Part IB, 2021, Paper 3

# Part IB, 2021, Paper 3

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Paper 3, Section II, F

commentDefine the terms connected and path-connected for a topological space. Prove that the interval $[0,1]$ is connected and that if a topological space is path-connected, then it is connected.

Let $X$ be an open subset of Euclidean space $\mathbb{R}^{n}$. Show that $X$ is connected if and only if $X$ is path-connected.

Let $X$ be a topological space with the property that every point has a neighbourhood homeomorphic to an open set in $\mathbb{R}^{n}$. Assume $X$ is connected; must $X$ be also pathconnected? Briefly justify your answer.

Consider the following subsets of $\mathbb{R}^{2}$ :

$\begin{gathered} A=\{(x, 0): x \in(0,1]\}, \quad B=\{(0, y): y \in[1 / 2,1]\}, \text { and } \\ C_{n}=\{(1 / n, y): y \in[0,1]\} \text { for } n \geqslant 1 \end{gathered}$

Let

$X=A \cup B \cup \bigcup_{n \geqslant 1} C_{n}$

with the subspace topology. Is $X$ path-connected? Is $X$ connected? Justify your answers.

Paper 3, Section II, G

commentLet $\gamma$ be a curve (not necessarily closed) in $\mathbb{C}$ and let $[\gamma]$ denote the image of $\gamma$. Let $\phi:[\gamma] \rightarrow \mathbb{C}$ be a continuous function and define

$f(z)=\int_{\gamma} \frac{\phi(\lambda)}{\lambda-z} d \lambda$

for $z \in \mathbb{C} \backslash[\gamma]$. Show that $f$ has a power series expansion about every $a \notin[\gamma]$.

Using Cauchy's Integral Formula, show that a holomorphic function has complex derivatives of all orders. [Properties of power series may be assumed without proof.] Let $f$ be a holomorphic function on an open set $U$ that contains the closed disc $\bar{D}(a, r)$. Obtain an integral formula for the derivative of $f$ on the open disc $D(a, r)$ in terms of the values of $f$ on the boundary of the disc.

Show that if holomorphic functions $f_{n}$ on an open set $U$ converge locally uniformly to a holomorphic function $f$ on $U$, then $f_{n}^{\prime}$ converges locally uniformly to $f^{\prime}$.

Let $D_{1}$ and $D_{2}$ be two overlapping closed discs. Let $f$ be a holomorphic function on some open neighbourhood of $D=D_{1} \cap D_{2}$. Show that there exist open neighbourhoods $U_{j}$ of $D_{j}$ and holomorphic functions $f_{j}$ on $U_{j}, j=1,2$, such that $f(z)=f_{1}(z)+f_{2}(z)$ on $U_{1} \cap U_{2}$.

Paper 3, Section I, B

commentFind the value of $A$ for which the function

$\phi(x, y)=x \cosh y \sin x+A y \sinh y \cos x$

satisfies Laplace's equation. For this value of $A$, find a complex analytic function of which $\phi$ is the real part.

Paper 3, Section II, 15D

comment(a) The energy density stored in the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$ is given by

$w=\frac{\epsilon_{0}}{2} \mathbf{E} \cdot \mathbf{E}+\frac{1}{2 \mu_{0}} \mathbf{B} \cdot \mathbf{B}$

Show that, in regions where no electric current flows,

$\frac{\partial w}{\partial t}+\boldsymbol{\nabla} \cdot \mathbf{S}=0$

for some vector field $\mathbf{S}$ that you should determine.

(b) The coordinates $x^{\prime \mu}=\left(c t^{\prime}, \mathbf{x}^{\prime}\right)$ in an inertial frame $\mathcal{S}^{\prime}$ are related to the coordinates $x^{\mu}=(c t, \mathbf{x})$ in an inertial frame $\mathcal{S}$ by a Lorentz transformation $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$, where

$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma v / c & 0 & 0 \\ -\gamma v / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

with $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$. Here $v$ is the relative velocity of $\mathcal{S}^{\prime}$ with respect to $\mathcal{S}$ in the x-direction.

In frame $\mathcal{S}^{\prime}$, there is a static electric field $\mathbf{E}^{\prime}\left(\mathbf{x}^{\prime}\right)$ with $\partial \mathbf{E}^{\prime} / \partial t^{\prime}=0$, and no magnetic field. Calculate the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ in frame $\mathcal{S}$. Show that the energy density in frame $\mathcal{S}$ is given in terms of the components of $\mathbf{E}^{\prime}$ by

$w=\frac{\epsilon_{0}}{2}\left[E_{x}^{\prime 2}+\left(\frac{c^{2}+v^{2}}{c^{2}-v^{2}}\right)\left(E_{y}^{\prime 2}+E_{z}^{\prime 2}\right)\right]$

Use the fact that $\partial w / \partial t^{\prime}=0$ to show that

$\frac{\partial w}{\partial t}+\nabla \cdot\left(w v \mathbf{e}_{x}\right)=0$

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

Paper 3, Section I, A

commentA two-dimensional flow $\mathbf{u}=(u, v)$ has a velocity field given by

$u=\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} \quad \text { and } \quad v=\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}$

(a) Show explicitly that this flow is incompressible and irrotational away from the origin.

(b) Find the stream function for this flow.

(c) Find the velocity potential for this flow.

Paper 3, Section II, A

commentA two-dimensional layer of viscous fluid lies between two rigid boundaries at $y=\pm L_{0}$. The boundary at $y=L_{0}$ oscillates in its own plane with velocity $\left(U_{0} \cos \omega t, 0\right)$, while the boundary at $y=-L_{0}$ oscillates in its own plane with velocity $\left(-U_{0} \cos \omega t, 0\right)$. Assume that there is no pressure gradient and that the fluid flows parallel to the boundary with velocity $(u(y, t), 0)$, where $u(y, t)$ can be written as $u(y, t)=\operatorname{Re}\left[U_{0} f(y) \exp (i \omega t)\right]$.

(a) By exploiting the symmetry of the system or otherwise, show that

$f(y)=\frac{\sinh [(1+i) \Delta \hat{y}]}{\sinh [(1+i) \Delta]}, \text { where } \hat{y}=\frac{y}{L_{0}} \text { and } \Delta=\left(\frac{\omega L_{0}^{2}}{2 \nu}\right)^{1 / 2}$

(b) Hence or otherwise, show that

where $\Delta_{\pm}=\Delta(1 \pm \hat{y})$.

(c) Show that, for $\Delta \ll 1$,

$u(y, t) \simeq \frac{U_{0} y}{L_{0}} \cos \omega t$

and briefly interpret this result physically.

$\begin{aligned} & \frac{u(y, t)}{U_{0}}=\frac{\cos \omega t\left[\cosh \Delta_{+} \cos \Delta_{-}-\cosh \Delta_{-} \cos \Delta_{+}\right]}{(\cosh 2 \Delta-\cos 2 \Delta)} \\ & +\frac{\sin \omega t\left[\sinh \Delta_{+} \sin \Delta_{-}-\sinh \Delta_{-} \sin \Delta_{+}\right]}{(\cosh 2 \Delta-\cos 2 \Delta)}, \end{aligned}$

Paper 3, Section I, E

commentState the local Gauss-Bonnet theorem for geodesic triangles on a surface. Deduce the Gauss-Bonnet theorem for closed surfaces. [Existence of a geodesic triangulation can be assumed.]

Let $S_{r} \subset \mathbb{R}^{3}$ denote the sphere with radius $r$ centred at the origin. Show that the Gauss curvature of $S_{r}$ is $1 / r^{2}$. An octant is any of the eight regions in $S_{r}$ bounded by arcs of great circles arising from the planes $x=0, y=0, z=0$. Verify directly that the local Gauss-Bonnet theorem holds for an octant. [You may assume that the great circles on $S_{r}$ are geodesics.]

Paper 3, Section II, E

commentLet $S \subset \mathbb{R}^{3}$ be an embedded smooth surface and $\gamma:[0,1] \rightarrow S$ a parameterised smooth curve on $S$. What is the energy of $\gamma$ ? By applying the Euler-Lagrange equations for stationary curves to the energy function, determine the differential equations for geodesics on $S$ explicitly in terms of a parameterisation of $S$.

If $S$ contains a straight line $\ell$, prove from first principles that each segment $[P, Q] \subset \ell$ (with some parameterisation) is a geodesic on $S$.

Let $H \subset \mathbb{R}^{3}$ be the hyperboloid defined by the equation $x^{2}+y^{2}-z^{2}=1$ and let $P=\left(x_{0}, y_{0}, z_{0}\right) \in H$. By considering appropriate isometries, or otherwise, display explicitly three distinct (as subsets of $H$ ) geodesics $\gamma: \mathbb{R} \rightarrow H$ through $P$ in the case when $z_{0} \neq 0$ and four distinct geodesics through $P$ in the case when $z_{0}=0$. Justify your answer.

Let $\gamma: \mathbb{R} \rightarrow H$ be a geodesic, with coordinates $\gamma(t)=(x(t), y(t), z(t))$. Clairaut's relation asserts $\rho(t) \sin \psi(t)$ is constant, where $\rho(t)=\sqrt{x(t)^{2}+y(t)^{2}}$ and $\psi(t)$ is the angle between $\dot{\gamma}(t)$ and the plane through the point $\gamma(t)$ and the $z$-axis. Deduce from Clairaut's relation that there exist infinitely many geodesics $\gamma(t)$ on $H$ which stay in the half-space $\{z>0\}$ for all $t \in \mathbb{R}$.

[You may assume that if $\gamma(t)$ satisfies the geodesic equations on $H$ then $\gamma$ is defined for all $t \in \mathbb{R}$ and the Euclidean norm $\|\dot{\gamma}(t)\|$ is constant. If you use a version of the geodesic equations for a surface of revolution, then that should be proved.]

Paper 3, Section I, G

commentLet $G$ be a finite group, and let $H$ be a proper subgroup of $G$ of index $n$.

Show that there is a normal subgroup $K$ of $G$ such that $|G / K|$ divides $n$ ! and $|G / K| \geqslant n$.

Show that if $G$ is non-abelian and simple, then $G$ is isomorphic to a subgroup of $A_{n}$.

Paper 3, Section II, 10G

commentLet $p$ be a non-zero element of a Principal Ideal Domain $R$. Show that the following are equivalent:

(i) $p$ is prime;

(ii) $p$ is irreducible;

(iii) $(p)$ is a maximal ideal of $R$;

(iv) $R /(p)$ is a field;

(v) $R /(p)$ is an Integral Domain.

Let $R$ be a Principal Ideal Domain, $S$ an Integral Domain and $\phi: R \rightarrow S$ a surjective ring homomorphism. Show that either $\phi$ is an isomorphism or $S$ is a field.

Show that if $R$ is a commutative ring and $R[X]$ is a Principal Ideal Domain, then $R$ is a field.

Let $R$ be an Integral Domain in which every two non-zero elements have a highest common factor. Show that in $R$ every irreducible element is prime.

Paper 3, Section II, 9E

comment(a) (i) State the rank-nullity theorem.

Let $U$ and $W$ be vector spaces. Write down the definition of their direct sum $U \oplus W$ and the inclusions $i: U \rightarrow U \oplus W, j: W \rightarrow U \oplus W$.

Now let $U$ and $W$ be subspaces of a vector space $V$. Define $l: U \cap W \rightarrow U \oplus W$ by $l(x)=i x-j x .$

Describe the quotient space $(U \oplus W) / \operatorname{Im}(l)$ as a subspace of $V$.

(ii) Let $V=\mathbb{R}^{5}$, and let $U$ be the subspace of $V$ spanned by the vectors

$\left(\begin{array}{c} 1 \\ 2 \\ -1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{c} -2 \\ 2 \\ 2 \\ 1 \\ -2 \end{array}\right)$

and $W$ the subspace of $V$ spanned by the vectors

$\left(\begin{array}{c} 3 \\ 2 \\ -3 \\ 1 \\ 3 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right),\left(\begin{array}{c} 1 \\ -4 \\ -1 \\ -2 \\ 1 \end{array}\right)$

Determine the dimension of $U \cap W$.

(b) Let $A, B$ be complex $n$ by $n$ matrices with $\operatorname{rank}(B)=k$.

Show that $\operatorname{det}(A+t B)$ is a polynomial in $t$ of degree at most $k$.

Show that if $k=n$ the polynomial is of degree precisely $n$.

Give an example where $k \geqslant 1$ but this polynomial is zero.

Paper 3 , Section I, H

commentConsider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ on a state space $I$.

(a) Define the notion of a communicating class. What does it mean for a communicating class to be closed?

(b) Taking $I=\{1, \ldots, 6\}$, find the communicating classes associated with the transition matrix $P$ given by

$P=\left(\begin{array}{cccccc} 0 & 0 & 0 & 0 & \frac{1}{4} & \frac{3}{4} \\ \frac{1}{4} & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{4} \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 & \frac{1}{2} & 0 \\ \frac{1}{4} & \frac{1}{2} & 0 & 0 & 0 & \frac{1}{4} \\ 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$

and identify which are closed.

(c) Find the expected time for the Markov chain with transition matrix $P$ above to reach 6 starting from 1 .

Paper 3, Section I, A

commentLet $f(\theta)$ be a $2 \pi$-periodic function with Fourier expansion

$f(\theta)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right)$

Find the Fourier coefficients $a_{n}$ and $b_{n}$ for

$f(\theta)=\left\{\begin{aligned} 1, & 0<\theta<\pi \\ -1, & \pi<\theta<2 \pi \end{aligned}\right.$

Hence, or otherwise, find the Fourier coefficients $A_{n}$ and $B_{n}$ for the $2 \pi$-periodic function $F$ defined by

$F(\theta)=\left\{\begin{array}{cc} \theta, & 0<\theta<\pi \\ 2 \pi-\theta, & \pi<\theta<2 \pi \end{array}\right.$

Use your answers to evaluate

$\sum_{r=0}^{\infty} \frac{(-1)^{r}}{2 r+1} \quad \text { and } \quad \sum_{r=0}^{\infty} \frac{1}{(2 r+1)^{2}}$

Paper 3, Section II, A

commentLet $P(x)$ be a solution of Legendre's equation with eigenvalue $\lambda$,

$\left(1-x^{2}\right) \frac{d^{2} P}{d x^{2}}-2 x \frac{d P}{d x}+\lambda P=0$

such that $P$ and its derivatives $P^{(k)}(x)=d^{k} P / d x^{k}, k=0,1,2, \ldots$, are regular at all points $x$ with $-1 \leqslant x \leqslant 1$.

(a) Show by induction that

$\left(1-x^{2}\right) \frac{d^{2}}{d x^{2}}\left[P^{(k)}\right]-2(k+1) x \frac{d}{d x}\left[P^{(k)}\right]+\lambda_{k} P^{(k)}=0$

for some constant $\lambda_{k}$. Find $\lambda_{k}$ explicitly and show that its value is negative when $k$ is sufficiently large, for a fixed value of $\lambda$.

(b) Write the equation for $P^{(k)}(x)$ in part (a) in self-adjoint form. Hence deduce that if $P^{(k)}(x)$ is not identically zero, then $\lambda_{k} \geqslant 0$.

[Hint: Establish a relation between integrals of the form $\int_{-1}^{1}\left[P^{(k+1)}(x)\right]^{2} f(x) d x$ and $\int_{-1}^{1}\left[P^{(k)}(x)\right]^{2} g(x) d x$ for certain functions $f(x)$ and $\left.g(x) .\right]$

(c) Use the results of parts (a) and (b) to show that if $P(x)$ is a non-zero, regular solution of Legendre's equation on $-1 \leqslant x \leqslant 1$, then $P(x)$ is a polynomial of degree $n$ and $\lambda=n(n+1)$ for some integer $n=0,1,2, \ldots$

Paper 3, Section II, B

commentThe functions $p_{0}, p_{1}, p_{2}, \ldots$ are generated by the formula

$p_{n}(x)=(-1)^{n} x^{-1 / 2} e^{x} \frac{d^{n}}{d x^{n}}\left(x^{n+1 / 2} e^{-x}\right), \quad 0 \leqslant x<\infty$

(a) Show that $p_{n}(x)$ is a monic polynomial of degree $n$. Write down the explicit forms of $p_{0}(x), p_{1}(x), p_{2}(x)$.

(b) Demonstrate the orthogonality of these polynomials with respect to the scalar product

$\langle f, g\rangle=\int_{0}^{\infty} x^{1 / 2} e^{-x} f(x) g(x) d x$

i.e. that $\left\langle p_{n}, p_{m}\right\rangle=0$ for $m \neq n$, and show that

$\left\langle p_{n}, p_{n}\right\rangle=n ! \Gamma\left(n+\frac{3}{2}\right)$

where $\Gamma(y)=\int_{0}^{\infty} x^{y-1} e^{-x} d x$.

(c) Assuming that a three-term recurrence relation in the form

$p_{n+1}(x)=\left(x-\alpha_{n}\right) p_{n}(x)-\beta_{n} p_{n-1}(x), \quad n=1,2, \ldots$

holds, find the explicit expressions for $\alpha_{n}$ and $\beta_{n}$ as functions of $n$.

[Hint: you may use the fact that $\Gamma(y+1)=y \Gamma(y) .]$

Paper 3, Section II, H

commentExplain what is meant by a two-person zero-sum game with $m \times n$ payoff matrix $A$, and define what is meant by an optimal strategy for each player. What are the relationships between the optimal strategies and the value of the game?

Suppose now that

$A=\left(\begin{array}{cccc} 0 & 1 & 1 & -4 \\ -1 & 0 & 2 & 2 \\ -1 & -2 & 0 & 3 \\ 4 & -2 & -3 & 0 \end{array}\right)$

Show that if strategy $p=\left(p_{1}, p_{2}, p_{3}, p_{4}\right)^{T}$ is optimal for player I, it must also be optimal for player II. What is the value of the game in this case? Justify your answer.

Explain why we must have $(A p)_{i} \leqslant 0$ for all $i$. Hence or otherwise, find the optimal strategy $p$ and prove that it is unique.

Paper 3, Section I, C

commentThe electron in a hydrogen-like atom moves in a spherically symmetric potential $V(r)=-K / r$ where $K$ is a positive constant and $r$ is the radial coordinate of spherical polar coordinates. The two lowest energy spherically symmetric normalised states of the electron are given by

$\chi_{1}(r)=\frac{1}{\sqrt{\pi} a^{3 / 2}} e^{-r / a} \quad \text { and } \quad \chi_{2}(r)=\frac{1}{4 \sqrt{2 \pi} a^{3 / 2}}\left(2-\frac{r}{a}\right) e^{-r / 2 a}$

where $a=\hbar^{2} / m K$ and $m$ is the mass of the electron. For any spherically symmetric function $f(r)$, the Laplacian is given by $\nabla^{2} f=\frac{d^{2} f}{d r^{2}}+\frac{2}{r} \frac{d f}{d r}$.

(i) Suppose that the electron is in the state $\chi(r)=\frac{1}{2} \chi_{1}(r)+\frac{\sqrt{3}}{2} \chi_{2}(r)$ and its energy is measured. Find the expectation value of the result.

(ii) Suppose now that the electron is in state $\chi(r)$ (as above) at time $t=0$. Let $R(t)$ be the expectation value of a measurement of the electron's radial position $r$ at time $t$. Show that the value of $R(t)$ oscillates sinusoidally about a constant level and determine the frequency of the oscillation.

Paper 3, Section II, $18 \mathrm{H}$

commentConsider the normal linear model $Y=X \beta+\varepsilon$ where $X$ is a known $n \times p$ design matrix with $n-2>p \geqslant 1, \beta \in \mathbb{R}^{p}$ is an unknown vector of parameters, and $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$ is a vector of normal errors with each component having variance $\sigma^{2}>0$. Suppose $X$ has full column rank.

(i) Write down the maximum likelihood estimators, $\hat{\beta}$ and $\hat{\sigma}^{2}$, for $\beta$ and $\sigma^{2}$ respectively. [You need not derive these.]

(ii) Show that $\hat{\beta}$ is independent of $\hat{\sigma}^{2}$.

(iii) Find the distributions of $\hat{\beta}$ and $n \hat{\sigma}^{2} / \sigma^{2}$.

(iv) Consider the following test statistic for testing the null hypothesis $H_{0}: \beta=0$ against the alternative $\beta \neq 0$ :

$T:=\frac{\|\hat{\beta}\|^{2}}{n \hat{\sigma}^{2}} .$

Let $\lambda_{1} \geqslant \lambda_{2} \geqslant \cdots \geqslant \lambda_{p}>0$ be the eigenvalues of $X^{T} X$. Show that under $H_{0}, T$ has the same distribution as

$\frac{\sum_{j=1}^{p} \lambda_{j}^{-1} W_{j}}{Z}$

where $Z \sim \chi_{n-p}^{2}$ and $W_{1}, \ldots, W_{p}$ are independent $\chi_{1}^{2}$ random variables, independent of $Z$.

[Hint: You may use the fact that $X=U D V^{T}$ where $U \in \mathbb{R}^{n \times p}$ has orthonormal columns, $V \in \mathbb{R}^{p \times p}$ is an orthogonal matrix and $D \in \mathbb{R}^{p \times p}$ is a diagonal matrix with $\left.D_{i i}=\sqrt{\lambda_{i}} .\right]$

(v) Find $\mathbb{E} T$ when $\beta \neq 0$. [Hint: If $R \sim \chi_{\nu}^{2}$ with $\nu>2$, then $\mathbb{E}(1 / R)=1 /(\nu-2)$.]

Paper 3, Section I, D

commentFind the function $y(x)$ that gives a stationary value of the functional

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

subject to the boundary conditions $y(0)=-1$ and $y(1)=e-e^{-1}-\frac{3}{2}$.