• # 2.I.1E

Define 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.]

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• # 2.II.10E

(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)) 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.]

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• # 2.I.7B

Suppose 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 $D$.

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• # 2.II.16B

A 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

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

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• # 2.I.4G

Let 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.

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• # 2.II.13G

(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} or if $a_{1}=a_{2}$ and $b_{1}. 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.

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• # 2.I.6G

Let $A$ 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 $A$.

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• # 2.II.15G

(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 $D U=U D$.

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• # 2.I.2C

Write 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.

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• # 2.II.11C

If $\mathbf{B}$ 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

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

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• # 2.I.5B

Applying 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 $A=Q R$.

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• # 2.II.14B

Let $f \in C[a, b]$ 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 $\gamma=\frac{x_{i}-x_{0}}{x_{n}-x_{0}}$.

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• # 2.I.8F

Explain 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 $v, w \in V$.

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• # 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 $\alpha^{*}\left(W^{\perp}\right) \subseteq W^{\perp}$.

An endomorphism of a complex inner-product space is said to be normal if it commutes with its adjoint. Prove the following facts about a normal endomorphism $\alpha$ of a finite-dimensional space $V$.

(i) $\alpha$ and $\alpha^{*}$ have the same kernel.

(ii) $\alpha$ and $\alpha^{*}$ have the same eigenvectors, with complex conjugate eigenvalues.

(iii) If $E_{\lambda}=\{x \in V: \alpha(x)=\lambda x\}$, then $\alpha\left(E_{\lambda}^{\perp}\right) \subseteq E_{\lambda}^{\perp}$.

(iv) There is an orthonormal basis of $V$ consisting of eigenvectors of $\alpha$.

Deduce that an endomorphism $\alpha$ is normal if and only if it can be written as a product $\beta \gamma$, where $\beta$ is Hermitian, $\gamma$ is unitary and $\beta$ and $\gamma$ commute with each other. [Hint: Given $\alpha$, define $\beta$ and $\gamma$ in terms of their effect on the basis constructed in (iv).]

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• # 2.I.9D

From the expressions

$L_{x}=y P_{z}-z P_{y}, \quad L_{y}=z P_{x}-x P_{z}, \quad L_{z}=x P_{y}-y P_{x}$

show that

$(x+i y) z$

is an eigenfunction of $\mathbf{L}^{2}$ and $L_{z}$, and compute the corresponding eigenvalues.

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• # 2.II.18D

Consider a quantum mechanical particle moving in an upside-down harmonic oscillator potential. Its wavefunction $\Psi(x, t)$ evolves according to the time-dependent Schrödinger equation,

$i \hbar \frac{\partial \Psi}{\partial t}=-\frac{\hbar^{2}}{2} \frac{\partial^{2} \Psi}{\partial x^{2}}-\frac{1}{2} x^{2} \Psi$

(a) Verify that

$\Psi(x, t)=A(t) e^{-B(t) x^{2}}$

is a solution of equation (1), provided that

$\frac{d A}{d t}=-i \hbar A B$

and

$\frac{d B}{d t}=-\frac{i}{2 \hbar}-2 i \hbar B^{2}$

(b) Verify that $B=\frac{1}{2 \hbar} \tan (\phi-i t)$ provides a solution to (3), where $\phi$ is an arbitrary real constant.

(c) The expectation value of an operator $\mathcal{O}$ at time $t$ is

$\langle\mathcal{O}\rangle(t) \equiv \int_{-\infty}^{\infty} d x \Psi^{*}(x, t) \mathcal{O} \Psi(x, t),$

where $\Psi(x, t)$ is the normalised wave function. Show that for $\Psi(x, t)$ given by (2),

$\left\langle x^{2}\right\rangle=\frac{1}{4 \operatorname{Re}(B)}, \quad\left\langle p^{2}\right\rangle=4 \hbar^{2}|B|^{2}\left\langle x^{2}\right\rangle$

Hence show that as $t \rightarrow \infty$,

$\left\langle x^{2}\right\rangle \approx\left\langle p^{2}\right\rangle \approx \frac{\hbar}{4 \sin 2 \phi} e^{2 t}$

[Hint: You may use

$\frac{\int_{-\infty}^{\infty} d x e^{-C x^{2}} x^{2}}{\int_{-\infty}^{\infty} d x e^{-C x^{2}}}=\frac{1}{2 C}$

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• # 2.II.12H

For ten steel ingots from a production process the following measures of hardness were obtained:

$73.2, \quad 74.3, \quad 75.4, \quad 73.8, \quad 74.4, \quad 76.7, \quad 76.1, \quad 73.0, \quad 74.6, \quad 74.1 .$

On the assumption that the variation is well described by a normal density function obtain an estimate of the process mean.

The manufacturer claims that he is supplying steel with mean hardness 75 . Derive carefully a (generalized) likelihood ratio test of this claim. Knowing that for the data above

$S_{X X}=\sum_{j=1}^{n}\left(X_{i}-\bar{X}\right)^{2}=12.824$

what is the result of the test at the $5 \%$ significance level?

$\left.\begin{array}{lll}{[\text { Distribution }} & t_{9} & t_{10} \\ 95 \% \text { percentile } & 1.83 & 1.81 \\ 97.5 \% \text { percentile } & 2.26 & 2.23\end{array}\right]$

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