Part IB, 2017, Paper 1

# Part IB, 2017, Paper 1

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Paper 1, Section II, G

commentWhat does it mean to say that a real-valued function on a metric space is uniformly continuous? Show that a continuous function on a closed interval in $\mathbb{R}$ is uniformly continuous.

What does it mean to say that a real-valued function on a metric space is Lipschitz? Show that if a function is Lipschitz then it is uniformly continuous.

Which of the following statements concerning continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ are true and which are false? Justify your answers.

(i) If $f$ is bounded then $f$ is uniformly continuous.

(ii) If $f$ is differentiable and $f^{\prime}$ is bounded, then $f$ is uniformly continuous.

(iii) There exists a sequence of uniformly continuous functions converging pointwise to $f$.

Paper 1, Section I, A

commentLet $F(z)=u(x, y)+i v(x, y)$ where $z=x+i y$. Suppose $F(z)$ is an analytic function of $z$ in a domain $\mathcal{D}$ of the complex plane.

Derive the Cauchy-Riemann equations satisfied by $u$ and $v$.

For $u=\frac{x}{x^{2}+y^{2}}$ find a suitable function $v$ and domain $\mathcal{D}$ such that $F=u+i v$ is analytic in $\mathcal{D}$.

Paper 1, Section II, A

comment(a) Let $f(z)$ be defined on the complex plane such that $z f(z) \rightarrow 0$ as $|z| \rightarrow \infty$ and $f(z)$ is analytic on an open set containing $\operatorname{Im}(z) \geqslant-c$, where $c$ is a positive real constant.

Let $C_{1}$ be the horizontal contour running from $-\infty-i c$ to $+\infty-i c$ and let

$F(\lambda)=\frac{1}{2 \pi i} \int_{C_{1}} \frac{f(z)}{z-\lambda} d z$

By evaluating the integral, show that $F(\lambda)$ is analytic for $\operatorname{Im}(\lambda)>-c$.

(b) Let $g(z)$ be defined on the complex plane such that $z g(z) \rightarrow 0$ as $|z| \rightarrow \infty$ with $\operatorname{Im}(z) \geqslant-c$. Suppose $g(z)$ is analytic at all points except $z=\alpha_{+}$and $z=\alpha_{-}$which are simple poles with $\operatorname{Im}\left(\alpha_{+}\right)>c$ and $\operatorname{Im}\left(\alpha_{-}\right)<-c$.

Let $C_{2}$ be the horizontal contour running from $-\infty+i c$ to $+\infty+i c$, and let

$\begin{gathered} H(\lambda)=\frac{1}{2 \pi i} \int_{C_{1}} \frac{g(z)}{z-\lambda} d z \\ J(\lambda)=-\frac{1}{2 \pi i} \int_{C_{2}} \frac{g(z)}{z-\lambda} d z . \end{gathered}$

(i) Show that $H(\lambda)$ is analytic for $\operatorname{Im}(\lambda)>-c$.

(ii) Show that $J(\lambda)$ is analytic for $\operatorname{Im}(\lambda)<c$.

(iii) Show that if $-c<\operatorname{Im}(\lambda)<c$ then $H(\lambda)+J(\lambda)=g(\lambda)$.

[You should be careful to make sure you consider all points in the required regions.]

Paper 1, Section II, C

commentWrite down Maxwell's equations for the electric field $\mathbf{E}(\mathbf{x}, t)$ and the magnetic field $\mathbf{B}(\mathbf{x}, t)$ in a vacuum. Deduce that both $\mathbf{E}$ and $\mathbf{B}$ satisfy a wave equation, and relate the wave speed $c$ to the physical constants $\epsilon_{0}$ and $\mu_{0}$.

Verify that there exist plane-wave solutions of the form

$\begin{aligned} &\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{e} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right] \\ &\mathbf{B}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{b} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right] \end{aligned}$

where $\mathbf{e}$ and $\mathbf{b}$ are constant complex vectors, $\mathbf{k}$ is a constant real vector and $\omega$ is a real constant. Derive the dispersion relation that relates the angular frequency $\omega$ of the wave to the wavevector $\mathbf{k}$, and give the algebraic relations between the vectors $\mathbf{e}, \mathbf{b}$ and $\mathbf{k}$ implied by Maxwell's equations.

Let $\mathbf{n}$ be a constant real unit vector. Suppose that a perfect conductor occupies the region $\mathbf{n} \cdot \mathbf{x}<0$ with a plane boundary $\mathbf{n} \cdot \mathbf{x}=0$. In the vacuum region $\mathbf{n} \cdot \mathbf{x}>0$, a plane electromagnetic wave of the above form, with $\mathbf{k} \cdot \mathbf{n}<0$, is incident on the plane boundary. Write down the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the surface of the conductor. Show that Maxwell's equations and the boundary conditions are satisfied if the solution in the vacuum region is the sum of the incident wave given above and a reflected wave of the form

$\begin{aligned} &\mathbf{E}^{\prime}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{e}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{x}-\omega t\right)}\right] \\ &\mathbf{B}^{\prime}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{b}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{x}-\omega t\right)}\right] \end{aligned}$

where

$\begin{aligned} &\mathbf{e}^{\prime}=-\mathbf{e}+2(\mathbf{n} \cdot \mathbf{e}) \mathbf{n} \\ &\mathbf{b}^{\prime}=\mathbf{b}-2(\mathbf{n} \cdot \mathbf{b}) \mathbf{n} \\ &\mathbf{k}^{\prime}=\mathbf{k}-2(\mathbf{n} \cdot \mathbf{k}) \mathbf{n} \end{aligned}$

Paper 1, Section I, D

commentFor each of the flows

(i) $\mathbf{u}=\left(2 x y, x^{2}+y^{2}\right)$

(ii) $\mathbf{u}=(-2 y,-2 x)$

determine whether or not the flow is incompressible and/or irrotational. Find the associated velocity potential and/or stream function when appropriate. For either one of the flows, sketch the streamlines of the flow, indicating the direction of the flow.

Paper 1, Section II, D

commentA layer of thickness $h$ of fluid of density $\rho$ and dynamic viscosity $\mu$ flows steadily down and parallel to a rigid plane inclined at angle $\alpha$ to the horizontal. Wind blows over the surface of the fluid and exerts a stress $S$ on the surface of the fluid in the upslope direction.

(a) Draw a diagram of this situation, including indications of the applied stresses and body forces, a suitable coordinate system and a representation of the expected velocity profile.

(b) Write down the equations and boundary conditions governing the flow, with a brief description of each, paying careful attention to signs. Solve these equations to determine the pressure and velocity fields.

(c) Determine the volume flux and show that there is no net flux if

$S=\frac{2}{3} \rho g h \sin \alpha$

Draw a sketch of the corresponding velocity profile.

(d) Determine the value of $S$ for which the shear stress on the rigid plane is zero and draw a sketch of the corresponding velocity profile.

Paper 1, Section I, G

commentGive the definition for the area of a hyperbolic triangle with interior angles $\alpha, \beta, \gamma$.

Let $n \geqslant 3$. Show that the area of a convex hyperbolic $n$-gon with interior angles $\alpha_{1}, \ldots, \alpha_{n}$ is $(n-2) \pi-\sum \alpha_{i}$.

Show that for every $n \geqslant 3$ and for every $A$ with $0<A<(n-2) \pi$ there exists a regular hyperbolic $n$-gon with area $A$.

Paper 1, Section II, 10E

comment(a) State Sylow's theorem.

(b) Let $G$ be a finite simple non-abelian group. Let $p$ be a prime number. Show that if $p$ divides $|G|$, then $|G|$ divides $n_{p} ! / 2$ where $n_{p}$ is the number of Sylow $p$-subgroups of $G$.

(c) Let $G$ be a group of order 48 . Show that $G$ is not simple. Find an example of $G$ which has no normal Sylow 2-subgroup.

Paper 1, Section I, F

commentState and prove the Steinitz Exchange Lemma.

Deduce that, for a subset $S$ of $\mathbb{R}^{n}$, any two of the following imply the third:

(i) $S$ is linearly independent

(ii) $S$ is spanning

(iii) $S$ has exactly $n$ elements

Let $e_{1}, e_{2}$ be a basis of $\mathbb{R}^{2}$. For which values of $\lambda$ do $\lambda e_{1}+e_{2}, e_{1}+\lambda e_{2}$ form a basis of $\mathbb{R}^{2} ?$

Paper 1, Section II, F

commentLet $U$ and $V$ be finite-dimensional real vector spaces, and let $\alpha: U \rightarrow V$ be a surjective linear map. Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate.

(i) There is a linear map $\beta: V \rightarrow U$ such that $\beta \alpha$ is the identity map on $U$.

(ii) There is a linear map $\beta: V \rightarrow U$ such that $\alpha \beta$ is the identity map on $V$.

(iii) There is a subspace $W$ of $U$ such that the restriction of $\alpha$ to $W$ is an isomorphism from $W$ to $V$.

(iv) If $X$ and $Y$ are subspaces of $U$ with $U=X \oplus Y$ then $V=\alpha(X) \oplus \alpha(Y)$.

(v) If $X$ and $Y$ are subspaces of $U$ with $V=\alpha(X) \oplus \alpha(Y)$ then $U=X \oplus Y$.

Paper 1, Section II, H

commentA rich and generous man possesses $n$ pounds. Some poor cousins arrive at his mansion. Being generous he decides to give them money. On day 1 , he chooses uniformly at random an integer between $n-1$ and 1 inclusive and gives it to the first cousin. Then he is left with $x$ pounds. On day 2 , he chooses uniformly at random an integer between $x-1$ and 1 inclusive and gives it to the second cousin and so on. If $x=1$ then he does not give the next cousin any money. His choices of the uniform numbers are independent. Let $X_{i}$ be his fortune at the end of day $i$.

Show that $X$ is a Markov chain and find its transition probabilities.

Let $\tau$ be the first time he has 1 pound left, i.e. $\tau=\min \left\{i \geqslant 1: X_{i}=1\right\}$. Show that

$\mathbb{E}[\tau]=\sum_{i=1}^{n-1} \frac{1}{i}$

Paper 1, Section II, 14B

comment(a)

(i) Compute the Fourier transform $\tilde{h}(k)$ of $h(x)=e^{-a|x|}$, where $a$ is a real positive constant.

(ii) Consider the boundary value problem

$-\frac{d^{2} u}{d x^{2}}+\omega^{2} u=e^{-|x|} \quad \text { on }-\infty<x<\infty$

with real constant $\omega \neq \pm 1$ and boundary condition $u(x) \rightarrow 0$ as $|x| \rightarrow \infty$.

Find the Fourier transform $\tilde{u}(k)$ of $u(x)$ and hence solve the boundary value problem. You should clearly state any properties of the Fourier transform that you use.

(b) Consider the wave equation

$v_{t t}=v_{x x} \quad \text { on } \quad-\infty<x<\infty \text { and } t>0$

with initial conditions

$v(x, 0)=f(x) \quad v_{t}(x, 0)=g(x) .$

Show that the Fourier transform $\tilde{v}(k, t)$ of the solution $v(x, t)$ with respect to the variable $x$ is given by

$\tilde{v}(k, t)=\tilde{f}(k) \cos k t+\frac{\tilde{g}(k)}{k} \sin k t$

where $\tilde{f}(k)$ and $\tilde{g}(k)$ are the Fourier transforms of the initial conditions. Starting from $\tilde{v}(k, t)$ derive d'Alembert's solution for the wave equation:

$v(x, t)=\frac{1}{2}(f(x-t)+f(x+t))+\frac{1}{2} \int_{x-t}^{x+t} g(\xi) d \xi$

You should state clearly any properties of the Fourier transform that you use.

Paper 1, Section II, E

commentConsider $\mathbb{R}$ and $\mathbb{R}^{2}$ with their usual Euclidean topologies.

(a) Show that a non-empty subset of $\mathbb{R}$ is connected if and only if it is an interval. Find a compact subset $K \subset \mathbb{R}$ for which $\mathbb{R} \backslash K$ has infinitely many connected components.

(b) Let $T$ be a countable subset of $\mathbb{R}^{2}$. Show that $\mathbb{R}^{2} \backslash T$ is path-connected. Deduce that $\mathbb{R}^{2}$ is not homeomorphic to $\mathbb{R}$.

Paper 1, Section I, C

commentGiven $n+1$ real points $x_{0}<x_{1}<\cdots<x_{n}$, define the Lagrange cardinal polynomials $\ell_{i}(x), i=0,1, \ldots, n$. Let $p(x)$ be the polynomial of degree $n$ that interpolates the function $f \in C^{n}\left[x_{0}, x_{n}\right]$ at these points. Express $p(x)$ in terms of the values $f_{i}=f\left(x_{i}\right)$, $i=0,1, \ldots, n$ and the Lagrange cardinal polynomials.

Define the divided difference $f\left[x_{0}, x_{1}, \ldots, x_{n}\right]$ and give an expression for it in terms of $f_{0}, f_{1}, \ldots, f_{n}$ and $x_{0}, x_{1}, \ldots, x_{n}$. Prove that

$f\left[x_{0}, x_{1}, \ldots, x_{n}\right]=\frac{1}{n !} f^{(n)}(\xi)$

for some number $\xi \in\left[x_{0}, x_{n}\right]$.

Paper 1, Section II, C

commentA three-stage explicit Runge-Kutta method for solving the autonomous ordinary differential equation

$\frac{d y}{d t}=f(y)$

is given by

$y_{n+1}=y_{n}+h\left(b_{1} k_{1}+b_{2} k_{2}+b_{3} k_{3}\right),$

where

$\begin{aligned} &k_{1}=f\left(y_{n}\right) \\ &k_{2}=f\left(y_{n}+h a_{1} k_{1}\right) \\ &k_{3}=f\left(y_{n}+h\left(a_{2} k_{1}+a_{3} k_{2}\right)\right) \end{aligned}$

and $h>0$ is the time-step. Derive sufficient conditions on the coefficients $b_{1}, b_{2}, b_{3}, a_{1}$, $a_{2}$ and $a_{3}$ for the method to be of third order.

Assuming that these conditions hold, verify that $-\frac{5}{2}$ belongs to the linear stability domain of the method.

Paper 1, Section I, H

commentSolve the following linear programming problem using the simplex method:

$\begin{array}{r} \max \left(x_{1}+2 x_{2}+x_{3}\right) \\ \text { subject to } \quad x_{1}, x_{2}, x_{3} \geqslant 0 \\ x_{1}+x_{2}+2 x_{3} \leqslant 10 \\ 2 x_{1}+x_{2}+3 x_{3} \leqslant 15 \end{array}$

Suppose we now subtract $\Delta \in[0,10]$ from the right hand side of the last two constraints. Find the new optimal value.

Paper 1, Section II, B

commentConsider the time-independent Schrödinger equation in one dimension for a particle of mass $m$ with potential $V(x)$.

(a) Show that if the potential is an even function then any non-degenerate stationary state has definite parity.

(b) A particle of mass $m$ is subject to the potential $V(x)$ given by

$V(x)=-\lambda(\delta(x-a)+\delta(x+a))$

where $\lambda$ and $a$ are real positive constants and $\delta(x)$ is the Dirac delta function.

Derive the conditions satisfied by the wavefunction $\psi(x)$ around the points $x=\pm a$.

Show (using a graphical method or otherwise) that there is a bound state of even parity for any $\lambda>0$, and that there is an odd parity bound state only if $\lambda>\hbar^{2} /(2 m a)$. [Hint: You may assume without proof that the functions $x \tanh x$ and $x \operatorname{coth} x$ are monotonically increasing for $x>0$.]

Paper 1, Section I, H

comment(a) State and prove the Rao-Blackwell theorem.

(b) Let $X_{1}, \ldots, X_{n}$ be an independent sample from $\operatorname{Poisson}(\lambda)$ with $\theta=e^{-\lambda}$ to be estimated. Show that $Y=1_{\{0\}}\left(X_{1}\right)$ is an unbiased estimator of $\theta$ and that $T=\sum_{i} X_{i}$ is a sufficient statistic.

What is $\mathbb{E}[Y \mid T] ?$

Paper 1, Section II, H

comment(a) Give the definitions of a sufficient and a minimal sufficient statistic $T$ for an unknown parameter $\theta$.

Let $X_{1}, X_{2}, \ldots, X_{n}$ be an independent sample from the geometric distribution with success probability $1 / \theta$ and mean $\theta>1$, i.e. with probability mass function

$p(m)=\frac{1}{\theta}\left(1-\frac{1}{\theta}\right)^{m-1} \text { for } m=1,2, \ldots$

Find a minimal sufficient statistic for $\theta$. Is your statistic a biased estimator of $\theta ?$

[You may use results from the course provided you state them clearly.]

(b) Define the bias of an estimator. What does it mean for an estimator to be unbiased?

Suppose that $Y$ has the truncated Poisson distribution with probability mass function

$p(y)=\left(e^{\theta}-1\right)^{-1} \cdot \frac{\theta^{y}}{y !} \quad \text { for } y=1,2, \ldots$

Show that the only unbiased estimator $T$ of $1-e^{-\theta}$ based on $Y$ is obtained by taking $T=0$ if $Y$ is odd and $T=2$ if $Y$ is even.

Is this a useful estimator? Justify your answer.

Paper 1, Section I, D

commentDerive the Euler-Lagrange equation for the function $u(x, y)$ that gives a stationary value of

$I[u]=\int_{\mathcal{D}} L\left(x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}\right) d x d y$

where $\mathcal{D}$ is a bounded domain in the $(x, y)$-plane and $u$ is fixed on the boundary $\partial \mathcal{D}$.

Find the equation satisfied by the function $u$ that gives a stationary value of

$I=\int_{\mathcal{D}}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}+k^{2} u^{2}\right] d x d y$

where $k$ is a constant and $u$ is prescribed on $\partial \mathcal{D}$.