Part IB, 2012, Paper 1

# Part IB, 2012, Paper 1

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Paper 1, Section II, E

commentState the inverse function theorem for a function $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$. Suppose $F$ is a differentiable bijection with $F^{-1}$ also differentiable. Show that the derivative of $F$ at any point in $\mathbb{R}^{n}$ is a linear isomorphism.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function such that the partial derivatives $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$ exist and are continuous. Assume there is a point $(a, b) \in \mathbb{R}^{2}$ for which $f(a, b)=0$ and $\frac{\partial f}{\partial x}(a, b) \neq 0$. Prove that there exist open sets $U \subset \mathbb{R}^{2}$ and $W \subset \mathbb{R}$ containing $(a, b)$ and $b$, respectively, such that for every $y \in W$ there exists a unique $x$ such that $(x, y) \in U$ and $f(x, y)=0$. Moreover, if we define $g: W \rightarrow \mathbb{R}$ by $g(y)=x$, prove that $g$ is differentiable with continuous derivative. Find the derivative of $g$ at $b$ in terms of $\frac{\partial f}{\partial x}(a, b)$ and $\frac{\partial f}{\partial y}(a, b)$.

Paper 1, Section I, A

commentFind a conformal transformation $\zeta=\zeta(z)$ that maps the domain $D, 0<\arg z<\frac{3 \pi}{2}$, on to the strip $0<\operatorname{Im}(\zeta)<1$.

Hence find a bounded harmonic function $\phi$ on $D$ subject to the boundary conditions $\phi=0, A$ on $\arg z=0, \frac{3 \pi}{2}$, respectively, where $A$ is a real constant.

Paper 1, Section II, 13A

commentUsing Cauchy's integral theorem, write down the value of a holomorphic function $f(z)$ where $|z|<1$ in terms of a contour integral around the unit circle, $\zeta=e^{i \theta} .$

By considering the point $1 / \bar{z}$, or otherwise, show that

$f(z)=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(\zeta) \frac{1-|z|^{2}}{|\zeta-z|^{2}} \mathrm{~d} \theta$

By setting $z=r e^{i \alpha}$, show that for any harmonic function $u(r, \alpha)$,

$u(r, \alpha)=\frac{1}{2 \pi} \int_{0}^{2 \pi} u(1, \theta) \frac{1-r^{2}}{1-2 r \cos (\alpha-\theta)+r^{2}} \mathrm{~d} \theta$

if $r<1$.

Assuming that the function $v(r, \alpha)$, which is the conjugate harmonic function to $u(r, \alpha)$, can be written as

$v(r, \alpha)=v(0)+\frac{1}{\pi} \int_{0}^{2 \pi} u(1, \theta) \frac{r \sin (\alpha-\theta)}{1-2 r \cos (\alpha-\theta)+r^{2}} \mathrm{~d} \theta$

deduce that

$f(z)=i v(0)+\frac{1}{2 \pi} \int_{0}^{2 \pi} u(1, \theta) \frac{\zeta+z}{\zeta-z} \mathrm{~d} \theta$

[You may use the fact that on the unit circle, $\zeta=1 / \bar{\zeta}$, and hence

$\left.\frac{\zeta}{\zeta-1 / \bar{z}}=-\frac{\bar{z}}{\bar{\zeta}-\bar{z}} \cdot\right]$

Paper 1, Section II, B

commentA sphere of radius a carries an electric charge $Q$ uniformly distributed over its surface. Calculate the electric field outside and inside the sphere. Also calculate the electrostatic potential outside and inside the sphere, assuming it vanishes at infinity. State the integral formula for the energy $U$ of the electric field and use it to evaluate $U$ as a function of $Q$

Relate $\frac{d U}{d Q}$ to the potential on the surface of the sphere and explain briefly the physical interpretation of the relation.

Paper 1, Section I, A

commentViscous fluid, with viscosity $\mu$ and density $\rho$ flows along a straight circular pipe of radius $R$. The average velocity of the flow is $U$. Define a Reynolds number for the flow.

The flow is driven by a constant pressure gradient $-G>0$ along the pipe and the velocity is parallel to the axis of the pipe with magnitude $u(r)$ that satisfies

$\frac{\mu}{r} \frac{\mathrm{d}}{\mathrm{d} r}\left(r \frac{\mathrm{d} u}{\mathrm{~d} r}\right)=-G,$

where $r$ is the radial distance from the axis.

State the boundary conditions on $u$ and find the velocity as a function of $r$ assuming that it is finite on the axis $r=0$. Hence, show that the shear stress $\tau$ at the pipe wall is independent of the viscosity. Why is this the case?

Paper 1, Section II, A

commentConsider inviscid, incompressible fluid flow confined to the $(x, y)$ plane. The fluid has density $\rho$, and gravity can be neglected. Using the conservation of volume flux, determine the velocity potential $\phi(r)$ of a point source of strength $m$, in terms of the distance $r$ from the source.

Two point sources each of strength $m$ are located at $\boldsymbol{x}_{+}=(0, a)$ and $\boldsymbol{x}_{-}=(0,-a)$. Find the velocity potential of the flow.

Show that the flow in the region $y \geqslant 0$ is equivalent to the flow due to a source at $\boldsymbol{x}_{+}$and a fixed boundary at $y=0 .$

Find the pressure on the boundary $y=0$ and hence determine the force on the boundary.

[Hint: you may find the substitution $x=a \tan \theta$ useful for the calculation of the pressure.]

Paper 1, Section I, G

commentDescribe a collection of charts which cover a circular cylinder of radius $R$. Compute the first fundamental form, and deduce that the cylinder is locally isometric to the plane.

Paper 1, Section II, G

commentLet $G$ be a finite group. What is a Sylow $p$-subgroup of $G$ ?

Assuming that a Sylow $p$-subgroup $H$ exists, and that the number of conjugates of $H$ is congruent to $1 \bmod p$, prove that all Sylow $p$-subgroups are conjugate. If $n_{p}$ denotes the number of Sylow $p$-subgroups, deduce that

$n_{p} \equiv 1 \quad \bmod p \quad \text { and } \quad n_{p}|| G \mid$

If furthermore $G$ is simple prove that either $G=H$ or

$|G| \mid n_{p} \text { ! }$

Deduce that a group of order $1,000,000$ cannot be simple.

Paper 1, Section I, F

commentDefine the notions of basis and dimension of a vector space. Prove that two finitedimensional real vector spaces with the same dimension are isomorphic.

In each case below, determine whether the set $S$ is a basis of the real vector space $V:$

(i) $V=\mathbb{C}$ is the complex numbers; $S=\{1, i\}$.

(ii) $V=\mathbb{R}[x]$ is the vector space of all polynomials in $x$ with real coefficients; $S=\{1,(x-1),(x-1)(x-2),(x-1)(x-2)(x-3), \ldots\} .$

(iii) $V=\{f:[0,1] \rightarrow \mathbb{R}\} ; S=\left\{\chi_{p} \mid p \in[0,1]\right\}$, where

$\chi_{p}(x)= \begin{cases}1 & x=p \\ 0 & x \neq p\end{cases}$

Paper 1, Section II, F

commentDefine what it means for two $n \times n$ matrices to be similar to each other. Show that if two $n \times n$ matrices are similar, then the linear transformations they define have isomorphic kernels and images.

If $A$ and $B$ are $n \times n$ real matrices, we define $[A, B]=A B-B A$. Let

$\begin{aligned} K_{A} &=\left\{X \in M_{n \times n}(\mathbb{R}) \mid[A, X]=0\right\} \\ L_{A} &=\left\{[A, X] \mid X \in M_{n \times n}(\mathbb{R})\right\} \end{aligned}$

Show that $K_{A}$ and $L_{A}$ are linear subspaces of $M_{n \times n}(\mathbb{R})$. If $A$ and $B$ are similar, show that $K_{A} \cong K_{B}$ and $L_{A} \cong L_{B}$.

Suppose that $A$ is diagonalizable and has characteristic polynomial

$\left(x-\lambda_{1}\right)^{m_{1}}\left(x-\lambda_{2}\right)^{m_{2}}$

where $\lambda_{1} \neq \lambda_{2}$. What are $\operatorname{dim} K_{A}$ and $\operatorname{dim} L_{A} ?$

Paper 1, Section II, 20H

commentA Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ has as its state space the integers, with

$p_{i, i+1}=p, \quad p_{i, i-1}=q=1-p$

and $p_{i j}=0$ otherwise. Assume $p>q$.

Let $T_{j}=\inf \left\{n \geqslant 1: X_{n}=j\right\}$ if this is finite, and $T_{j}=\infty$ otherwise. Let $V_{0}$ be the total number of hits on 0 , and let $V_{0}(n)$ be the total number of hits on 0 within times $0, \ldots, n-1$. Let

$\begin{aligned} h_{i} &=P\left(V_{0}>0 \mid X_{0}=i\right) \\ r_{i}(n) &=E\left[V_{0}(n) \mid X_{0}=i\right] \\ r_{i} &=E\left[V_{0} \mid X_{0}=i\right] \end{aligned}$

(i) Quoting an appropriate theorem, find, for every $i$, the value of $h_{i}$.

(ii) Show that if $\left(x_{i}, i \in \mathbb{Z}\right)$ is any non-negative solution to the system of equations

$\begin{aligned} x_{0} &=1+q x_{1}+p x_{-1}, \\ x_{i} &=q x_{i-1}+p x_{i+1}, \quad \text { for all } i \neq 0 \end{aligned}$

then $x_{i} \geqslant r_{i}(n)$ for all $i$ and $n$.

(iii) Show that $P\left(V_{0}\left(T_{1}\right) \geqslant k \mid X_{0}=1\right)=q^{k}$ and $E\left[V_{0}\left(T_{1}\right) \mid X_{0}=1\right]=q / p$.

(iv) Explain why $r_{i+1}=(q / p) r_{i}$ for $i>0$.

(v) Find $r_{i}$ for all $i$.

Paper 1, Section II, C

commentConsider the regular Sturm-Liouville (S-L) system

$(\mathcal{L} y)(x)-\lambda \omega(x) y(x)=0, \quad a \leqslant x \leqslant b$

where

$(\mathcal{L} y)(x):=-\left[p(x) y^{\prime}(x)\right]^{\prime}+q(x) y(x)$

with $\omega(x)>0$ and $p(x)>0$ for all $x$ in $[a, b]$, and the boundary conditions on $y$ are

$\left\{\begin{array}{l} A_{1} y(a)+A_{2} y^{\prime}(a)=0 \\ B_{1} y(b)+B_{2} y^{\prime}(b)=0 \end{array}\right.$

Show that with these boundary conditions, $\mathcal{L}$ is self-adjoint. By considering $y \mathcal{L} y$, or otherwise, show that the eigenvalue $\lambda$ can be written as

$\lambda=\frac{\int_{a}^{b}\left(p y^{\prime 2}+q y^{2}\right) d x-\left[p y y^{\prime}\right]_{a}^{b}}{\int_{a}^{b} \omega y^{2} d x}$

Now suppose that $a=0$ and $b=\ell$, that $p(x)=1, q(x) \geqslant 0$ and $\omega(x)=1$ for all $x \in[0, \ell]$, and that $A_{1}=1, A_{2}=0, B_{1}=k \in \mathbb{R}^{+}$and $B_{2}=1$. Show that the eigenvalues of this regular S-L system are strictly positive. Assuming further that $q(x)=0$, solve the system explicitly, and with the aid of a graph, show that there exist infinitely many eigenvalues $\lambda_{1}<\lambda_{2}<\cdots<\lambda_{n}<\cdots$. Describe the behaviour of $\lambda_{n}$ as $n \rightarrow \infty$.

Paper 1, Section II, F

commentA topological space $X$ is said to be normal if each point of $X$ is a closed subset of $X$ and for each pair of closed sets $C_{1}, C_{2} \subset X$ with $C_{1} \cap C_{2}=\emptyset$ there are open sets $U_{1}, U_{2} \subset X$ so that $C_{i} \subset U_{i}$ and $U_{1} \cap U_{2}=\emptyset$. In this case we say that the $U_{i}$ separate the $C_{i}$.

Show that a compact Hausdorff space is normal. [Hint: first consider the case where $C_{2}$ is a point.]

For $C \subset X$ we define an equivalence relation $\sim_{C}$ on $X$ by $x \sim_{C} y$ for all $x, y \in C$, $x \sim_{C} x$ for $x \notin C$. If $C, C_{1}$ and $C_{2}$ are pairwise disjoint closed subsets of a normal space $X$, show that $C_{1}$ and $C_{2}$ may be separated by open subsets $U_{1}$ and $U_{2}$ such that $U_{i} \cap C=\emptyset$. Deduce that the quotient space $X / \sim_{C}$ is also normal.

Paper 1, Section I, D

comment$A=\left[\begin{array}{cccc} 1 & a & a^{2} & a^{3} \\ a^{3} & 1 & a & a^{2} \\ a^{2} & a^{3} & 1 & a \\ a & a^{2} & a^{3} & 1 \end{array}\right], \quad b=\left[\begin{array}{l} \gamma \\ 0 \\ 0 \\ 0 \end{array}\right], \quad \gamma=1-a^{4} \neq 0$

Find the LU factorization of the matrix $A$ and use it to solve the system $A x=b$ via forward and backward substitution. [Other methods of solution are not acceptable.]

Paper 1, Section II, D

commentFor a numerical method for solving $y^{\prime}=f(t, y)$, define the linear stability domain, and state when such a method is A-stable.

Determine all values of the real parameter $a$ for which the Runge-Kutta method

$\begin{aligned} k_{1} &=f\left(t_{n}+\left(\frac{1}{2}-a\right) h, y_{n}+\left(\frac{1}{4} h k_{1}+\left(\frac{1}{4}-a\right) h k_{2}\right)\right) \\ k_{2} &=f\left(t_{n}+\left(\frac{1}{2}+a\right) h, y_{n}+\left(\left(\frac{1}{4}+a\right) h k_{1}+\frac{1}{4} h k_{2}\right)\right) \\ y_{n+1} &=y_{n}+\frac{1}{2} h\left(k_{1}+k_{2}\right) \end{aligned}$

is A-stable.

Paper 1, Section I, 8H

commentState the Lagrangian sufficiency theorem.

Use Lagrange multipliers to find the optimal values of $x_{1}$ and $x_{2}$ in the problem: maximize $x_{1}^{2}+x_{2} \quad$ subject to $\quad x_{1}^{2}+\frac{1}{2} x_{2}^{2} \leqslant b_{1}, \quad x_{1} \geqslant b_{2}$ and $x_{1}, x_{2} \geqslant 0$ for all values of $b_{1}, b_{2}$ such that $b_{1}-b_{2}^{2} \geqslant 0$.

Paper 1, Section II, C

commentShow that if the energy levels are discrete, the general solution of the Schrödinger equation

$i \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \nabla^{2} \psi+V(\mathbf{x}) \psi$

is a linear superposition of stationary states

$\psi(\mathbf{x}, t)=\sum_{n=1}^{\infty} a_{n} \chi_{n}(\mathbf{x}) \exp \left(-i E_{n} t / \hbar\right)$

where $\chi_{n}(\mathbf{x})$ is a solution of the time-independent Schrödinger equation and $a_{n}$ are complex coefficients. Can this general solution be considered to be a stationary state? Justify your answer.

A linear operator $\hat{O}$ acts on the orthonormal energy eigenfunctions $\chi_{n}$ as follows:

$\begin{aligned} &\hat{O} \chi_{1}=\chi_{1}+\chi_{2} \\ &\hat{O} \chi_{2}=\chi_{1}+\chi_{2} \\ &\hat{O} \chi_{n}=0, \quad n \geqslant 3 \end{aligned}$

Obtain the eigenvalues of $\hat{O}$. Hence, find the normalised eigenfunctions of $\hat{O}$. In an experiment a measurement is made of $\hat{O}$ at $t=0$ yielding an eigenvalue of 2 . What is the probability that a measurement at some later time $t$ will yield an eigenvalue of 2 ?

Paper 1, Section I, H

commentDescribe the generalised likelihood ratio test and the type of statistical question for which it is useful.

Suppose that $X_{1}, \ldots, X_{n}$ are independent and identically distributed random variables with the Gamma $(2, \lambda)$ distribution, having density function $\lambda^{2} x \exp (-\lambda x), x \geqslant 0$. Similarly, $Y_{1}, \ldots, Y_{n}$ are independent and identically distributed with the Gamma $(2, \mu)$ distribution. It is desired to test the hypothesis $H_{0}: \lambda=\mu$ against $H_{1}: \lambda \neq \mu$. Derive the generalised likelihood ratio test and express it in terms of $R=\sum_{i} X_{i} / \sum_{i} Y_{i}$.

Let $F_{\nu_{1}, \nu_{2}}^{(1-\alpha)}$ denote the value that a random variable having the $F_{\nu_{1}, \nu_{2}}$ distribution exceeds with probability $\alpha$. Explain how to decide the outcome of a size $0.05$ test when $n=5$ by knowing only the value of $R$ and the value $F_{\nu_{1}, \nu_{2}}^{(1-\alpha)}$, for some $\nu_{1}, \nu_{2}$ and $\alpha$, which you should specify.

[You may use the fact that the $\chi_{k}^{2}$ distribution is equivalent to the $\operatorname{Gamma}(k / 2,1 / 2)$ distribution.]

Paper 1, Section II, H

commentState and prove the Neyman-Pearson lemma.

A sample of two independent observations, $\left(x_{1}, x_{2}\right)$, is taken from a distribution with density $f(x ; \theta)=\theta x^{\theta-1}, 0 \leqslant x \leqslant 1$. It is desired to test $H_{0}: \theta=1$ against $H_{1}: \theta=2$. Show that the best test of size $\alpha$ can be expressed using the number $c$ such that

$1-c+c \log c=\alpha .$

Is this the uniformly most powerful test of size $\alpha$ for testing $H_{0}$ against $H_{1}: \theta>1 ?$

Suppose that the prior distribution of $\theta$ is $P(\theta=1)=4 \gamma /(1+4 \gamma), P(\theta=2)=$ $1 /(1+4 \gamma)$, where $1>\gamma>0$. Find the test of $H_{0}$ against $H_{1}$ that minimizes the probability of error.

Let $w(\theta)$ denote the power function of this test at $\theta(\geqslant 1)$. Show that

$w(\theta)=1-\gamma^{\theta}+\gamma^{\theta} \log \gamma^{\theta}$

Paper 1, Section I, B

commentState how to find the stationary points of a $C^{1}$ function $f(x, y)$ restricted to the circle $x^{2}+y^{2}=1$, using the method of Lagrange multipliers. Explain why, in general, the method of Lagrange multipliers works, in the case where there is just one constraint.

Find the stationary points of $x^{4}+2 y^{3}$ restricted to the circle $x^{2}+y^{2}=1$.