• # 3.I.1A

Define 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.

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• # 3.II.11A

Prove 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.

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• # 3.I.8G

Inviscid 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 $t>0$.

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• # 3.II.18G

State 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, $\nabla \phi=\frac{\partial \phi}{\partial r} \mathbf{e}_{r}+\frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{e}_{\theta}$.]

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• # 3.I.3B

State 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) $(z-3)$.]

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• # 3.II.13B

If $X$ 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.

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• # 3.I.4B

State 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.

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• # 3.II.14B

Describe 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 $\pi$.

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• # 3.I $7 \mathrm{C} \quad$

Determine 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 $W$ ?

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• # 3.II.17C

Let $V$ 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.

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• # 3.I.2G

Laplace'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}

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• # 3.II.12H

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

$f(x)= \begin{cases}0, & 0 \leqslant x

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.

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• # 3.I.6E

Given $f \in C^{n+1}[a, b]$, 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 $f$.

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• # 3.II.16E

Let the monic polynomials $p_{n}, n \geqslant 0$, be orthogonal with respect to the weight function $w(x)>0, a, 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 $(a, b)$.

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• # 3.II.15D

Consider 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} the optimal objective function value is $18-5 p$.

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• # 3.I.9B

Let $A$ 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.]

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• # 3.II.19B

Let $J_{1}$ 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.]

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• # 3.II.20F

A quantum system has a complete set of orthonormalised energy eigenfunctions $\psi_{n}(x)$ with corresponding energy eigenvalues $E_{n}, n=1,2,3, \ldots$

(a) If the time-dependent wavefunction $\psi(x, t)$ is, at $t=0$,

$\psi(x, 0)=\sum_{n=1}^{\infty} a_{n} \psi_{n}(x)$

determine $\psi(x, t)$ for all $t>0$.

(b) A linear operator $\mathcal{S}$ acts on the energy eigenfunctions as follows:

\begin{aligned} &\mathcal{S} \psi_{1}=7 \psi_{1}+24 \psi_{2} \\ &\mathcal{S} \psi_{2}=24 \psi_{1}-7 \psi_{2} \\ &\mathcal{S} \psi_{n}=0, \quad n \geqslant 3 \end{aligned}

Find the eigenvalues of $\mathcal{S}$. At time $t=0, \mathcal{S}$ is measured and its lowest eigenvalue is found. At time $t>0, \mathcal{S}$ is measured again. Show that the probability for obtaining the lowest eigenvalue again is

$\frac{1}{625}(337+288 \cos (\omega t))$

where $\omega=\left(E_{1}-E_{2}\right) / \hbar$.

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• # 3.I.10F

A particle of rest mass $m$ and four-momentum $P=(E / c, \mathbf{p})$ is detected by an observer with four-velocity $U$, satisfying $U \cdot U=c^{2}$, where the product of two four-vectors $P=\left(p^{0}, \mathbf{p}\right)$ and $Q=\left(q^{0}, \mathbf{q}\right)$ is $P \cdot Q=p^{0} q^{0}-\mathbf{p} \cdot \mathbf{q}$.

Show that the speed of the detected particle in the observer's rest frame is

$v=c \sqrt{1-\frac{P \cdot P c^{2}}{(P \cdot U)^{2}}}$

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