# Part IB, 2002

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$3 . \mathrm{II} . 11 \mathrm{E} \quad$

commentShow that if $a, b$ and $c$ are non-negative numbers, and $a \leqslant b+c$, then

$\frac{a}{1+a} \leqslant \frac{b}{1+b}+\frac{c}{1+c}$

Deduce that if $(X, d)$ is a metric space, then $d(x, y) /[1+d(x, y)]$ is a metric on $X$.

Let $D=\{z \in \mathbb{C}:|z|<1\}$ and $K_{n}=\{z \in D:|z| \leqslant(n-1) / n\}$. Let $\mathcal{F}$ be the class of continuous complex-valued functions on $D$ and, for $f$ and $g$ in $\mathcal{F}$, define

$\sigma(f, g)=\sum_{n=2}^{\infty} \frac{1}{2^{n}} \frac{\|f-g\|_{n}}{1+\|f-g\|_{n}}$

where $\|f-g\|_{n}=\sup \left\{|f(z)-g(z)|: z \in K_{n}\right\}$. Show that the series for $\sigma(f, g)$ converges, and that $\sigma$ is a metric on $\mathcal{F}$.

For $|z|<1$, let $s_{k}(z)=1+z+z^{2}+\cdots+z^{k}$ and $s(z)=1+z+z^{2}+\cdots$. Show that for $n \geqslant 2,\left\|s_{k}-s\right\|_{n}=n\left(1-\frac{1}{n}\right)^{k+1}$. By considering the sums for $2 \leqslant n \leqslant N$ and $n>N$ separately, show that for each $N$,

$\sigma\left(s_{k}, s\right) \leqslant \sum_{n=2}^{N}\left\|s_{k}-s\right\|_{n}+2^{-N},$

and deduce that $\sigma\left(s_{k}, s\right) \rightarrow 0$ as $k \rightarrow \infty$.

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 $\{x+i y: x>0\}$.

2.I.1E

commentDefine what is meant by (i) a complete metric space, and (ii) a totally bounded metric space.

Give an example of a metric space that is complete but not totally bounded. Give an example of a metric space that is totally bounded but not complete.

Give an example of a continuous function that maps a complete metric space onto a metric space that is not complete. Give an example of a continuous function that maps a totally bounded metric space onto a metric space that is not totally bounded.

[You need not justify your examples.]

2.II.10E

comment(a) Let $f$ be a map of a complete metric space $(X, d)$ into itself, and suppose that there exists some $k$ in $(0,1)$, and some positive integer $N$, such that $d\left(f^{N}(x), f^{N}(y)\right) \leqslant$ $k d(x, y)$ for all distinct $x$ and $y$ in $X$, where $f^{m}$ is the $m$ th iterate of $f$. Show that $f$ has a unique fixed point in $X$.

(b) Let $f$ be a map of a compact metric space $(X, d)$ into itself such that $d(f(x), f(y))<d(x, y)$ for all distinct $x$ and $y$ in $X$. By considering the function $d(f(x), x)$, or otherwise, show that $f$ has a unique fixed point in $X$.

(c) Suppose that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ satisfies $|f(x)-f(y)|<|x-y|$ for every distinct $x$ and $y$ in $\mathbb{R}^{n}$. Suppose that for some $x$, the orbit $O(x)=\left\{x, f(x), f^{2}(x), \ldots\right\}$ is bounded. Show that $f$ maps the closure of $O(x)$ into itself, and deduce that $f$ has a unique fixed point in $\mathbb{R}^{n}$.

[The Contraction Mapping Theorem may be used without proof providing that it is correctly stated.]

3.I.1E

commentLet $f: \mathbb{R} \rightarrow \mathbb{R}^{2}$ be defined by $f=(u, v)$, where $u$ and $v$ are defined by $u(0)=v(0)=0$ and, for $t \neq 0, u(t)=t^{2} \sin (1 / t)$ and $v(t)=t^{2} \cos (1 / t)$. Show that $f$ is differentiable on $\mathbb{R}$.

Show that for any real non-zero $a,\left\|f^{\prime}(a)-f^{\prime}(0)\right\|>1$, where we regard $f^{\prime}(a)$ as the vector $\left(u^{\prime}(a), v^{\prime}(a)\right)$ in $\mathbb{R}^{2}$.

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.]

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 $-b c \neq 0$.]

2.I.7B

commentSuppose that $f$ is analytic, and that $|f(z)|^{2}$ is constant in an open disk $D$. Use the Cauchy-Riemann equations to show that $f(z)$ is constant in $D$.

2.II.16B

commentA function $f(z)$ has an isolated singularity at $a$, with Laurent expansion

$f(z)=\sum_{n=-\infty}^{\infty} c_{n}(z-a)^{n}$

(a) Define res $(f, a)$, the residue of $f$ at the point $a$.

(b) Prove that if $a$ is a pole of order $k+1$, then

$\operatorname{res}(f, a)=\lim _{z \rightarrow a} \frac{h^{(k)}(z)}{k !}, \quad \text { where } \quad h(z)=(z-a)^{k+1} f(z) .$

(c) Using the residue theorem and the formula above show that

$\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{k+1}}=\pi \frac{(2 k) !}{(k !)^{2}} 4^{-k}, \quad k \geq 1$

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

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

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.

3.I.8C

commentState and prove Kelvin's circulation theorem.

Consider a planar flow in the unbounded region outside a cylinder for which the vorticity vanishes everywhere at time $t=0$. What may be deduced about the circulation around closed loops in the fluid at time $t$ :

(i) that do not enclose the cylinder;

(ii) that enclose the cylinder?

Give a brief justification for your answer in each case.

3.II.18C

commentUse Euler's equation to derive Bernoulli's theorem for the steady flow of an inviscid fluid of uniform density $\rho$ in the absence of body forces.

Such a fluid flows steadily through a long cylindrical elastic tube having circular cross-section. The variable $z$ measures distance downstream along the axis of the tube. The tube wall has thickness $h(z)$, so that if the external radius of the tube is $r(z)$, its internal radius is $r(z)-h(z)$, where $h(z) \geqslant 0$ is a given slowly-varying function that tends to zero as $z \rightarrow \pm \infty$. The elastic tube wall exerts a pressure $p(z)$ on the fluid given as

$p(z)=p_{0}+k[r(z)-R]$

where $p_{0}, k$ and $R$ are positive constants. Far upstream, $r$ has the constant value $R$, the fluid pressure has the constant value $p_{0}$, and the fluid velocity $u$ has the constant value $V$. Assume that gravity is negligible and that $h(z)$ varies sufficiently slowly that the velocity may be taken as uniform across the tube at each value of $z$. Use mass conservation and Bernoulli's theorem to show that $u(z)$ satisfies

$\frac{h}{R}=1-\left(\frac{V}{u}\right)^{1 / 2}+\frac{1}{4} \lambda\left[1-\left(\frac{u}{V}\right)^{2}\right], \quad \text { where } \quad \lambda=\frac{2 \rho V^{2}}{k R}$

Sketch a graph of $h / R$ against $u / V$. Show that if $h(z)$ exceeds a critical value $h_{c}(\lambda)$, no such flow is possible and find $h_{c}(\lambda)$.

Show that if $h<h_{c}(\lambda)$ everywhere, then for given $h$ the equation has two positive solutions for $u$. Explain how the given value of $\lambda$ determines which solution should be chosen.

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 $t$ ?

2.I.4G

commentLet the function $f=u+i v$ be analytic in the complex plane $\mathbb{C}$ with $u, v$ real-valued.

Prove that, if $u v$ is bounded above everywhere on $\mathbb{C}$, then $f$ is constant.

2.II.13G

comment(a) Given a topology $\mathcal{T}$ on $X$, a collection $\mathcal{B} \subseteq \mathcal{T}$ is called a basis for $\mathcal{T}$ if every non-empty set in $\mathcal{T}$ is a union of sets in $\mathcal{B}$. Prove that a collection $\mathcal{B}$ is a basis for some topology if it satisfies:

(i) the union of all sets in $\mathcal{B}$ is $X$;

(ii) if $x \in B_{1} \cap B_{2}$ for two sets $B_{1}$ and $B_{2}$ in $\mathcal{B}$, then there is a set $B \in \mathcal{B}$ with $x \in B \subset B_{1} \cap B_{2}$.

(b) On $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ consider the dictionary order given by

$\left(a_{1}, b_{1}\right)<\left(a_{2}, b_{2}\right)$

if $a_{1}<a_{2}$ or if $a_{1}=a_{2}$ and $b_{1}<b_{2}$. Given points $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{2}$ let

$\langle\mathbf{x}, \mathbf{y}\rangle=\left\{\mathbf{z} \in \mathbb{R}^{2}: \mathbf{x}<\mathbf{z}<\mathbf{y}\right\}$

Show that the sets $\langle\mathbf{x}, \mathbf{y}\rangle$ for $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{2}$ form a basis of a topology.

(c) Show that this topology on $\mathbb{R}^{2}$ does not have a countable basis.

3.I.3G

commentLet $f: X \rightarrow Y$ be a continuous map between topological spaces. Let

$G_{f}=\{(x, f(x)): x \in X\} .$

(a) Show that if $Y$ is Hausdorff, then $G_{f}$ is closed in $X \times Y$.

(b) Show that if $X$ is compact, then $G_{f}$ is also compact.

3.II.13G

comment(a) Let $f$ and $g$ be two analytic functions on a domain $D$ and let $\gamma \subset D$ be a simple closed curve homotopic in $D$ to a point. If $|g(z)|<|f(z)|$ for every $z$ in $\gamma$, prove that $\gamma$ encloses the same number of zeros of $f$ as of $f+g$.

(b) Let $g$ be an analytic function on the disk $|z|<1+\epsilon$, for some $\epsilon>0$. Suppose that $g$ maps the closed unit disk into the open unit disk (both centred at 0 ). Prove that $g$ has exactly one fixed point in the open unit disk.

(c) Prove that, if $|a|<1$, then

$z^{m}\left(\frac{z-a}{1-\bar{a} z}\right)^{n}-a$

has $m+n$ zeros in $|z|<1$.

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.

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 $c$.

3.I.4E

commentState Euler's formula for a graph $\mathcal{G}$ with $F$ faces, $E$ edges and $V$ vertices on the surface of a sphere.

Suppose that every face in $\mathcal{G}$ has at least three edges, and that at least three edges meet at every vertex of $\mathcal{G}$. Let $F_{n}$ be the number of faces of $\mathcal{G}$ that have exactly $n$ edges $(n \geqslant 3)$, and let $V_{m}$ be the number of vertices at which exactly $m$ edges meet $(m \geqslant 3)$. By expressing $6 F-\sum_{n} n F_{n}$ in terms of the $V_{j}$, or otherwise, show that every convex polyhedron has at least four faces each of which is a triangle, a quadrilateral or a pentagon.

3.II.14E

commentShow that every isometry of Euclidean space $\mathbb{R}^{3}$ is a composition of reflections in planes

What is the smallest integer $N$ such that every isometry $f$ of $\mathbb{R}^{3}$ with $f(0)=0$ can be expressed as the composition of at most $N$ reflections? Give an example of an isometry that needs this number of reflections and justify your answer.

Describe (geometrically) all twelve orientation-reversing isometries of a regular tetrahedron.

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 $\operatorname{deg}\left(m_{\alpha}\right)>1$.

2.I.6G

commentLet $A$ be a complex $4 \times 4$ matrix such that $A^{3}=A^{2}$. What are the possible minimal polynomials of $A$ ? If $A$ is not diagonalisable and $A^{2} \neq 0$, list all possible Jordan normal forms of $A$.

2.II.15G

comment(a) A complex $n \times n$ matrix is said to be unipotent if $U-I$ is nilpotent, where $I$ is the identity matrix. Show that $U$ is unipotent if and only if 1 is the only eigenvalue of $U$.

(b) Let $T$ be an invertible complex matrix. By considering the Jordan normal form of $T$ show that there exists an invertible matrix $P$ such that

$P T P^{-1}=D_{0}+N$

where $D_{0}$ is an invertible diagonal matrix, $N$ is an upper triangular matrix with zeros in the diagonal and $D_{0} N=N D_{0}$.

(c) Set $D=P^{-1} D_{0} P$ and show that $U=D^{-1} T$ is unipotent.

(d) Conclude that any invertible matrix $T$ can be written as $T=D U$ where $D$ is diagonalisable, $U$ is unipotent and $D U=U D$.

3.I $77 \mathrm{~F} \quad$

commentWhich of the following statements are true, and which false? Give brief justifications for your answers.

(a) If $U$ and $W$ are subspaces of a vector space $V$, then $U \cap W$ is always a subspace of $V$.

(b) If $U$ and $W$ are distinct subspaces of a vector space $V$, then $U \cup W$ is never a subspace of $V$.

(c) If $U, W$ and $X$ are subspaces of a vector space $V$, then $U \cap(W+X)=$ $(U \cap W)+(U \cap X)$.

(d) If $U$ is a subspace of a finite-dimensional space $V$, then there exists a subspace $W$ such that $U \cap W=\{0\}$ and $U+W=V$.

3.II.17F

commentDefine the determinant of an $n \times n$ matrix $A$, and prove from your definition that if $A^{\prime}$ is obtained from $A$ by an elementary row operation (i.e. by adding a scalar multiple of the $i$ th row of $A$ to the $j$ th row, for some $j \neq i$ ), then $\operatorname{det} A^{\prime}=\operatorname{det} A$.

Prove also that if $X$ is a $2 n \times 2 n$ matrix of the form

$\left(\begin{array}{ll} A & B \\ O & C \end{array}\right)$

where $O$ denotes the $n \times n$ zero matrix, then $\operatorname{det} X=\operatorname{det} A$ det $C$. Explain briefly how the $2 n \times 2 n$ matrix

$\left(\begin{array}{ll} B & I \\ O & A \end{array}\right)$

can be transformed into the matrix

$\left(\begin{array}{cc} B & I \\ -A B & O \end{array}\right)$

by a sequence of elementary row operations. Hence or otherwise prove that $\operatorname{det} A B=$ $\operatorname{det} A \operatorname{det} B$.

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 $\lambda_{i}$.

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 $\lambda=n(n+1) .]$

2.I.2C

commentWrite down the transformation law for the components of a second-rank tensor $A_{i j}$ explaining the meaning of the symbols that you use.

A tensor is said to have cubic symmetry if its components are unchanged by rotations of $\pi / 2$ about each of the three co-ordinate axes. Find the most general secondrank tensor having cubic symmetry.

2.II.11C

commentIf $\mathbf{B}$ is a vector, and

$T_{i j}=\alpha B_{i} B_{j}+\beta B_{k} B_{k} \delta_{i j}$

show for arbitrary scalars $\alpha$ and $\beta$ that $T_{i j}$ is a symmetric second-rank tensor.

Find the eigenvalues and eigenvectors of $T_{i j}$.

Suppose now that $\mathbf{B}$ depends upon position $\mathbf{x}$ and that $\nabla \cdot \mathbf{B}=0$. Find constants $\alpha$ and $\beta$ such that

$\frac{\partial}{\partial x_{j}} T_{i j}=[(\nabla \times \mathbf{B}) \times \mathbf{B}]_{i} .$

Hence or otherwise show that if $\mathbf{B}$ vanishes everywhere on a surface $S$ that encloses a volume $V$ then

$\int_{V}(\nabla \times \mathbf{B}) \times \mathbf{B} d V=0$

3.I.2A

commentWrite down the wave equation for the displacement $y(x, t)$ of a stretched string with constant mass density and tension. Obtain the general solution in the form

$y(x, t)=f(x+c t)+g(x-c t)$

where $c$ is the wave velocity. For a solution in the region $0 \leqslant x<\infty$, with $y(0, t)=0$ and $y \rightarrow 0$ as $x \rightarrow \infty$, show that

$E=\int_{0}^{\infty}\left[\frac{1}{2}\left(\frac{\partial y}{\partial t}\right)^{2}+\frac{1}{2} c^{2}\left(\frac{\partial y}{\partial x}\right)^{2}\right] d x$

is constant in time. Express $E$ in terms of the general solution in this case.

3.II.12A

commentConsider the real Sturm-Liouville problem

$\mathcal{L} y(x)=-\left(p(x) y^{\prime}\right)^{\prime}+q(x) y=\lambda r(x) y,$

with the boundary conditions $y(a)=y(b)=0$, where $p, q$ and $r$ are continuous and positive on $[a, b]$. Show that, with suitable choices of inner product and normalisation, the eigenfunctions $y_{n}(x), \quad n=1,2,3 \ldots$, form an orthonormal set.

Hence show that the corresponding Green's function $G(x, \xi)$ satisfying

$(\mathcal{L}-\mu r(x)) G(x, \xi)=\delta(x-\xi),$

where $\mu$ is not an eigenvalue, is

$G(x, \xi)=\sum_{n=1}^{\infty} \frac{y_{n}(x) y_{n}(\xi)}{\lambda_{n}-\mu}$

where $\lambda_{n}$ is the eigenvalue corresponding to $y_{n}$.

Find the Green's function in the case where

$\mathcal{L} y \equiv y^{\prime \prime},$

with boundary conditions $y(0)=y(\pi)=0$, and deduce, by suitable choice of $\mu$, that

$\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8} .$

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

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

2.I.5B

commentApplying the Gram-Schmidt orthogonalization, compute a "skinny"

QR-factorization of the matrix

$A=\left[\begin{array}{lll} 1 & 1 & 2 \\ 1 & 3 & 6 \\ 1 & 1 & 0 \\ 1 & 3 & 4 \end{array}\right],$

i.e. find a $4 \times 3$ matrix $Q$ with orthonormal columns and an upper triangular $3 \times 3$ matrix $R$ such that $A=Q R$.

2.II.14B

commentLet $f \in C[a, b]$ and let $n+1$ distinct points $x_{0}, \ldots, x_{n} \in[a, b]$ be given.

(a) Define the divided difference $f\left[x_{0}, \ldots, x_{n}\right]$ of order $n$ in terms of interpolating polynomials. Prove that it is a symmetric function of the variables $x_{i}, i=0, \ldots, n$.

(b) Prove the recurrence relation

$f\left[x_{0}, \ldots, x_{n}\right]=\frac{f\left[x_{1}, \ldots, x_{n}\right]-f\left[x_{0}, \ldots, x_{n-1}\right]}{x_{n}-x_{0}}$

(c) Hence or otherwise deduce that, for any $i \neq j$, we have

$f\left[x_{0}, \ldots, x_{n}\right]=\frac{f\left[x_{0}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{n}\right]-f\left[x_{0}, \ldots, x_{j-1}, x_{j+1}, \ldots, x_{n}\right]}{x_{j}-x_{i}} .$

(d) From the formulas above, show that, for any $i=1, \ldots, n-1$,

$f\left[x_{0}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{n}\right]=\gamma f\left[x_{0}, \ldots, x_{n-1}\right]+(1-\gamma) f\left[x_{1}, \ldots, x_{n}\right],$

where $\gamma=\frac{x_{i}-x_{0}}{x_{n}-x_{0}}$.

3.I.6B

commentFor numerical integration, a quadrature formula

$\int_{a}^{b} f(x) d x \approx \sum_{i=0}^{n} a_{i} f\left(x_{i}\right)$

is applied which is exact on $\mathcal{P}_{n}$, i.e., for all polynomials of degree $n$.

Prove that such a formula is exact for all $f \in \mathcal{P}_{2 n+1}$ if and only if $x_{i}, i=0, \ldots, n$, are the zeros of an orthogonal polynomial $p_{n+1} \in \mathcal{P}_{n+1}$ which satisfies $\int_{a}^{b} p_{n+1}(x) r(x) d x=0$ for all $r \in \mathcal{P}_{n}$. [You may assume that $p_{n+1}$ has $(n+1)$ distinct zeros.]

3.II.16B

comment(a) Consider a system of linear equations $A x=b$ with a non-singular square $n \times n$ matrix $A$. To determine its solution $x=x^{*}$ we apply the iterative method

$x^{k+1}=H x^{k}+v .$

Here $v \in \mathbb{R}^{n}$, while the matrix $H \in \mathbb{R}^{n \times n}$ is such that $x^{*}=H x^{*}+v$ implies $A x^{*}=b$. The initial vector $x^{0} \in \mathbb{R}^{n}$ is arbitrary. Prove that, if the matrix $H$ possesses $n$ linearly independent eigenvectors $w_{1}, \ldots, w_{n}$ whose corresponding eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$ satisfy $\max _{i}\left|\lambda_{i}\right|<1$, then the method converges for any choice of $x^{0}$, i.e. $x^{k} \rightarrow x^{*}$ as $k \rightarrow \infty$.

(b) Describe the Jacobi iteration method for solving $A x=b$. Show directly from the definition of the method that, if the matrix $A$ is strictly diagonally dominant by rows, i.e.

$\left|a_{i i}\right|^{-1} \sum_{j=1, j \neq i}^{n}\left|a_{i j}\right| \leq \gamma<1, \quad i=1, \ldots, n,$

then the method converges.

$3 . \mathrm{I} . 5 \mathrm{H} \quad$

commentConsider a two-person zero-sum game with a payoff matrix

$\left(\begin{array}{ll} 3 & b \\ 5 & 2 \end{array}\right)$

where $0<b<\infty$. Here, the $(i, j)$ entry of the matrix indicates the payoff to player one if he chooses move $i$ and player two move $j$. Suppose player one chooses moves 1 and 2 with probabilities $p$ and $1-p, 0 \leq p \leq 1$. Write down the maximization problem for the optimal strategy and solve it for each value of $b$.

3.II.15H

commentConsider the following linear programming problem

$\begin{array}{ll} \text { maximise } & -2 x_{1}+3 x_{2} \\ \text { subject to } & x_{1}-x_{2} \geq 1 \\ & 4 x_{1}-x_{2} \geq 10 \\ & x_{2} \leq 6 \\ & x_{i} \geq 0, i=1,2 \end{array}$

Write down the Phase One problem for (1) and solve it.

By using the solution of the Phase One problem as an initial basic feasible solution for the Phase Two simplex algorithm, solve (1), i.e., find the optimal tableau and read the optimal solution $\left(x_{1}, x_{2}\right)$ and optimal value from it.

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?

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 $P^{T} B P$.

2.I.8F

commentExplain what is meant by a sesquilinear form on a complex vector space $V$. If $\phi$ and $\psi$ are two such forms, and $\phi(v, v)=\psi(v, v)$ for all $v \in V$, prove that $\phi(v, w)=\psi(v, w)$ for all $v, w \in V$. Deduce that if $\alpha: V \rightarrow V$ is a linear map satisfying $\phi(\alpha(v), \alpha(v))=\phi(v, v)$ for all $v \in V$, then $\phi(\alpha(v), \alpha(w))=\phi(v, w)$ for all $v, w \in V$.

2.II.17F

Define the adjoint $\alpha^{*}$ of an endomorphism $\alpha$ of a complex inner-product space $V$. Show that if $W$ is a subspace of $V$, then $\alpha(W) \subseteq W$ if and only if $\alpha^{*}\left(W^{\perp}\right) \subseteq W^{\perp}$