Part IB, 2006, Paper 1

# Part IB, 2006, Paper 1

### Jump to course

1.II.11F

commentLet $a_{n}$ and $b_{n}$ be sequences of real numbers for $n \geqslant 1$ such that $\left|a_{n}\right| \leqslant c / n^{1+\epsilon}$ and $\left|b_{n}\right| \leqslant c / n^{1+\epsilon}$ for all $n \geqslant 1$, for some constants $c>0$ and $\epsilon>0$. Show that the series

$f(x)=\sum_{n \geqslant 1} a_{n} \cos n x+\sum_{n \geqslant 1} b_{n} \sin n x$

converges uniformly to a continuous function on the real line. Show that $f$ is periodic in the sense that $f(x+2 \pi)=f(x)$.

Now suppose that $\left|a_{n}\right| \leqslant c / n^{2+\epsilon}$ and $\left|b_{n}\right| \leqslant c / n^{2+\epsilon}$ for all $n \geqslant 1$, for some constants $c>0$ and $\epsilon>0$. Show that $f$ is differentiable on the real line, with derivative

$f^{\prime}(x)=-\sum_{n \geqslant 1} n a_{n} \sin n x+\sum_{n \geqslant 1} n b_{n} \cos n x .$

[You may assume the convergence of standard series.]

1.I.3D

commentLet $L$ be the Laplace operator, i.e., $L(g)=g_{x x}+g_{y y}$. Prove that if $f: \Omega \rightarrow \mathbf{C}$ is analytic in a domain $\Omega$, then

$L\left(|f(z)|^{2}\right)=4\left|f^{\prime}(z)\right|^{2}, \quad z \in \Omega .$

1.II.13D

commentBy integrating round the contour involving the real axis and the line $\operatorname{Im}(z)=2 \pi$, or otherwise, evaluate

$\int_{-\infty}^{\infty} \frac{e^{a x}}{1+e^{x}} d x, \quad 0<a<1 .$

Explain why the given restriction on the value $a$ is necessary.

1.II.16G

commentThree concentric conducting spherical shells of radii $a, b$ and $c(a<b<c)$ carry charges $q,-2 q$ and $3 q$ respectively. Find the electric field and electric potential at all points of space.

Calculate the total energy of the electric field.

1.I.5A

commentUse the Euler equation for the motion of an inviscid fluid to derive the vorticity equation in the form

$D \boldsymbol{\omega} / D t=\boldsymbol{\omega} \cdot \nabla \mathbf{u} .$

Give a physical interpretation of the terms in this equation and deduce that irrotational flows remain irrotational.

In a plane flow the vorticity at time $t=0$ has the uniform value $\boldsymbol{\omega}_{0} \neq \mathbf{0}$. Find the vorticity everywhere at times $t>0$.

1.II.17A

commentA point source of fluid of strength $m$ is located at $\mathbf{x}_{s}=(0,0, a)$ in inviscid fluid of density $\rho$. Gravity is negligible. The fluid is confined to the region $z \geqslant 0$ by the fixed boundary $z=0$. Write down the equation and boundary conditions satisfied by the velocity potential $\phi$. Find $\phi$.

[Hint: consider the flow generated in unbounded fluid by the source $m$ together with an 'image source' of equal strength at $\overline{\mathbf{x}}_{s}=(0,0,-a)$.]

Use Bernoulli's theorem, which may be stated without proof, to find the fluid pressure everywhere on $z=0$. Deduce the magnitude of the hydrodynamic force on the boundary $z=0$. Determine whether the boundary is attracted toward the source or repelled from it.

1.I $2 \mathrm{H} \quad$

commentDefine the hyperbolic metric in the upper half-plane model $H$ of the hyperbolic plane. How does one define the hyperbolic area of a region in $H$ ? State the Gauss-Bonnet theorem for hyperbolic triangles.

Let $R$ be the region in $H$ defined by

$0<x<\frac{1}{2}, \quad \sqrt{1-x^{2}}<y<1$

Calculate the hyperbolic area of $R$.

1.II.10E

commentFind all subgroups of indices $2,3,4$ and 5 in the alternating group $A_{5}$ on 5 letters. You may use any general result that you choose, provided that you state it clearly, but you must justify your answers.

[You may take for granted the fact that $A_{4}$ has no subgroup of index 2.]

1.I.1H

commentDefine what is meant by the minimal polynomial of a complex $n \times n$ matrix, and show that it is unique. Deduce that the minimal polynomial of a real $n \times n$ matrix has real coefficients.

For $n>2$, find an $n \times n$ matrix with minimal polynomial $(t-1)^{2}(t+1)$.

1.II.9H

commentLet $U, V$ be finite-dimensional vector spaces, and let $\theta$ be a linear map of $U$ into $V$. Define the rank $r(\theta)$ and the nullity $n(\theta)$ of $\theta$, and prove that

$r(\theta)+n(\theta)=\operatorname{dim} U$

Now let $\theta, \phi$ be endomorphisms of a vector space $U$. Define the endomorphisms $\theta+\phi$ and $\theta \phi$, and prove that

$\begin{aligned} r(\theta+\phi) & \leqslant r(\theta)+r(\phi) \\ n(\theta \phi) & \leqslant n(\theta)+n(\phi) . \end{aligned}$

Prove that equality holds in both inequalities if and only if $\theta+\phi$ is an isomorphism and $\theta \phi$ is zero.

1.II.19C

commentExplain what is meant by a stopping time of a Markov chain $\left(X_{n}\right)_{n \geq 0}$. State the strong Markov property.

Show that, for any state $i$, the probability, starting from $i$, that $\left(X_{n}\right)_{n \geq 0}$ makes infinitely many visits to $i$ can take only the values 0 or 1 .

Show moreover that, if

$\sum_{n=0}^{\infty} \mathbb{P}_{i}\left(X_{n}=i\right)=\infty$

then $\left(X_{n}\right)_{n \geq 0}$ makes infinitely many visits to $i$ with probability $1 .$

1.II.14A

commentDefine a second rank tensor. Show from your definition that if $M_{i j}$ is a second rank tensor then $M_{i i}$ is a scalar.

A rigid body consists of a thin flat plate of material having density $\rho(\mathbf{x})$ per unit area, where $\mathbf{x}$ is the position vector. The body occupies a region $D$ of the $(x, y)$-plane; its thickness in the $z$-direction is negligible. The moment of inertia tensor of the body is given as

$M_{i j}=\int_{D}\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \rho d S$

Show that the $z$-direction is an eigenvector of $M_{i j}$ and write down an integral expression for the corresponding eigenvalue $M_{\perp}$.

Hence or otherwise show that if the remaining eigenvalues of $M_{i j}$ are $M_{1}$ and $M_{2}$ then

$M_{\perp}=M_{1}+M_{2} .$

Find $M_{i j}$ for a circular disc of radius $a$ and uniform density having its centre at the origin.

1.II.12F

comment(i) Define the product topology on $X \times Y$ for topological spaces $X$ and $Y$, proving that your definition does define a topology.

(ii) Let $X$ be the logarithmic spiral defined in polar coordinates by $r=e^{\theta}$, where $-\infty<\theta<\infty$. Show that $X$ (with the subspace topology from $\mathbf{R}^{2}$ ) is homeomorphic to the real line.

1.I.6D

comment(a) Perform the LU-factorization with column pivoting of the matrix

$A=\left[\begin{array}{rrr} 2 & 1 & 1 \\ 4 & 1 & 0 \\ -2 & 2 & 1 \end{array}\right]$

(b) Explain briefly why every nonsingular matrix $A$ admits an LU-factorization with column pivoting.

1.I.8C

commentState the Lagrangian sufficiency theorem.

Let $p \in(1, \infty)$ and let $a_{1}, \ldots, a_{n} \in \mathbb{R}$. Maximize

$\sum_{i=1}^{n} a_{i} x_{i}$

subject to

$\sum_{i=1}^{n}\left|x_{i}\right|^{p} \leqslant 1, \quad x_{1}, \ldots, x_{n} \in \mathbb{R}$

1.II.15B

commentLet $V_{1}(x)$ and $V_{2}(x)$ be two real potential functions of one space dimension, and let $a$ be a positive constant. Suppose also that $V_{1}(x) \leqslant V_{2}(x) \leqslant 0$ for all $x$ and that $V_{1}(x)=V_{2}(x)=0$ for all $x$ such that $|x| \geqslant a$. Consider an incoming beam of particles described by the plane wave $\exp (i k x)$, for some $k>0$, scattering off one of the potentials $V_{1}(x)$ or $V_{2}(x)$. Let $p_{i}$ be the probability that a particle in the beam is reflected by the potential $V_{i}(x)$. Is it necessarily the case that $p_{1} \leqslant p_{2}$ ? Justify your answer carefully, either by giving a rigorous proof or by presenting a counterexample with explicit calculations of $p_{1}$ and $p_{2}$.

1.I.4B

commentA ball of clay of mass $m$ travels at speed $v$ in the laboratory frame towards an identical ball at rest. After colliding head-on, the balls stick together, moving in the same direction as the first ball was moving before the collision. Calculate the mass $m^{\prime}$ and speed $v^{\prime}$ of the combined lump, justifying your answers carefully.

1.II.18C

commentLet $X$ be a random variable whose distribution depends on an unknown parameter $\theta$. Explain what is meant by a sufficient statistic $T(X)$ for $\theta$.

In the case where $X$ is discrete, with probability mass function $f(x \mid \theta)$, explain, with justification, how a sufficient statistic may be found.

Assume now that $X=\left(X_{1}, \ldots, X_{n}\right)$, where $X_{1}, \ldots, X_{n}$ are independent nonnegative random variables with common density function

$f(x \mid \theta)= \begin{cases}\lambda e^{-\lambda(x-\theta)} & \text { if } x \geqslant \theta \\ 0 & \text { otherwise }\end{cases}$

Here $\theta \geq 0$ is unknown and $\lambda$ is a known positive parameter. Find a sufficient statistic for $\theta$ and hence obtain an unbiased estimator $\hat{\theta}$ for $\theta$ of variance $(n \lambda)^{-2}$.

[You may use without proof the following facts: for independent exponential random variables $X$ and $Y$, having parameters $\lambda$ and $\mu$ respectively, $X$ has mean $\lambda^{-1}$ and variance $\lambda^{-2}$ and $\min \{X, Y\}$ has exponential distribution of parameter $\lambda+\mu$.]