Part IB, 2011, Paper 3

# Part IB, 2011, Paper 3

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Paper 3, Section I, $2 \mathrm{E}$

commentSuppose $f$ is a uniformly continuous mapping from a metric space $X$ to a metric space $Y$. Prove that $f\left(x_{n}\right)$ is a Cauchy sequence in $Y$ for every Cauchy sequence $x_{n}$ in $X$.

Let $f$ be a continuous mapping between metric spaces and suppose that $f$ has the property that $f\left(x_{n}\right)$ is a Cauchy sequence whenever $x_{n}$ is a Cauchy sequence. Is it true that $f$ must be uniformly continuous? Justify your answer.

Paper 3, Section II, E

commentConsider a map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$.

Assume $f$ is differentiable at $x$ and let $D_{x} f$ denote the derivative of $f$ at $x$. Show that

$D_{x} f(v)=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t}$

for any $v \in \mathbb{R}^{n}$.

Assume now that $f$ is such that for some fixed $x$ and for every $v \in \mathbb{R}^{n}$ the limit

$\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t}$

exists. Is it true that $f$ is differentiable at $x ?$ Justify your answer.

Let $M_{k}$ denote the set of all $k \times k$ real matrices which is identified with $\mathbb{R}^{k^{2}}$. Consider the function $f: M_{k} \rightarrow M_{k}$ given by $f(A)=A^{3}$. Explain why $f$ is differentiable. Show that the derivative of $f$ at the matrix $A$ is given by

$D_{A} f(H)=H A^{2}+A H A+A^{2} H$

for any matrix $H \in M_{k}$. State carefully the inverse function theorem and use it to prove that there exist open sets $U$ and $V$ containing the identity matrix such that given $B \in V$ there exists a unique $A \in U$ such that $A^{3}=B$.

Paper 3, Section II, E

commentLet $g: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function such that

$\int_{\Gamma} g(z) d z=0$

for any closed curve $\Gamma$ which is the boundary of a rectangle in $\mathbb{C}$ with sides parallel to the real and imaginary axes. Prove that $g$ is analytic.

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be continuous. Suppose in addition that $f$ is analytic at every point $z \in \mathbb{C}$ with non-zero imaginary part. Show that $f$ is analytic at every point in $\mathbb{C} .$

Let $\mathbb{H}$ be the upper half-plane of complex numbers $z$ with positive imaginary part $\Im(z)>0$. Consider a continuous function $F: \mathbb{H} \cup \mathbb{R} \rightarrow \mathbb{C}$ such that $F$ is analytic on $\mathbb{H}$ and $F(\mathbb{R}) \subset \mathbb{R}$. Define $f: \mathbb{C} \rightarrow \mathbb{C}$ by

$f(z)= \begin{cases}F(z) & \text { if } \Im(z) \geqslant 0 \\ \overline{F(\bar{z})} & \text { if } \Im(z) \leqslant 0\end{cases}$

Show that $f$ is analytic.

Paper 3, Section I, D

commentWrite down the function $\psi(u, v)$ that satisfies

$\frac{\partial^{2} \psi}{\partial u^{2}}+\frac{\partial^{2} \psi}{\partial v^{2}}=0, \quad \psi\left(-\frac{1}{2}, v\right)=-1, \quad \psi\left(\frac{1}{2}, v\right)=1$

The circular arcs $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ in the complex $z$-plane are defined by

$|z+1|=1, z \neq 0 \text { and }|z-1|=1, z \neq 0,$

respectively. You may assume without proof that the mapping from the complex $z$-plane to the complex $\zeta$-plane defined by

$\zeta=\frac{1}{z}$

takes $\mathcal{C}_{1}$ to the line $u=-\frac{1}{2}$ and $\mathcal{C}_{2}$ to the line $u=\frac{1}{2}$, where $\zeta=u+i v$, and that the region $\mathcal{D}$ in the $z$-plane exterior to both the circles $|z+1|=1$ and $|z-1|=1$ maps to the region in the $\zeta$-plane given by $-\frac{1}{2}<u<\frac{1}{2}$.

Use the above mapping to solve the problem

$\nabla^{2} \phi=0 \quad \text { in } \mathcal{D}, \quad \phi=-1 \text { on } \mathcal{C}_{1} \text { and } \phi=1 \text { on } \mathcal{C}_{2}$

Paper 3, Section II, C

commentShow, using the vacuum Maxwell equations, that for any volume $V$ with surface $S$,

$\frac{d}{d t} \int_{V}\left(\frac{\epsilon_{0}}{2} \mathbf{E} \cdot \mathbf{E}+\frac{1}{2 \mu_{0}} \mathbf{B} \cdot \mathbf{B}\right) d V=\int_{S}\left(-\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}\right) \cdot \mathbf{d} \mathbf{S}$

What is the interpretation of this equation?

A uniform straight wire, with a circular cross section of radius $r$, has conductivity $\sigma$ and carries a current $I$. Calculate $\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$ at the surface of the wire, and hence find the flux of $\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$ into unit length of the wire. Relate your result to the resistance of the wire, and the rate of energy dissipation.

Paper 3, Section II, $18 \mathrm{D}$

commentWater of constant density $\rho$ flows steadily through a long cylindrical tube, the wall of which is elastic. The exterior radius of the tube at a distance $z$ along the tube, $r(z)$, is determined by the pressure in the tube, $p(z)$, according to

$r(z)=r_{0}+b\left(p(z)-p_{0}\right)$

where $r_{0}$ and $p_{0}$ are the radius and pressure far upstream $(z \rightarrow-\infty)$, and $b$ is a positive constant.

The interior radius of the tube is $r(z)-h(z)$, where $h(z)$, the thickness of the wall, is a given slowly varying function of $z$ which is zero at both ends of the pipe. The velocity of the water in the pipe is $u(z)$ and the water enters the pipe at velocity $V$.

Show that $u(z)$ satisfies

$H=1-v^{-\frac{1}{2}}+\frac{1}{4} k\left(1-v^{2}\right)$

where $H=\frac{h}{r_{0}}, v=\frac{u}{V}$ and $k=\frac{2 b \rho V^{2}}{r_{0}}$. Sketch the graph of $H$ against $v$.

Let $H_{m}$ be the maximum value of $H$ in the tube. Show that the flow is only possible if $H_{m}$ does not exceed a certain critical value $H_{c}$. Find $H_{c}$ in terms of $k$.

Show that, under conditions to be determined (which include a condition on the value of $k)$, the water can leave the pipe with speed less than $V$.

Paper 3, Section I, F

commentLet $R(x, \theta)$ denote anti-clockwise rotation of the Euclidean plane $\mathbb{R}^{2}$ through an angle $\theta$ about a point $x$.

Show that $R(x, \theta)$ is a composite of two reflexions.

Assume $\theta, \phi \in(0, \pi)$. Show that the composite $R(y, \phi) \cdot R(x, \theta)$ is also a rotation $R(z, \psi)$. Find $z$ and $\psi$.

Paper 3, Section II, F

commentSuppose that $\eta(u)=(f(u), 0, g(u))$ is a unit speed curve in $\mathbb{R}^{3}$. Show that the corresponding surface of revolution $S$ obtained by rotating this curve about the $z$-axis has Gaussian curvature $K=-\left(d^{2} f / d u^{2}\right) / f$.

Paper 3, Section I, $1 F$

commentSuppose that $A$ is an integral domain containing a field $K$ and that $A$ is finitedimensional as a $K$-vector space. Prove that $A$ is a field.

Paper 3, Section II, F

commentSuppose that $A$ is a matrix over $\mathbb{Z}$. What does it mean to say that $A$ can be brought to Smith normal form?

Show that the structure theorem for finitely generated modules over $\mathbb{Z}$ (which you should state) follows from the existence of Smith normal forms for matrices over $\mathbb{Z}$.

Bring the matrix $\left(\begin{array}{cc}-4 & -6 \\ 2 & 2\end{array}\right)$ to Smith normal form.

Suppose that $M$ is the $\mathbb{Z}$-module with generators $e_{1}, e_{2}$, subject to the relations

$-4 e_{1}+2 e_{2}=-6 e_{1}+2 e_{2}=0$

Describe $M$ in terms of the structure theorem.

Paper 3, Section II, G

comment(i) Let $A$ be an $n \times n$ complex matrix and $f(X)$ a polynomial with complex coefficients. By considering the Jordan normal form of $A$ or otherwise, show that if the eigenvalues of $A$ are $\lambda_{1}, \ldots, \lambda_{n}$ then the eigenvalues of $f(A)$ are $f\left(\lambda_{1}\right), \ldots, f\left(\lambda_{n}\right)$.

(ii) Let $B=\left(\begin{array}{llll}a & d & c & b \\ b & a & d & c \\ c & b & a & d \\ d & c & b & a\end{array}\right)$. Write $B$ as $B=f(A)$ for a polynomial $f$ with $A=\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right)$, and find the eigenvalues of $B$

[Hint: compute the powers of $A$.]

Paper 3, Section I, H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain with state space $S$.

(i) What does it mean to say that $\left(X_{n}\right)_{n} \geqslant 0$ has the strong Markov property? Your answer should include the definition of the term stopping time.

(ii) Show that

$\mathbb{P}\left(X_{n}=i \text { at least } k \text { times } \mid X_{0}=i\right)=\left[\mathbb{P}\left(X_{n}=i \text { at least once } \mid X_{0}=i\right)\right]^{k}$

for a state $i \in S$. You may use without proof the fact that $\left(X_{n}\right)_{n \geqslant 0}$ has the strong Markov property.

Paper 3, Section I, A

commentThe Fourier transform $\tilde{h}(k)$ of the function $h(x)$ is defined by

$\tilde{h}(k)=\int_{-\infty}^{\infty} h(x) e^{-i k x} d x$

(i) State the inverse Fourier transform formula expressing $h(x)$ in terms of $\widetilde{h}(k)$.

(ii) State the convolution theorem for Fourier transforms.

(iii) Find the Fourier transform of the function $f(x)=e^{-|x|}$. Hence show that the convolution of the function $f(x)=e^{-|x|}$ with itself is given by the integral expression

$\frac{2}{\pi} \int_{-\infty}^{\infty} \frac{e^{i k x}}{\left(1+k^{2}\right)^{2}} d k$

Paper 3, Section II, A

commentA uniform stretched string of length $L$, density per unit length $\mu$ and tension $T=\mu c^{2}$ is fixed at both ends. Its transverse displacement is given by $y(x, t)$ for $0 \leqslant x \leqslant L$. The motion of the string is resisted by the surrounding medium with a resistive force per unit length of $-2 k \mu \frac{\partial y}{\partial t}$.

(i) Show that the equation of motion of the string is

$\frac{\partial^{2} y}{\partial t^{2}}+2 k \frac{\partial y}{\partial t}-c^{2} \frac{\partial^{2} y}{\partial x^{2}}=0$

provided that the transverse motion can be regarded as small.

(ii) Suppose now that $k=\frac{\pi c}{L}$. Find the displacement of the string for $t \geqslant 0$ given the initial conditions

$y(x, 0)=A \sin \left(\frac{\pi x}{L}\right) \quad \text { and } \quad \frac{\partial y}{\partial t}(x, 0)=0$

(iii) Sketch the transverse displacement at $x=\frac{L}{2}$ as a function of time for $t \geqslant 0$.

Paper 3, Section I, 3G

commentLet $X, Y$ be topological spaces, and suppose $Y$ is Hausdorff.

(i) Let $f, g: X \rightarrow Y$ be two continuous maps. Show that the set

$E(f, g):=\{x \in X \mid f(x)=g(x)\} \subset X$

is a closed subset of $X$.

(ii) Let $W$ be a dense subset of $X$. Show that a continuous map $f: X \rightarrow Y$ is determined by its restriction $\left.f\right|_{W}$ to $W$.

Paper 3, Section II, B

commentA Gaussian quadrature formula provides an approximation to the integral

$\int_{-1}^{1}\left(1-x^{2}\right) f(x) d x \approx \sum_{k=1}^{\nu} b_{k} f\left(c_{k}\right)$

which is exact for all $f(x)$ that are polynomials of degree $\leqslant(2 \nu-1)$.

Write down explicit expressions for the $b_{k}$ in terms of integrals, and explain why it is necessary that the $c_{k}$ are the zeroes of a (monic) polynomial $p_{\nu}$ of degree $\nu$ that satisfies $\int_{-1}^{1}\left(1-x^{2}\right) p_{\nu}(x) q(x) d x=0$ for any polynomial $q(x)$ of degree less than $\nu .$

The first such polynomials are $p_{0}=1, p_{1}=x, p_{2}=x^{2}-1 / 5, p_{3}=x^{3}-3 x / 7$. Show that the Gaussian quadrature formulae for $\nu=2,3$ are

$\begin{array}{ll} \nu=2: & \frac{2}{3}\left[f\left(-\frac{1}{\sqrt{5}}\right)+f\left(\frac{1}{\sqrt{5}}\right)\right] \\ \nu=3: & \frac{14}{45}\left[f\left(-\sqrt{\frac{3}{7}}\right)+f\left(\sqrt{\frac{3}{7}}\right)\right]+\frac{32}{45} f(0) \end{array}$

Verify the result for $\nu=3$ by considering $f(x)=1, x^{2}, x^{4}$.

Paper 3, Section II, H

comment(i) What does it mean to say a set $C \subseteq \mathbb{R}^{n}$ is convex?

(ii) What does it mean to say $z$ is an extreme point of a convex set $C ?$

Let $A$ be an $m \times n$ matrix, where $n>m$. Let $b$ be an $m \times 1$ vector, and let

$C=\left\{x \in \mathbb{R}^{n}: A x=b, x \geqslant 0\right\}$

where the inequality is interpreted component-wise.

(iii) Show that $C$ is convex.

(iv) Let $z=\left(z_{1}, \ldots, z_{n}\right)^{T}$ be a point in $C$ with the property that at least $m+1$ indices $i$ are such that $z_{i}>0$. Show that $z$ is not an extreme point of $C$. [Hint: If $r>m$, then any set of $r$ vectors in $\mathbb{R}^{m}$ is linearly dependent.]

(v) Now suppose that every set of $m$ columns of $A$ is linearly independent. Let $z=\left(z_{1}, \ldots, z_{n}\right)^{T}$ be a point in $C$ with the property that at most $m$ indices $i$ are such that $z_{i}>0$. Show that $z$ is an extreme point of $C$.

Paper 3, Section I, C

commentA particle of mass $m$ and energy $E$, incident from $x=-\infty$, scatters off a delta function potential at $x=0$. The time independent Schrödinger equation is

$-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+U \delta(x) \psi=E \psi$

where $U$ is a positive constant. Find the reflection and transmission probabilities.

Paper 3, Section II, C

commentFor an electron in a hydrogen atom, the stationary state wavefunctions are of the form $\psi(r, \theta, \phi)=R(r) Y_{l m}(\theta, \phi)$, where in suitable units $R$ obeys the radial equation

$\frac{d^{2} R}{d r^{2}}+\frac{2}{r} \frac{d R}{d r}-\frac{l(l+1)}{r^{2}} R+2\left(E+\frac{1}{r}\right) R=0 .$

Explain briefly how the terms in this equation arise.

This radial equation has bound state solutions of energy $E=E_{n}$, where $E_{n}=-\frac{1}{2 n^{2}}(n=1,2,3, \ldots)$. Show that when $l=n-1$, there is a solution of the form $R(r)=r^{\alpha} e^{-r / n}$, and determine $\alpha$. Find the expectation value $\langle r\rangle$ in this state.

What is the total degeneracy of the energy level with energy $E_{n}$ ?

Paper 3, Section II, H

commentConsider the general linear model

$Y=X \beta+\epsilon$

where $X$ is a known $n \times p$ matrix, $\beta$ is an unknown $p \times 1$ vector of parameters, and $\epsilon$ is an $n \times 1$ vector of independent $N\left(0, \sigma^{2}\right)$ random variables with unknown variance $\sigma^{2}$. Assume the $p \times p$ matrix $X^{T} X$ is invertible.

(i) Derive the least squares estimator $\widehat{\beta}$ of $\beta$.

(ii) Derive the distribution of $\widehat{\beta}$. Is $\widehat{\beta}$ an unbiased estimator of $\beta$ ?

(iii) Show that $\frac{1}{\sigma^{2}}\|Y-X \widehat{\beta}\|^{2}$ has the $\chi^{2}$ distribution with $k$ degrees of freedom, where $k$ is to be determined.

(iv) Let $\tilde{\beta}$ be an unbiased estimator of $\beta$ of the form $\tilde{\beta}=C Y$ for some $p \times n$ matrix $C$. By considering the matrix $\mathbb{E}\left[(\widehat{\beta}-\widetilde{\beta})(\widehat{\beta}-\beta)^{T}\right]$ or otherwise, show that $\widehat{\beta}$ and $\widehat{\beta}-\widetilde{\beta}$ are independent.

[You may use standard facts about the multivariate normal distribution as well as results from linear algebra, including the fact that $I-X\left(X^{T} X\right)^{-1} X^{T}$ is a projection matrix of rank $n-p$, as long as they are carefully stated.]

Paper 3, Section I, D

commentFind, using a Lagrange multiplier, the four stationary points in $\mathbb{R}^{3}$ of the function $x^{2}+y^{2}+z^{2}$ subject to the constraint $x^{2}+2 y^{2}-z^{2}=1$. By considering the situation geometrically, or otherwise, identify the nature of the constrained stationary points.

How would your answers differ if, instead, the stationary points of the function $x^{2}+2 y^{2}-z^{2}$ were calculated subject to the constraint $x^{2}+y^{2}+z^{2}=1 ?$