Part IB, 2013, Paper 4

# Part IB, 2013, Paper 4

### Jump to course

Paper 4, Section I, $3 F$

commentState and prove the chain rule for differentiable mappings $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ and $G: \mathbb{R}^{m} \rightarrow \mathbb{R}^{k}$.

Suppose now $F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ has image lying on the unit circle in $\mathbb{R}^{2}$. Prove that the determinant $\operatorname{det}\left(\left.D F\right|_{x}\right)$ vanishes for every $x \in \mathbb{R}^{2}$.

Paper 4, Section II, F

commentState the contraction mapping theorem.

A metric space $(X, d)$ is bounded if $\{d(x, y) \mid x, y \in X\}$ is a bounded subset of $\mathbb{R}$. Suppose $(X, d)$ is complete and bounded. Let $\operatorname{Maps}(X, X)$ denote the set of continuous $\operatorname{maps}$ from $X$ to itself. For $f, g \in \operatorname{Maps}(X, X)$, let

$\delta(f, g)=\sup _{x \in X} d(f(x), g(x))$

Prove that $(\operatorname{Maps}(X, X), \delta)$ is a complete metric space. Is the subspace $\mathcal{C} \subset \operatorname{Maps}(X, X)$ of contraction mappings a complete subspace?

Let $\tau: \mathcal{C} \rightarrow X$ be the map which associates to any contraction its fixed point. Prove that $\tau$ is continuous.

Paper 4, Section I, E

commentState Rouché's theorem. How many roots of the polynomial $z^{8}+3 z^{7}+6 z^{2}+1$ are contained in the annulus $1<|z|<2$ ?

Paper 4, Section II, D

commentLet $C_{1}$ and $C_{2}$ be the circles $x^{2}+y^{2}=1$ and $5 x^{2}-4 x+5 y^{2}=0$, respectively, and let $D$ be the (finite) region between the circles. Use the conformal mapping

$w=\frac{z-2}{2 z-1}$

to solve the following problem:

$\nabla^{2} \phi=0 \text { in } D \text { with } \phi=1 \text { on } C_{1} \text { and } \phi=2 \text { on } C_{2}$

Paper 4, Section I, D

commentThe infinite plane $z=0$ is earthed and the infinite plane $z=d$ carries a charge of $\sigma$ per unit area. Find the electrostatic potential between the planes.

Show that the electrostatic energy per unit area (of the planes $z=$ constant) between the planes can be written as either $\frac{1}{2} \sigma^{2} d / \epsilon_{0}$ or $\frac{1}{2} \epsilon_{0} V^{2} / d$, where $V$ is the potential at $z=d$.

The distance between the planes is now increased by $\alpha d$, where $\alpha$ is small. Show that the change in the energy per unit area is $\frac{1}{2} \sigma V \alpha$ if the upper plane $(z=d)$ is electrically isolated, and is approximately $-\frac{1}{2} \sigma V \alpha$ if instead the potential on the upper plane is maintained at $V$. Explain briefly how this difference can be accounted for.

Paper 4, Section II, A

commentThe axisymmetric, irrotational flow generated by a solid sphere of radius $a$ translating at velocity $U$ in an inviscid, incompressible fluid is represented by a velocity potential $\phi(r, \theta)$. Assume the fluid is at rest far away from the sphere. Explain briefly why $\nabla^{2} \phi=0$.

By trying a solution of the form $\phi(r, \theta)=f(r) g(\theta)$, show that

$\phi=-\frac{U a^{3} \cos \theta}{2 r^{2}}$

and write down the fluid velocity.

Show that the total kinetic energy of the fluid is $k M U^{2} / 4$ where $M$ is the mass of the sphere and $k$ is the ratio of the density of the fluid to the density of the sphere.

A heavy sphere (i.e. $k<1$ ) is released from rest in an inviscid fluid. Determine its speed after it has fallen a distance $h$ in terms of $M, k, g$ and $h$.

Note, in spherical polars:

$\begin{gathered} \boldsymbol{\nabla} \phi=\frac{\partial \phi}{\partial r} \mathbf{e}_{\mathbf{r}}+\frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{e}_{\theta} \\ \nabla^{2} \phi=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \phi}{\partial \theta}\right) \end{gathered}$

Paper 4, Section II, F

commentLet $\eta$ be a smooth curve in the $x z$-plane $\eta(s)=(f(s), 0, g(s))$, with $f(s)>0$ for every $s \in \mathbb{R}$ and $f^{\prime}(s)^{2}+g^{\prime}(s)^{2}=1$. Let $S$ be the surface obtained by rotating $\eta$ around the $z$-axis. Find the first fundamental form of $S$.

State the equations for a curve $\gamma:(a, b) \rightarrow S$ parametrised by arc-length to be a geodesic.

A parallel on $S$ is the closed circle swept out by rotating a single point of $\eta$. Prove that for every $n \in \mathbb{Z}_{>0}$ there is some $\eta$ for which exactly $n$ parallels are geodesics. Sketch possible such surfaces $S$ in the cases $n=1$ and $n=2$.

If every parallel is a geodesic, what can you deduce about $S$ ? Briefly justify your answer.

Paper 4, Section I, $2 G$

commentLet $p$ be a prime number, and $G$ be a non-trivial finite group whose order is a power of $p$. Show that the size of every conjugacy class in $G$ is a power of $p$. Deduce that the centre $Z$ of $G$ has order at least $p$.

Paper 4, Section II, 11G

commentLet $R$ be an integral domain, and $M$ be a finitely generated $R$-module.

(i) Let $S$ be a finite subset of $M$ which generates $M$ as an $R$-module. Let $T$ be a maximal linearly independent subset of $S$, and let $N$ be the $R$-submodule of $M$ generated by $T$. Show that there exists a non-zero $r \in R$ such that $r x \in N$ for every $x \in M$.

(ii) Now assume $M$ is torsion-free, i.e. $r x=0$ for $r \in R$ and $x \in M$ implies $r=0$ or $x=0$. By considering the map $M \rightarrow N$ mapping $x$ to $r x$ for $r$ as in (i), show that every torsion-free finitely generated $R$-module is isomorphic to an $R$-submodule of a finitely generated free $R$-module.

Paper 4, Section I, E

commentWhat is a quadratic form on a finite dimensional real vector space $V$ ? What does it mean for two quadratic forms to be isomorphic (i.e. congruent)? State Sylvester's law of inertia and explain the definition of the quantities which appear in it. Find the signature of the quadratic form on $\mathbb{R}^{3}$ given by $q(\mathbf{v})=\mathbf{v}^{T} A \mathbf{v}$, where

$A=\left(\begin{array}{ccc} -2 & 1 & 6 \\ 1 & -1 & -3 \\ 6 & -3 & 1 \end{array}\right)$

Paper 4, Section II, E

commentWhat does it mean for an $n \times n$ matrix to be in Jordan form? Show that if $A \in M_{n \times n}(\mathbb{C})$ is in Jordan form, there is a sequence $\left(A_{m}\right)$ of diagonalizable $n \times n$ matrices which converges to $A$, in the sense that the $(i j)$ th component of $A_{m}$ converges to the $(i j)$ th component of $A$ for all $i$ and $j$. [Hint: A matrix with distinct eigenvalues is diagonalizable.] Deduce that the same statement holds for all $A \in M_{n \times n}(\mathbb{C})$.

Let $V=M_{2 \times 2}(\mathbb{C})$. Given $A \in V$, define a linear map $T_{A}: V \rightarrow V$ by $T_{A}(B)=A B+B A$. Express the characteristic polynomial of $T_{A}$ in terms of the trace and determinant of $A$. [Hint: First consider the case where $A$ is diagonalizable.]

Paper 4, Section I, H

commentSuppose $P$ is the transition matrix of an irreducible recurrent Markov chain with state space $I$. Show that if $x$ is an invariant measure and $x_{k}>0$ for some $k \in I$, then $x_{j}>0$ for all $j \in I$.

Let

$\gamma_{j}^{k}=p_{k j}+\sum_{t=1}^{\infty} \sum_{i_{1} \neq k, \ldots, i_{t} \neq k} p_{k i_{t}} p_{i_{t} i_{t-1}} \cdots p_{i_{1} j}$

Give a meaning to $\gamma_{j}^{k}$ and explain why $\gamma_{k}^{k}=1$.

Suppose $x$ is an invariant measure with $x_{k}=1$. Prove that $x_{j} \geqslant \gamma_{j}^{k}$ for all $j$.

Paper 4, Section I, C

commentShow that the general solution of the wave equation

$\frac{1}{c^{2}} \frac{\partial^{2} y}{\partial t^{2}}-\frac{\partial^{2} y}{\partial x^{2}}=0$

can be written in the form

$y(x, t)=f(c t-x)+g(c t+x) .$

For the boundary conditions

$y(0, t)=y(L, t)=0, \quad t>0,$

find the relation between $f$ and $g$ and show that they are $2 L$-periodic. Hence show that

$E(t)=\frac{1}{2} \int_{0}^{L}\left(\frac{1}{c^{2}}\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right) d x$

is independent of $t$.

Paper 4, Section II, C

commentFind the inverse Fourier transform $G(x)$ of the function

$g(k)=e^{-a|k|}, \quad a>0, \quad-\infty<k<\infty .$

Assuming that appropriate Fourier transforms exist, determine the solution $\psi(x, y)$ of

$\nabla^{2} \psi=0, \quad-\infty<x<\infty, \quad 0<y<1$

with the following boundary conditions

$\psi(x, 0)=\delta(x), \quad \psi(x, 1)=\frac{1}{\pi} \frac{1}{x^{2}+1}$

Here $\delta(x)$ is the Dirac delta-function.

Paper 4, Section II, G

commentLet $X$ be a topological space. A connected component of $X$ means an equivalence class with respect to the equivalence relation on $X$ defined as:

$x \sim y \Longleftrightarrow x, y \text { belong to some connected subspace of } X .$

(i) Show that every connected component is a connected and closed subset of $X$.

(ii) If $X, Y$ are topological spaces and $X \times Y$ is the product space, show that every connected component of $X \times Y$ is a direct product of connected components of $X$ and $Y$.

Paper 4, Section I, C

commentFor a continuous function $f$, and $k+1$ distinct points $\left\{x_{0}, x_{1}, \ldots, x_{k}\right\}$, define the divided difference $f\left[x_{0}, \ldots, x_{k}\right]$ of order $k$.

Given $n+1$ points $\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}$, let $p_{n} \in \mathbb{P}_{n}$ be the polynomial of degree $n$ that interpolates $f$ at these points. Prove that $p_{n}$ can be written in the Newton form

$p_{n}(x)=f\left(x_{0}\right)+\sum_{k=1}^{n} f\left[x_{0}, \ldots, x_{k}\right] \prod_{i=0}^{k-1}\left(x-x_{i}\right)$

Paper 4, Section II, 20H

commentGiven real numbers $a$ and $b$, consider the problem $\mathrm{P}$ of minimizing

$f(x)=a x_{11}+2 x_{12}+3 x_{13}+b x_{21}+4 x_{22}+x_{23}$

subject to $x_{i j} \geqslant 0$ and

$\begin{array}{r} x_{11}+x_{12}+x_{13}=5 \\ x_{21}+x_{22}+x_{23}=5 \\ x_{11}+x_{21}=3 \\ x_{12}+x_{22}=3 \\ x_{13}+x_{23}=4 \end{array}$

List all the basic feasible solutions, writing them as $2 \times 3$ matrices $\left(x_{i j}\right)$.

Let $f(x)=\sum_{i j} c_{i j} x_{i j}$. Suppose there exist $\lambda_{i}, \mu_{j}$ such that

$\lambda_{i}+\mu_{j} \leqslant c_{i j} \text { for all } i \in\{1,2\}, j \in\{1,2,3\}$

Prove that if $x$ and $x^{\prime}$ are both feasible for $\mathrm{P}$ and $\lambda_{i}+\mu_{j}=c_{i j}$ whenever $x_{i j}>0$, then $f(x) \leqslant f\left(x^{\prime}\right) .$

Let $x^{*}$ be the initial feasible solution that is obtained by formulating $\mathrm{P}$ as a transportation problem and using a greedy method that starts in the upper left of the matrix $\left(x_{i j}\right)$. Show that if $a+2 \leqslant b$ then $x^{*}$ minimizes $f$.

For what values of $a$ and $b$ is one step of the transportation algorithm sufficient to pivot from $x^{*}$ to a solution that maximizes $f$ ?

Paper 4, Section I, B

commentThe components of the three-dimensional angular momentum operator $\hat{\mathbf{L}}$ are defined as follows:

$\hat{L}_{x}=-i \hbar\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}\right) \quad \hat{L}_{y}=-i \hbar\left(z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}\right) \quad \hat{L}_{z}=-i \hbar\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right)$

Given that the wavefunction

$\psi=(f(x)+i y) z$

is an eigenfunction of $\hat{L}_{z}$, find all possible values of $f(x)$ and the corresponding eigenvalues of $\psi$. Letting $f(x)=x$, show that $\psi$ is an eigenfunction of $\hat{\mathbf{L}}^{2}$ and calculate the corresponding eigenvalue.

Paper 4, Section II, H

commentExplain the notion of a sufficient statistic.

Suppose $X$ is a random variable with distribution $F$ taking values in $\{1, \ldots, 6\}$, with $P(X=i)=p_{i}$. Let $x_{1}, \ldots, x_{n}$ be a sample from $F$. Suppose $n_{i}$ is the number of these $x_{j}$ that are equal to $i$. Use a factorization criterion to explain why $\left(n_{1}, \ldots, n_{6}\right)$ is sufficient for $\theta=\left(p_{1}, \ldots, p_{6}\right)$.

Let $H_{0}$ be the hypothesis that $p_{i}=1 / 6$ for all $i$. Derive the statistic of the generalized likelihood ratio test of $H_{0}$ against the alternative that this is not a good fit.

Assuming that $n_{i} \approx n / 6$ when $H_{0}$ is true and $n$ is large, show that this test can be approximated by a chi-squared test using a test statistic

$T=-n+\frac{6}{n} \sum_{i=1}^{6} n_{i}^{2}$

Suppose $n=100$ and $T=8.12$. Would you reject $H_{0} ?$ Explain your answer.

Paper 4, Section II, A

commentDerive the Euler-Lagrange equation for the integral

$\int_{a}^{b} f\left(x, y, y^{\prime}, y^{\prime \prime}\right) d x$

where prime denotes differentiation with respect to $x$, and both $y$ and $y^{\prime}$ are specified at $x=a, b$.

Find $y(x)$ that extremises the integral

$\int_{0}^{\pi}\left(y+\frac{1}{2} y^{2}-\frac{1}{2} y^{\prime \prime 2}\right) d x$

subject to $y(0)=-1, y^{\prime}(0)=0, y(\pi)=\cosh \pi$ and $y^{\prime}(\pi)=\sinh \pi$.

Show that your solution is a global maximum. You may use the result that

$\int_{0}^{\pi} \phi^{2}(x) d x \leqslant \int_{0}^{\pi} \phi^{\prime 2}(x) d x$

for any (suitably differentiable) function $\phi$ which satisfies $\phi(0)=0$ and $\phi(\pi)=0$.