Part IB, 2008, Paper 1

Analysis II

• 1.II.11F

  Analysis II | Part IB, 2008

  State and prove the Contraction Mapping Theorem.

  Let $(X, d)$ be a nonempty complete metric space and $f: X \rightarrow X$ a mapping such that, for some $k>0$, the $k$ th iterate $f^{k}$ of $f$ (that is, $f$ composed with itself $k$ times) is a contraction mapping. Show that $f$ has a unique fixed point.

  Now let $X$ be the space of all continuous real-valued functions on $[0,1]$, equipped with the uniform norm $\|h\|_{\infty}=\sup \{|h(t)|: t \in[0,1]\}$, and let $\phi: \mathbb{R} \times[0,1] \rightarrow \mathbb{R}$ be a continuous function satisfying the Lipschitz condition

  $$|\phi(x, t)-\phi(y, t)| \leqslant M|x-y|$$

  for all $t \in[0,1]$ and all $x, y \in \mathbb{R}$, where $M$ is a constant. Let $F: X \rightarrow X$ be defined by

  $$F(h)(t)=g(t)+\int_{0}^{t} \phi(h(s), s) d s$$

  where $g$ is a fixed continuous function on $[0,1]$. Show by induction on $n$ that

  $$\left|F^{n}(h)(t)-F^{n}(k)(t)\right| \leqslant \frac{M^{n} t^{n}}{n !}\|h-k\|_{\infty}$$

  for all $h, k \in X$ and all $t \in[0,1]$. Deduce that the integral equation

  $$f(t)=g(t)+\int_{0}^{t} \phi(f(s), s) d s$$

  has a unique continuous solution $f$ on $[0,1]$.

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Complex Analysis or Complex Methods

• 1.I.3C

  Complex Analysis or Complex Methods | Part IB, 2008

  Given that $f(z)$ is an analytic function, show that the mapping $w=f(z)$

  (a) preserves angles between smooth curves intersecting at $z$ if $f^{\prime}(z) \neq 0$;

  (b) has Jacobian given by $\left|f^{\prime}(z)\right|^{2}$.

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• 1.II.13C

  Complex Analysis or Complex Methods | Part IB, 2008

  By a suitable choice of contour show the following:

  (a)

  $$\int_{0}^{\infty} \frac{x^{1 / n}}{1+x^{2}} d x=\frac{\pi}{2 \cos (\pi / 2 n)}$$

  where $n>1$,

  (b)

  $$\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x=\frac{\pi^{2}}{2 \sqrt{2}}$$

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Electromagnetism

• 1.II.16B

  Electromagnetism | Part IB, 2008

  Suppose that the current density $\mathbf{J}(\mathbf{r})$ is constant in time but the charge density $\rho(\mathbf{r}, t)$ is not.

  (i) Show that $\rho$ is a linear function of time:

  $$\rho(\mathbf{r}, t)=\rho(\mathbf{r}, 0)+\dot{\rho}(\mathbf{r}, 0) t$$

  where $\dot{\rho}(\mathbf{r}, 0)$ is the time derivative of $\rho$ at time $t=0$.

  (ii) The magnetic induction due to a current density $\mathbf{J}(\mathbf{r})$ can be written as

  $$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} d V^{\prime}$$

  Show that this can also be written as

  $$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \nabla \times \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d V^{\prime}$$

  (iii) Assuming that $\mathbf{J}$ vanishes at infinity, show that the curl of the magnetic field in (1) can be written as

  $$\nabla \times \mathbf{B}(\mathbf{r})=\mu_{0} \mathbf{J}(\mathbf{r})+\frac{\mu_{0}}{4 \pi} \nabla \int \frac{\nabla^{\prime} \cdot \mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d V^{\prime}$$

  [You may find useful the identities $\nabla \times(\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^{2} \mathbf{A}$ and also $\left.\nabla^{2}\left(1 /\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)=-4 \pi \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) .\right]$

  (iv) Show that the second term on the right hand side of (2) can be expressed in terms of the time derivative of the electric field in such a way that $\mathbf{B}$ itself obeys Ampère's law with Maxwell's displacement current term, i.e. $\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\mu_{0} \epsilon_{0} \partial \mathbf{E} / \partial t$.

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Fluid Dynamics

• 1.I.5B

  Fluid Dynamics | Part IB, 2008

  Verify that the two-dimensional flow given in Cartesian coordinates by

  $$\mathbf{u}=\left(\mathrm{e}^{y} \sinh x,-\mathrm{e}^{y} \cosh x\right)$$

  satisfies $\nabla \cdot \mathbf{u}=0$. Find the stream function $\psi(x, y)$. Sketch the streamlines.

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• 1.II.17B

  Fluid Dynamics | Part IB, 2008

  Two incompressible fluids flow in infinite horizontal streams, the plane of contact being $z=0$, with $z$ positive upwards. The flow is given by

  $$\mathbf{U}(\mathbf{r})= \begin{cases}U_{2} \hat{\mathbf{e}}_{x}, & z>0 \\ U_{1} \hat{\mathbf{e}}_{x}, & z<0\end{cases}$$

  where $\hat{\mathbf{e}}_{x}$ is the unit vector in the positive $x$ direction. The upper fluid has density $\rho_{2}$ and pressure $p_{0}-g \rho_{2} z$, the lower has density $\rho_{1}$ and pressure $p_{0}-g \rho_{1} z$, where $p_{0}$ is a constant and $g$ is the acceleration due to gravity.

  (i) Consider a perturbation to the flat surface $z=0$ of the form

  $$z \equiv \zeta(x, y, t)=\zeta_{0} e^{i(k x+\ell y)+s t} .$$

  State the kinematic boundary conditions on the velocity potentials $\phi_{i}$ that hold on the interface in the two domains, and show by linearising in $\zeta$ that they reduce to

  $$\frac{\partial \phi_{i}}{\partial z}=\frac{\partial \zeta}{\partial t}+U_{i} \frac{\partial \zeta}{\partial x} \quad(z=0, i=1,2) .$$

  (ii) State the dynamic boundary condition on the perturbed interface, and show by linearising in $\zeta$ that it reduces to

  $$\rho_{1}\left(U_{1} \frac{\partial \phi_{1}}{\partial x}+\frac{\partial \phi_{1}}{\partial t}+g \zeta\right)=\rho_{2}\left(U_{2} \frac{\partial \phi_{2}}{\partial x}+\frac{\partial \phi_{2}}{\partial t}+g \zeta\right) \quad(z=0)$$

  (iii) Use the velocity potentials

  $$\phi_{1}=U_{1} x+A_{1} e^{q z} e^{i(k x+\ell y)+s t}, \quad \phi_{2}=U_{2} x+A_{2} e^{-q z} e^{i(k x+\ell y)+s t},$$

  where $q=\sqrt{k^{2}+\ell^{2}}$, and the conditions in (i) and (ii) to perform a stability analysis. Show that the relation between $s, k$ and $\ell$ is

  $$s=-i k \frac{\rho_{1} U_{1}+\rho_{2} U_{2}}{\rho_{1}+\rho_{2}} \pm\left[\frac{k^{2} \rho_{1} \rho_{2}\left(U_{1}-U_{2}\right)^{2}}{\left(\rho_{1}+\rho_{2}\right)^{2}}-\frac{q g\left(\rho_{1}-\rho_{2}\right)}{\rho_{1}+\rho_{2}}\right]^{1 / 2} .$$

  Find the criterion for instability.

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Geometry

• 1.I.2G

  Geometry | Part IB, 2008

  Show that any element of $S O(3, \mathbb{R})$ is a rotation, and that it can be written as the product of two reflections.

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Groups, Rings and Modules

• 1.II.10G

  Groups, Rings and Modules | Part IB, 2008

  (i) Show that $A_{4}$ is not simple.

  (ii) Show that the group Rot $(D)$ of rotational symmetries of a regular dodecahedron is a simple group of order 60 .

  (iii) Show that $\operatorname{Rot}(D)$ is isomorphic to $A_{5}$.

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Linear Algebra

• 1.I.1E

  Linear Algebra | Part IB, 2008

  Let $A$ be an $n \times n$ matrix over $\mathbb{C}$. What does it mean to say that $\lambda$ is an eigenvalue of $A$ ? Show that $A$ has at least one eigenvalue. For each of the following statements, provide a proof or a counterexample as appropriate.

  (i) If $A$ is Hermitian, all eigenvalues of $A$ are real.

  (ii) If all eigenvalues of $A$ are real, $A$ is Hermitian.

  (iii) If all entries of $A$ are real and positive, all eigenvalues of $A$ have positive real part.

  (iv) If $A$ and $B$ have the same trace and determinant then they have the same eigenvalues.

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• 1.II.9E

  Linear Algebra | Part IB, 2008

  Let $A$ be an $m \times n$ matrix of real numbers. Define the row rank and column rank of $A$ and show that they are equal.

  Show that if a matrix $A^{\prime}$ is obtained from $A$ by elementary row and column operations then $\operatorname{rank}\left(A^{\prime}\right)=\operatorname{rank}(A)$.

  Let $P, Q$ and $R$ be $n \times n$ matrices. Show that the $2 n \times 2 n$ matrices $\left(\begin{array}{cc}P Q & 0 \\ Q & Q R\end{array}\right)$ and $\left(\begin{array}{cc}0 & P Q R \\ Q & 0\end{array}\right)$ have the same rank.

  Hence, or otherwise, prove that

  $$\operatorname{rank}(P Q)+\operatorname{rank}(Q R) \leqslant \operatorname{rank}(Q)+\operatorname{rank}(P Q R)$$

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Markov Chains

• 1.II.19H

  Markov Chains | Part IB, 2008

  The village green is ringed by a fence with $N$ fenceposts, labelled $0,1, \ldots, N-1$. The village idiot is given a pot of paint and a brush, and started at post 0 with instructions to paint all the posts. He paints post 0 , and then chooses one of the two nearest neighbours, 1 or $N-1$, with equal probability, moving to the chosen post and painting it. After painting a post, he chooses with equal probability one of the two nearest neighbours, moves there and paints it (regardless of whether it is already painted). Find the distribution of the last post unpainted.

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Methods

• 1.II.14D

  Methods | Part IB, 2008

  Write down the Euler-Lagrange equation for the variational problem for $y(x)$ that extremizes the integral $I$ defined as

  $$I=\int_{x_{1}}^{x_{2}} f\left(x, y, y^{\prime}\right) d x$$

  with boundary conditions $y\left(x_{1}\right)=y_{1}, y\left(x_{2}\right)=y_{2}$, where $y_{1}$ and $y_{2}$ are positive constants such that $y_{2}>y_{1}$, with $x_{2}>x_{1}$. Find a first integral of the equation when $f$ is independent of $y$, i.e. $f=f\left(x, y^{\prime}\right)$.

  A light ray moves in the $(x, y)$ plane from $\left(x_{1}, y_{1}\right)$ to $\left(x_{2}, y_{2}\right)$ with speed $c(x)$ taking a time $T$. Show that the equation of the path that makes $T$ an extremum satisfies

  $$\frac{d y}{d x}=\frac{c(x)}{\sqrt{k^{2}-c^{2}(x)}}$$

  where $k$ is a constant and write down an integral relating $k, x_{1}, x_{2}, y_{1}$ and $y_{2}$.

  When $c(x)=a x$ where $a$ is a constant and $k=a x_{2}$, show that the path is given by

  $$\left(y_{2}-y\right)^{2}=x_{2}^{2}-x^{2} .$$

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Metric and Topological Spaces

• 1.II.12F

  Metric and Topological Spaces | Part IB, 2008

  Write down the definition of a topology on a set $X$.

  For each of the following families $\mathcal{T}$ of subsets of $\mathbb{Z}$, determine whether $\mathcal{T}$ is a topology on $\mathbb{Z}$. In the cases where the answer is 'yes', determine also whether $(\mathbb{Z}, \mathcal{T})$ is a Hausdorff space and whether it is compact.

  (a) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : either $U$ is finite or $0 \in U\}$.

  (b) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : either $\mathbb{Z} \backslash U$ is finite or $0 \notin U\}$.

  (c) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : there exists $k>0$ such that, for all $n, n \in U \Leftrightarrow n+k \in U\}$.

  (d) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : for all $n \in U$, there exists $k>0$ such that $\{n+k m: m \in \mathbb{Z}\} \subseteq U\}$.

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Numerical Analysis

• 1.I.6D

  Numerical Analysis | Part IB, 2008

  Show that if $A=L D L^{T}$, where $L \in \mathbb{R}^{m \times m}$ is a lower triangular matrix with all elements on the main diagonal being unity and $D \in \mathbb{R}^{m \times m}$ is a diagonal matrix with positive elements, then $A$ is positive definite. Find $L$ and the corresponding $D$ when

  $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 3 \end{array}\right]$$

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Optimization

• 1.I.8H

  Optimization | Part IB, 2008

  State the Lagrangian Sufficiency Theorem for the maximization over $x$ of $f(x)$ subject to the constraint $g(x)=b$.

  For each $p>0$, solve

  $$\max \sum_{i=1}^{d} x_{i}^{p} \quad \text { subject to } \sum_{i=1}^{d} x_{i}=1, \quad x_{i} \geqslant 0 .$$

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Quantum Mechanics

• 1.II.15A

  Quantum Mechanics | Part IB, 2008

  The radial wavefunction $g(r)$ for the hydrogen atom satisfies the equation

  $$-\frac{\hbar^{2}}{2 m r^{2}} \frac{d}{d r}\left(r^{2} \frac{d g(r)}{d r}\right)-\frac{e^{2} g(r)}{4 \pi \epsilon_{0} r}+\hbar^{2} \frac{\ell(\ell+1)}{2 m r^{2}} g(r)=E g(r)$$

  With reference to the general form for the time-independent Schrödinger equation, explain the origin of each term. What are the allowed values of $\ell$ ?

  The lowest-energy bound-state solution of $(*)$, for given $\ell$, has the form $r^{\alpha} e^{-\beta r}$. Find $\alpha$ and $\beta$ and the corresponding energy $E$ in terms of $\ell$.

  A hydrogen atom makes a transition between two such states corresponding to $\ell+1$ and $\ell$. What is the frequency of the emitted photon?

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Special Relativity

• 1.I.4C

  Special Relativity | Part IB, 2008

  In an inertial frame $S$ a photon of energy $E$ is observed to travel at an angle $\theta$ relative to the $x$-axis. The inertial frame $S^{\prime}$ moves relative to $S$ at velocity $v$ in the $x$ direction and the $x^{\prime}$-axis of $S^{\prime}$ is taken parallel to the $x$-axis of $S$. Observed in $S^{\prime}$, the photon has energy $E^{\prime}$ and travels at an angle $\theta^{\prime}$ relative to the $x^{\prime}$-axis. Show that

  $$E^{\prime}=\frac{E(1-\beta \cos \theta)}{\sqrt{1-\beta^{2}}}, \quad \cos \theta^{\prime}=\frac{\cos \theta-\beta}{1-\beta \cos \theta},$$

  where $\beta=v / c$.

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Statistics

• 1.II.18H

  Statistics | Part IB, 2008

  Suppose that $X_{1}, \ldots, X_{n}$ is a sample of size $n$ with common $N\left(\mu_{X}, 1\right)$ distribution, and $Y_{1}, \ldots, Y_{n}$ is an independent sample of size $n$ from a $N\left(\mu_{Y}, 1\right)$ distribution.

  (i) Find (with careful justification) the form of the size- $\alpha$ likelihood-ratio test of the null hypothesis $H_{0}: \mu_{Y}=0$ against alternative $H_{1}:\left(\mu_{X}, \mu_{Y}\right)$ unrestricted.

  (ii) Find the form of the size- $\alpha$ likelihood-ratio test of the hypothesis

  $$H_{0}: \mu_{X} \geqslant A, \mu_{Y}=0$$

  against $H_{1}:\left(\mu_{X}, \mu_{Y}\right)$ unrestricted, where $A$ is a given constant.

  Compare the critical regions you obtain in (i) and (ii) and comment briefly.

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