Part IB, 2018, Paper 4

# Part IB, 2018, Paper 4

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Paper 4, Section I, F

commentState the Bolzano-Weierstrass theorem in $\mathbb{R}$. Use it to deduce the BolzanoWeierstrass theorem in $\mathbb{R}^{n}$.

Let $D$ be a closed, bounded subset of $\mathbb{R}^{n}$, and let $f: D \rightarrow \mathbb{R}$ be a function. Let $\mathcal{S}$ be the set of points in $D$ where $f$ is discontinuous. For $\rho>0$ and $z \in \mathbb{R}^{n}$, let $B_{\rho}(z)$ denote the ball $\left\{x \in \mathbb{R}^{n}:\|x-z\|<\rho\right\}$. Prove that for every $\epsilon>0$, there exists $\delta>0$ such that $|f(x)-f(y)|<\epsilon$ whenever $x \in D, y \in D \backslash \cup_{z \in \mathcal{S}} B_{\epsilon}(z)$ and $\|x-y\|<\delta$.

(If you use the fact that a continuous function on a compact metric space is uniformly continuous, you must prove it.)

Paper 4, Section II, F

comment(a) Define what it means for a metric space $(X, d)$ to be complete. Give a metric $d$ on the interval $I=(0,1]$ such that $(I, d)$ is complete and such that a subset of $I$ is open with respect to $d$ if and only if it is open with respect to the Euclidean metric on $I$. Be sure to prove that $d$ has the required properties.

(b) Let $(X, d)$ be a complete metric space.

(i) If $Y \subset X$, show that $Y$ taken with the subspace metric is complete if and only if $Y$ is closed in $X$.

(ii) Let $f: X \rightarrow X$ and suppose that there is a number $\lambda \in(0,1)$ such that $d(f(x), f(y)) \leqslant \lambda d(x, y)$ for every $x, y \in X$. Show that there is a unique point $x_{0} \in X$ such that $f\left(x_{0}\right)=x_{0}$.

Deduce that if $\left(a_{n}\right)$ is a sequence of points in $X$ converging to a point $a \neq x_{0}$, then there are integers $\ell$ and $m \geqslant \ell$ such that $f\left(a_{m}\right) \neq a_{n}$ for every $n \geqslant \ell$.

Paper 4, Section I, F

comment(a) Let $\Omega \subset \mathbb{C}$ be open, $a \in \Omega$ and suppose that $D_{\rho}(a)=\{z \in \mathbb{C}:|z-a| \leqslant \rho\} \subset \Omega$. Let $f: \Omega \rightarrow \mathbb{C}$ be analytic.

State the Cauchy integral formula expressing $f(a)$ as a contour integral over $C=\partial D_{\rho}(a)$. Give, without proof, a similar expression for $f^{\prime}(a)$.

If additionally $\Omega=\mathbb{C}$ and $f$ is bounded, deduce that $f$ must be constant.

(b) If $g=u+i v: \mathbb{C} \rightarrow \mathbb{C}$ is analytic where $u, v$ are real, and if $u^{2}(z)-u(z) \geqslant v^{2}(z)$ for all $z \in \mathbb{C}$, show that $g$ is constant.

Paper 4, Section II, A

comment(a) Find the Laplace transform of

$y(t)=\frac{e^{-a^{2} / 4 t}}{\sqrt{\pi t}}$

for $a \in \mathbb{R}, a \neq 0$.

[You may use without proof that

$\left.\int_{0}^{\infty} \exp \left(-c^{2} x^{2}-\frac{c^{2}}{x^{2}}\right) d x=\frac{\sqrt{\pi}}{2|c|} e^{-2 c^{2}} .\right]$

(b) By using the Laplace transform, show that the solution to

$\begin{aligned} \frac{\partial^{2} u}{\partial x^{2}} &=\frac{\partial u}{\partial t} \quad-\infty<x<\infty, \quad t>0 \\ u(x, 0) &=f(x) \\ u(x, t) \quad \text { bounded, } \end{aligned}$

can be written as

$u(x, t)=\int_{-\infty}^{\infty} K(|x-\xi|, t) f(\xi) d \xi$

for some $K(|x-\xi|, t)$ to be determined.

[You may use without proof that a particular solution to

$y^{\prime \prime}(x)-s y(x)+f(x)=0$

is given by

$\left.y(x)=\frac{e^{-\sqrt{s} x}}{2 \sqrt{s}} \int_{0}^{x} e^{\sqrt{s} \xi} f(\xi) d \xi-\frac{e^{\sqrt{s} x}}{2 \sqrt{s}} \int_{0}^{x} e^{-\sqrt{s} \xi} f(\xi) d \xi .\right]$

Paper 4, Section I, $7 \mathrm{C}$

commentShow that Maxwell's equations imply the conservation of charge.

A conducting medium has $\mathbf{J}=\sigma \mathbf{E}$ where $\sigma$ is a constant. Show that any charge density decays exponentially in time, at a rate to be determined.

Paper 4, Section II, D

commentA deep layer of inviscid fluid is initially confined to the region $0<x<a, 0<y<a$, $z<0$ in Cartesian coordinates, with $z$ directed vertically upwards. An irrotational disturbance is caused to the fluid so that its upper surface takes position $z=\eta(x, y, t)$. Determine the linear normal modes of the system and the dispersion relation between the frequencies of the normal modes and their wavenumbers.

If the interface is initially displaced to position $z=\epsilon \cos \frac{3 \pi x}{a} \cos \frac{4 \pi y}{a}$ and released from rest, where $\epsilon$ is a small constant, determine its position for subsequent times. How far below the surface will the velocity have decayed to $1 / e$ times its surface value?

Paper 4, Section II, G

commentA Möbius strip in $\mathbb{R}^{3}$ is parametrized by

$\sigma(u, v)=(Q(u, v) \sin u, Q(u, v) \cos u, v \cos (u / 2))$

for $(u, v) \in U=(0,2 \pi) \times \mathbb{R}$, where $Q \equiv Q(u, v)=2-v \sin (u / 2)$. Show that the Gaussian curvature is

$K=\frac{-1}{\left(v^{2} / 4+Q^{2}\right)^{2}}$

at $(u, v) \in U$

Paper 4, Section I, G

comment(a) Show that every automorphism $\alpha$ of the dihedral group $D_{6}$ is equal to conjugation by an element of $D_{6}$; that is, there is an $h \in D_{6}$ such that

$\alpha(g)=h g h^{-1}$

for all $g \in D_{6}$.

(b) Give an example of a non-abelian group $G$ with an automorphism which is not equal to conjugation by an element of $G$.

Paper 4, Section II, G

comment(a) State the classification theorem for finitely generated modules over a Euclidean domain.

(b) Deduce the existence of the rational canonical form for an $n \times n$ matrix $A$ over a field $F$.

(c) Compute the rational canonical form of the matrix

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

Paper 4, Section I, E

commentDefine a quadratic form on a finite dimensional real vector space. What does it mean for a quadratic form to be positive definite?

Find a basis with respect to which the quadratic form

$x^{2}+2 x y+2 y^{2}+2 y z+3 z^{2}$

is diagonal. Is this quadratic form positive definite?

Paper 4, Section II, E

commentLet $V$ be a finite dimensional inner-product space over $\mathbb{C}$. What does it mean to say that an endomorphism of $V$ is self-adjoint? Prove that a self-adjoint endomorphism has real eigenvalues and may be diagonalised.

An endomorphism $\alpha: V \rightarrow V$ is called positive definite if it is self-adjoint and satisfies $\langle\alpha(x), x\rangle>0$ for all non-zero $x \in V$; it is called negative definite if $-\alpha$ is positive definite. Characterise the property of being positive definite in terms of eigenvalues, and show that the sum of two positive definite endomorphisms is positive definite.

Show that a self-adjoint endomorphism $\alpha: V \rightarrow V$ has all eigenvalues in the interval $[a, b]$ if and only if $\alpha-\lambda I$ is positive definite for all $\lambda<a$ and negative definite for all $\lambda>b$.

Let $\alpha, \beta: V \rightarrow V$ be self-adjoint endomorphisms whose eigenvalues lie in the intervals $[a, b]$ and $[c, d]$ respectively. Show that all of the eigenvalues of $\alpha+\beta$ lie in the interval $[a+c, b+d]$.

Paper 4, Section I, H

commentLet $P=\left(p_{i j}\right)_{i, j \in S}$ be the transition matrix for an irreducible Markov chain on the finite state space $S$.

(a) What does it mean to say that a distribution $\pi$ is the invariant distribution for the chain?

(b) What does it mean to say that the chain is in detailed balance with respect to a distribution $\pi$ ? Show that if the chain is in detailed balance with respect to a distribution $\pi$ then $\pi$ is the invariant distribution for the chain.

(c) A symmetric random walk on a connected finite graph is the Markov chain whose state space is the set of vertices of the graph and whose transition probabilities are

$p_{i j}= \begin{cases}1 / D_{i} & \text { if } j \text { is adjacent to } i \\ 0 & \text { otherwise }\end{cases}$

where $D_{i}$ is the number of vertices adjacent to vertex $i$. Show that the random walk is in detailed balance with respect to its invariant distribution.

Paper 4, Section I, A

commentBy using separation of variables, solve Laplace's equation

$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 \quad 0<x<1, \quad 0<y<1$

subject to

$\begin{array}{ll} u(0, y)=0 & 0 \leqslant y \leqslant 1 \\ u(1, y)=0 & 0 \leqslant y \leqslant 1 \\ u(x, 0)=0 & 0 \leqslant x \leqslant 1 \\ u(x, 1)=2 \sin (3 \pi x) & 0 \leqslant x \leqslant 1 \end{array}$

Paper 4, Section II, 17C

commentLet $\Omega$ be a bounded region in the plane, with smooth boundary $\partial \Omega$. Green's second identity states that for any smooth functions $u, v$ on $\Omega$

$\int_{\Omega}\left(u \nabla^{2} v-v \nabla^{2} u\right) \mathrm{d} x \mathrm{~d} y=\oint_{\partial \Omega} u(\mathbf{n} \cdot \nabla v)-v(\mathbf{n} \cdot \nabla u) \mathrm{d} s$

where $\mathbf{n}$ is the outward pointing normal to $\partial \Omega$. Using this identity with $v$ replaced by

$G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right)=\frac{1}{2 \pi} \ln \left(\left\|\mathbf{x}-\mathbf{x}_{0}\right\|\right)=\frac{1}{4 \pi} \ln \left(\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right)$

and taking care of the singular point $(x, y)=\left(x_{0}, y_{0}\right)$, show that if $u$ solves the Poisson equation $\nabla^{2} u=-\rho$ then

$\begin{aligned} u(\mathbf{x})=-\int_{\Omega} G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right) \rho\left(\mathbf{x}_{0}\right) \mathrm{d} x_{0} \mathrm{~d} y_{0} \\ &+\oint_{\partial \Omega}\left(u\left(\mathbf{x}_{0}\right) \mathbf{n} \cdot \nabla G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right)-G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right) \mathbf{n} \cdot \nabla u\left(\mathbf{x}_{0}\right)\right) \mathrm{d} s \end{aligned}$

at any $\mathbf{x}=(x, y) \in \Omega$, where all derivatives are taken with respect to $\mathbf{x}_{0}=\left(x_{0}, y_{0}\right)$.

In the case that $\Omega$ is the unit disc $\|\mathbf{x}\| \leqslant 1$, use the method of images to show that the solution to Laplace's equation $\nabla^{2} u=0$ inside $\Omega$, subject to the boundary condition

$u(1, \theta)=\delta(\theta-\alpha),$

is

$u(r, \theta)=\frac{1}{2 \pi} \frac{1-r^{2}}{1+r^{2}-2 r \cos (\theta-\alpha)}$

where $(r, \theta)$ are polar coordinates in the disc and $\alpha$ is a constant.

[Hint: The image of a point $\mathbf{x}_{0} \in \Omega$ is the point $\mathbf{y}_{0}=\mathbf{x}_{0} /\left\|\mathbf{x}_{0}\right\|^{2}$, and then

$\left\|\mathbf{x}-\mathbf{x}_{0}\right\|=\left\|\mathbf{x}_{0}\right\|\left\|\mathbf{x}-\mathbf{y}_{0}\right\|$

for all $\mathbf{x} \in \partial \Omega .]$

Paper 4, Section II, E

commentLet $X=\{2,3,4,5,6,7,8, \ldots\}$ and for each $n \in X$ let

$U_{n}=\{d \in X \mid d \text { divides } n\} .$

Prove that the set of unions of the sets $U_{n}$ forms a topology on $X$.

Prove or disprove each of the following:

(i) $X$ is Hausdorff;

(ii) $X$ is compact.

If $Y$ and $Z$ are topological spaces, $Y$ is the union of closed subspaces $A$ and $B$, and $f: Y \rightarrow Z$ is a function such that both $\left.f\right|_{A}: A \rightarrow Z$ and $\left.f\right|_{B}: B \rightarrow Z$ are continuous, show that $f$ is continuous. Hence show that $X$ is path-connected.

Paper 4 , Section I, D

comment$A=\left[\begin{array}{cccc} 1 & 2 & 1 & 2 \\ 2 & 5 & 5 & 6 \\ 1 & 5 & 13 & 14 \\ 2 & 6 & 14 & \lambda \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 3 \\ 7 \\ \mu \end{array}\right]$

where $\lambda$ and $\mu$ are real parameters. Find the $L U$ factorisation of the matrix $A$. For what values of $\lambda$ does the equation $A x=b$ have a unique solution for $x$ ?

For $\lambda=20$, use the $L U$ decomposition with forward and backward substitution to determine a value for $\mu$ for which a solution to $A x=b$ exists. Find the most general solution to the equation in this case.

Paper 4, Section II, H

commentGiven a network with a source $A$, a sink $B$, and capacities on directed edges, define a cut. What is meant by the capacity of a cut? State the max-flow min-cut theorem. If the capacities of edges are integral, what can be said about the maximum flow?

Consider an $m \times n$ matrix $A$ in which each entry is either 0 or 1 . We say that a set of lines (rows or columns of the matrix) covers the matrix if each 1 belongs to some line of the set. We say that a set of 1 's is independent if no pair of 1 's of the set lie in the same line. Use the max-flow min-cut theorem to show that the maximal number of independent 1's equals the minimum number of lines that cover the matrix.

Paper 4, Section I, B

commentA particle moving in one space dimension with wavefunction $\Psi(x, t)$ obeys the timedependent Schrödinger equation. Write down the probability density $\rho$ and current density $j$ in terms of the wavefunction and show that they obey the equation

$\frac{\partial j}{\partial x}+\frac{\partial \rho}{\partial t}=0$

Evaluate $j(x, t)$ in the case that

$\Psi(x, t)=\left(A e^{i k x}+B e^{-i k x}\right) e^{-i E t / \hbar}$

where $E=\hbar^{2} k^{2} / 2 m$, and $A$ and $B$ are constants, which may be complex.

Paper 4, Section II, H

commentThere is widespread agreement amongst the managers of the Reliable Motor Company that the number $X$ of faulty cars produced in a month has a binomial distribution

$P(X=s)=\left(\begin{array}{c} n \\ s \end{array}\right) p^{s}(1-p)^{n-s} \quad(s=0,1, \ldots, n ; \quad 0 \leqslant p \leqslant 1)$

where $n$ is the total number of cars produced in a month. There is, however, some dispute about the parameter $p$. The general manager has a prior distribution for $p$ which is uniform, while the more pessimistic production manager has a prior distribution with density $2 p$, both on the interval $[0,1]$.

In a particular month, $s$ faulty cars are produced. Show that if the general manager's loss function is $(\hat{p}-p)^{2}$, where $\hat{p}$ is her estimate and $p$ the true value, then her best estimate of $p$ is

$\hat{p}=\frac{s+1}{n+2}$

The production manager has responsibilities different from those of the general manager, and a different loss function given by $(1-p)(\hat{p}-p)^{2}$. Find his best estimate of $p$ and show that it is greater than that of the general manager unless $s \geqslant \frac{1}{2} n$.

[You may use the fact that for non-negative integers $\alpha, \beta$,

$\left.\int_{0}^{1} p^{\alpha}(1-p)^{\beta} d p=\frac{\alpha ! \beta !}{(\alpha+\beta+1) !}\right]$

Paper 4, Section II, B

comment(a) A two-dimensional oscillator has action

$S=\int_{t_{0}}^{t_{1}}\left\{\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \dot{y}^{2}-\frac{1}{2} \omega^{2} x^{2}-\frac{1}{2} \omega^{2} y^{2}\right\} d t$

Find the equations of motion as the Euler-Lagrange equations associated with $S$, and use them to show that

$J=\dot{x} y-\dot{y} x$

is conserved. Write down the general solution of the equations of motion in terms of $\sin \omega t$ and $\cos \omega t$, and evaluate $J$ in terms of the coefficients that arise in the general solution.

(b) Another kind of oscillator has action

$\widetilde{S}=\int_{t_{0}}^{t_{1}}\left\{\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \dot{y}^{2}-\frac{1}{4} \alpha x^{4}-\frac{1}{2} \beta x^{2} y^{2}-\frac{1}{4} \alpha y^{4}\right\} d t$

where $\alpha$ and $\beta$ are real constants. Find the equations of motion and use these to show that in general $J=\dot{x} y-\dot{y} x$ is not conserved. Find the special value of the ratio $\beta / \alpha$ for which $J$ is conserved. Explain what is special about the action $\widetilde{S}$in this case, and state the interpretation of $J$.