5)$ admits a solution for $\hat{\phi}$ of the form

$\hat{\phi}(r, z)=A J_{0}\left(k_{n} r\right) Z(z)$

where $A$ is an arbitrary amplitude, $J_{0}(x)$ satisfies the equation

$\frac{d^{2} J_{0}}{d x^{2}}+\frac{1}{x} \frac{d J_{0}}{d x}+J_{0}=0$

the wavenumber $k_{n}, n=1,2, \ldots$ is such that $x_{n}=k_{n} R$ is one of the zeros of the function $d J_{0} / d x$, and the function $Z(z)$ should be determined explicitly.

(c) Show that the frequency $\sigma_{n}$ of the $n$-th mode is given by

$\sigma_{n}^{2}=\frac{g}{h} \Psi\left(k_{n} h\right)$

where the function $\Psi(x)$ is to be determined.

[Hint: In cylindrical coordinates $(r, \theta, z)$,

$\left.\nabla^{2}=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}+\frac{\partial^{2}}{\partial z^{2}} \cdot\right]$

Paper 2, Section II, F

commentLet $H=\{z=x+i y \in \mathbb{C}: y>0\}$ be the hyperbolic half-plane with the metric $g_{H}=\left(d x^{2}+d y^{2}\right) / y^{2}$. Define the length of a continuously differentiable curve in $H$ with respect to $g_{H}$.

What are the hyperbolic lines in $H$ ? Show that for any two distinct points $z, w$ in $H$, the infimum $\rho(z, w)$ of the lengths (with respect to $g_{H}$ ) of curves from $z$ to $w$ is attained by the segment $[z, w]$ of the hyperbolic line with an appropriate parameterisation.

The 'hyperbolic Pythagoras theorem' asserts that if a hyperbolic triangle $A B C$ has angle $\pi / 2$ at $C$ then

$\cosh c=\cosh a \cosh b,$

where $a, b, c$ are the lengths of the sides $B C, A C, A B$, respectively.

Let $l$ and $m$ be two hyperbolic lines in $H$ such that

$\inf \{\rho(z, w): z \in l, w \in m\}=d>0$

Prove that the distance $d$ is attained by the points of intersection with a hyperbolic line $h$ that meets each of $l, m$ orthogonally. Give an example of two hyperbolic lines $l$ and $m$ such that the infimum of $\rho(z, w)$ is not attained by any $z \in l, w \in m$.

[You may assume that every Möbius transformation that maps H onto itself is an isometry of $\left.g_{H} \cdot\right]$

Paper 2, Section I, G

commentAssume a group $G$ acts transitively on a set $\Omega$ and that the size of $\Omega$ is a prime number. Let $H$ be a normal subgroup of $G$ that acts non-trivially on $\Omega$.

Show that any two $H$-orbits of $\Omega$ have the same size. Deduce that the action of $H$ on $\Omega$ is transitive.

Let $\alpha \in \Omega$ and let $G_{\alpha}$ denote the stabiliser of $\alpha$ in $G$. Show that if $H \cap G_{\alpha}$ is trivial, then there is a bijection $\theta: H \rightarrow \Omega$ under which the action of $G_{\alpha}$ on $H$ by conjugation corresponds to the action of $G_{\alpha}$ on $\Omega$.

Paper 2, Section II, G

commentState Gauss' lemma. State and prove Eisenstein's criterion.

Define the notion of an algebraic integer. Show that if $\alpha$ is an algebraic integer, then $\{f \in \mathbb{Z}[X]: f(\alpha)=0\}$ is a principal ideal generated by a monic, irreducible polynomial.

Let $f=X^{4}+2 X^{3}-3 X^{2}-4 X-11$. Show that $\mathbb{Q}[X] /(f)$ is a field. Show that $\mathbb{Z}[X] /(f)$ is an integral domain, but not a field. Justify your answers.

Paper 2, Section II, F

commentLet $V$ be a finite-dimensional vector space over a field. Show that an endomorphism $\alpha$ of $V$ is idempotent, i.e. $\alpha^{2}=\alpha$, if and only if $\alpha$ is a projection onto its image.

Determine whether the following statements are true or false, giving a proof or counterexample as appropriate:

(i) If $\alpha^{3}=\alpha^{2}$, then $\alpha$ is idempotent.

(ii) The condition $\alpha(1-\alpha)^{2}=0$ is equivalent to $\alpha$ being idempotent.

(iii) If $\alpha$ and $\beta$ are idempotent and such that $\alpha+\beta$ is also idempotent, then $\alpha \beta=0$.

(iv) If $\alpha$ and $\beta$ are idempotent and $\alpha \beta=0$, then $\alpha+\beta$ is also idempotent.

Paper 2, Section I, H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain with state space $\{1,2\}$ and transition matrix

$P=\left(\begin{array}{cc} 1-\alpha & \alpha \\ \beta & 1-\beta \end{array}\right)$

where $\alpha, \beta \in(0,1]$. Compute $\mathbb{P}\left(X_{n}=1 \mid X_{0}=1\right)$. Find the value of $\mathbb{P}\left(X_{n}=1 \mid X_{0}=2\right)$.

Paper 2, Section I, B

commentFind the Fourier transform of the function

$f(x)= \begin{cases}A, & |x| \leqslant 1 \\ 0, & |x|>1\end{cases}$

Determine the convolution of the function $f(x)$ with itself.

State the convolution theorem for Fourier transforms. Using it, or otherwise, determine the Fourier transform of the function

$g(x)= \begin{cases}B(2-|x|), & |x| \leqslant 2 \\ 0, & |x|>2\end{cases}$

Paper 2, Section II, A

comment(i) The solution to the equation

$\frac{d}{d x}\left(x \frac{d F}{d x}\right)+\alpha^{2} x F=0$

that is regular at the origin is $F(x)=C J_{0}(\alpha x)$, where $\alpha$ is a real, positive parameter, $J_{0}$ is a Bessel function, and $C$ is an arbitrary constant. The Bessel function has infinitely many zeros: $J_{0}\left(\gamma_{k}\right)=0$ with $\gamma_{k}>0$, for $k=1,2, \ldots$. Show that

$\int_{0}^{1} J_{0}(\alpha x) J_{0}(\beta x) x d x=\frac{\beta J_{0}(\alpha) J_{0}^{\prime}(\beta)-\alpha J_{0}(\beta) J_{0}^{\prime}(\alpha)}{\alpha^{2}-\beta^{2}}, \quad \alpha \neq \beta$

(where $\alpha$ and $\beta$ are real and positive) and deduce that

$\int_{0}^{1} J_{0}\left(\gamma_{k} x\right) J_{0}\left(\gamma_{\ell} x\right) x d x=0, \quad k \neq \ell ; \quad \int_{0}^{1}\left(J_{0}\left(\gamma_{k} x\right)\right)^{2} x d x=\frac{1}{2}\left(J_{0}^{\prime}\left(\gamma_{k}\right)\right)^{2}$

[Hint: For the second identity, consider $\alpha=\gamma_{k}$ and $\beta=\gamma_{k}+\epsilon$ with $\epsilon$ small.]

(ii) The displacement $z(r, t)$ of the membrane of a circular drum of unit radius obeys

$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial z}{\partial r}\right)=\frac{\partial^{2} z}{\partial t^{2}}, \quad z(1, t)=0$

where $r$ is the radial coordinate on the membrane surface, $t$ is time (in certain units), and the displacement is assumed to have no angular dependence. At $t=0$ the drum is struck, so that

$z(r, 0)=0, \quad \frac{\partial z}{\partial t}(r, 0)=\left\{\begin{array}{cc} U, & r<b \\ 0, & r>b \end{array}\right.$

where $U$ and $b<1$ are constants. Show that the subsequent motion is given by

$z(r, t)=\sum_{k=1}^{\infty} C_{k} J_{0}\left(\gamma_{k} r\right) \sin \left(\gamma_{k} t\right) \quad \text { where } \quad C_{k}=-2 b U \frac{J_{0}^{\prime}\left(\gamma_{k} b\right)}{\gamma_{k}^{2}\left(J_{0}^{\prime}\left(\gamma_{k}\right)\right)^{2}}$

Paper 2, Section II, C

commentConsider a multistep method for numerical solution of the differential equation $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$ :

$\mathbf{y}_{n+2}-\mathbf{y}_{n+1}=h\left[(1+\alpha) \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)+\beta \mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)-(\alpha+\beta) \mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)\right],$

where $n=0,1, \ldots$, and $\alpha$ and $\beta$ are constants.

(a) Define the order of a method for numerically solving an ODE.

(b) Show that in general an explicit method of the form $(*)$ has order 1 . Determine the values of $\alpha$ and $\beta$ for which this multistep method is of order 3 .

(c) Show that the multistep method (*) is convergent.

Paper 2, Section II, H

commentState and prove the Lagrangian sufficiency theorem.

Solve, using the Lagrangian method, the optimization problem

$\begin{array}{ll} \operatorname{maximise} & x+y+2 a \sqrt{1+z} \\ \text { subject to } & x+\frac{1}{2} y^{2}+z=b \\ & x, z \geqslant 0 \end{array}$

where the constants $a$ and $b$ satisfy $a \geqslant 1$ and $b \geqslant 1 / 2$.

[You need not prove that your solution is unique.]

Paper 2, Section II, A

comment(a) The potential $V(x)$ for a particle of mass $m$ in one dimension is such that $V \rightarrow 0$ rapidly as $x \rightarrow \pm \infty$. Let $\psi(x)$ be a wavefunction for the particle satisfying the time-independent Schrödinger equation with energy $E$.

Suppose $\psi$ has the asymptotic behaviour

$\psi(x) \sim A e^{i k x}+B e^{-i k x} \quad(x \rightarrow-\infty), \quad \psi(x) \sim C e^{i k x} \quad(x \rightarrow+\infty)$

where $A, B, C$ are complex coefficients. Explain, in outline, how the probability current $j(x)$ is used in the interpretation of such a solution as a scattering process and how the transmission and reflection probabilities $P_{\mathrm{tr}}$ and $P_{\text {ref }}$ are found.

Now suppose instead that $\psi(x)$ is a bound state solution. Write down the asymptotic behaviour in this case, relating an appropriate parameter to the energy $E$.

(b) Consider the potential

$V(x)=-\frac{\hbar^{2}}{m} \frac{a^{2}}{\cosh ^{2} a x}$

where $a$ is a real, positive constant. Show that

$\psi(x)=N e^{i k x}(a \tanh a x-i k)$

where $N$ is a complex coefficient, is a solution of the time-independent Schrödinger equation for any real $k$ and find the energy $E$. Show that $\psi$ represents a scattering process for which $P_{\text {ref }}=0$, and find $P_{\mathrm{tr}}$ explicitly.

Now let $k=i \lambda$ in the formula for $\psi$ above. Show that this defines a bound state if a certain real positive value of $\lambda$ is chosen and find the energy of this solution.

Paper 2, Section II, H

commentConsider the general linear model $Y=X \beta^{0}+\varepsilon$ where $X$ is a known $n \times p$ design matrix with $p \geqslant 2, \beta^{0} \in \mathbb{R}^{p}$ is an unknown vector of parameters, and $\varepsilon \in \mathbb{R}^{n}$ is a vector of stochastic errors with $\mathbb{E}\left(\varepsilon_{i}\right)=0, \operatorname{var}\left(\varepsilon_{i}\right)=\sigma^{2}>0$ and $\operatorname{cov}\left(\varepsilon_{i}, \varepsilon_{j}\right)=0$ for all $i, j=1, \ldots, n$ with $i \neq j$. Suppose $X$ has full column rank.

(a) Write down the least squares estimate $\hat{\beta}$ of $\beta^{0}$ and show that it minimises the least squares objective $S(\beta)=\|Y-X \beta\|^{2}$ over $\beta \in \mathbb{R}^{p}$.

(b) Write down the variance-covariance matrix $\operatorname{cov}(\hat{\beta})$.

(c) Let $\tilde{\beta} \in \mathbb{R}^{p}$ minimise $S(\beta)$ over $\beta \in \mathbb{R}^{p}$ subject to $\beta_{p}=0$. Let $Z$ be the $n \times(p-1)$ submatrix of $X$ that excludes the final column. Write $\operatorname{down} \operatorname{cov}(\tilde{\beta})$.

(d) Let $P$ and $P_{0}$ be $n \times n$ orthogonal projections onto the column spaces of $X$ and $Z$ respectively. Show that for all $u \in \mathbb{R}^{n}, u^{T} P u \geqslant u^{T} P_{0} u$.

(e) Show that for all $x \in \mathbb{R}^{p}$,

$\operatorname{var}\left(x^{T} \tilde{\beta}\right) \leqslant \operatorname{var}\left(x^{T} \hat{\beta}\right) .$

[Hint: Argue that $x=X^{T} u$ for some $u \in \mathbb{R}^{n}$.]

Paper 2, Section I, D

commentFind the stationary points of the function $\phi=x y z$ subject to the constraint $x+a^{2} y^{2}+z^{2}=b^{2}$, with $a, b>0$. What are the maximum and minimum values attained by $\phi$, subject to this constraint, if we further restrict to $x \geqslant 0$ ?