Part IB, 2015, Paper 3

# Part IB, 2015, Paper 3

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Paper 3, Section I, G

commentDefine what is meant by a uniformly continuous function $f$ on a subset $E$ of a metric space. Show that every continuous function on a closed, bounded interval is uniformly continuous. [You may assume the Bolzano-Weierstrass theorem.]

Suppose that a function $g:[0, \infty) \rightarrow \mathbb{R}$ is continuous and tends to a finite limit at $\infty$. Is $g$ necessarily uniformly continuous on $[0, \infty) ?$ Give a proof or a counterexample as appropriate.

Paper 3, Section II, G

commentDefine what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ to be differentiable at $x \in \mathbb{R}^{n}$ with derivative $D f(x)$.

State and prove the chain rule for the derivative of $g \circ f$, where $g: \mathbb{R}^{m} \rightarrow \mathbb{R}^{p}$ is a differentiable function.

Now let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a differentiable function and let $g(x)=f(x, c-x)$ where $c$ is a constant. Show that $g$ is differentiable and find its derivative in terms of the partial derivatives of $f$. Show that if $D_{1} f(x, y)=D_{2} f(x, y)$ holds everywhere in $\mathbb{R}^{2}$, then $f(x, y)=h(x+y)$ for some differentiable function $h .$

Paper 3, Section II, G

commentState the argument principle.

Let $U \subset \mathbb{C}$ be an open set and $f: U \rightarrow \mathbb{C}$ a holomorphic injective function. Show that $f^{\prime}(z) \neq 0$ for each $z$ in $U$ and that $f(U)$ is open.

Stating clearly any theorems that you require, show that for each $a \in U$ and a sufficiently small $r>0$,

$g(w)=\frac{1}{2 \pi i} \int_{|z-a|=r} \frac{z f^{\prime}(z)}{f(z)-w} d z$

defines a holomorphic function on some open disc $D$ about $f(a)$.

Show that $g$ is the inverse for the restriction of $f$ to $g(D)$.

Paper 3, Section I, B

commentFind the Fourier transform of the function

$f(x)=\frac{1}{1+x^{2}}, \quad x \in \mathbb{R}$

using an appropriate contour integration. Hence find the Fourier transform of its derivative, $f^{\prime}(x)$, and evaluate the integral

$I=\int_{-\infty}^{\infty} \frac{4 x^{2}}{\left(1+x^{2}\right)^{4}} d x$

Paper 3, Section II, A

commentA charge density $\rho=\lambda / r$ fills the region of 3-dimensional space $a<r<b$, where $r$ is the radial distance from the origin and $\lambda$ is a constant. Compute the electric field in all regions of space in terms of $Q$, the total charge of the region. Sketch a graph of the magnitude of the electric field versus $r$ (assuming that $Q>0$ ).

Now let $\Delta=b-a \rightarrow 0$. Derive the surface charge density $\sigma$ in terms of $\Delta, a$ and $\lambda$ and explain how a finite surface charge density may be obtained in this limit. Sketch the magnitude of the electric field versus $r$ in this limit. Comment on any discontinuities, checking a standard result involving $\sigma$ for this particular case.

A second shell of equal and opposite total charge is centred on the origin and has a radius $c<a$. Sketch the electric potential of this system, assuming that it tends to 0 as $r \rightarrow \infty$.

Paper 3, Section II, B

commentA source of sound induces a travelling wave of pressure above the free surface of a fluid located in the $z<0$ domain as

$p=p_{a t m}+p_{0} \cos (k x-\omega t),$

with $p_{0} \ll p_{a t m}$. Here $k$ and $\omega$ are fixed real numbers. We assume that the flow induced in the fluid is irrotational.

(i) State the linearized equation of motion for the fluid and the free surface, $z=h(x, t)$, with all boundary conditions.

(ii) Solve for the velocity potential, $\phi(x, z, t)$, and the height of the free surface, $h(x, t)$. Verify that your solutions are dimensionally correct.

(iii) Interpret physically the behaviour of the solution when $\omega^{2}=g k$.

Paper 3, Section $I$, F

commentState the sine rule for spherical triangles.

Let $\Delta$ be a spherical triangle with vertices $A, B$, and $C$, with angles $\alpha, \beta$ and $\gamma$ at the respective vertices. Let $a, b$, and $c$ be the lengths of the edges $B C, A C$ and $A B$ respectively. Show that $b=c$ if and only if $\beta=\gamma$. [You may use the cosine rule for spherical triangles.] Show that this holds if and only if there exists a reflection $M$ such that $M(A)=A, M(B)=C$ and $M(C)=B$.

Are there equilateral triangles on the sphere? Justify your answer.

Paper 3, Section II, F

commentLet $T: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ be a Möbius transformation on the Riemann sphere $\mathbb{C}_{\infty}$.

(i) Show that $T$ has either one or two fixed points.

(ii) Show that if $T$ is a Möbius transformation corresponding to (under stereographic projection) a rotation of $S^{2}$ through some fixed non-zero angle, then $T$ has two fixed points, $z_{1}, z_{2}$, with $z_{2}=-1 / \bar{z}_{1}$.

(iii) Suppose $T$ has two fixed points $z_{1}, z_{2}$ with $z_{2}=-1 / \bar{z}_{1}$. Show that either $T$ corresponds to a rotation as in (ii), or one of the fixed points, say $z_{1}$, is attractive, i.e. $T^{n} z \rightarrow z_{1}$ as $n \rightarrow \infty$ for any $z \neq z_{2}$.

Paper 3, Section I, F

commentState two equivalent conditions for a commutative ring to be Noetherian, and prove they are equivalent. Give an example of a ring which is not Noetherian, and explain why it is not Noetherian.

Paper 3, Section II, F

commentCan a group of order 55 have 20 elements of order 11? If so, give an example. If not, give a proof, including the proof of any statements you need.

Let $G$ be a group of order $p q$, with $p$ and $q$ primes, $p>q$. Suppose furthermore that $q$ does not divide $p-1$. Show that $G$ is cyclic.

Paper 3, Section II, E

commentLet $A_{1}, A_{2}, \ldots, A_{k}$ be $n \times n$ matrices over a field $\mathbb{F}$. We say $A_{1}, A_{2}, \ldots, A_{k}$ are simultaneously diagonalisable if there exists an invertible matrix $P$ such that $P^{-1} A_{i} P$ is diagonal for all $1 \leqslant i \leqslant k$. We say the matrices are commuting if $A_{i} A_{j}=A_{j} A_{i}$ for all $i, j$.

(i) Suppose $A_{1}, A_{2}, \ldots, A_{k}$ are simultaneously diagonalisable. Prove that they are commuting.

(ii) Define an eigenspace of a matrix. Suppose $B_{1}, B_{2}, \ldots, B_{k}$ are commuting $n \times n$ matrices over a field $\mathbb{F}$. Let $E$ denote an eigenspace of $B_{1}$. Prove that $B_{i}(E) \leqslant E$ for all $i$.

(iii) Suppose $B_{1}, B_{2}, \ldots, B_{k}$ are commuting diagonalisable matrices. Prove that they are simultaneously diagonalisable.

(iv) Are the $2 \times 2$ diagonalisable matrices over $\mathbb{C}$ simultaneously diagonalisable? Explain your answer.

Paper 3, Section I, H

commentDefine what is meant by a communicating class and a closed class in a Markov chain.

A Markov chain $\left(X_{n}: n \geqslant 0\right)$ with state space $\{1,2,3,4\}$ has transition matrix

$P=\left(\begin{array}{cccc} \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{array}\right)$

Write down the communicating classes for this Markov chain and state whether or not each class is closed.

If $X_{0}=2$, let $N$ be the smallest $n$ such that $X_{n} \neq 2$. Find $\mathbb{P}(N=n)$ for $n=1,2, \ldots$ and $\mathbb{E}(N)$. Describe the evolution of the chain if $X_{0}=2$.

Paper 3, Section I, $7 \mathrm{C}$

comment(a) From the defining property of the $\delta$ function,

$\int_{-\infty}^{\infty} \delta(x) f(x) d x=f(0)$

for any function $f$, prove that

(i) $\delta(-x)=\delta(x)$

(ii) $\delta(a x)=|a|^{-1} \delta(x)$ for $a \in \mathbb{R}, a \neq 0$,

(iii) If $g: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto g(x)$ is smooth and has isolated zeros $x_{i}$ where the derivative $g^{\prime}\left(x_{i}\right) \neq 0$, then

$\delta[g(x)]=\sum_{i} \frac{\delta\left(x-x_{i}\right)}{\left|g^{\prime}\left(x_{i}\right)\right|}$

(b) Show that the function $\gamma(x)$ defined by

$\gamma(x)=\lim _{s \rightarrow 0} \frac{e^{x / s}}{s\left(1+e^{x / s}\right)^{2}}$

is the $\delta(x)$ function.

Paper 3, Section II, C

comment(i) Consider the Poisson equation $\nabla^{2} \psi(\mathbf{r})=f(\mathbf{r})$ with forcing term $f$ on the infinite domain $\mathbb{R}^{3}$ with $\lim _{|\mathbf{r}| \rightarrow \infty} \psi=0$. Derive the Green's function $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-1 /\left(4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)$ for this equation using the divergence theorem. [You may assume without proof that the divergence theorem is valid for the Green's function.]

(ii) Consider the Helmholtz equation

$\tag{†} \nabla^{2} \psi(\mathbf{r})+k^{2} \psi(\mathbf{r})=f(\mathbf{r})$

where $k$ is a real constant. A Green's function $g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ for this equation can be constructed from $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ of (i) by assuming $g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=U(r) G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ where $r=\left|\mathbf{r}-\mathbf{r}^{\prime}\right|$ and $U(r)$ is a regular function. Show that $\lim _{r \rightarrow 0} U(r)=1$ and that $U$ satisfies the equation

$\tag{‡} \frac{d^{2} U}{d r^{2}}+k^{2} U(r)=0$

(iii) Take the Green's function with the specific solution $U(r)=e^{i k r}$ to Eq. ($‡$) and consider the Helmholtz equation $(†)$ on the semi-infinite domain $z>0, x, y \in \mathbb{R}$. Use the method of images to construct a Green's function for this problem that satisfies the boundary conditions

$\frac{\partial g}{\partial z^{\prime}}=0 \text { on } z^{\prime}=0 \quad \text { and } \quad \lim _{|\mathbf{r}| \rightarrow \infty} g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=0$

(iv) A solution to the Helmholtz equation on a bounded domain can be constructed in complete analogy to that of the Poisson equation using the Green's function in Green's 3rd identity

$\psi(\mathbf{r})=\int_{\partial V}\left[\psi\left(\mathbf{r}^{\prime}\right) \frac{\partial g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)}{\partial n^{\prime}}-g\left(\mathbf{r}, \mathbf{r}^{\prime}\right) \frac{\partial \psi\left(\mathbf{r}^{\prime}\right)}{\partial n^{\prime}}\right] d S^{\prime}+\int_{V} f\left(\mathbf{r}^{\prime}\right) g\left(\mathbf{r}, \mathbf{r}^{\prime}\right) d V^{\prime},$

where $V$ denotes the volume of the domain, $\partial V$ its boundary and $\partial / \partial n^{\prime}$ the outgoing normal derivative on the boundary. Now consider the homogeneous Helmholtz equation $\nabla^{2} \psi(\mathbf{r})+k^{2} \psi(\mathbf{r})=0$ on the domain $z>0, x, y \in \mathbb{R}$ with boundary conditions $\psi(\mathbf{r})=0$ at $|\mathbf{r}| \rightarrow \infty$ and

$\left.\frac{\partial \psi}{\partial z}\right|_{z=0}= \begin{cases}0 & \text { for } \rho>a \\ A & \text { for } \rho \leqslant a\end{cases}$

where $\rho=\sqrt{x^{2}+y^{2}}$ and $A$ and $a$ are real constants. Construct a solution in integral form to this equation using cylindrical coordinates $(z, \rho, \varphi)$ with $x=\rho \cos \varphi, y=\rho \sin \varphi$.

Paper 3, Section I, $3 \mathrm{E}$

commentDefine what it means for a topological space $X$ to be (i) connected (ii) path-connected.

Prove that any path-connected space $X$ is connected. [You may assume the interval $[0,1]$ is connected. $]$

Give a counterexample (without justification) to the converse statement.

Paper 3, Section II, D

commentDefine the QR factorization of an $m \times n$ matrix $A$. Explain how it can be used to solve the least squares problem of finding the vector $x^{*} \in \mathbb{R}^{n}$ which minimises $\|\mathrm{A} x-b\|$, where $b \in \mathbb{R}^{m}, m>n$, and $\|\cdot\|$ is the Euclidean norm.

Explain how to construct $Q$ and $R$ by the Gram-Schmidt procedure. Why is this procedure not useful for numerical factorization of large matrices?

Let

$A=\left[\begin{array}{rrr} 5 & 6 & -14 \\ 5 & 4 & 4 \\ -5 & 2 & -8 \\ 5 & 12 & -18 \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 1 \\ 1 \\ 0 \end{array}\right]$

Using the Gram-Schmidt procedure find a QR decomposition of A. Hence solve the least squares problem giving both $x^{*}$ and $\left\|\mathrm{A} x^{*}-b\right\|$.

Paper 3, Section II, H

commentConsider the linear programming problem $P$ :

$\text { minimise } c^{T} x \text { subject to } A x \geqslant b, x \geqslant 0,$

where $x$ and $c$ are in $\mathbb{R}^{n}, A$ is a real $m \times n$ matrix, $b$ is in $\mathbb{R}^{m}$ and ${ }^{T}$ denotes transpose. Derive the dual linear programming problem $D$. Show from first principles that the dual of $D$ is $P$.

Suppose that $c^{T}=(6,10,11), b^{T}=(1,1,3)$ and $A=\left(\begin{array}{lll}1 & 3 & 8 \\ 1 & 1 & 2 \\ 2 & 4 & 4\end{array}\right)$. Write down the dual $D$ and find the optimal solution of the dual using the simplex algorithm. Hence, or otherwise, find the optimal solution $x^{*}=\left(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}\right)$ of $P$.

Suppose that $c$ is changed to $\tilde{c}=\left(6+\varepsilon_{1}, 10+\varepsilon_{2}, 11+\varepsilon_{3}\right)$. Give necessary and sufficient conditions for $x^{*}$ still to be the optimal solution of $P$. If $\varepsilon_{1}=\varepsilon_{2}=0$, find the range of values for $\varepsilon_{3}$ for which $x^{*}$ is still the optimal solution of $P$.

Paper 3, Section $I$, D

commentA quantum-mechanical system has normalised energy eigenstates $\chi_{1}$ and $\chi_{2}$ with non-degenerate energies $E_{1}$ and $E_{2}$ respectively. The observable $A$ has normalised eigenstates,

$\begin{aligned} \phi_{1} &=C\left(\chi_{1}+2 \chi_{2}\right), & & \text { eigenvalue }=a_{1} \\ \phi_{2} &=C\left(2 \chi_{1}-\chi_{2}\right), & & \text { eigenvalue }=a_{2} \end{aligned}$

where $C$ is a positive real constant. Determine $C$.

Initially, at time $t=0$, the state of the system is $\phi_{1}$. Write down an expression for $\psi(t)$, the state of the system with $t \geqslant 0$. What is the probability that a measurement of energy at time $t$ will yield $E_{2}$ ?

For the same initial state, determine the probability that a measurement of $A$ at time $t>0$ will yield $a_{1}$ and the probability that it will yield $a_{2}$.

Paper 3, Section II, D

commentDefine the angular momentum operators $\hat{L}_{i}$ for a particle in three dimensions in terms of the position and momentum operators $\hat{x}_{i}$ and $\hat{p}_{i}=-i \hbar \frac{\partial}{\partial x_{i}}$. Write down an expression for $\left[\hat{L}_{i}, \hat{L}_{j}\right]$ and use this to show that $\left[\hat{L}^{2}, \hat{L}_{i}\right]=0$ where $\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}$. What is the significance of these two commutation relations?

Let $\psi(x, y, z)$ be both an eigenstate of $\hat{L}_{z}$ with eigenvalue zero and an eigenstate of $\hat{L}^{2}$ with eigenvalue $\hbar^{2} l(l+1)$. Show that $\left(\hat{L}_{x}+i \hat{L}_{y}\right) \psi$ is also an eigenstate of both $\hat{L}_{z}$ and $\hat{L}^{2}$ and determine the corresponding eigenvalues.

Find real constants $A$ and $B$ such that

$\phi(x, y, z)=\left(A z^{2}+B y^{2}-r^{2}\right) e^{-r}, \quad r^{2}=x^{2}+y^{2}+z^{2},$

is an eigenfunction of $\hat{L}_{z}$ with eigenvalue zero and an eigenfunction of $\hat{L}^{2}$ with an eigenvalue which you should determine. [Hint: You might like to show that $\left.\hat{L}_{i} f(r)=0 .\right]$

Paper 3, Section II, H

comment(a) Suppose that $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables, each with density $f(x)=\theta \exp (-\theta x), x>0$ for some unknown $\theta>0$. Use the generalised likelihood ratio to obtain a size $\alpha$ test of $H_{0}: \theta=1$ against $H_{1}: \theta \neq 1$.

(b) A die is loaded so that, if $p_{i}$ is the probability of face $i$, then $p_{1}=p_{2}=\theta_{1}$, $p_{3}=p_{4}=\theta_{2}$ and $p_{5}=p_{6}=\theta_{3}$. The die is thrown $n$ times and face $i$ is observed $x_{i}$ times. Write down the likelihood function for $\theta=\left(\theta_{1}, \theta_{2}, \theta_{3}\right)$ and find the maximum likelihood estimate of $\theta$.

Consider testing whether or not $\theta_{1}=\theta_{2}=\theta_{3}$ for this die. Find the generalised likelihood ratio statistic $\Lambda$ and show that

$2 \log _{e} \Lambda \approx T, \quad \text { where } T=\sum_{i=1}^{3} \frac{\left(o_{i}-e_{i}\right)^{2}}{e_{i}}$

where you should specify $o_{i}$ and $e_{i}$ in terms of $x_{1}, \ldots, x_{6}$. Explain how to obtain an approximate size $0.05$ test using the value of $T$. Explain what you would conclude (and why ) if $T=2.03$.

Paper 3, Section $I$, A

comment(a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex.

(b) Define the Legendre transform $f^{*}(p)$ of a convex function $f(x)$, where $x \in \mathbb{R}$. Show that $f^{*}(p)$ is a convex function.

(c) Find the Legendre transform $f^{*}(p)$ of the function $f(x)=e^{x}$, and the domain of $f^{*}$.