Part IB, 2002, Paper 1

# Part IB, 2002, Paper 1

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1.I.1E

commentSuppose that for each $n=1,2, \ldots$, the function $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ is uniformly continuous on $\mathbb{R}$.

(a) If $f_{n} \rightarrow f$ pointwise on $\mathbb{R}$ is $f$ necessarily continuous on $\mathbb{R}$ ?

(b) If $f_{n} \rightarrow f$ uniformly on $\mathbb{R}$ is $f$ necessarily continuous on $\mathbb{R}$ ?

In each case, give a proof or a counter-example (with justification).

1.II.10E

commentSuppose that $(X, d)$ is a metric space that has the Bolzano-Weierstrass property (that is, any sequence has a convergent subsequence). Let $\left(Y, d^{\prime}\right)$ be any metric space, and suppose that $f$ is a continuous map of $X$ onto $Y$. Show that $\left(Y, d^{\prime}\right)$ also has the Bolzano-Weierstrass property.

Show also that if $f$ is a bijection of $X$ onto $Y$, then $f^{-1}: Y \rightarrow X$ is continuous.

By considering the map $x \mapsto e^{i x}$ defined on the real interval $[-\pi / 2, \pi / 2]$, or otherwise, show that there exists a continuous choice of arg $z$ for the complex number $z$ lying in the right half-plane $\{x+i y: x>0\}$.

1.I.7B

commentUsing contour integration around a rectangle with vertices

$-x, x, x+i y,-x+i y,$

prove that, for all real $y$,

$\int_{-\infty}^{+\infty} e^{-(x+i y)^{2}} d x=\int_{-\infty}^{+\infty} e^{-x^{2}} d x$

Hence derive that the function $f(x)=e^{-x^{2} / 2}$ is an eigenfunction of the Fourier transform

$\widehat{f}(y)=\int_{-\infty}^{+\infty} f(x) e^{-i x y} d x$

i.e. $\widehat{f}$ is a constant multiple of $f$.

1.II.16B

comment(a) Show that if $f$ is an analytic function at $z_{0}$ and $f^{\prime}\left(z_{0}\right) \neq 0$, then $f$ is conformal at $z_{0}$, i.e. it preserves angles between paths passing through $z_{0}$.

(b) Let $D$ be the disc given by $|z+i|<\sqrt{2}$, and let $H$ be the half-plane given by $y>0$, where $z=x+i y$. Construct a map of the domain $D \cap H$ onto $H$, and hence find a conformal mapping of $D \cap H$ onto the disc $\{z:|z|<1\}$. [Hint: You may find it helpful to consider a mapping of the form $(a z+b) /(c z+d)$, where ad $-b c \neq 0$.]

1.I.6C

commentA fluid flow has velocity given in Cartesian co-ordinates as $\mathbf{u}=(k t y, 0,0)$ where $k$ is a constant and $t$ is time. Show that the flow is incompressible. Find a stream function and determine an equation for the streamlines at time $t$.

At $t=0$ the points along the straight line segment $x=0,0 \leqslant y \leqslant a, z=0$ are marked with dye. Show that at any later time the marked points continue to form a segment of a straight line. Determine the length of this line segment at time $t$ and the angle that it makes with the $x$-axis.

1.II.15C

commentState the unsteady form of Bernoulli's theorem.

A spherical bubble having radius $R_{0}$ at time $t=0$ is located with its centre at the origin in unbounded fluid. The fluid is inviscid, has constant density $\rho$ and is everywhere at rest at $t=0$. The pressure at large distances from the bubble has the constant value $p_{\infty}$, and the pressure inside the bubble has the constant value $p_{\infty}-\triangle p$. In consequence the bubble starts to collapse so that its radius at time $t$ is $R(t)$. Find the velocity everywhere in the fluid in terms of $R(t)$ at time $t$ and, assuming that surface tension is negligible, show that $R$ satisfies the equation

$R \ddot{R}+\frac{3}{2} \dot{R}^{2}=-\frac{\triangle p}{\rho}$

Find the total kinetic energy of the fluid in terms of $R(t)$ at time $t$. Hence or otherwise obtain a first integral of the above equation.

1.I.4E

commentShow that any finite group of orientation-preserving isometries of the Euclidean plane is cyclic.

Show that any finite group of orientation-preserving isometries of the hyperbolic plane is cyclic.

[You may assume that given any non-empty finite set $E$ in the hyperbolic plane, or the Euclidean plane, there is a unique smallest closed disc that contains E. You may also use any general fact about the hyperbolic plane without proof providing that it is stated carefully.]

1.II.13E

commentLet $\mathbb{H}=\{x+i y \in \mathbb{C}: y>0\}$, and let $\mathbb{H}$ have the hyperbolic metric $\rho$ derived from the line element $|d z| / y$. Let $\Gamma$ be the group of Möbius maps of the form $z \mapsto(a z+b) /(c z+d)$, where $a, b, c$ and $d$ are real and $a d-b c=1$. Show that every $g$ in $\Gamma$ is an isometry of the metric space $(\mathbb{H}, \rho)$. For $z$ and $w$ in $\mathbb{H}$, let

$h(z, w)=\frac{|z-w|^{2}}{\operatorname{Im}(z) \operatorname{Im}(w)}$

Show that for every $g$ in $\Gamma, h(g(z), g(w))=h(z, w)$. By considering $z=i y$, where $y>1$, and $w=i$, or otherwise, show that for all $z$ and $w$ in $\mathbb{H}$,

$\cosh \rho(z, w)=1+\frac{|z-w|^{2}}{2 \operatorname{Im}(z) \operatorname{Im}(w)}$

By considering points $i, i y$, where $y>1$ and $s+i t$, where $s^{2}+t^{2}=1$, or otherwise, derive Pythagoras' Theorem for hyperbolic geometry in the form $\cosh a \cosh b=\cosh c$, where $a, b$ and $c$ are the lengths of sides of a right-angled triangle whose hypotenuse has length $c$.

1.I.5G

commentDefine $f: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$ by

$f(a, b, c)=(a+3 b-c, 2 b+c,-4 b-c)$

Find the characteristic polynomial and the minimal polynomial of $f$. Is $f$ diagonalisable? Are $f$ and $f^{2}$ linearly independent endomorphisms of $\mathbb{C}^{3}$ ? Justify your answers.

1.II.14G

commentLet $\alpha$ be an endomorphism of a vector space $V$ of finite dimension $n$.

(a) What is the dimension of the vector space of linear endomorphisms of $V$ ? Show that there exists a non-trivial polynomial $p(X)$ such that $p(\alpha)=0$. Define what is meant by the minimal polynomial $m_{\alpha}$ of $\alpha$.

(b) Show that the eigenvalues of $\alpha$ are precisely the roots of the minimal polynomial of $\alpha$.

(c) Let $W$ be a subspace of $V$ such that $\alpha(W) \subseteq W$ and let $\beta$ be the restriction of $\alpha$ to $W$. Show that $m_{\beta}$ divides $m_{\alpha}$.

(d) Give an example of an endomorphism $\alpha$ and a subspace $W$ as in (c) not equal to $V$ for which $m_{\alpha}=m_{\beta}$, and $\operatorname{deg}\left(m_{\alpha}\right)>1$.

1.I.2A

commentFind the Fourier sine series for $f(x)=x$, on $0 \leqslant x<L$. To which value does the series converge at $x=\frac{3}{2} L$ ?

Now consider the corresponding cosine series for $f(x)=x$, on $0 \leqslant x<L$. Sketch the cosine series between $x=-2 L$ and $x=2 L$. To which value does the series converge at $x=\frac{3}{2} L$ ? [You do not need to determine the cosine series explicitly.]

1.II.11A

commentThe potential $\Phi(r, \vartheta)$, satisfies Laplace's equation everywhere except on a sphere of unit radius and $\Phi \rightarrow 0$ as $r \rightarrow \infty$. The potential is continuous at $r=1$, but the derivative of the potential satisfies

$\lim _{r \rightarrow 1^{+}} \frac{\partial \Phi}{\partial r}-\lim _{r \rightarrow 1^{-}} \frac{\partial \Phi}{\partial r}=V \cos ^{2} \vartheta$

where $V$ is a constant. Use the method of separation of variables to find $\Phi$ for both $r>1$ and $r<1$.

[The Laplacian in spherical polar coordinates for axisymmetric systems is

$\nabla^{2} \equiv \frac{1}{r^{2}}\left(\frac{\partial}{\partial r} r^{2} \frac{\partial}{\partial r}\right)+\frac{1}{r^{2} \sin \vartheta}\left(\frac{\partial}{\partial \vartheta} \sin \vartheta \frac{\partial}{\partial \vartheta}\right)$

You may assume that the equation

$\left(\left(1-x^{2}\right) y^{\prime}\right)^{\prime}+\lambda y=0$

has polynomial solutions of degree $n$, which are regular at $x=\pm 1$, if and only if $\lambda=n(n+1) .]$

1.I.8F

commentDefine the rank and signature of a symmetric bilinear form $\phi$ on a finite-dimensional real vector space. (If your definitions involve a matrix representation of $\phi$, you should explain why they are independent of the choice of representing matrix.)

Let $V$ be the space of all $n \times n$ real matrices (where $n \geqslant 2$ ), and let $\phi$ be the bilinear form on $V$ defined by

$\phi(A, B)=\operatorname{tr} A B-\operatorname{tr} A \operatorname{tr} B$

Find the rank and signature of $\phi$.

[Hint: You may find it helpful to consider the subspace of symmetric matrices having trace zero, and a suitable complement for this subspace.]

1.II.17F

commentLet $A$ and $B$ be $n \times n$ real symmetric matrices, such that the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ is positive definite. Show that it is possible to find an invertible matrix $P$ such that $P^{T} A P=I$ and $P^{T} B P$ is diagonal. Show also that the diagonal entries of the matrix $P^{T} B P$ may be calculated directly from $A$ and $B$, without finding the matrix $P$. If

$A=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{ccc} 4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{array}\right)$

find the diagonal entries of $P^{T} B P$.

1.I.9D

commentConsider a quantum mechanical particle of mass $m$ moving in one dimension, in a potential well

$V(x)=\left\{\begin{array}{cr} \infty, & x<0 \\ 0, & 0<x<a \\ V_{0}, & x>a \end{array}\right.$

Sketch the ground state energy eigenfunction $\chi(x)$ and show that its energy is $E=\frac{\hbar^{2} k^{2}}{2 m}$, where $k$ satisfies

$\tan k a=-\frac{k}{\sqrt{\frac{2 m V_{0}}{\hbar^{2}}-k^{2}}} .$

[Hint: You may assume that $\chi(0)=0 .]$

1.II.18D

commentA quantum mechanical particle of mass $M$ moves in one dimension in the presence of a negative delta function potential

$V=-\frac{\hbar^{2}}{2 M \Delta} \delta(x),$

where $\Delta$ is a parameter with dimensions of length.

(a) Write down the time-independent Schrödinger equation for energy eigenstates $\chi(x)$, with energy $E$. By integrating this equation across $x=0$, show that the gradient of the wavefunction jumps across $x=0$ according to

$\lim _{\epsilon \rightarrow 0}\left(\frac{d \chi}{d x}(\epsilon)-\frac{d \chi}{d x}(-\epsilon)\right)=-\frac{1}{\Delta} \chi(0)$

[You may assume that $\chi$ is continuous across $x=0 .$ ]

(b) Show that there exists a negative energy solution and calculate its energy.

(c) Consider a double delta function potential

$V(x)=-\frac{\hbar^{2}}{2 M \Delta}[\delta(x+a)+\delta(x-a)] .$

For sufficiently small $\Delta$, this potential yields a negative energy solution of odd parity, i.e. $\chi(-x)=-\chi(x)$. Show that its energy is given by

$E=-\frac{\hbar^{2}}{2 M} \lambda^{2}, \quad \text { where } \quad \tanh \lambda a=\frac{\lambda \Delta}{1-\lambda \Delta}$

[You may again assume $\chi$ is continuous across $x=\pm a$.]

1.II.12H

commentSuppose we ask 50 men and 150 women whether they are early risers, late risers, or risers with no preference. The data are given in the following table.

$\begin{array}{lcccc} & \text { Early risers } & \text { Late risers } & \text { No preference } & \text { Totals } \\ \text { Men } & 17 & 22 & 11 & 50 \\ \text { Women } & 43 & 78 & 29 & 150 \\ \text { Totals } & 60 & 100 & 40 & 200\end{array}$

Derive carefully a (generalized) likelihood ratio test of independence of classification. What is the result of applying this test at the $0.01$ level?

$\left[\begin{array}{lccccc}\text { Distribution } & \chi_{1}^{2} & \chi_{2}^{2} & \chi_{3}^{2} & \chi_{5}^{2} & \chi_{6}^{2} \\ 99 \% \text { percentile } & 6.63 & 9.21 & 11.34 & 15.09 & 16.81\end{array}\right]$