Part IB, 2002, Paper 4

# Part IB, 2002, Paper 4

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4.I.1E

comment(a) Let $(X, d)$ be a metric space containing the point $x_{0}$, and let

$U=\left\{x \in X: d\left(x, x_{0}\right)<1\right\}, \quad K=\left\{x \in X: d\left(x, x_{0}\right) \leqslant 1\right\}$

Is $U$ necessarily the largest open subset of $K$ ? Is $K$ necessarily the smallest closed set that contains $U$ ? Justify your answers.

(b) Let $X$ be a normed space with norm $\|\cdot\|$, and let

$U=\{x \in X:\|x\|<1\}, \quad K=\{x \in X:\|x\| \leqslant 1\}$

Is $U$ necessarily the largest open subset of $K$ ? Is $K$ necessarily the smallest closed set that contains $U$ ? Justify your answers.

4.II.10E

comment(a) Let $V$ be a finite-dimensional real vector space, and let $\left\|_{1}\right\|_{1}$ and $\|\cdot\|_{2}$ be two norms on $V$. Show that a function $f: V \rightarrow \mathbb{R}$ is differentiable at a point $a$ in $V$ with respect to $\|\cdot\|_{1}$ if and only if it is differentiable at $a$ with respect to $\|\cdot\|_{2}$, and that if this is so then the derivative $f^{\prime}(a)$ of $f$ is independent of the norm used. [You may assume that all norms on a finite-dimensional vector space are equivalent.]

(b) Let $V_{1}, V_{2}$ and $V_{3}$ be finite-dimensional normed real vector spaces with $V_{j}$ having norm $\|\cdot\|_{j}, j=1,2,3$, and let $f: V_{1} \times V_{2} \rightarrow V_{3}$ be a continuous bilinear mapping. Show that $f$ is differentiable at any point $(a, b)$ in $V_{1} \times V_{2}$, and that $f^{\prime}(a, b)(h, k)=$ $f(h, b)+f(a, k) . \quad$ [You may assume that $\left(\|u\|_{1}^{2}+\|v\|_{2}^{2}\right)^{1 / 2}$ is a norm on $V_{1} \times V_{2}$, and that $\left\{(x, y) \in V_{1} \times V_{2}:\|x\|_{1}=1,\|y\|_{2}=1\right\}$ is compact.]

4.I.8B

commentLet $f$ be a function such that $\int_{-\infty}^{+\infty}|f(x)|^{2} d x<\infty$. Prove that

$\int_{-\infty}^{+\infty} f(x+k) \overline{f(x+l)} d x=0 \quad \text { for all integers } k \text { and } l \text { with } k \neq l,$

if and only if

$\int_{-\infty}^{+\infty}|\widehat{f}(t)|^{2} e^{-i m t} d t=0 \quad \text { for all integers } m \neq 0$

where $\widehat{f}$ is the Fourier transform of $f$.

4.II.17B

comment(a) Using the inequality $\sin \theta \geq 2 \theta / \pi$ for $0 \leq \theta \leq \frac{\pi}{2}$, show that, if $f$ is continuous for large $|z|$, and if $f(z) \rightarrow 0$ as $z \rightarrow \infty$, then

$\lim _{R \rightarrow \infty} \int_{\Gamma_{R}} f(z) e^{i \lambda z} d z=0 \quad \text { for } \quad \lambda>0$

where $\Gamma_{R}=R e^{i \theta}, 0 \leq \theta \leq \pi$.

(b) By integrating an appropriate function $f(z)$ along the contour formed by the semicircles $\Gamma_{R}$ and $\Gamma_{r}$ in the upper half-plane with the segments of the real axis $[-R,-r]$ and $[r, R]$, show that

$\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}$

4.I $. 7 \mathrm{C} \quad$

commentIf $\mathbf{u}$ is given in Cartesian co-ordinates as $\mathbf{u}=(-\Omega y, \Omega x, 0)$, with $\Omega$ a constant, verify that

$\mathbf{u} \cdot \nabla \mathbf{u}=\nabla\left(-\frac{1}{2} \mathbf{u}^{2}\right)$

When incompressible fluid is placed in a stationary cylindrical container of radius $a$ with its axis vertical, the depth of the fluid is $h$. Assuming that the free surface does not reach the bottom of the container, use cylindrical polar co-ordinates to find the equation of the free surface when the fluid and the container rotate steadily about this axis with angular velocity $\Omega$.

Deduce the angular velocity at which the free surface first touches the bottom of the container.

4.II.16C

commentUse Euler's equation to show that in a planar flow of an inviscid fluid the vorticity $\boldsymbol{\omega}$ satisfies

$\frac{D \omega}{D t}=0$

Write down the velocity field associated with a point vortex of strength $\kappa$ in unbounded fluid.

Consider now the flow generated in unbounded fluid by two point vortices of strengths $\kappa_{1}$ and $\kappa_{2}$ at $\mathbf{x}_{1}(t)=\left(x_{1}, y_{1}\right)$ and $\mathbf{x}_{2}(t)=\left(x_{2}, y_{2}\right)$, respectively. Show that in the subsequent motion the quantity

$\mathbf{q}=\kappa_{1} \mathbf{x}_{1}+\kappa_{2} \mathbf{x}_{2}$

remains constant. Show also that the separation of the vortices, $\left|\mathbf{x}_{2}-\mathbf{x}_{1}\right|$, remains constant.

Suppose finally that $\kappa_{1}=\kappa_{2}$ and that the vortices are placed at time $t=0$ at positions $(a, 0)$ and $(-a, 0)$. What are the positions of the vortices at time $t$ ?

4.I.4G

comment(a) Let $X$ be a topological space and suppose $X=C \cup D$, where $C$ and $D$ are disjoint nonempty open subsets of $X$. Show that, if $Y$ is a connected subset of $X$, then $Y$ is entirely contained in either $C$ or $D$.

(b) Let $X$ be a topological space and let $\left\{A_{n}\right\}$ be a sequence of connected subsets of $X$ such that $A_{n} \cap A_{n+1} \neq \emptyset$, for $n=1,2,3, \ldots$. Show that $\bigcup_{n \geqslant 1} A_{n}$ is connected.

4.II.13G

commentA function $f$ is said to be analytic at $\infty$ if there exists a real number $r>0$ such that $f$ is analytic for $|z|>r$ and $\lim _{z \rightarrow 0} f(1 / z)$ is finite (i.e. $f(1 / z)$ has a removable singularity at $z=0)$. $f$ is said to have a pole at $\infty$ if $f(1 / z)$ has a pole at $z=0$. Suppose that $f$ is a meromorphic function on the extended plane $\mathbb{C}_{\infty}$, that is, $f$ is analytic at each point of $\mathbb{C}_{\infty}$ except for poles.

(a) Show that if $f$ has a pole at $z=\infty$, then there exists $r>0$ such that $f(z)$ has no poles for $r<|z|<\infty$.

(b) Show that the number of poles of $f$ is finite.

(c) By considering the Laurent expansions around the poles show that $f$ is in fact a rational function, i.e. of the form $p / q$, where $p$ and $q$ are polynomials.

(d) Deduce that the only bijective meromorphic maps of $\mathbb{C}_{\infty}$ onto itself are the Möbius maps.

4.I $. 6 \mathrm{~F} \quad$

commentDefine the rank and nullity of a linear map between finite-dimensional vector spaces.

State the rank-nullity formula.

Let $\alpha: U \rightarrow V$ and $\beta: V \rightarrow W$ be linear maps. Prove that

$\operatorname{rank}(\alpha)+\operatorname{rank}(\beta)-\operatorname{dim} V \leqslant \operatorname{rank}(\beta \alpha) \leqslant \min \{\operatorname{rank}(\alpha), \operatorname{rank}(\beta)\}$

Part IB

4.II.15F

commentDefine the dual space $V^{*}$ of a finite-dimensional real vector space $V$, and explain what is meant by the basis of $V^{*}$ dual to a given basis of $V$. Explain also what is meant by the statement that the second dual $V^{* *}$ is naturally isomorphic to $V$.

Let $V_{n}$ denote the space of real polynomials of degree at most $n$. Show that, for any real number $x$, the function $e_{x}$ mapping $p$ to $p(x)$ is an element of $V_{n}^{*}$. Show also that, if $x_{1}, x_{2}, \ldots, x_{n+1}$ are distinct real numbers, then $\left\{e_{x_{1}}, e_{x_{2}}, \ldots, e_{x_{n+1}}\right\}$ is a basis of $V_{n}^{*}$, and find the basis of $V_{n}$ dual to it.

Deduce that, for any $(n+1)$ distinct points $x_{1}, \ldots, x_{n+1}$ of the interval $[-1,1]$, there exist scalars $\lambda_{1}, \ldots, \lambda_{n+1}$ such that

$\int_{-1}^{1} p(t) d t=\sum_{i=1}^{n+1} \lambda_{i} p\left(x_{i}\right)$

for all $p \in V_{n}$. For $n=4$ and $\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=\left(-1,-\frac{1}{2}, 0, \frac{1}{2}, 1\right)$, find the corresponding scalars $\lambda_{i}$.

4.I.2A

commentUse the method of Lagrange multipliers to find the largest volume of a rectangular parallelepiped that can be inscribed in the ellipsoid

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 .$

4.II.11A

commentA function $y(x)$ is chosen to make the integral

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

stationary, subject to given values of $y(a), y^{\prime}(a), y(b)$ and $y^{\prime}(b)$. Derive an analogue of the Euler-Lagrange equation for $y(x)$.

Solve this equation for the case where

$f=x^{4} y^{\prime \prime 2}+4 y^{2} y^{\prime}$

in the interval $[0,1]$ and

$x^{2} y(x) \rightarrow 0, \quad x y(x) \rightarrow 1$

as $x \rightarrow 0$, whilst

$y(1)=2, \quad y^{\prime}(1)=0$

4.I.5H

commentState and prove the max flow/min cut theorem. In your answer you should define clearly the following terms: flow, maximal flow, cut, capacity.

4.II.14H

commentA gambler at a horse race has an amount $b>0$ to bet. The gambler assesses $p_{i}$, the probability that horse $i$ will win, and knows that $s_{i} \geq 0$ has been bet on horse $i$ by others, for $i=1,2, \ldots, n$. The total amount bet on the race is shared out in proportion to the bets on the winning horse, and so the gambler's optimal strategy is to choose $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ so that it maximizes

$\sum_{i=1}^{n} \frac{p_{i} x_{i}}{s_{i}+x_{i}} \quad \text { subject to } \sum_{i=1}^{n} x_{i}=b, \quad x_{1}, \ldots, x_{n} \geq 0$

where $x_{i}$ is the amount the gambler bets on horse $i$. Show that the optimal solution to (1) also solves the following problem:

$\text { minimize } \sum_{i=1}^{n} \frac{p_{i} s_{i}}{s_{i}+x_{i}} \quad \text { subject to } \sum_{i=1}^{n} x_{i}=b, \quad x_{1}, \ldots, x_{n} \geq 0$

Assume that $p_{1} / s_{1} \geq p_{2} / s_{2} \geq \ldots \geq p_{n} / s_{n}$. Applying the Lagrangian sufficiency theorem, prove that the optimal solution to (1) satisfies

$\frac{p_{1} s_{1}}{\left(s_{1}+x_{1}\right)^{2}}=\ldots=\frac{p_{k} s_{k}}{\left(s_{k}+x_{k}\right)^{2}}, \quad x_{k+1}=\ldots=x_{n}=0$

with maximal possible $k \in\{1,2, \ldots, n\}$.

[You may use the fact that for all $\lambda<0$, the minimum of the function $x \mapsto \frac{p s}{s+x}-\lambda x$ on the non-negative axis $0 \leq x<\infty$ is attained at

$\left.x(\lambda)=\left(\sqrt{\frac{p s}{-\lambda}}-s\right)^{+} \equiv \max \left(\sqrt{\frac{p s}{-\lambda}}-s, 0\right) .\right]$

Deduce that if $b$ is small enough, the gambler's optimal strategy is to bet on the horses for which the ratio $p_{i} / s_{i}$ is maximal. What is his expected gain in this case?

4.I $. 9$ D

commentA particle with mass $M$ is observed to be at rest. It decays into a particle of mass $m<M$, and a massless particle. Calculate the energies and momenta of both final particles.

4.II.18D

commentA javelin of length $2 \mathrm{~m}$ is thrown horizontally and lengthwise into a shed of length $1.5 \mathrm{~m}$ at a speed of $0.8 c$, where $c$ is the speed of light.

(a) What is the length of the javelin in the rest frame of the shed?

(b) What is the length of the shed in the rest frame of the javelin?

(c) Draw a space-time diagram in the rest frame coordinates $(c t, x)$ of the shed, showing the world lines of both ends of the javelin, and of the front and back of the shed. Draw a second space-time diagram in the rest frame coordinates $\left(c t^{\prime}, x^{\prime}\right)$ of the javelin, again showing the world lines of both ends of the javelin and of the front and back of the shed.

(d) Clearly mark the space-time events corresponding to (A) the trailing end of the javelin entering the shed, and (B) the leading end of the javelin hitting the back of the shed. Give the corresponding $(c t, x)$ and $\left(c t^{\prime}, x^{\prime}\right)$ coordinates for both (A) and (B). Are these two events space-like, null or time-like separated? How does the javelin fit inside the shed, even though it is initially longer than the shed in its own rest frame?

4.I.3H

commentFrom each of 100 concrete mixes six sample blocks were taken and subjected to strength tests, the number out of the six blocks failing the test being recorded in the following table:

$\begin{array}{lrrrrrrr} \text { No. } x \text { failing strength tests } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \text { No. of mixes with } x \text { failures } & 53 & 32 & 12 & 2 & 1 & 0 & 0 \end{array}$

On the assumption that the probability of failure is the same for each block, obtain an unbiased estimate of this probability and explain how to find a $95 \%$ confidence interval for it.

4.II.12H

commentExplain what is meant by a prior distribution, a posterior distribution, and a Bayes estimator. Relate the Bayes estimator to the posterior distribution for both quadratic and absolute error loss functions.

Suppose $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables from a distribution uniform on $(\theta-1, \theta+1)$, and that the prior for $\theta$ is uniform on $(20,50)$.

Calculate the posterior distribution for $\theta$, given $\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$, and find the point estimate for $\theta$ under both quadratic and absolute error loss function.