Part IB, 2019, Paper 3

# Part IB, 2019, Paper 3

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Paper 3, Section I, $2 E$

comment(a) Let $A \subset \mathbb{R}$. What does it mean for a function $f: A \rightarrow \mathbb{R}$ to be uniformly continuous?

(b) Which of the following functions are uniformly continuous? Briefly justify your answers.

(i) $f(x)=x^{2}$ on $\mathbb{R}$.

(ii) $f(x)=\sqrt{x}$ on $[0, \infty)$.

(iii) $f(x)=\cos (1 / x)$ on $[1, \infty)$.

Paper 3, Section II, E

comment(a) Carefully state the Picard-Lindelöf theorem on solutions to ordinary differential equations.

(b) Let $X=C\left([1, b], \mathbb{R}^{n}\right)$ be the set of continuous functions from a closed interval $[1, b]$ to $\mathbb{R}^{n}$, and let $\|\cdot\|$ be a norm on $\mathbb{R}^{n}$.

(i) Let $f \in X$. Show that for any $c \in[0, \infty)$ the norm

$\|f\|_{c}=\sup _{t \in[1, b]}\left\|f(t) t^{-c}\right\|$

is Lipschitz equivalent to the usual sup norm on $X$.

(ii) Assume that $F:[1, b] \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is continuous and Lipschitz in the second variable, i.e. there exists $M>0$ such that

$\|F(t, x)-F(t, y)\| \leqslant M\|x-y\|$

for all $t \in[1, b]$ and all $x, y \in \mathbb{R}^{n}$. Define $\varphi: X \rightarrow X$ by

$\varphi(f)(t)=\int_{1}^{t} F(l, f(l)) d l$

for $t \in[1, b]$.

Show that there is a choice of $c$ such that $\varphi$ is a contraction on $\left(X,\|\cdot\|_{c}\right)$. Deduce that for any $y_{0} \in \mathbb{R}^{n}$, the differential equation

$D f(t)=F(t, f(t))$

has a unique solution on $[1, b]$ with $f(1)=y_{0}$.

Paper 3, Section II, F

commentDefine the winding number $n(\gamma, w)$ of a closed path $\gamma:[a, b] \rightarrow \mathbb{C}$ around a point $w \in \mathbb{C}$ which does not lie on the image of $\gamma$. [You do not need to justify its existence.]

If $f$ is a meromorphic function, define the order of a zero $z_{0}$ of $f$ and of a pole $w_{0}$ of $f$. State the Argument Principle, and explain how it can be deduced from the Residue Theorem.

How many roots of the polynomial

$z^{4}+10 z^{3}+4 z^{2}+10 z+5$

lie in the right-hand half plane?

Paper 3, Section I, D

commentBy considering the transformation $w=i(1-z) /(1+z)$, find a solution to Laplace's equation $\nabla^{2} \phi=0$ inside the unit disc $D \subset \mathbb{C}$, subject to the boundary conditions

$\left.\phi\right|_{|z|=1}= \begin{cases}\phi_{0} & \text { for } \arg (z) \in(0, \pi) \\ -\phi_{0} & \text { for } \arg (z) \in(\pi, 2 \pi)\end{cases}$

where $\phi_{0}$ is constant. Give your answer in terms of $(x, y)=(\operatorname{Re} z, \operatorname{Im} z)$.

Paper 3, Section II, A

commentThe electric and magnetic fields $\mathbf{E}, \mathbf{B}$ in an inertial frame $\mathcal{S}$ are related to the fields $\mathbf{E}^{\prime}, \mathbf{B}^{\prime}$ in a frame $\mathcal{S}^{\prime}$ by a Lorentz transformation. Given that $\mathcal{S}^{\prime}$ moves in the $x$-direction with speed $v$ relative to $\mathcal{S}$, and that

$E_{y}^{\prime}=\gamma\left(E_{y}-v B_{z}\right), \quad B_{z}^{\prime}=\gamma\left(B_{z}-\left(v / c^{2}\right) E_{y}\right),$

write down equations relating the remaining field components and define $\gamma$. Use your answers to show directly that $\mathbf{E}^{\prime} \cdot \mathbf{B}^{\prime}=\mathbf{E} \cdot \mathbf{B}$.

Give an expression for an additional, independent, Lorentz-invariant function of the fields, and check that it is invariant for the special case when $E_{y}=E$ and $B_{y}=B$ are the only non-zero components in the frame $\mathcal{S}$.

Now suppose in addition that $c B=\lambda E$ with $\lambda$ a non-zero constant. Show that the angle $\theta$ between the electric and magnetic fields in $\mathcal{S}^{\prime}$ is given by

$\cos \theta=f(\beta)=\frac{\lambda\left(1-\beta^{2}\right)}{\left\{\left(1+\lambda^{2} \beta^{2}\right)\left(\lambda^{2}+\beta^{2}\right)\right\}^{1 / 2}}$

where $\beta=v / c$. By considering the behaviour of $f(\beta)$ as $\beta$ approaches its limiting values, show that the relative velocity of the frames can be chosen so that the angle takes any value in one of the ranges $0 \leqslant \theta<\pi / 2$ or $\pi / 2<\theta \leqslant \pi$, depending on the sign of $\lambda$.

Paper 3, Section II, C

commentA cubic box of side $2 h$, enclosing the region $0<x<2 h, 0<y<2 h,-h<z<h$, contains equal volumes of two incompressible fluids that remain distinct. The system is initially at rest, with the fluid of density $\rho_{1}$ occupying the region $0<z<h$ and the fluid of density $\rho_{2}$ occupying the region $-h<z<0$, and with gravity $(0,0,-g)$. The interface between the fluids is then slightly perturbed. Derive the linearized equations and boundary conditions governing small disturbances to the initial state.

In the case $\rho_{2}>\rho_{1}$, show that the angular frequencies $\omega$ of the normal modes are given by

$\omega^{2}=\left(\frac{\rho_{2}-\rho_{1}}{\rho_{1}+\rho_{2}}\right) g k \tanh (k h)$

and express the allowable values of the wavenumber $k$ in terms of $h$. Identify the lowestfrequency non-trivial mode $(\mathrm{s})$. Comment on the limit $\rho_{1} \ll \rho_{2}$. What physical behaviour is expected in the case $\rho_{1}>\rho_{2}$ ?

Paper 3, Section I, E

commentState a formula for the area of a spherical triangle with angles $\alpha, \beta, \gamma$.

Let $n \geqslant 3$. What is the area of a convex spherical $n$-gon with interior angles $\alpha_{1}, \ldots, \alpha_{n}$ ? Justify your answer.

Find the range of possible values for the interior angle of a regular convex spherical $n-g \mathrm{gn}$

Paper 3, Section II, E

commentDefine a geodesic triangulation of an abstract closed smooth surface. Define the Euler number of a triangulation, and state the Gauss-Bonnet theorem for closed smooth surfaces. Given a vertex in a triangulation, its valency is defined to be the number of edges incident at that vertex.

(a) Given a triangulation of the torus, show that the average valency of a vertex of the triangulation is 6 .

(b) Consider a triangulation of the sphere.

(i) Show that the average valency of a vertex is strictly less than 6 .

(ii) A triangulation can be subdivided by replacing one triangle $\Delta$ with three sub-triangles, each one with vertices two of the original ones, and a fixed interior point of $\Delta$.

Using this, or otherwise, show that there exist triangulations of the sphere with average vertex valency arbitrarily close to 6 .

(c) Suppose $S$ is a closed abstract smooth surface of everywhere negative curvature. Show that the average vertex valency of a triangulation of $S$ is bounded above and below.

Paper 3, Section I, $1 G$

commentProve that the ideal $(2,1+\sqrt{-13})$ in $\mathbb{Z}[\sqrt{-13}]$ is not principal.

Paper 3, Section II, G

commentLet $\omega=\frac{1}{2}(-1+\sqrt{-3})$.

(a) Prove that $\mathbb{Z}[\omega]$ is a Euclidean domain.

(b) Deduce that $\mathbb{Z}[\omega]$ is a unique factorisation domain, stating carefully any results from the course that you use.

(c) By working in $\mathbb{Z}[\omega]$, show that whenever $x, y \in \mathbb{Z}$ satisfy

$x^{2}-x+1=y^{3}$

then $x$ is not congruent to 2 modulo 3 .

Paper 3, Section II, F

commentIf $q$ is a quadratic form on a finite-dimensional real vector space $V$, what is the associated symmetric bilinear form $\varphi(\cdot, \cdot)$ ? Prove that there is a basis for $V$ with respect to which the matrix for $\varphi$ is diagonal. What is the signature of $q$ ?

If $R \leqslant V$ is a subspace such that $\varphi(r, v)=0$ for all $r \in R$ and all $v \in V$, show that $q^{\prime}(v+R)=q(v)$ defines a quadratic form on the quotient vector space $V / R$. Show that the signature of $q^{\prime}$ is the same as that of $q$.

If $e, f \in V$ are vectors such that $\varphi(e, e)=0$ and $\varphi(e, f)=1$, show that there is a direct sum decomposition $V=\operatorname{span}(e, f) \oplus U$ such that the signature of $\left.q\right|_{U}$ is the same as that of $q$.

Paper 3, Section I, H

commentSuppose that $\left(X_{n}\right)$ is a Markov chain with state space $S$.

(a) Give the definition of a communicating class.

(b) Give the definition of the period of a state $a \in S$.

(c) Show that if two states communicate then they have the same period.

Paper 3, Section I, D

commentDefine the discrete Fourier transform of a sequence $\left\{x_{0}, x_{1}, \ldots, x_{N-1}\right\}$ of $N$ complex numbers.

Compute the discrete Fourier transform of the sequence

$x_{n}=\frac{1}{N}\left(1+e^{2 \pi i n / N}\right)^{N-1} \quad \text { for } n=0, \ldots, N-1 .$

Paper 3, Section II, D

commentBy differentiating the expression $\psi(t)=H(t) \sin (\alpha t) / \alpha$, where $\alpha$ is a constant and $H(t)$ is the Heaviside step function, show that

$\frac{d^{2} \psi}{d t^{2}}+\alpha^{2} \psi=\delta(t)$

where $\delta(t)$ is the Dirac $\delta$-function.

Hence, by taking a Fourier transform with respect to the spatial variables only, derive the retarded Green's function for the wave operator $\partial_{t}^{2}-c^{2} \nabla^{2}$ in three spatial dimensions.

[You may use that

$\frac{1}{2 \pi} \int_{\mathbb{R}^{3}} e^{i \mathbf{k} \cdot(\mathbf{x}-\mathbf{y})} \frac{\sin (k c t)}{k c} d^{3} k=-\frac{i}{c|\mathbf{x}-\mathbf{y}|} \int_{-\infty}^{\infty} e^{i k|\mathbf{x}-\mathbf{y}|} \sin (k c t) d k$

without proof.]

Thus show that the solution to the homogeneous wave equation $\partial_{t}^{2} u-c^{2} \nabla^{2} u=0$, subject to the initial conditions $u(\mathbf{x}, 0)=0$ and $\partial_{t} u(\mathbf{x}, 0)=f(\mathbf{x})$, may be expressed as

$u(\mathbf{x}, t)=\langle f\rangle t$

where $\langle f\rangle$ is the average value of $f$ on a sphere of radius $c t$ centred on $\mathbf{x}$. Interpret this result.

Paper 3, Section I, $3 G$

commentLet $X$ be a metric space.

(a) What does it mean for $X$ to be compact? What does it mean for $X$ to be sequentially compact?

(b) Prove that if $X$ is compact then $X$ is sequentially compact.

Paper 3, Section II, C

comment(a) Let $w(x)$ be a positive weight function on the interval $[a, b]$. Show that

$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) w(x) d x$

defines an inner product on $C[a, b]$.

(b) Consider the sequence of polynomials $p_{n}(x)$ defined by the three-term recurrence relation

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

where

$p_{0}(x)=1, \quad p_{1}(x)=x-\alpha_{0},$

and the coefficients $\alpha_{n}$ (for $\left.n \geqslant 0\right)$ and $\beta_{n}$ (for $\left.n \geqslant 1\right)$ are given by

$\alpha_{n}=\frac{\left\langle p_{n}, x p_{n}\right\rangle}{\left\langle p_{n}, p_{n}\right\rangle}, \quad \beta_{n}=\frac{\left\langle p_{n}, p_{n}\right\rangle}{\left\langle p_{n-1}, p_{n-1}\right\rangle}$

Prove that this defines a sequence of monic orthogonal polynomials on $[a, b]$.

(c) The Hermite polynomials $H e_{n}(x)$ are orthogonal on the interval $(-\infty, \infty)$ with weight function $e^{-x^{2} / 2}$. Given that

$H e_{n}(x)=(-1)^{n} e^{x^{2} / 2} \frac{d^{n}}{d x^{n}}\left(e^{-x^{2} / 2}\right)$

deduce that the Hermite polynomials satisfy a relation of the form $(*)$ with $\alpha_{n}=0$ and $\beta_{n}=n$. Show that $\left\langle H e_{n}, H e_{n}\right\rangle=n ! \sqrt{2 \pi}$.

(d) State, without proof, how the properties of the Hermite polynomial $\operatorname{He}_{N}(x)$, for some positive integer $N$, can be used to estimate the integral

$\int_{-\infty}^{\infty} f(x) e^{-x^{2} / 2} d x$

where $f(x)$ is a given function, by the method of Gaussian quadrature. For which polynomials is the quadrature formula exact?

Paper 3, Section II, H

comment(a) Suppose that $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^{m}$, with $n \geqslant m$. What does it mean for $x \in \mathbb{R}^{n}$ to be a basic feasible solution of the equation $A x=b ?$

Assume that the $m$ rows of $A$ are linearly independent, every set of $m$ columns is linearly independent, and every basic solution has exactly $m$ non-zero entries. Prove that the extreme points of $\mathcal{X}(b)=\{x \geqslant 0: A x=b\}$ are the basic feasible solutions of $A x=b$. [Here, $x \geqslant 0$ means that each of the coordinates of $x$ are at least 0 .]

(b) Use the simplex method to solve the linear program

$\begin{array}{cl} \max & 4 x_{1}+3 x_{2}+7 x_{3} \\ \text { s.t. } & x_{1}+3 x_{2}+x_{3} \leqslant 14 \\ & 4 x_{1}+3 x_{2}+2 x_{3} \leqslant 5 \\ & -x_{1}+x_{2}-x_{3} \geqslant-2 \\ & x_{1}, x_{2}, x_{3} \geqslant 0 \end{array}$

Paper 3, Section $I$, B

commentConsider a quantum mechanical particle moving in two dimensions with Cartesian coordinates $x, y$. Show that, for wavefunctions with suitable decay as $x^{2}+y^{2} \rightarrow \infty$, the operators

$x \quad \text { and } \quad-i \hbar \frac{\partial}{\partial x}$

are Hermitian, and similarly

$y \text { and }-i \hbar \frac{\partial}{\partial y}$

are Hermitian.

Show that if $F$ and $G$ are Hermitian operators, then

$\frac{1}{2}(F G+G F)$

is Hermitian. Deduce that

$L=-i \hbar\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right) \quad \text { and } \quad D=-i \hbar\left(x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y}+1\right)$

are Hermitian. Show that

$[L, D]=0 .$

Paper 3, Section II, B

commentConsider a particle of unit mass in a one-dimensional square well potential

$V(x)=0 \text { for } 0 \leqslant x \leqslant \pi,$

with $V$ infinite outside. Find all the stationary states $\psi_{n}(x)$ and their energies $E_{n}$, and write down the general normalized solution of the time-dependent Schrödinger equation in terms of these.

The particle is initially constrained by a barrier to be in the ground state in the narrower square well potential

$V(x)=0 \quad \text { for } \quad 0 \leqslant x \leqslant \frac{\pi}{2}$

with $V$ infinite outside. The barrier is removed at time $t=0$, and the wavefunction is instantaneously unchanged. Show that the particle is now in a superposition of stationary states of the original potential well, and calculate the probability that an energy measurement will yield the result $E_{n}$.

Paper 3, Section II, H

commentSuppose that $X_{1}, \ldots, X_{n}$ are i.i.d. $N\left(\mu, \sigma^{2}\right)$. Let

$\bar{X}=\frac{1}{n} \sum_{i=1}^{n} X_{i} \quad \text { and } \quad S_{X X}=\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}$

(a) Compute the distributions of $\bar{X}$ and $S_{X X}$ and show that $\bar{X}$ and $S_{X X}$ are independent.

(b) Write down the distribution of $\sqrt{n}(\bar{X}-\mu) / \sqrt{S_{X X} /(n-1)}$.

(c) For $\alpha \in(0,1)$, find a $100(1-\alpha) \%$ confidence interval in each of the following situations: (i) for $\mu$ when $\sigma^{2}$ is known; (ii) for $\mu$ when $\sigma^{2}$ is not known; (iii) for $\sigma^{2}$ when $\mu$ is not known.

(d) Suppose that $\widetilde{X}_{1}, \ldots, \widetilde{X}_{\widetilde{n}}$ are i.i.d. $N\left(\widetilde{\mu}, \widetilde{\sigma}^{2}\right)$. Explain how you would use the $F$ test to test the hypothesis $H_{1}: \sigma^{2}>\tilde{\sigma}^{2}$ against the hypothesis $H_{0}: \sigma^{2}=\tilde{\sigma}^{2}$. Does the $F$ test depend on whether $\mu, \widetilde{\mu}$ are known?

Paper 3, Section I, A

commentThe function $f$ with domain $x>0$ is defined by $f(x)=\frac{1}{a} x^{a}$, where $a>1$. Verify that $f$ is convex, using an appropriate sufficient condition.

Determine the Legendre transform $f^{*}$ of $f$, specifying clearly its domain of definition, and find $\left(f^{*}\right)^{*}$.

Show that

$\frac{x^{r}}{r}+\frac{y^{s}}{s} \geqslant x y$

where $x, y>0$ and $r$ and $s$ are positive real numbers such that $\frac{1}{r}+\frac{1}{s}=1$.