# Part IB, 2001

### Jump to course

1.I.1A

commentDefine uniform continuity for functions defined on a (bounded or unbounded) interval in $\mathbb{R}$.

Is it true that a real function defined and uniformly continuous on $[0,1]$ is necessarily bounded?

Is it true that a real function defined and with a bounded derivative on $[1, \infty)$ is necessarily uniformly continuous there?

Which of the following functions are uniformly continuous on $[1, \infty)$ :

(i) $x^{2}$;

(ii) $\sin \left(x^{2}\right)$;

(iii) $\frac{\sin x}{x}$ ?

Justify your answers.

1.II.10A

commentShow that each of the functions below is a metric on the set of functions $x(t) \in$ $C[a, b]$ :

$\begin{gathered} d_{1}(x, y)=\sup _{t \in[a, b]}|x(t)-y(t)| \\ d_{2}(x, y)=\left\{\int_{a}^{b}|x(t)-y(t)|^{2} d t\right\}^{1 / 2} \end{gathered}$

Is the space complete in the $d_{1}$ metric? Justify your answer.

Show that the set of functions

$x_{n}(t)= \begin{cases}0, & -1 \leqslant t<0 \\ n t, & 0 \leqslant t<1 / n \\ 1, & 1 / n \leqslant t \leqslant 1\end{cases}$

is a Cauchy sequence with respect to the $d_{2}$ metric on $C[-1,1]$, yet does not tend to a limit in the $d_{2}$ metric in this space. Hence, deduce that this space is not complete in the $d_{2}$ metric.

2.I.1A

commentState and prove the contraction mapping theorem.

Let $A=\{x, y, z\}$, let $d$ be the discrete metric on $A$, and let $d^{\prime}$ be the metric given by: $d^{\prime}$ is symmetric and

$\begin{gathered} d^{\prime}(x, y)=2, d^{\prime}(x, z)=2, d^{\prime}(y, z)=1 \\ d^{\prime}(x, x)=d^{\prime}(y, y)=d^{\prime}(z, z)=0 \end{gathered}$

Verify that $d^{\prime}$ is a metric, and that it is Lipschitz equivalent to $d$.

Define an appropriate function $f: A \rightarrow A$ such that $f$ is a contraction in the $d^{\prime}$ metric, but not in the $d$ metric.

2.II.10A

commentDefine total boundedness for metric spaces.

Prove that a metric space has the Bolzano-Weierstrass property if and only if it is complete and totally bounded.

3.I.1A

commentDefine what is meant by a norm on a real vector space.

(a) Prove that two norms on a vector space (not necessarily finite-dimensional) give rise to equivalent metrics if and only if they are Lipschitz equivalent.

(b) Prove that if the vector space $V$ has an inner product, then for all $x, y \in V$,

$\|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2},$

in the induced norm.

Hence show that the norm on $\mathbb{R}^{2}$ defined by $\|x\|=\max \left(\left|x_{1}\right|,\left|x_{2}\right|\right)$, where $x=\left(x_{1}, x_{2}\right) \in$ $\mathbb{R}^{2}$, cannot be induced by an inner product.

3.II.11A

commentProve that if all the partial derivatives of $f: \mathbb{R}^{p} \rightarrow \mathbb{R}$ (with $p \geqslant 2$ ) exist in an open set containing $(0,0, \ldots, 0)$ and are continuous at this point, then $f$ is differentiable at $(0,0, \ldots, 0)$.

Let

$g(x)= \begin{cases}x^{2} \sin (1 / x), & x \neq 0 \\ 0, & x=0\end{cases}$

and

$f(x, y)=g(x)+g(y) .$

At which points of the plane is the partial derivative $f_{x}$ continuous?

At which points is the function $f(x, y)$ differentiable? Justify your answers.

4.I.1A

commentLet $f$ be a mapping of a metric space $(X, d)$ into itself such that $d(f(x), f(y))<$ $d(x, y)$ for all distinct $x, y$ in $\mathrm{X}$.

Show that $f(x)$ and $d(x, f(x))$ are continuous functions of $x$.

Now suppose that $(X, d)$ is compact and let

$h=\inf _{x \in X} d(x, f(x))$

Show that we cannot have $h>0$.

[You may assume that a continuous function on a compact metric space is bounded and attains its bounds.]

Deduce that $f$ possesses a fixed point, and that it is unique.

4.II.10A

commentLet $\left\{f_{n}\right\}$ be a pointwise convergent sequence of real-valued functions on a closed interval $[a, b]$. Prove that, if for every $x \in[a, b]$, the sequence $\left\{f_{n}(x)\right\}$ is monotonic in $n$, and if all the functions $f_{n}, n=1,2, \ldots$, and $f=\lim f_{n}$ are continuous, then $f_{n} \rightarrow f$ uniformly on $[a, b]$.

By considering a suitable sequence of functions $\left\{f_{n}\right\}$ on $[0,1)$, show that if the interval is not closed but all other conditions hold, the conclusion of the theorem may fail.

$2 . \mathrm{I} . 7 \mathrm{E} \quad$

commentA complex function is defined for every $z \in V$, where $V$ is a non-empty open subset of $\mathbb{C}$, and it possesses a derivative at every $z \in V$. Commencing from a formal definition of derivative, deduce the Cauchy-Riemann equations.

1.I.7E

commentState the Cauchy integral formula.

Assuming that the function $f(z)$ is analytic in the disc $|z|<1$, prove that, for every $0<r<1$, it is true that

$\frac{d^{n} f(0)}{d z^{n}}=\frac{n !}{2 \pi i} \int_{|\xi|=r} \frac{f(\xi)}{\xi^{n+1}} d \xi, \quad n=0,1, \ldots$

[Taylor's theorem may be used if clearly stated.]

1.II.16E

commentLet the function $F$ be integrable for all real arguments $x$, such that

$\int_{-\infty}^{\infty}|F(x)| d x<\infty$

and assume that the series

$f(\tau)=\sum_{n=-\infty}^{\infty} F(2 n \pi+\tau)$

converges uniformly for all $0 \leqslant \tau \leqslant 2 \pi$.

Prove the Poisson summation formula

$f(\tau)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} \hat{F}(n) e^{i n \tau}$

where $\hat{F}$ is the Fourier transform of $F$. [Hint: You may show that

$\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{-i m x} f(x) d x=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-i m x} F(x) d x$

or, alternatively, prove that $f$ is periodic and express its Fourier expansion coefficients explicitly in terms of $\hat{F}$.]

Letting $F(x)=e^{-|x|}$, use the Poisson summation formula to evaluate the sum

$\sum_{n=-\infty}^{\infty} \frac{1}{1+n^{2}}$

2.II.16E

commentLet $R$ be a rational function such that $\lim _{z \rightarrow \infty}\{z R(z)\}=0$. Assuming that $R$ has no real poles, use the residue calculus to evaluate

$\int_{-\infty}^{\infty} R(x) d x$

Given that $n \geqslant 1$ is an integer, evaluate

$\int_{0}^{\infty} \frac{d x}{1+x^{2 n}}$

4.I.8F

commentConsider a conformal mapping of the form

$f(z)=\frac{a+b z}{c+d z}, \quad z \in \mathbb{C}$

where $a, b, c, d \in \mathbb{C}$, and $a d \neq b c$. You may assume $b \neq 0$. Show that any such $f(z)$ which maps the unit circle onto itself is necessarily of the form

$f(z)=e^{i \psi} \frac{a+z}{1+\bar{a} z} .$

[Hint: Show that it is always possible to choose $b=1$.]

4.II.17F

commentState Jordan's Lemma.

Consider the integral

$I=\oint_{C} d z \frac{z \sin (x z)}{\left(a^{2}+z^{2}\right) \sin \pi z}$

for real $x$ and $a$. The rectangular contour $C$ runs from $+\infty+i \epsilon$ to $-\infty+i \epsilon$, to $-\infty-i \epsilon$, to $+\infty-i \epsilon$ and back to $+\infty+i \epsilon$, where $\epsilon$ is infinitesimal and positive. Perform the integral in two ways to show that

$\sum_{n=-\infty}^{\infty}(-1)^{n} \frac{n \sin n x}{a^{2}+n^{2}}=-\pi \frac{\sinh a x}{\sinh a \pi}$

for $|x|<\pi$.

1.I.6G

commentDetermine the pressure at a depth $z$ below the surface of a static fluid of density $\rho$ subject to gravity $g$. A rigid body having volume $V$ is fully submerged in such a fluid. Calculate the buoyancy force on the body.

An iceberg of uniform density $\rho_{I}$ is observed to float with volume $V_{I}$ protruding above a large static expanse of seawater of density $\rho_{w}$. What is the total volume of the iceberg?

1.II.15G

commentA fluid motion has velocity potential $\phi(x, y, t)$ given by

$\phi=\epsilon y \cos (x-t)$

where $\epsilon$ is a constant. Find the corresponding velocity field $\mathbf{u}(x, y, t)$. Determine $\nabla \cdot \mathbf{u}$.

The time-average of a quantity $\psi(x, y, t)$ is defined as $\frac{1}{2 \pi} \int_{0}^{2 \pi} \psi(x, y, t) d t$.

Show that the time-average of this velocity field at every point $(x, y)$ is zero.

Write down an expression for the fluid acceleration and find the time-average acceleration at $(x, y)$.

Suppose now that $|\epsilon| \ll 1$. The material particle at $(0,0)$ at time $t=0$ is marked with dye. Write down equations for its subsequent motion and verify that its position $(x, y)$ at time $t>0$ is given (correct to terms of order $\epsilon^{2}$ ) as

$\begin{aligned} &x=\epsilon^{2}\left(\frac{1}{2} t-\frac{1}{4} \sin 2 t\right) \\ &y=\epsilon \sin t \end{aligned}$

Deduce the time-average velocity of the dyed particle correct to this order.

3.I.8G

commentInviscid incompressible fluid occupies the region $y>0$, which is bounded by a rigid barrier along $y=0$. At time $t=0$, a line vortex of strength $\kappa$ is placed at position $(a, b)$. By considering the flow due to an image vortex at $(a,-b)$, or otherwise, determine the velocity potential in the fluid.

Derive the position of the original vortex at time $t>0$.

3.II.18G

commentState the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid.

A circular cylinder of radius $a$ is immersed in unbounded inviscid fluid of uniform density $\rho$. The cylinder moves in a prescribed direction perpendicular to its axis, with speed $U$. Use cylindrical polar coordinates, with the direction $\theta=0$ parallel to the direction of the motion, to find the velocity potential in the fluid.

If $U$ depends on time $t$ and gravity is negligible, determine the pressure field in the fluid at time $t$. Deduce the fluid force per unit length on the cylinder.

[In cylindrical polar coordinates, $\nabla \phi=\frac{\partial \phi}{\partial r} \mathbf{e}_{r}+\frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{e}_{\theta}$.]

4.I.7G

commentStarting from the Euler equation, derive the vorticity equation for the motion of an inviscid incompressible fluid under a conservative body force, and give a physical interpretation of each term in the equation. Deduce that in a flow field of the form $\mathbf{u}=(u(x, y, t), v(x, y, t), 0)$ the vorticity of a material particle is conserved.

Find the vorticity for such a flow in terms of the stream function $\psi$. Deduce that if the flow is steady, there must be a function $f$ such that

$\nabla^{2} \psi=f(\psi)$

4.II.16G

commentA long straight canal has rectangular cross-section with a horizontal bottom and width $w(x)$ that varies slowly with distance $x$ downstream. Far upstream, $w$ has a constant value $W$, the water depth has a constant value $H$, and the velocity has a constant value $U$. Assuming that the water velocity is steady and uniform across the channel, use mass conservation and Bernoulli's theorem, which should be stated carefully, to show that the water depth $h(x)$ satisfies

$\left(\frac{W}{w}\right)^{2}=\left(1+\frac{2}{F}\right)\left(\frac{h}{H}\right)^{2}-\frac{2}{F}\left(\frac{h}{H}\right)^{3} \text { where } F=\frac{U^{2}}{g H}$

Deduce that for a given value of $F$, a flow of this kind can exist only if $w(x)$ is everywhere greater than or equal to a critical value $w_{c}$, which is to be determined in terms of $F$.

Suppose that $w(x)>w_{c}$ everywhere. At locations where the channel width exceeds $W$, determine graphically, or otherwise, under what circumstances the water depth exceeds $H .$

2.I.4B

commentDefine the terms connected and path connected for a topological space. If a topological space $X$ is path connected, prove that it is connected.

Consider the following subsets of $\mathbb{R}^{2}$ :

$\begin{gathered} I=\{(x, 0): 0 \leq x \leq 1\}, \quad A=\left\{(0, y): \frac{1}{2} \leq y \leq 1\right\}, \text { and } \\ J_{n}=\left\{\left(n^{-1}, y\right): 0 \leq y \leq 1\right\} \quad \text { for } n \geq 1 \end{gathered}$

Let

$X=A \cup I \cup \bigcup_{n \geq 1} J_{n}$

with the subspace (metric) topology. Prove that $X$ is connected.

[You may assume that any interval in $\mathbb{R}$ (with the usual topology) is connected.]

2.II.13A

commentState Liouville's Theorem. Prove it by considering

$\int_{|z|=R} \frac{f(z) d z}{(z-a)(z-b)}$

and letting $R \rightarrow \infty$.

Prove that, if $g(z)$ is a function analytic on all of $\mathbb{C}$ with real and imaginary parts $u(z)$ and $v(z)$, then either of the conditions:

$\text { (i) } u+v \geqslant 0 \text { for all } z \text {; or (ii) } u v \geqslant 0 \text { for all } z \text {, }$

implies that $g(z)$ is constant.

3.I.3B

commentState a version of Rouché's Theorem. Find the number of solutions (counted with multiplicity) of the equation

$z^{4}=a(z-1)\left(z^{2}-1\right)+\frac{1}{2}$

inside the open disc $\{z:|z|<\sqrt{2}\}$, for the cases $a=\frac{1}{3}, 12$ and 5 .

[Hint: For the case $a=5$, you may find it helpful to consider the function $\left(z^{2}-1\right)(z-$ 2) $(z-3)$.]

3.II.13B

commentIf $X$ and $Y$ are topological spaces, describe the open sets in the product topology on $X \times Y$. If the topologies on $X$ and $Y$ are induced from metrics, prove that the same is true for the product.

What does it mean to say that a topological space is compact? If the topologies on $X$ and $Y$ are compact, prove that the same is true for the product.

4.I.4A

commentLet $f(z)$ be analytic in the $\operatorname{disc}|z|<R$. Assume the formula

$f^{\prime}\left(z_{0}\right)=\frac{1}{2 \pi i} \int_{|z|=r} \frac{f(z) d z}{\left(z-z_{0}\right)^{2}}, \quad 0 \leqslant\left|z_{0}\right|<r<R .$

By combining this formula with a complex conjugate version of Cauchy's Theorem, namely

$0=\int_{|z|=r} \overline{f(z)} d \bar{z},$

prove that

$f^{\prime}(0)=\frac{1}{\pi r} \int_{0}^{2 \pi} u(\theta) e^{-i \theta} d \theta$

where $u(\theta)$ is the real part of $f\left(r e^{i \theta}\right)$.

4.II.13B

commentLet $\Delta^{*}=\{z: 0<|z|<r\}$ be a punctured disc, and $f$ an analytic function on $\Delta^{*}$. What does it mean to say that $f$ has the origin as (i) a removable singularity, (ii) a pole, and (iii) an essential singularity? State criteria for (i), (ii), (iii) to occur, in terms of the Laurent series for $f$ at 0 .

Suppose now that the origin is an essential singularity for $f$. Given any $w \in \mathbb{C}$, show that there exists a sequence $\left(z_{n}\right)$ of points in $\Delta^{*}$ such that $z_{n} \rightarrow 0$ and $f\left(z_{n}\right) \rightarrow w$. [You may assume the fact that an isolated singularity is removable if the function is bounded in some open neighbourhood of the singularity.]

State the Open Mapping Theorem. Prove that if $f$ is analytic and injective on $\Delta^{*}$, then the origin cannot be an essential singularity. By applying this to the function $g(1 / z)$, or otherwise, deduce that if $g$ is an injective analytic function on $\mathbb{C}$, then $g$ is linear of the form $a z+b$, for some non-zero complex number $a$. [Here, you may assume that $g$ injective implies that its derivative $g^{\prime}$ is nowhere vanishing.]

Part IB

1.I.4B

commentWrite down the Riemannian metric on the disc model $\Delta$ of the hyperbolic plane. What are the geodesics passing through the origin? Show that the hyperbolic circle of radius $\rho$ centred on the origin is just the Euclidean circle centred on the origin with Euclidean radius $\tanh (\rho / 2)$.

Write down an isometry between the upper half-plane model $H$ of the hyperbolic plane and the disc model $\Delta$, under which $i \in H$ corresponds to $0 \in \Delta$. Show that the hyperbolic circle of radius $\rho$ centred on $i$ in $H$ is a Euclidean circle with centre $i \cosh \rho$ and of radius $\sinh \rho$.

1.II.13B

commentDescribe geometrically the stereographic projection map $\phi$ from the unit sphere $S^{2}$ to the extended complex plane $\mathbb{C}_{\infty}=\mathbb{C} \cup \infty$, and find a formula for $\phi$. Show that any rotation of $S^{2}$ about the $z$-axis corresponds to a Möbius transformation of $\mathbb{C}_{\infty}$. You are given that the rotation of $S^{2}$ defined by the matrix

$\left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{array}\right)$

corresponds under $\phi$ to a Möbius transformation of $\mathbb{C}_{\infty}$; deduce that any rotation of $S^{2}$ about the $x$-axis also corresponds to a Möbius transformation.

Suppose now that $u, v \in \mathbb{C}$ correspond under $\phi$ to distinct points $P, Q \in S^{2}$, and let $d$ denote the angular distance from $P$ to $Q$ on $S^{2}$. Show that $-\tan ^{2}(d / 2)$ is the cross-ratio of the points $u, v,-1 / \bar{u},-1 / \bar{v}$, taken in some order (which you should specify). [You may assume that the cross-ratio is invariant under Möbius transformations.]

3.I.4B

commentState and prove the Gauss-Bonnet theorem for the area of a spherical triangle.

Suppose $\mathbf{D}$ is a regular dodecahedron, with centre the origin. Explain how each face of $\mathbf{D}$ gives rise to a spherical pentagon on the 2 -sphere $S^{2}$. For each such spherical pentagon, calculate its angles and area.

3.II.14B

commentDescribe the hyperbolic lines in the upper half-plane model $H$ of the hyperbolic plane. The group $G=\mathrm{SL}(2, \mathbb{R}) /\{\pm I\}$ acts on $H$ via Möbius transformations, which you may assume are isometries of $H$. Show that $G$ acts transitively on the hyperbolic lines. Find explicit formulae for the reflection in the hyperbolic line $L$ in the cases (i) $L$ is a vertical line $x=a$, and (ii) $L$ is the unit semi-circle with centre the origin. Verify that the composite $T$ of a reflection of type (ii) followed afterwards by one of type (i) is given by $T(z)=-z^{-1}+2 a$.

Suppose now that $L_{1}$ and $L_{2}$ are distinct hyperbolic lines in the hyperbolic plane, with $R_{1}, R_{2}$ denoting the corresponding reflections. By considering different models of the hyperbolic plane, or otherwise, show that

(a) $R_{1} R_{2}$ has infinite order if $L_{1}$ and $L_{2}$ are parallel or ultraparallel, and

(b) $R_{1} R_{2}$ has finite order if and only if $L_{1}$ and $L_{2}$ meet at an angle which is a rational multiple of $\pi$.

1.I $. 5 \mathrm{C} \quad$

commentDetermine for which values of $x \in \mathbb{C}$ the matrix

$M=\left(\begin{array}{ccc} x & 1 & 1 \\ 1-x & 0 & -1 \\ 2 & 2 x & 1 \end{array}\right)$

is invertible. Determine the rank of $M$ as a function of $x$. Find the adjugate and hence the inverse of $M$ for general $x$.

1.II.14C

comment(a) Find a matrix $M$ over $\mathbb{C}$ with both minimal polynomial and characteristic polynomial equal to $(x-2)^{3}(x+1)^{2}$. Furthermore find two matrices $M_{1}$ and $M_{2}$ over $\mathbb{C}$ which have the same characteristic polynomial, $(x-3)^{5}(x-1)^{2}$, and the same minimal polynomial, $(x-3)^{2}(x-1)^{2}$, but which are not conjugate to one another. Is it possible to find a third such matrix, $M_{3}$, neither conjugate to $M_{1}$ nor to $M_{2}$ ? Justify your answer.

(b) Suppose $A$ is an $n \times n$ matrix over $\mathbb{R}$ which has minimal polynomial of the form $\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)$ for distinct roots $\lambda_{1} \neq \lambda_{2}$ in $\mathbb{R}$. Show that the vector space $V=\mathbb{R}^{n}$ on which $A$ defines an endomorphism $\alpha: V \rightarrow V$ decomposes as a direct sum into $V=\operatorname{ker}\left(\alpha-\lambda_{1} \iota\right) \oplus \operatorname{ker}\left(\alpha-\lambda_{2} \iota\right)$, where $\iota$ is the identity.

[Hint: Express $v \in V$ in terms of $\left(\alpha-\lambda_{1} \iota\right)(v)$ and $\left.\left(\alpha-\lambda_{2} \iota\right)(v) .\right]$

Now suppose that $A$ has minimal polynomial $\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right) \ldots\left(x-\lambda_{m}\right)$ for distinct $\lambda_{1}, \ldots, \lambda_{m} \in \mathbb{R}$. By induction or otherwise show that

$V=\operatorname{ker}\left(\alpha-\lambda_{1} \iota\right) \oplus \operatorname{ker}\left(\alpha-\lambda_{2} \iota\right) \oplus \ldots \oplus \operatorname{ker}\left(\alpha-\lambda_{m} \iota\right)$

Use this last statement to prove that an arbitrary matrix $A \in M_{n \times n}(\mathbb{R})$ is diagonalizable if and only if all roots of its minimal polynomial lie in $\mathbb{R}$ and have multiplicity $1 .$

2.I $6 \mathrm{C} \quad$

commentShow that right multiplication by $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in M_{2 \times 2}(\mathbb{C})$ defines a linear transformation $\rho_{A}: M_{2 \times 2}(\mathbb{C}) \rightarrow M_{2 \times 2}(\mathbb{C})$. Find the matrix representing $\rho_{A}$ with respect to the basis

$\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right)$

of $M_{2 \times 2}(\mathbb{C})$. Prove that the characteristic polynomial of $\rho_{A}$ is equal to the square of the characteristic polynomial of $A$, and that $A$ and $\rho_{A}$ have the same minimal polynomial.

2.II.15C

commentDefine the dual $V^{*}$ of a vector space $V$. Given a basis $\left\{v_{1}, \ldots, v_{n}\right\}$ of $V$ define its dual and show it is a basis of $V^{*}$. For a linear transformation $\alpha: V \rightarrow W$ define the dual $\alpha^{*}: W^{*} \rightarrow V^{*}$.

Explain (with proof) how the matrix representing $\alpha: V \rightarrow W$ with respect to given bases of $V$ and $W$ relates to the matrix representing $\alpha^{*}: W^{*} \rightarrow V^{*}$ with respect to the corresponding dual bases of $V^{*}$ and $W^{*}$.

Prove that $\alpha$ and $\alpha^{*}$ have the same rank.

Suppose that $\alpha$ is an invertible endomorphism. Prove that $\left(\alpha^{*}\right)^{-1}=\left(\alpha^{-1}\right)^{*}$.

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

commentDetermine the dimension of the subspace $W$ of $\mathbb{R}^{5}$ spanned by the vectors

$\left(\begin{array}{r} 1 \\ 2 \\ 2 \\ -1 \\ 1 \end{array}\right),\left(\begin{array}{r} 4 \\ 2 \\ -2 \\ 6 \\ -2 \end{array}\right),\left(\begin{array}{l} 4 \\ 5 \\ 3 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 5 \\ 4 \\ 0 \\ 5 \\ -1 \end{array}\right)$

Write down a $5 \times 5$ matrix $M$ which defines a linear map $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$ whose image is $W$ and which contains $(1,1,1,1,1)^{T}$ in its kernel. What is the dimension of the space of all linear maps $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$ with $(1,1,1,1,1)^{T}$ in the kernel, and image contained in $W$ ?

3.II.17C

commentLet $V$ be a vector space over $\mathbb{R}$. Let $\alpha: V \rightarrow V$ be a nilpotent endomorphism of $V$, i.e. $\alpha^{m}=0$ for some positive integer $m$. Prove that $\alpha$ can be represented by a strictly upper-triangular matrix (with zeros along the diagonal). [You may wish to consider the subspaces $\operatorname{ker}\left(\alpha^{j}\right)$ for $j=1, \ldots, m$.]

Show that if $\alpha$ is nilpotent, then $\alpha^{n}=0$ where $n$ is the dimension of $V$. Give an example of a $4 \times 4$ matrix $M$ such that $M^{4}=0$ but $M^{3} \neq 0$.

Let $A$ be a nilpotent matrix and $I$ the identity matrix. Prove that $I+A$ has all eigenvalues equal to 1 . Is the same true of $(I+A)(I+B)$ if $A$ and $B$ are nilpotent? Justify your answer.

4.I $6 \mathrm{C} \quad$

commentFind the Jordan normal form $J$ of the matrix

$M=\left(\begin{array}{rrrr} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -1 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right)$

and determine both the characteristic and the minimal polynomial of $M$.

Find a basis of $\mathbb{C}^{4}$ such that $J$ (the Jordan normal form of $M$ ) is the matrix representing the endomorphism $M: \mathbb{C}^{4} \rightarrow \mathbb{C}^{4}$ in this basis. Give a change of basis matrix $P$ such that $P^{-1} M P=J$.

4.II.15C

commentLet $A$ and $B$ be $n \times n$ matrices over $\mathbb{C}$. Show that $A B$ and $B A$ have the same characteristic polynomial. [Hint: Look at $\operatorname{det}(C B C-x C)$ for $C=A+y I$, where $x$ and $y$ are scalar variables.]

Show by example that $A B$ and $B A$ need not have the same minimal polynomial.

Suppose that $A B$ is diagonalizable, and let $p(x)$ be its minimal polynomial. Show that the minimal polynomial of $B A$ must divide $x p(x)$. Using this and the first part of the question prove that $(A B)^{2}$ and $(B A)^{2}$ are conjugate.

1.I.2H

commentThe even function $f(x)$ has the Fourier cosine series

$f(x)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \cos n x$

in the interval $-\pi \leqslant x \leqslant \pi$. Show that

$\frac{1}{\pi} \int_{-\pi}^{\pi}(f(x))^{2} d x=\frac{1}{2} a_{0}^{2}+\sum_{n=1}^{\infty} a_{n}^{2}$

Find the Fourier cosine series of $x^{2}$ in the same interval, and show that

$\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\frac{\pi^{4}}{90}$

1.II.11H

commentUse the substitution $y=x^{p}$ to find the general solution of

$\mathcal{L}_{x} y \equiv \frac{d^{2} y}{d x^{2}}-\frac{2}{x^{2}} y=0$

Find the Green's function $G(x, \xi), 0<\xi<\infty$, which satisfies

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

for $x>0$, subject to the boundary conditions $G(x, \xi) \rightarrow 0$ as $x \rightarrow 0$ and as $x \rightarrow \infty$, for each fixed $\xi$.

Hence, find the solution of the equation

$\mathcal{L}_{x} y= \begin{cases}1, & 0 \leqslant x<1, \\ 0, & x>1\end{cases}$

subject to the same boundary conditions.

Verify that both forms of your solution satisfy the appropriate equation and boundary conditions, and match at $x=1$.

2.I.2G

commentShow that the symmetric and antisymmetric parts of a second-rank tensor are themselves tensors, and that the decomposition of a tensor into symmetric and antisymmetric parts is unique.

For the tensor $A$ having components

$A=\left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{array}\right)$

find the scalar $a$, vector $\mathbf{p}$ and symmetric traceless tensor $B$ such that

$A \mathbf{x}=a \mathbf{x}+\mathbf{p} \wedge \mathbf{x}+B \mathbf{x}$

for every vector $\mathbf{x}$.

2.II.11G

commentExplain what is meant by an isotropic tensor.

Show that the fourth-rank tensor

$A_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$

is isotropic for arbitrary scalars $\alpha, \beta$ and $\gamma$.

Assuming that the most general isotropic tensor of rank 4 has the form $(*)$, or otherwise, evaluate

$B_{i j k l}=\int_{r<a} x_{i} x_{j} \frac{\partial^{2}}{\partial x_{k} \partial x_{l}}\left(\frac{1}{r}\right) d V$

where $\mathbf{x}$ is the position vector and $r=|\mathbf{x}|$.

3.I.2G

commentLaplace's equation in the plane is given in terms of plane polar coordinates $r$ and $\theta$ in the form

$\nabla^{2} \phi \equiv \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}=0$

In each of the cases

$\text { (i) } 0 \leqslant r \leqslant 1, \text { and (ii) } 1 \leqslant r<\infty \text {, }$

find the general solution of Laplace's equation which is single-valued and finite.

Solve also Laplace's equation in the annulus $a \leqslant r \leqslant b$ with the boundary conditions

$\begin{aligned} &\phi=1 \quad \text { on } \quad r=a \text { for all } \theta \\ &\phi=2 \quad \text { on } \quad r=b \text { for all } \theta . \end{aligned}$

3.II.12H

commentFind the Fourier sine series representation on the interval $0 \leqslant x \leqslant l$ of the function

$f(x)= \begin{cases}0, & 0 \leqslant x<a \\ 1, & a \leqslant x \leqslant b \\ 0, & b<x \leqslant l\end{cases}$

The motion of a struck string is governed by the equation

$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}, \quad \text { for } \quad 0 \leqslant x \leqslant l \quad \text { and } \quad t \geqslant 0$

subject to boundary conditions $y=0$ at $x=0$ and $x=l$ for $t \geqslant 0$, and to the initial conditions $y=0$ and $\frac{\partial y}{\partial t}=\delta\left(x-\frac{1}{4} l\right)$ at $t=0$.

Obtain the solution $y(x, t)$ for this motion. Evaluate $y(x, t)$ for $t=\frac{1}{2} l / c$, and sketch it clearly.

4.I.2H

commentThe Legendre polynomial $P_{n}(x)$ satisfies

$\left(1-x^{2}\right) P_{n}^{\prime \prime}-2 x P_{n}^{\prime}+n(n+1) P_{n}=0, \quad n=0,1, \ldots,-1 \leqslant x \leqslant 1 .$

Show that $R_{n}(x)=P_{n}^{\prime}(x)$ obeys an equation which can be recast in Sturm-Liouville form and has the eigenvalue $(n-1)(n+2)$. What is the orthogonality relation for $R_{n}(x), R_{m}(x)$ for $n \neq m$ ?

4.II.11H

commentA curve $y(x)$ in the $x y$-plane connects the points $(\pm a, 0)$ and has a fixed length $l, 2 a<l<\pi a$. Find an expression for the area $A$ of the surface of the revolution obtained by rotating $y(x)$ about the $x$-axis.

Show that the area $A$ has a stationary value for

$y=\frac{1}{k}(\cosh k x-\cosh k a),$

where $k$ is a constant such that

$l k=2 \sinh k a .$

Show that the latter equation admits a unique positive solution for $k$.

2.I.5E

commentFind an LU factorization of the matrix

$A=\left(\begin{array}{rrrr} 2 & -1 & 3 & 2 \\ -4 & 3 & -4 & -2 \\ 4 & -2 & 3 & 6 \\ -6 & 5 & -8 & 1 \end{array}\right)$

and use it to solve the linear system $A \mathbf{x}=\mathbf{b}$, where

$\mathbf{b}=\left(\begin{array}{r} -2 \\ 2 \\ 4 \\ 11 \end{array}\right)$

2.II.14E

comment(a) Let $B$ be an $n \times n$ positive-definite, symmetric matrix. Define the Cholesky factorization of $B$ and prove that it is unique.

(b) Let $A$ be an $m \times n$ matrix, $m \geqslant n$, such that $\operatorname{rank} A=n$. Prove the uniqueness of the "skinny QR factorization"

$A=Q R,$

where the matrix $Q$ is $m \times n$ with orthonormal columns, while $R$ is an $n \times n$ upper-triangular matrix with positive diagonal elements.

[Hint: Show that you may choose $R$ as a matrix that features in the Cholesky factorization of $B=A^{T} A$.]

3.I.6E

commentGiven $f \in C^{n+1}[a, b]$, let the $n$ th-degree polynomial $p$ interpolate the values $f\left(x_{i}\right)$, $i=0,1, \ldots, n$, where $x_{0}, x_{1}, \ldots, x_{n} \in[a, b]$ are distinct. Given $x \in[a, b]$, find the error $f(x)-p(x)$ in terms of a derivative of $f$.

3.II.16E

commentLet the monic polynomials $p_{n}, n \geqslant 0$, be orthogonal with respect to the weight function $w(x)>0, a<x<b$, where the degree of each $p_{n}$ is exactly $n$.

(a) Prove that each $p_{n}, n \geqslant 1$, has $n$ distinct zeros in the interval $(a, b)$.

(b) Suppose that the $p_{n}$ satisfy the three-term recurrence relation

$p_{n}(x)=\left(x-a_{n}\right) p_{n-1}(x)-b_{n}^{2} p_{n-2}(x), \quad n \geqslant 2$

where $p_{0}(x) \equiv 1, p_{1}(x)=x-a_{1}$. Set

$A_{n}=\left(\begin{array}{ccccc} a_{1} & b_{2} & 0 & \cdots & 0 \\ b_{2} & a_{2} & b_{3} & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & b_{n-1} & a_{n-1} & b_{n} \\ 0 & \cdots & 0 & b_{n} & a_{n} \end{array}\right), \quad n \geqslant 2 .$

Prove that $p_{n}(x)=\operatorname{det}\left(x I-A_{n}\right), n \geqslant 2$, and deduce that all the eigenvalues of $A_{n}$ reside in $(a, b)$.

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

commentLet $a_{1}, \ldots, a_{n}$ be given constants, not all equal.

Use the Lagrangian sufficiency theorem, which you should state clearly, without proof, to minimize $\sum_{i=1}^{n} x_{i}^{2}$ subject to the two constraints $\sum_{i=1}^{n} x_{i}=1, \sum_{i=1}^{n} a_{i} x_{i}=0 .$

3.II.15D

commentConsider the following linear programming problem,

$\begin{array}{lrl} \operatorname{minimize} \quad(3-p) x_{1}+p x_{2} & \\ \text { subject to } & 2 x_{1}+x_{2} & \geqslant 8 \\ x_{1}+3 x_{2} & \geqslant 9 \\ x_{1} & \leqslant 6 \\ x_{1}, x_{2} & \geqslant 0 \end{array}$

Formulate the problem in a suitable way for solution by the two-phase simplex method.

Using the two-phase simplex method, show that if $2 \leqslant p \leqslant \frac{9}{4}$ then the optimal solution has objective function value $9-p$, while if $\frac{9}{4}<p \leqslant 3$ the optimal objective function value is $18-5 p$.

4.I.5D

commentExplain what is meant by a two-person zero-sum game with payoff matrix $A=\left[a_{i j}\right]$. Write down a set of sufficient conditions for a pair of strategies to be optimal for such a game.

A fair coin is tossed and the result is shown to player I, who must then decide to 'pass' or 'bet'. If he passes, he must pay player II $£ 1$. If he bets, player II, who does not know the result of the coin toss, may either 'fold' or 'call the bet'. If player II folds, she pays player I $£ 1$. If she calls the bet and the toss was a head, she pays player I $£ 2$; if she calls the bet and the toss was a tail, player I must pay her $£ 2$.

Formulate this as a two-person zero-sum game and find optimal strategies for the two players. Show that the game has value $\frac{1}{3}$.

[Hint: Player I has four possible moves and player II two.]

4.II.14D

commentDumbledore Publishers must decide how many copies of the best-selling "History of Hogwarts" to print in the next two months to meet demand. It is known that the demands will be for 40 thousand and 60 thousand copies in the first and second months respectively, and these demands must be met on time. At the beginning of the first month, a supply of 10 thousand copies is available, from existing stock. During each month, Dumbledore can produce up to 40 thousand copies, at a cost of 400 galleons per thousand copies. By having employees work overtime, up to 150 thousand additional copies can be printed each month, at a cost of 450 galleons per thousand copies. At the end of each month, after production and the current month's demand has been satisfied, a holding cost of 20 galleons per thousand copies is incurred.

Formulate a transportation problem, with 5 supply points and 3 demand points, to minimize the sum of production and holding costs during the two month period, and solve it.

[You may assume that copies produced during a month can be used to meet demand in that month.]

1.I.8B

commentLet $q(x, y)=a x^{2}+b x y+c y^{2}$ be a binary quadratic form with integer coefficients. Define what is meant by the discriminant $d$ of $q$, and show that $q$ is positive-definite if and only if $a>0>d$. Define what it means for the form $q$ to be reduced. For any integer $d<0$, we define the class number $h(d)$ to be the number of positive-definite reduced binary quadratic forms (with integer coefficients) with discriminant $d$. Show that $h(d)$ is always finite (for negative $d)$. Find $h(-39)$, and exhibit the corresponding reduced forms.

1.II.17B

commentLet $\phi$ be a symmetric bilinear form on a finite dimensional vector space $V$ over a field $k$ of characteristic $\neq 2$. Prove that the form $\phi$ may be diagonalized, and interpret the rank $r$ of $\phi$ in terms of the resulting diagonal form.

For $\phi$ a symmetric bilinear form on a real vector space $V$ of finite dimension $n$, define the signature $\sigma$ of $\phi$, proving that it is well-defined. A subspace $U$ of $V$ is called null if $\left.\phi\right|_{U} \equiv 0$; show that $V$ has a null subspace of dimension $n-\frac{1}{2}(r+|\sigma|)$, but no null subspace of higher dimension.

Consider now the quadratic form $q$ on $\mathbb{R}^{5}$ given by

$2\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}+x_{5} x_{1}\right)$

Write down the matrix $A$ for the corresponding symmetric bilinear form, and calculate $\operatorname{det} A$. Hence, or otherwise, find the rank and signature of $q$.

2.I.8B

commentLet $V$ be a finite-dimensional vector space over a field $k$. Describe a bijective correspondence between the set of bilinear forms on $V$, and the set of linear maps of $V$ to its dual space $V^{*}$. If $\phi_{1}, \phi_{2}$ are non-degenerate bilinear forms on $V$, prove that there exists an isomorphism $\alpha: V \rightarrow V$ such that $\phi_{2}(u, v)=\phi_{1}(u, \alpha v)$ for all $u, v \in V$. If furthermore both $\phi_{1}, \phi_{2}$ are symmetric, show that $\alpha$ is self-adjoint (i.e. equals its adjoint) with respect to $\phi_{1}$.

2.II.17B

commentSuppose $p$ is an odd prime and $a$ an integer coprime to $p$. Define the Legendre symbol $\left(\frac{a}{p}\right)$, and state (without proof) Euler's criterion for its calculation.

For $j$ any positive integer, we denote by $r_{j}$ the (unique) integer with $\left|r_{j}\right| \leq(p-1) / 2$ and $r_{j} \equiv a j \bmod p$. Let $l$ be the number of integers $1 \leq j \leq(p-1) / 2$ for which $r_{j}$ is negative. Prove that

$\left(\frac{a}{p}\right)=(-1)^{l} .$

Hence determine the odd primes for which 2 is a quadratic residue.

Suppose that $p_{1}, \ldots, p_{m}$ are primes congruent to 7 modulo 8 , and let

$N=8\left(p_{1} \cdots p_{m}\right)^{2}-1$

Show that 2 is a quadratic residue for any prime dividing $N$. Prove that $N$ is divisible by some prime $p \equiv 7 \bmod 8$. Hence deduce that there are infinitely many primes congruent to 7 modulo 8 .

3.I.9B

commentLet $A$ be the Hermitian matrix

$\left(\begin{array}{rrr} 1 & i & 2 i \\ -i & 3 & -i \\ -2 i & i & 5 \end{array}\right)$

Explaining carefully the method you use, find a diagonal matrix $D$ with rational entries, and an invertible (complex) matrix $T$ such that $T^{*} D T=A$, where $T^{*}$ here denotes the conjugated transpose of $T$.

Explain briefly why we cannot find $T, D$ as above with $T$ unitary.

[You may assume that if a monic polynomial $t^{3}+a_{2} t^{2}+a_{1} t+a_{0}$ with integer coefficients has all its roots rational, then all its roots are in fact integers.]

3.II.19B

commentLet $J_{1}$ denote the $2 \times 2$ matrix $\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right)$. Suppose that $T$ is a $2 \times 2$ uppertriangular real matrix with strictly positive diagonal entries and that $J_{1}^{-1} T J_{1} T^{-1}$ is orthogonal. Verify that $J_{1} T=T J_{1}$.

Prove that any real invertible matrix $A$ has a decomposition $A=B C$, where $B$ is an orthogonal matrix and $C$ is an upper-triangular matrix with strictly positive diagonal entries.

Let $A$ now denote a $2 n \times 2 n$ real matrix, and $A=B C$ be the decomposition of the previous paragraph. Let $K$ denote the $2 n \times 2 n$ matrix with $n$ copies of $J_{1}$ on the diagonal, and zeros elsewhere, and suppose that $K A=A K$. Prove that $K^{-1} C K C^{-1}$ is orthogonal. From this, deduce that the entries of $K^{-1} C K C^{-1}$ are zero, apart from $n$ orthogonal $2 \times 2$ blocks $E_{1}, \ldots, E_{n}$ along the diagonal. Show that each $E_{i}$ has the form $J_{1}{ }^{-1} C_{i} J_{1} C_{i}^{-1}$, for some $2 \times 2$ upper-triangular matrix $C_{i}$ with strictly positive diagonal entries. Deduce that $K C=C K$ and $K B=B K$.

[Hint: The invertible $2 n \times 2 n$ matrices $S$ with $2 \times 2$ blocks $S_{1}, \ldots, S_{n}$ along the diagonal, but with all other entries below the diagonal zero, form a group under matrix multiplication.]

1.I $. 9 \mathrm{~F} \quad$

commentA quantum mechanical particle of mass $m$ and energy $E$ encounters a potential step,

$V(x)= \begin{cases}0, & x<0 \\ V_{0}, & x \geqslant 0\end{cases}$

Calculate the probability $P$ that the particle is reflected in the case $E>V_{0}$.

If $V_{0}$ is positive, what is the limiting value of $P$ when $E$ tends to $V_{0}$ ? If $V_{0}$ is negative, what is the limiting value of $P$ as $V_{0}$ tends to $-\infty$ for fixed $E$ ?

1.II.18F

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

$V(x)= \begin{cases}0, & -a<x<a \\ \infty, & \text { elsewhere }\end{cases}$

(a) Verify that the set of normalised energy eigenfunctions are

$\psi_{n}(x)=\sqrt{\frac{1}{a}} \sin \left(\frac{n \pi(x+a)}{2 a}\right), \quad n=1,2, \ldots$

and evaluate the corresponding energy eigenvalues $E_{n}$.

(b) At time $t=0$ the wavefunction for the particle is only nonzero in the positive half of the well,

$\psi(x)= \begin{cases}\sqrt{\frac{2}{a}} \sin \left(\frac{\pi x}{a}\right), & 0<x<a \\ 0, & \text { elsewhere }\end{cases}$

Evaluate the expectation value of the energy, first using

$\langle E\rangle=\int_{-a}^{a} \psi H \psi d x$

and secondly using

$\langle E\rangle=\sum_{n}\left|a_{n}\right|^{2} E_{n},$

where the $a_{n}$ are the expansion coefficients in

$\psi(x)=\sum_{n} a_{n} \psi_{n}(x)$

Hence, show that

$1=\frac{1}{2}+\frac{8}{\pi^{2}} \sum_{p=0}^{\infty} \frac{(2 p+1)^{2}}{\left[(2 p+1)^{2}-4\right]^{2}}$

2.I $. 9 \mathrm{~F} \quad$

commentConsider a solution $\psi(x, t)$ of the time-dependent Schrödinger equation for a particle of mass $m$ in a potential $V(x)$. The expectation value of an operator $\mathcal{O}$ is defined as

$\langle\mathcal{O}\rangle=\int d x \psi^{*}(x, t) \mathcal{O} \psi(x, t)$

Show that

$\frac{d}{d t}\langle x\rangle=\frac{\langle p\rangle}{m},$

where

$p=\frac{\hbar}{i} \frac{\partial}{\partial x},$

and that

$\frac{d}{d t}\langle p\rangle=\left\langle-\frac{\partial V}{\partial x}(x)\right\rangle$

[You may assume that $\psi(x, t)$ vanishes as $x \rightarrow \pm \infty .]$

2.II.18F

(a) Write down the angular momentum operators $L_{1}, L_{2}, L_{3}$