Part IB, 2010, Paper 2

# Part IB, 2010, Paper 2

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Paper 2, Section I, G

commentLet $c>1$ be a real number, and let $F_{c}$ be the space of sequences $\mathbf{a}=\left(a_{1}, a_{2}, \ldots\right)$ of real numbers $a_{i}$ with $\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ convergent. Show that $\|\mathbf{a}\|_{c}=\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ defines a norm on $F_{c}$.

Let $F$ denote the space of sequences a with $\left|a_{i}\right|$ bounded; show that $F \subset F_{c}$. If $c^{\prime}>c$, show that the norms on $F$ given by restricting to $F$ the norms $\|\cdot\|_{c}$ on $F_{c}$ and $\|\cdot\|_{c^{\prime}}$ on $F_{c^{\prime}}$ are not Lipschitz equivalent.

By considering sequences of the form $\mathbf{a}^{(n)}=\left(a, a^{2}, \ldots, a^{n}, 0,0, \ldots\right)$ in $F$, for $a$ an appropriate real number, or otherwise, show that $F$ (equipped with the norm $\|.\|_{c}$ ) is not complete.

Paper 2, Section II, G

commentSuppose the functions $f_{n}(n=1,2, \ldots)$ are defined on the open interval $(0,1)$ and that $f_{n}$ tends uniformly on $(0,1)$ to a function $f$. If the $f_{n}$ are continuous, show that $f$ is continuous. If the $f_{n}$ are differentiable, show by example that $f$ need not be differentiable.

Assume now that each $f_{n}$ is differentiable and the derivatives $f_{n}^{\prime}$ converge uniformly on $(0,1)$. For any given $c \in(0,1)$, we define functions $g_{c, n}$ by

$g_{c, n}(x)= \begin{cases}\frac{f_{n}(x)-f_{n}(c)}{x-c} & \text { for } x \neq c, \\ f_{n}^{\prime}(c) & \text { for } x=c .\end{cases}$

Show that each $g_{c, n}$ is continuous. Using the general principle of uniform convergence (the Cauchy criterion) and the Mean Value Theorem, or otherwise, prove that the functions $g_{c, n}$ converge uniformly to a continuous function $g_{c}$ on $(0,1)$, where

$g_{c}(x)=\frac{f(x)-f(c)}{x-c} \quad \text { for } x \neq c$

Deduce that $f$ is differentiable on $(0,1)$.

Paper 2, Section II, A

comment(a) Prove that a complex differentiable map, $f(z)$, is conformal, i.e. preserves angles, provided a certain condition holds on the first complex derivative of $f(z)$.

(b) Let $D$ be the region

$D:=\{z \in \mathbb{C}:|z-1|>1 \text { and }|z-2|<2\}$

Draw the region $D$. It might help to consider the two sets

$\begin{aligned} &C(1):=\{z \in \mathbb{C}:|z-1|=1\} \\ &C(2):=\{z \in \mathbb{C}:|z-2|=2\} \end{aligned}$

(c) For the transformations below identify the images of $D$.

Step 1: The first map is $f_{1}(z)=\frac{z-1}{z}$,

Step 2: The second map is the composite $f_{2} f_{1}$ where $f_{2}(z)=\left(z-\frac{1}{2}\right) i$,

Step 3: The third map is the composite $f_{3} f_{2} f_{1}$ where $f_{3}(z)=e^{2 \pi z}$.

(d) Write down the inverse map to the composite $f_{3} f_{2} f_{1}$, explaining any choices of branch.

[The composite $f_{2} f_{1}$ means $f_{2}\left(f_{1}(z)\right)$.]

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

commentWrite down Maxwell's equations for electromagnetic fields in a non-polarisable and non-magnetisable medium.

Show that the homogenous equations (those not involving charge or current densities) can be solved in terms of vector and scalar potentials $\mathbf{A}$ and $\phi$.

Then re-express the inhomogeneous equations in terms of $\mathbf{A}, \phi$ and $f=\nabla \cdot \mathbf{A}+c^{-2} \dot{\phi}$. Show that the potentials can be chosen so as to set $f=0$ and hence rewrite the inhomogeneous equations as wave equations for the potentials. [You may assume that the inhomogeneous wave equation $\nabla^{2} \psi-c^{-2} \ddot{\psi}=\sigma(\mathbf{x}, t)$ always has a solution $\psi(\mathbf{x}, t)$ for any given $\sigma(\mathbf{x}, t)$.]

Paper 2, Section II, C

commentA steady current $I_{2}$ flows around a loop $\mathcal{C}_{2}$ of a perfectly conducting narrow wire. Assuming that the gauge condition $\nabla \cdot \mathbf{A}=0$ holds, the vector potential at points away from the loop may be taken to be

$\mathbf{A}(\mathbf{r})=\frac{\mu_{0} I_{2}}{4 \pi} \oint_{\mathcal{C}_{2}} \frac{d \mathbf{r}_{2}}{\left|\mathbf{r}-\mathbf{r}_{2}\right|}$

First verify that the gauge condition is satisfied here. Then obtain the Biot-Savart formula for the magnetic field

$\mathbf{B}(\mathbf{r})=\frac{\mu_{0} I_{2}}{4 \pi} \oint_{\mathcal{C}_{2}} \frac{d \mathbf{r}_{2} \times\left(\mathbf{r}-\mathbf{r}_{2}\right)}{\left|\mathbf{r}-\mathbf{r}_{2}\right|^{3}}$

Next suppose there is a similar but separate loop $\mathcal{C}_{1}$ with current $I_{1}$. Show that the magnetic force exerted on loop $\mathcal{C}_{1}$ by loop $\mathcal{C}_{2}$ is

$\mathbf{F}_{12}=\frac{\mu_{0} I_{1} I_{2}}{4 \pi} \oint_{\mathcal{C}_{1}} \oint_{\mathcal{C}_{2}} d \mathbf{r}_{1} \times\left(d \mathbf{r}_{2} \times \frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|^{3}}\right)$

Is this consistent with Newton's third law? Justify your answer.

Paper 2, Section I, B

commentWrite down an expression for the velocity field of a line vortex of strength $\kappa$.

Consider $N$ identical line vortices of strength $\kappa$ arranged at equal intervals round a circle of radius $a$. Show that the vortices all move around the circle at constant angular velocity $(N-1) \kappa /\left(4 \pi a^{2}\right)$.

Paper 2, Section II, F

commentSuppose that $a>0$ and that $S \subset \mathbb{R}^{3}$ is the half-cone defined by $z^{2}=a\left(x^{2}+y^{2}\right)$, $z>0$. By using an explicit smooth parametrization of $S$, calculate the curvature of $S$.

Describe the geodesics on $S$. Show that for $a=3$, no geodesic intersects itself, while for $a>3$ some geodesic does so.

Paper 2, Section I, $2 \mathrm{H}$

commentGive the definition of conjugacy classes in a group $G$. How many conjugacy classes are there in the symmetric group $S_{4}$ on four letters? Briefly justify your answer.

Paper 2, Section II, H

commentFor ideals $I, J$ of a ring $R$, their product $I J$ is defined as the ideal of $R$ generated by the elements of the form $x y$ where $x \in I$ and $y \in J$.

(1) Prove that, if a prime ideal $P$ of $R$ contains $I J$, then $P$ contains either $I$ or $J$.

(2) Give an example of $R, I$ and $J$ such that the two ideals $I J$ and $I \cap J$ are different from each other.

(3) Prove that there is a natural bijection between the prime ideals of $R / I J$ and the prime ideals of $R /(I \cap J)$.

Paper 2, Section I, F

commentSuppose that $\phi$ is an endomorphism of a finite-dimensional complex vector space.

(i) Show that if $\lambda$ is an eigenvalue of $\phi$, then $\lambda^{2}$ is an eigenvalue of $\phi^{2}$.

(ii) Show conversely that if $\mu$ is an eigenvalue of $\phi^{2}$, then there is an eigenvalue $\lambda$ of $\phi$ with $\lambda^{2}=\mu$.

Paper 2, Section II, F

comment(i) Show that two $n \times n$ complex matrices $A, B$ are similar (i.e. there exists invertible $P$ with $A=P^{-1} B P$ ) if and only if they represent the same linear map $\mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ with respect to different bases.

(ii) Explain the notion of Jordan normal form of a square complex matrix.

(iii) Show that any square complex matrix $A$ is similar to its transpose.

(iv) If $A$ is invertible, describe the Jordan normal form of $A^{-1}$ in terms of that of $A$.

Justify your answers.

Paper 2, Section II, E

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a simple, symmetric random walk on the integers $\{\ldots,-1,0,1, \ldots\}$, with $X_{0}=0$ and $\mathbb{P}\left(X_{n+1}=i \pm 1 \mid X_{n}=i\right)=1 / 2$. For each integer $a \geqslant 1$, let $T_{a}=\inf \left\{n \geqslant 0: X_{n}=a\right\}$. Show that $T_{a}$ is a stopping time.

Define a random variable $Y_{n}$ by the rule

$Y_{n}= \begin{cases}X_{n} & \text { if } n<T_{a} \\ 2 a-X_{n} & \text { if } n \geqslant T_{a}\end{cases}$

Show that $\left(Y_{n}\right)_{n} \geqslant 0$ is also a simple, symmetric random walk.

Let $M_{n}=\max _{0 \leqslant i \leqslant n} X_{n}$. Explain why $\left\{M_{n} \geqslant a\right\}=\left\{T_{a} \leqslant n\right\}$ for $a \geqslant 0$. By using the process $\left(Y_{n}\right)_{n \geqslant 0}$ constructed above, show that, for $a \geqslant 0$,

$\mathbb{P}\left(M_{n} \geqslant a, X_{n} \leqslant a-1\right)=\mathbb{P}\left(X_{n} \geqslant a+1\right),$

and thus

$\mathbb{P}\left(M_{n} \geqslant a\right)=\mathbb{P}\left(X_{n} \geqslant a\right)+\mathbb{P}\left(X_{n} \geqslant a+1\right)$

Hence compute

$\mathbb{P}\left(M_{n}=a\right)$

when $a$ and $n$ are positive integers with $n \geqslant a$. [Hint: if $n$ is even, then $X_{n}$ must be even, and if $n$ is odd, then $X_{n}$ must be odd.]

Paper 2, Section I, A

commentConsider the initial value problem

$\mathcal{L} x(t)=f(t), \quad x(0)=0, \quad \dot{x}(0)=0, \quad t \geqslant 0,$

where $\mathcal{L}$ is a second-order linear operator involving differentiation with respect to $t$. Explain briefly how to solve this by using a Green's function.

Now consider

$\ddot{x}(t)= \begin{cases}a & 0 \leqslant t \leqslant T \\ 0 & T<t<\infty\end{cases}$

where $a$ is a constant, subject to the same initial conditions. Solve this using the Green's function, and explain how your answer is related to a problem in Newtonian dynamics.

Paper 2, Section II, B

commentExplain briefly the use of the method of characteristics to solve linear first-order partial differential equations.

Use the method to solve the problem

$(x-y) \frac{\partial u}{\partial x}+(x+y) \frac{\partial u}{\partial y}=\alpha u$

where $\alpha$ is a constant, with initial condition $u(x, 0)=x^{2}, x \geqslant 0$.

By considering your solution explain:

(i) why initial conditions cannot be specified on the whole $x$-axis;

(ii) why a single-valued solution in the entire plane is not possible if $\alpha \neq 2$.

Paper 2, Section I, H

commentOn the set $\mathbb{Q}$ of rational numbers, the 3 -adic metric $d_{3}$ is defined as follows: for $x, y \in \mathbb{Q}$, define $d_{3}(x, x)=0$ and $d_{3}(x, y)=3^{-n}$, where $n$ is the integer satisfying $x-y=3^{n} u$ where $u$ is a rational number whose denominator and numerator are both prime to 3 .

(1) Show that this is indeed a metric on $\mathbb{Q}$.

(2) Show that in $\left(\mathbb{Q}, d_{3}\right)$, we have $3^{n} \rightarrow 0$ as $n \rightarrow \infty$ while $3^{-n} \nrightarrow \infty$ as $n \rightarrow \infty$. Let $d$ be the usual metric $d(x, y)=|x-y|$ on $\mathbb{Q}$. Show that neither the identity map $(\mathbb{Q}, d) \rightarrow\left(\mathbb{Q}, d_{3}\right)$ nor its inverse is continuous.

Paper 2, Section II, C

commentConsider the initial value problem for an autonomous differential equation

$y^{\prime}(t)=f(y(t)), \quad y(0)=y_{0} \text { given }$

and its approximation on a grid of points $t_{n}=n h, n=0,1,2, \ldots$. Writing $y_{n}=y\left(t_{n}\right)$, it is proposed to use one of two Runge-Kutta schemes defined by

$y_{n+1}=y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right)$

where $k_{1}=h f\left(y_{n}\right)$ and

$k_{2}= \begin{cases}h f\left(y_{n}+k_{1}\right) & \text { scheme I } \\ h f\left(y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right)\right) & \text { scheme II }\end{cases}$

What is the order of each scheme? Determine the $A$-stability of each scheme.

Paper 2, Section I, E

commentConsider the function $\phi$ defined by

$\phi(b)=\inf \left\{x^{2}+y^{4}: x+2 y=b\right\}$

Use the Lagrangian sufficiency theorem to evaluate $\phi(3)$. Compute the derivative $\phi^{\prime}(3)$.

Paper 2, Section II, D

commentA particle of mass $m$ moves in a one-dimensional potential defined by

$V(x)= \begin{cases}\infty & \text { for } x<0 \\ 0 & \text { for } 0 \leqslant x \leqslant a \\ V_{0} & \text { for } a<x\end{cases}$

where $a$ and $V_{0}$ are positive constants. Defining $c=\left[2 m\left(V_{0}-E\right)\right]^{1 / 2} / \hbar$ and $k=$ $(2 m E)^{1 / 2} / \hbar$, show that for any allowed positive value $E$ of the energy with $E<V_{0}$ then

$c+k \cot k a=0$

Find the minimum value of $V_{0}$ for this equation to have a solution.

Find the normalized wave function for the particle. Write down an expression for the expectation value of $x$ in terms of two integrals, which you need not evaluate. Given that

$\langle x\rangle=\frac{1}{2 k}(k a-\tan k a),$

discuss briefly the possibility of $\langle x\rangle$ being greater than $a$. [Hint: consider the graph of - ka cot $k a$ against $k a .]$

Paper 2, Section I, E

commentA washing powder manufacturer wants to determine the effectiveness of a television advertisement. Before the advertisement is shown, a pollster asks 100 randomly chosen people which of the three most popular washing powders, labelled $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$, they prefer. After the advertisement is shown, another 100 randomly chosen people (not the same as before) are asked the same question. The results are summarized below.

\begin{tabular}{c|ccc} & $\mathrm{A}$ & $\mathrm{B}$ & $\mathrm{C}$ \ \hline before & 36 & 47 & 17 \ after & 44 & 33 & 23 \end{tabular}

Derive and carry out an appropriate test at the $5 \%$ significance level of the hypothesis that the advertisement has had no effect on people's preferences.

[You may find the following table helpful:

$\left.\begin{array}{c|cccccc} & \chi_{1}^{2} & \chi_{2}^{2} & \chi_{3}^{2} & \chi_{4}^{2} & \chi_{5}^{2} & \chi_{6}^{2} \\ \hline 95 \text { percentile } & 3.84 & 5.99 & 7.82 & 9.49 & 11.07 & 12.59\end{array} \cdot\right]$

Paper 2, Section II, D

commentDescribe briefly the method of Lagrange multipliers for finding the stationary points of a function $f(x, y)$ subject to a constraint $\phi(x, y)=0$.

A tent manufacturer wants to maximize the volume of a new design of tent, subject only to a constant weight (which is directly proportional to the amount of fabric used). The models considered have either equilateral-triangular or semi-circular vertical crosssection, with vertical planar ends in both cases and with floors of the same fabric. Which shape maximizes the volume for a given area $A$ of fabric?

[Hint: $(2 \pi)^{-1 / 2} 3^{-3 / 4}(2+\pi)<1 .$ ]