Part IB, 2013, Paper 2

Analysis II

• Paper 2, Section I, F

  Analysis II | Part IB, 2013

  Let $\mathcal{C}[a, b]$ denote the vector space of continuous real-valued functions on the interval $[a, b]$, and let $\mathcal{C}^{\prime}[a, b]$ denote the subspace of continuously differentiable functions.

  Show that $\|f\|_{1}=\max |f|+\max \left|f^{\prime}\right|$ defines a norm on $\mathcal{C}^{\prime}[a, b]$. Show furthermore that the map $\Phi: f \mapsto f^{\prime}((a+b) / 2)$ takes the closed unit ball $\left\{\|f\|_{1} \leqslant 1\right\} \subset \mathcal{C}^{\prime}[a, b]$ to a bounded subset of $\mathbb{R}$.

  If instead we had used the norm $\|f\|_{0}=\max |f|$ restricted from $\mathcal{C}[a, b]$ to $\mathcal{C}^{\prime}[a, b]$, would $\Phi$ take the closed unit ball $\left\{\|f\|_{0} \leqslant 1\right\} \subset \mathcal{C}^{\prime}[a, b]$ to a bounded subset of $\mathbb{R}$ ? Justify your answer.

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• Paper 2, Section II, F

  Analysis II | Part IB, 2013

  Let $f: U \rightarrow \mathbb{R}$ be continuous on an open set $U \subset \mathbb{R}^{2}$. Suppose that on $U$ the partial derivatives $D_{1} f, D_{2} f, D_{1} D_{2} f$ and $D_{2} D_{1} f$ exist and are continuous. Prove that $D_{1} D_{2} f=D_{2} D_{1} f$ on $U$.

  If $f$ is infinitely differentiable, and $m \in \mathbb{N}$, what is the maximum number of distinct $m$-th order partial derivatives that $f$ may have on $U$ ?

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

  $$f(x, y)= \begin{cases}\frac{x^{2} y^{2}}{x^{4}+y^{4}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0)\end{cases}$$

  Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

  $$g(x, y)= \begin{cases}\frac{x y\left(x^{4}-y^{4}\right)}{x^{4}+y^{4}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0)\end{cases}$$

  For each of $f$ and $g$, determine whether they are (i) differentiable, (ii) infinitely differentiable at the origin. Briefly justify your answers.

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Complex Analysis or Complex Methods

• Paper 2, Section II, 13D Let

  Complex Analysis or Complex Methods | Part IB, 2013

  $$I=\oint_{C} \frac{e^{i z^{2} / \pi}}{1+e^{-2 z}} d z$$

  where $C$ is the rectangle with vertices at $\pm R$ and $\pm R+i \pi$, traversed anti-clockwise.

  (i) Show that $I=\frac{\pi(1+i)}{\sqrt{2}}$.

  (ii) Assuming that the contribution to $I$ from the vertical sides of the rectangle is negligible in the limit $R \rightarrow \infty$, show that

  $$\int_{-\infty}^{\infty} e^{i x^{2} / \pi} d x=\frac{\pi(1+i)}{\sqrt{2}}$$

  (iii) Justify briefly the assumption that the contribution to $I$ from the vertical sides of the rectangle is negligible in the limit $R \rightarrow \infty$.

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Electromagnetism

• Paper 2, Section I, D

  Electromagnetism | Part IB, 2013

  Use Maxwell's equations to obtain the equation of continuity

  $$\frac{\partial \rho}{\partial t}+\nabla \cdot \mathbf{J}=0$$

  Show that, for a body made from material of uniform conductivity $\sigma$, the charge density at any fixed internal point decays exponentially in time. If the body is finite and isolated, explain how this result can be consistent with overall charge conservation.

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• Paper 2, Section II, D

  Electromagnetism | Part IB, 2013

  Starting with the expression

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

  for the magnetic vector potential at the point $r$ due to a current distribution of density $\mathbf{J}(\mathbf{r})$, obtain the Biot-Savart law for the magnetic field due to a current $I$ flowing in a simple loop $C$ :

  $$\mathbf{B}(\mathbf{r})=-\frac{\mu_{0} I}{4 \pi} \oint_{C} \frac{d \mathbf{r}^{\prime} \times\left(\mathbf{r}^{\prime}-\mathbf{r}\right)}{\left|\mathbf{r}^{\prime}-\mathbf{r}\right|^{3}} \quad(\mathbf{r} \notin C) .$$

  Verify by direct differentiation that this satisfies $\boldsymbol{\nabla} \times \mathbf{B}=\mathbf{0}$. You may use without proof the identity $\boldsymbol{\nabla} \times(\mathbf{a} \times \mathbf{v})=\mathbf{a}(\boldsymbol{\nabla} \cdot \mathbf{v})-(\mathbf{a} \cdot \boldsymbol{\nabla}) \mathbf{v}$, where $\mathbf{a}$ is a constant vector and $\mathbf{v}$ is a vector field.

  Given that $C$ is planar, and is described in cylindrical polar coordinates by $z=0$, $r=f(\theta)$, show that the magnetic field at the origin is

  $$\widehat{\mathbf{z}} \frac{\mu_{0} I}{4 \pi} \oint \frac{d \theta}{f(\theta)}$$

  If $C$ is the ellipse $r(1-e \cos \theta)=\ell$, find the magnetic field at the focus due to a current $I$.

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Fluid Dynamics

• Paper 2, Section I, A

  Fluid Dynamics | Part IB, 2013

  An incompressible, inviscid fluid occupies the region beneath the free surface $y=\eta(x, t)$ and moves with a velocity field determined by the velocity potential $\phi(x, y, t) .$ Gravity acts in the $-y$ direction. You may assume Bernoulli's integral of the equation of motion:

  $$\frac{p}{\rho}+\frac{\partial \phi}{\partial t}+\frac{1}{2}|\nabla \phi|^{2}+g y=F(t)$$

  Give the kinematic and dynamic boundary conditions that must be satisfied by $\phi$ on $y=\eta(x, t)$.

  In the absence of waves, the fluid has constant uniform velocity $U$ in the $x$ direction. Derive the linearised form of the boundary conditions for small amplitude waves.

  Assume that the free surface and velocity potential are of the form:

  $$\begin{aligned} \eta &=a e^{i(k x-\omega t)} \\ \phi &=U x+i b e^{k y} e^{i(k x-\omega t)} \end{aligned}$$

  (where implicitly the real parts are taken). Show that

  $$(\omega-k U)^{2}=g k$$

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Geometry

• Paper 2, Section II, F

  Geometry | Part IB, 2013

  Let $A$ and $B$ be disjoint circles in $\mathbb{C}$. Prove that there is a Möbius transformation which takes $A$ and $B$ to two concentric circles.

  A collection of circles $X_{i} \subset \mathbb{C}, 0 \leqslant i \leqslant n-1$, for which

  1. $X_{i}$ is tangent to $A, B$ and $X_{i+1}$, where indices are $\bmod n$;

  2. the circles are disjoint away from tangency points;

  is called a constellation on $(A, B)$. Prove that for any $n \geqslant 2$ there is some pair $(A, B)$ and a constellation on $(A, B)$ made up of precisely $n$ circles. Draw a picture illustrating your answer.

  Given a constellation on $(A, B)$, prove that the tangency points $X_{i} \cap X_{i+1}$ for $0 \leqslant i \leqslant n-1$ all lie on a circle. Moreover, prove that if we take any other circle $Y_{0}$ tangent to $A$ and $B$, and then construct $Y_{i}$ for $i \geqslant 1$ inductively so that $Y_{i}$ is tangent to $A, B$ and $Y_{i-1}$, then we will have $Y_{n}=Y_{0}$, i.e. the chain of circles will again close up to form a constellation.

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Groups, Rings and Modules

• Paper 2, Section I, G

  Groups, Rings and Modules | Part IB, 2013

  Show that every Euclidean domain is a PID. Define the notion of a Noetherian ring, and show that $\mathbb{Z}[i]$ is Noetherian by using the fact that it is a Euclidean domain.

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• Paper 2, Section II, G

  Groups, Rings and Modules | Part IB, 2013

  (i) State the structure theorem for finitely generated modules over Euclidean domains.

  (ii) Let $\mathbb{C}[X]$ be the polynomial ring over the complex numbers. Let $M$ be a $\mathbb{C}[X]$ module which is 4-dimensional as a $\mathbb{C}$-vector space and such that $(X-2)^{4} \cdot x=0$ for all $x \in M$. Find all possible forms we obtain when we write $M \cong \bigoplus_{i=1}^{m} \mathbb{C}[X] /\left(P_{i}^{n_{i}}\right)$ for irreducible $P_{i} \in \mathbb{C}[X]$ and $n_{i} \geqslant 1$.

  (iii) Consider the quotient ring $M=\mathbb{C}[X] /\left(X^{3}+X\right)$ as a $\mathbb{C}[X]$-module. Show that $M$ is isomorphic as a $\mathbb{C}[X]$-module to the direct sum of three copies of $\mathbb{C}$. Give the isomorphism and its inverse explicitly.

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Linear Algebra

• Paper 2, Section I, E

  Linear Algebra | Part IB, 2013

  If $A$ is an $n \times n$ invertible Hermitian matrix, let

  $$U_{A}=\left\{U \in M_{n \times n}(\mathbb{C}) \mid \bar{U}^{T} A U=A\right\}$$

  Show that $U_{A}$ with the operation of matrix multiplication is a group, and that det $U$ has norm 1 for any $U \in U_{A}$. What is the relation between $U_{A}$ and the complex Hermitian form defined by $A$ ?

  If $A=I_{n}$ is the $n \times n$ identity matrix, show that any element of $U_{A}$ is diagonalizable.

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• Paper 2, Section II, E

  Linear Algebra | Part IB, 2013

  Define what it means for a set of vectors in a vector space $V$ to be linearly dependent. Prove from the definition that any set of $n+1$ vectors in $\mathbb{R}^{n}$ is linearly dependent.

  Using this or otherwise, prove that if $V$ has a finite basis consisting of $n$ elements, then any basis of $V$ has exactly $n$ elements.

  Let $V$ be the vector space of bounded continuous functions on $\mathbb{R}$. Show that $V$ is infinite dimensional.

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Markov Chains

• Paper 2, Section II, H

  Markov Chains | Part IB, 2013

  (i) Suppose $\left(X_{n}\right)_{n \geqslant 0}$ is an irreducible Markov chain and $f_{i j}=P\left(X_{n}=j\right.$ for some $\left.n \geqslant 1 \mid X_{0}=i\right)$. Prove that $f_{i i} \geqslant f_{i j} f_{j i}$ and that

  $$\sum_{n=0}^{\infty} P_{i}\left(X_{n}=i\right)=\sum_{n=1}^{\infty} f_{i i}^{n-1}$$

  (ii) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a symmetric random walk on the $\mathbb{Z}^{2}$ lattice. Prove that $\left(X_{n}\right)_{n \geqslant 0}$ is recurrent. You may assume, for $n \geqslant 1$,

  $$1 / 2<2^{-2 n} \sqrt{n}\left(\begin{array}{c} 2 n \\ n \end{array}\right)<1$$

  (iii) A princess and monster perform independent random walks on the $\mathbb{Z}^{2}$ lattice. The trajectory of the princess is the symmetric random walk $\left(X_{n}\right)_{n \geqslant 0}$. The monster's trajectory, denoted $\left(Z_{n}\right)_{n \geqslant 0}$, is a sleepy version of an independent symmetric random walk $\left(Y_{n}\right)_{n \geqslant 0}$. Specifically, given an infinite sequence of integers $0=n_{0}<n_{1}<\cdots$, the monster sleeps between these times, so $Z_{n_{i}+1}=\cdots=Z_{n_{i+1}}=Y_{i+1}$. Initially, $X_{0}=(100,0)$ and $Z_{0}=Y_{0}=(0,100)$. The princess is captured if and only if at some future time she and the monster are simultaneously at $(0,0)$.

  Compare the capture probabilities for an active monster, who takes $n_{i+1}=n_{i}+1$ for all $i$, and a sleepy monster, who takes $n_{i}$ spaced sufficiently widely so that

  $$P\left(X_{k}=(0,0) \text { for some } k \in\left\{n_{i}+1, \ldots, n_{i+1}\right\}\right)>1 / 2$$

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Methods

• Paper 2, Section I, B

  Methods | Part IB, 2013

  Consider the equation

  $$x u_{x}+(x+y) u_{y}=1$$

  subject to the Cauchy data $u(1, y)=y$. Using the method of characteristics, obtain a solution to this equation.

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• Paper 2, Section II, B

  Methods | Part IB, 2013

  The steady-state temperature distribution $u(x)$ in a uniform rod of finite length satisfies the boundary value problem

  $$\begin{gathered} -D \frac{d^{2}}{d x^{2}} u(x)=f(x), \quad 0<x<l \\ u(0)=0, \quad u(l)=0 \end{gathered}$$

  where $D>0$ is the (constant) diffusion coefficient. Determine the Green's function $G(x, \xi)$ for this problem. Now replace the above homogeneous boundary conditions with the inhomogeneous boundary conditions $u(0)=\alpha, \quad u(l)=\beta$ and give a solution to the new boundary value problem. Hence, obtain the steady-state solution for the following problem with the specified boundary conditions:

  $$\begin{aligned} &-D \frac{\partial^{2}}{\partial x^{2}} u(x, t)+\frac{\partial}{\partial t} u(x, t)=x, \quad 0<x<1 \\ &u(0, t)=1 / D, \quad u(1, t)=2 / D, \quad t>0 \end{aligned}$$

  [You may assume that a steady-state solution exists.]

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Metric and Topological Spaces

• Paper 2, Section I, G

  Metric and Topological Spaces | Part IB, 2013

  Let $X$ be a topological space. Prove or disprove the following statements.

  (i) If $X$ is discrete, then $X$ is compact if and only if it is a finite set.

  (ii) If $Y$ is a subspace of $X$ and $X, Y$ are both compact, then $Y$ is closed in $X$.

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Numerical Analysis

• Paper 2, Section II, C

  Numerical Analysis | Part IB, 2013

  Explain briefly what is meant by the convergence of a numerical method for solving the ordinary differential equation

  $$y^{\prime}(t)=f(t, y), \quad t \in[0, T], \quad y(0)=y_{0} .$$

  Prove from first principles that if the function $f$ is sufficiently smooth and satisfies the Lipschitz condition

  $$|f(t, x)-f(t, y)| \leqslant L|x-y|, \quad x, y \in \mathbb{R}, \quad t \in[0, T],$$

  for some $L>0$, then the backward Euler method

  $$y_{n+1}=y_{n}+h f\left(t_{n+1}, y_{n+1}\right)$$

  converges and find the order of convergence.

  Find the linear stability domain of the backward Euler method.

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Optimization

• Paper 2, Section I, H

  Optimization | Part IB, 2013

  Given a network with a source $A$, a sink $B$, and capacities on directed arcs, define what is meant by a minimum cut.

  The $m$ streets and $n$ intersections of a town are represented by sets of edges $E$ and vertices $V$ of a connected graph. A city planner wishes to make all streets one-way while ensuring it possible to drive away from each intersection along at least $k$ different streets.

  Use a theorem about min-cut and max-flow to prove that the city planner can achieve his goal provided that the following is true:

  $$d(U) \geqslant k|U| \text { for all } U \subseteq V,$$

  where $|U|$ is the size of $U$ and $d(U)$ is the number edges with at least one end in $U$. How could the planner find street directions that achieve his goal?

  [Hint: Consider a network having nodes $A, B$, nodes $a_{1}, \ldots, a_{m}$ for the streets and nodes $b_{1}, \ldots, b_{n}$ for the intersections. There are directed arcs from $A$ to each and from each $b_{i}$ to $B$. From each $a_{i}$ there are two further arcs, directed towards $b_{j}$ and $b_{j^{\prime}}$ that correspond to endpoints of street $i .]$

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Quantum Mechanics

• Paper 2, Section II, B

  Quantum Mechanics | Part IB, 2013

  (i) Consider a particle of mass $m$ confined to a one-dimensional potential well of depth $U>0$ and potential

  $$V(x)=\left\{\begin{array}{cl} -U, & |x|<l \\ 0, & |x|>l \end{array}\right.$$

  If the particle has energy $E$ where $-U \leqslant E<0$, show that for even states

  $$\alpha \tan \alpha l=\beta$$

  where $\alpha=\left[\frac{2 m}{\hbar^{2}}(U+E)\right]^{1 / 2}$ and $\beta=\left[-\frac{2 m}{\hbar^{2}} E\right]^{1 / 2}$.

  (ii) A particle of mass $m$ that is incident from the left scatters off a one-dimensional potential given by

  $$V(x)=k \delta(x)$$

  where $\delta(x)$ is the Dirac delta. If the particle has energy $E>0$ and $k>0$, obtain the reflection and transmission coefficients $R$ and $T$, respectively. Confirm that $R+T=1$.

  For the case $k<0$ and $E<0$ show that the energy of the only even parity bound state of the system is given by

  $$E=-\frac{m k^{2}}{2 \hbar^{2}}$$

  Use part (i) to verify this result by taking the limit $U \rightarrow \infty, l \rightarrow 0$ with $U l$ fixed.

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Statistics

• Paper 2, Section I, H

  Statistics | Part IB, 2013

  State and prove the Rao-Blackwell theorem.

  Individuals in a population are independently of three types $\{0,1,2\}$, with unknown probabilities $p_{0}, p_{1}, p_{2}$ where $p_{0}+p_{1}+p_{2}=1$. In a random sample of $n$ people the $i$ th person is found to be of type $x_{i} \in\{0,1,2\}$.

  Show that an unbiased estimator of $\theta=p_{0} p_{1} p_{2}$ is

  $$\hat{\theta}= \begin{cases}1, & \text { if }\left(x_{1}, x_{2}, x_{3}\right)=(0,1,2) \\ 0, & \text { otherwise. }\end{cases}$$

  Suppose that $n_{i}$ of the individuals are of type $i$. Find an unbiased estimator of $\theta$, say $\theta^{*}$, such that $\operatorname{var}\left(\theta^{*}\right)<\theta(1-\theta)$.

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Variational Principles

• Paper 2, Section II, A

  Variational Principles | Part IB, 2013

  Starting from the Euler-Lagrange equation, show that a condition for

  $$\int f\left(y, y^{\prime}\right) d x$$

  to be stationary is

  $$f-y^{\prime} \frac{\partial f}{\partial y^{\prime}}=\text { constant }$$

  In the half-plane $y>0$, light has speed $c(y)=y+c_{0}$ where $c_{0}>0$. Find the equation for a light ray between $(-a, 0)$ and $(a, 0)$. Sketch the solution.

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