Part IB, 2003, Paper 4

# Part IB, 2003, Paper 4

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4.I.1F

commentExplain what it means for a sequence of functions $\left(f_{n}\right)$ to converge uniformly to a function $f$ on an interval. If $\left(f_{n}\right)$ is a sequence of continuous functions converging uniformly to $f$ on a finite interval $[a, b]$, show that

$\int_{a}^{b} f_{n}(x) d x \longrightarrow \int_{a}^{b} f(x) d x \quad \text { as } n \rightarrow \infty$

Let $f_{n}(x)=x \exp (-x / n) / n^{2}, x \geqslant 0$. Does $f_{n} \rightarrow 0$ uniformly on $[0, \infty) ?$ Does $\int_{0}^{\infty} f_{n}(x) d x \rightarrow 0$ ? Justify your answers.

4.II.10F

commentLet $\left(f_{n}\right)_{n \geqslant 1}$ be a sequence of continuous complex-valued functions defined on a set $E \subseteq \mathbb{C}$, and converging uniformly on $E$ to a function $f$. Prove that $f$ is continuous on $E$.

State the Weierstrass $M$-test for uniform convergence of a series $\sum_{n=1}^{\infty} u_{n}(z)$ of complex-valued functions on a set $E$.

Now let $f(z)=\sum_{n=1}^{\infty} u_{n}(z)$, where

$u_{n}(z)=n^{-2} \sec (\pi z / 2 n) .$

Prove carefully that $f$ is continuous on $\mathbb{C} \backslash \mathbb{Z}$.

[You may assume the inequality $|\cos z| \geqslant|\cos (\operatorname{Re} z)| \cdot]$

4.I.8B

commentFind the Laurent series centred on 0 for the function

$f(z)=\frac{1}{(z-1)(z-2)}$

in each of the domains (a) $|z|<1$, (b) $1<|z|<2$, (c) $|z|>2$.

4.II.17B

commentLet

$f(z)=\frac{z^{m}}{1+z^{n}}, \quad n>m+1, \quad m, n \in \mathbb{N},$

and let $C_{R}$ be the boundary of the domain

$D_{R}=\left\{z=r e^{i \theta}: 0<r<R, \quad 0<\theta<\frac{2 \pi}{n}\right\}, \quad R>1 .$

(a) Using the residue theorem, determine

$\int_{C_{R}} f(z) d z$

(b) Show that the integral of $f(z)$ along the circular part $\gamma_{R}$ of $C_{R}$ tends to 0 as $R \rightarrow \infty$.

(c) Deduce that

$\int_{0}^{\infty} \frac{x^{m}}{1+x^{n}} d x=\frac{\pi}{n \sin \frac{\pi(m+1)}{n}}$

4.I.7C

commentInviscid fluid issues vertically downwards at speed $u_{0}$ from a circular tube of radius a. The fluid falls onto a horizontal plate a distance $H$ below the end of the tube, where it spreads out axisymmetrically.

Show that while the fluid is falling freely it has speed

$u=u_{0}\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{1 / 2}$

and occupies a circular jet of radius

$R=a\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{-1 / 4},$

where $z$ is the height above the plate and $g$ is the acceleration due to gravity.

Show further that along the plate, at radial distances $r \gg a$ (i.e. far from the falling jet), where the fluid is flowing almost horizontally, it does so as a film of height $h(r)$, where

$\frac{a^{4}}{4 r^{2} h^{2}}=1+\frac{2 g}{u_{0}^{2}}(H-h)$

4.II.16C

commentDefine the terms irrotational flow and incompressible flow. The two-dimensional flow of an incompressible fluid is given in terms of a streamfunction $\psi(x, y)$ as

$\mathbf{u}=(u, v)=\left(\frac{\partial \psi}{\partial y},-\frac{\partial \psi}{\partial x}\right)$

in Cartesian coordinates $(x, y)$. Show that the line integral

$\int_{\mathbf{x}_{1}}^{\mathbf{x}_{\mathbf{2}}} \mathbf{u} \cdot \mathbf{n} d l=\psi\left(\mathbf{x}_{\mathbf{2}}\right)-\psi\left(\mathbf{x}_{\mathbf{1}}\right)$

along any path joining the points $\mathbf{x}_{\mathbf{1}}$ and $\mathbf{x}_{\mathbf{2}}$, where $\mathbf{n}$ is the unit normal to the path. Describe how this result is related to the concept of mass conservation.

Inviscid, incompressible fluid is contained in the semi-infinite channel $x>0$, $0<y<1$, which has rigid walls at $x=0$ and at $y=0,1$, apart from a small opening at the origin through which the fluid is withdrawn with volume flux $m$ per unit distance in the third dimension. Show that the streamfunction for irrotational flow in the channel can be chosen (up to an additive constant) to satisfy the equation

$\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}=0$

and boundary conditions

if it is assumed that the flow at infinity is uniform. Solve the boundary-value problem above using separation of variables to obtain

$\psi=-m y+\frac{2 m}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin n \pi y e^{-n \pi x}$

$\begin{aligned} & \psi=0 \quad \text { on } y=0, x>0, \\ & \psi=-m \quad \text { on } x=0,0<y<1, \\ & \psi=-m \quad \text { on } y=1, x>0, \\ & \psi \rightarrow-m y \quad \text { as } x \rightarrow \infty, \end{aligned}$

4.I.4E

comment(a) State and prove Morera's Theorem.

(b) Let $D$ be a domain and for each $n \in \mathbb{N}$ let $f_{n}: D \rightarrow \mathbb{C}$ be an analytic function. Suppose that $f: D \rightarrow \mathbb{C}$ is another function and that $f_{n} \rightarrow f$ uniformly on $D$. Prove that $f$ is analytic.

4.II.13E

comment(a) State the residue theorem and use it to deduce the principle of the argument, in a form that involves winding numbers.

(b) Let $p(z)=z^{5}+z$. Find all $z$ such that $|z|=1$ and $\operatorname{Im}(p(z))=0$. Calculate $\operatorname{Re}(p(z))$ for each such $z$. [It will be helpful to set $z=e^{i \theta}$. You may use the addition formulae $\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$ and $\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$.]

(c) Let $\gamma:[0,2 \pi] \rightarrow \mathbb{C}$ be the closed path $\theta \mapsto e^{i \theta}$. Use your answer to (b) to give a rough sketch of the path $p \circ \gamma$, paying particular attention to where it crosses the real axis.

(d) Hence, or otherwise, determine for every real $t$ the number of $z$ (counted with multiplicity) such that $|z|<1$ and $p(z)=t$. (You need not give rigorous justifications for your calculations.)

4.I.6G

commentLet $\alpha$ be an endomorphism of a finite-dimensional real vector space $U$ such that $\alpha^{2}=\alpha$. Show that $U$ can be written as the direct sum of the kernel of $\alpha$ and the image of $\alpha$. Hence or otherwise, find the characteristic polynomial of $\alpha$ in terms of the dimension of $U$ and the rank of $\alpha$. Is $\alpha$ diagonalizable? Justify your answer.

4.II.15G

commentLet $\alpha \in L(U, V)$ be a linear map between finite-dimensional vector spaces. Let

$\begin{gathered} M^{l}(\alpha)=\{\beta \in L(V, U): \beta \alpha=0\} \quad \text { and } \\ M^{r}(\alpha)=\{\beta \in L(V, U): \alpha \beta=0\} . \end{gathered}$

(a) Prove that $M^{l}(\alpha)$ and $M^{r}(\alpha)$ are subspaces of $L(V, U)$ of dimensions

$\begin{gathered} \operatorname{dim} M^{l}(\alpha)=(\operatorname{dim} V-\operatorname{rank} \alpha) \operatorname{dim} U \quad \text { and } \\ \operatorname{dim} M^{r}(\alpha)=\operatorname{dim} \operatorname{ker}(\alpha) \operatorname{dim} V \end{gathered}$

[You may use the result that there exist bases in $U$ and $V$ so that $\alpha$ is represented by

$\left(\begin{array}{cc} I_{r} & 0 \\ 0 & 0 \end{array}\right)$

where $I_{r}$ is the $r \times r$ identity matrix and $r$ is the rank of $\left.\alpha .\right]$

(b) Let $\Phi: L(U, V) \rightarrow L\left(V^{*}, U^{*}\right)$ be given by $\Phi(\alpha)=\alpha^{*}$, where $\alpha^{*}$ is the dual map induced by $\alpha$. Prove that $\Phi$ is an isomorphism. [You may assume that $\Phi$ is linear, and you may use the result that a finite-dimensional vector space and its dual have the same dimension.]

(c) Prove that

$\Phi\left(M^{l}(\alpha)\right)=M^{r}\left(\alpha^{*}\right) \quad \text { and } \quad \Phi\left(M^{r}(\alpha)\right)=M^{l}\left(\alpha^{*}\right)$

[You may use the results that $(\beta \alpha)^{*}=\alpha^{*} \beta^{*}$ and that $\beta^{* *}$ can be identified with $\beta$ under the canonical isomorphism between a vector space and its double dual.]

(d) Conclude that $\operatorname{rank}(\alpha)=\operatorname{rank}\left(\alpha^{*}\right)$.

4.I.2D

commentConsider the wave equation in a spherically symmetric coordinate system

$\frac{\partial^{2} u(r, t)}{\partial t^{2}}=c^{2} \Delta u(r, t)$

where $\Delta u=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r u)$ is the spherically symmetric Laplacian operator.

(a) Show that the general solution to the equation above is

$u(r, t)=\frac{1}{r}[f(r+c t)+g(r-c t)]$

where $f(x), g(x)$ are arbitrary functions.

(b) Using separation of variables, determine the wave field $u(r, t)$ in response to a pulsating source at the origin $u(0, t)=A \sin \omega t$.

4.II.11D

commentThe velocity potential $\phi(r, \theta)$ for inviscid flow in two dimensions satisfies the Laplace equation

$\Delta \phi=\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right] \phi(r, \theta)=0$

(a) Using separation of variables, derive the general solution to the equation above that is single-valued and finite in each of the domains (i) $0 \leqslant r \leqslant a$; (ii) $a \leqslant r<\infty$.

(b) Assuming $\phi$ is single-valued, solve the Laplace equation subject to the boundary conditions $\frac{\partial \phi}{\partial r}=0$ at $r=a$, and $\frac{\partial \phi}{\partial r} \rightarrow U \cos \theta$ as $r \rightarrow \infty$. Sketch the lines of constant potential.

4.I.5H

commentState and prove the Lagrangian sufficiency theorem for a general optimization problem with constraints.

4.II.14H

commentUse the two-phase simplex method to solve the problem

$\begin{array}{llllll} \operatorname{minimize} & 5 x_{1}-12 x_{2}+13 x_{3} & & \\ \text { subject to } & 4 x_{1}+5 x_{2} & & \leq & 9 \\ & 6 x_{1}+4 x_{2}+ & x_{3} & \geq & 12 \\ & 3 x_{1}+2 x_{2}-x_{3} & \leq & 3 \\ & x_{i} \geq 0, & i=1,2,3 \end{array}$

4.I.9A

commentProve that the two-dimensional Lorentz transformation can be written in the form

$\begin{aligned} x^{\prime} &=x \cosh \phi-c t \sinh \phi \\ c t^{\prime} &=-x \sinh \phi+c t \cosh \phi \end{aligned}$

where $\tanh \phi=v / c$. Hence, show that

$\begin{aligned} &x^{\prime}+c t^{\prime}=e^{-\phi}(x+c t) \\ &x^{\prime}-c t^{\prime}=e^{\phi}(x-c t) \end{aligned}$

Given that frame $S^{\prime}$ has speed $v$ with respect to $S$ and $S^{\prime \prime}$ has speed $v^{\prime}$ with respect to $S^{\prime}$, use this formalism to find the speed $v^{\prime \prime}$ of $S^{\prime \prime}$ with respect to $S$.

[Hint: rotation through a hyperbolic angle $\phi$, followed by rotation through $\phi^{\prime}$, is equivalent to rotation through $\phi+\phi^{\prime}$.]

4.II.18A

commentA pion of rest mass $M$ decays at rest into a muon of rest mass $m<M$ and a neutrino of zero rest mass. What is the speed $u$ of the muon?

In the pion rest frame $S$, the muon moves in the $y$-direction. A moving observer, in a frame $S^{\prime}$ with axes parallel to those in the pion rest frame, wishes to take measurements of the decay along the $x$-axis, and notes that the pion has speed $v$ with respect to the $x$-axis. Write down the four-dimensional Lorentz transformation relating $S^{\prime}$ to $S$ and determine the momentum of the muon in $S^{\prime}$. Hence show that in $S^{\prime}$ the direction of motion of the muon makes an angle $\theta$ with respect to the $y$-axis, where

$\tan \theta=\frac{M^{2}+m^{2}}{M^{2}-m^{2}} \frac{v}{\left(c^{2}-v^{2}\right)^{1 / 2}} .$

4.I.3H

commentThe following table contains a distribution obtained in 320 tosses of 6 coins and the corresponding expected frequencies calculated with the formula for the binomial distribution for $p=0.5$ and $n=6$.

\begin{tabular}{l|rrrrrrr} No. heads & 0 & 1 & 2 & 3 & 4 & 5 & 6 \ \hline Observed frequencies & 3 & 21 & 85 & 110 & 62 & 32 & 7 \ Expected frequencies & 5 & 30 & 75 & 100 & 75 & 30 & 5 \end{tabular}

Conduct a goodness-of-fit test at the $0.05$ level for the null hypothesis that the coins are all fair.

[Hint:

$\left.\begin{array}{lcccc}\text { Distribution } & \chi_{5}^{2} & \chi_{6}^{2} & \chi_{7}^{2} \\ 95 \% \text { percentile } & 11.07 & 12.59 & 14.07 & \end{array}\right]$

4.II.12H $\quad$

commentState and prove the Rao-Blackwell theorem.

Suppose that $X_{1}, \ldots, X_{n}$ are independent random variables uniformly distributed over $(\theta, 3 \theta)$. Find a two-dimensional sufficient statistic $T(X)$ for $\theta$. Show that an unbiased estimator of $\theta$ is $\hat{\theta}=X_{1} / 2$.

Find an unbiased estimator of $\theta$ which is a function of $T(X)$ and whose mean square error is no more than that of $\hat{\theta}$.