Part IB, 2004, Paper 1

# Part IB, 2004, Paper 1

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1.I.4G

commentDefine what it means for a sequence of functions $F_{n}:(0,1) \rightarrow \mathbb{R}$, where $n=1,2, \ldots$, to converge uniformly to a function $F$.

For each of the following sequences of functions on $(0,1)$, find the pointwise limit function. Which of these sequences converge uniformly? Justify your answers.

(i) $F_{n}(x)=\frac{1}{n} e^{x}$

(ii) $F_{n}(x)=e^{-n x^{2}}$

(iii) $F_{n}(x)=\sum_{i=0}^{n} x^{i}$

1.II.15G

commentState the axioms for a norm on a vector space. Show that the usual Euclidean norm on $\mathbb{R}^{n}$,

$\|x\|=\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)^{1 / 2}$

satisfies these axioms.

Let $U$ be any bounded convex open subset of $\mathbb{R}^{n}$ that contains 0 and such that if $x \in U$ then $-x \in U$. Show that there is a norm on $\mathbb{R}^{n}$, satisfying the axioms, for which $U$ is the set of points in $\mathbb{R}^{n}$ of norm less than 1 .

1.I.5A

commentDetermine the poles of the following functions and calculate their residues there. (i) $\frac{1}{z^{2}+z^{4}}$, (ii) $\frac{e^{1 / z^{2}}}{z-1}$, (iii) $\frac{1}{\sin \left(e^{z}\right)}$.

1.II.16A

commentLet $p$ and $q$ be two polynomials such that

$q(z)=\prod_{l=1}^{m}\left(z-\alpha_{l}\right)$

where $\alpha_{1}, \ldots, \alpha_{m}$ are distinct non-real complex numbers and $\operatorname{deg} p \leqslant m-1$. Using contour integration, determine

$\int_{-\infty}^{\infty} \frac{p(x)}{q(x)} e^{i x} d x$

carefully justifying all steps.

1.I.7B

commentWrite down Maxwell's equations and show that they imply the conservation of charge.

In a conducting medium of conductivity $\sigma$, where $\mathbf{J}=\sigma \mathbf{E}$, show that any charge density decays in time exponentially at a rate to be determined.

1.II.18B

commentInside a volume $D$ there is an electrostatic charge density $\rho(\mathbf{r})$, which induces an electric field $\mathbf{E}(\mathbf{r})$ with associated electrostatic potential $\phi(\mathbf{r})$. The potential vanishes on the boundary of $D$. The electrostatic energy is

$W=\frac{1}{2} \int_{D} \rho \phi d^{3} \mathbf{r}$

Derive the alternative form

$W=\frac{\epsilon_{0}}{2} \int_{D} E^{2} d^{3} \mathbf{r}$

A capacitor consists of three identical and parallel thin metal circular plates of area $A$ positioned in the planes $z=-H, z=a$ and $z=H$, with $-H<a<H$, with centres on the $z$ axis, and at potentials $0, V$ and 0 respectively. Find the electrostatic energy stored, verifying that expressions (1) and (2) give the same results. Why is the energy minimal when $a=0$ ?

1.I.9C

commentFrom the general mass-conservation equation, show that the velocity field $\mathbf{u}(\mathbf{x})$ of an incompressible fluid is solenoidal, i.e. that $\nabla \cdot \mathbf{u}=0$.

Verify that the two-dimensional flow

$\mathbf{u}=\left(\frac{y}{x^{2}+y^{2}}, \frac{-x}{x^{2}+y^{2}}\right)$

is solenoidal and find a streamfunction $\psi(x, y)$ such that $\mathbf{u}=(\partial \psi / \partial y,-\partial \psi / \partial x)$.

1.II.20C

commentA layer of water of depth $h$ flows along a wide channel with uniform velocity $(U, 0)$, in Cartesian coordinates $(x, y)$, with $x$ measured downstream. The bottom of the channel is at $y=-h$, and the free surface of the water is at $y=0$. Waves are generated on the free surface so that it has the new position $y=\eta(x, t)=a e^{i(\omega t-k x)}$.

Write down the equation and the full nonlinear boundary conditions for the velocity potential $\phi$ (for the perturbation velocity) and the motion of the free surface.

By linearizing these equations about the state of uniform flow, show that

where $g$ is the acceleration due to gravity.

Hence, determine the dispersion relation for small-amplitude surface waves

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

1.I.3G

commentUsing the Riemannian metric

$d s^{2}=\frac{d x^{2}+d y^{2}}{y^{2}}$

define the length of a curve and the area of a region in the upper half-plane $H=\{x+i y: y>0\}$.

Find the hyperbolic area of the region $\{(x, y) \in H: 0<x<1, y>1\}$.

1.II.14G

commentShow that for every hyperbolic line $L$ in the hyperbolic plane $H$ there is an isometry of $H$ which is the identity on $L$ but not on all of $H$. Call it the reflection $R_{L}$.

Show that every isometry of $H$ is a composition of reflections.

1.II.13F

commentState the structure theorem for finitely generated abelian groups. Prove that a finitely generated abelian group $A$ is finite if and only if there exists a prime $p$ such that $A / p A=0$.

Show that there exist abelian groups $A \neq 0$ such that $A / p A=0$ for all primes $p$. Prove directly that your example of such an $A$ is not finitely generated.

1.I.1H

commentSuppose that $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r+1}\right\}$ is a linearly independent set of distinct elements of a vector space $V$ and $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r}, \mathbf{f}_{r+1}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$. Prove that $\mathbf{f}_{r+1}, \ldots, \mathbf{f}_{m}$ may be reordered, as necessary, so that $\left\{\mathbf{e}_{1}, \ldots \mathbf{e}_{r+1}, \mathbf{f}_{r+2}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$.

Suppose that $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}\right\}$ is a linearly independent set of distinct elements of $V$ and that $\left\{\mathbf{f}_{1}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$. Show that $n \leqslant m$.

1.II.12H

commentLet $U$ and $W$ be subspaces of the finite-dimensional vector space $V$. Prove that both the sum $U+W$ and the intersection $U \cap W$ are subspaces of $V$. Prove further that

$\operatorname{dim} U+\operatorname{dim} W=\operatorname{dim}(U+W)+\operatorname{dim}(U \cap W)$

Let $U, W$ be the kernels of the maps $A, B: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2}$ given by the matrices $A$ and $B$ respectively, where

$A=\left(\begin{array}{rrrr} 1 & 2 & -1 & -3 \\ -1 & 1 & 2 & -4 \end{array}\right), \quad B=\left(\begin{array}{rrrr} 1 & -1 & 2 & 0 \\ 0 & 1 & 2 & -4 \end{array}\right)$

Find a basis for the intersection $U \cap W$, and extend this first to a basis of $U$, and then to a basis of $U+W$.

1.I.11H

commentLet $P=\left(P_{i j}\right)$ be a transition matrix. What does it mean to say that $P$ is (a) irreducible, $(b)$ recurrent?

Suppose that $P$ is irreducible and recurrent and that the state space contains at least two states. Define a new transition matrix $\tilde{P}$ by

$\tilde{P}_{i j}=\left\{\begin{array}{lll} 0 & \text { if } & i=j \\ \left(1-P_{i i}\right)^{-1} P_{i j} & \text { if } & i \neq j \end{array}\right.$

Prove that $\tilde{P}$ is also irreducible and recurrent.

1.II.22H

commentConsider the Markov chain with state space $\{1,2,3,4,5,6\}$ and transition matrix

$\left(\begin{array}{cccccc} 0 & 0 & \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & 0 \\ \frac{1}{3} & 0 & \frac{1}{3} & 0 & 0 & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ \frac{1}{4} & 0 & \frac{1}{2} & 0 & 0 & \frac{1}{4} \end{array}\right) \text {. }$

Determine the communicating classes of the chain, and for each class indicate whether it is open or closed.

Suppose that the chain starts in state 2 ; determine the probability that it ever reaches state 6 .

Suppose that the chain starts in state 3 ; determine the probability that it is in state 6 after exactly $n$ transitions, $n \geqslant 1$.

1.I.6B

commentWrite down the general isotropic tensors of rank 2 and 3 .

According to a theory of magnetostriction, the mechanical stress described by a second-rank symmetric tensor $\sigma_{i j}$ is induced by the magnetic field vector $B_{i}$. The stress is linear in the magnetic field,

$\sigma_{i j}=A_{i j k} B_{k},$

where $A_{i j k}$ is a third-rank tensor which depends only on the material. Show that $\sigma_{i j}$ can be non-zero only in anisotropic materials.

1.II.17B

commentThe equation governing small amplitude waves on a string can be written as

$\frac{\partial^{2} y}{\partial t^{2}}=\frac{\partial^{2} y}{\partial x^{2}}$

The end points $x=0$ and $x=1$ are fixed at $y=0$. At $t=0$, the string is held stationary in the waveform,

$y(x, 0)=x(1-x) \quad \text { in } \quad 0 \leq x \leq 1 .$

The string is then released. Find $y(x, t)$ in the subsequent motion.

Given that the energy

$\int_{0}^{1}\left[\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right] d x$

is constant in time, show that

$\sum_{\substack{n \text { odd } \\ n \geqslant 1}} \frac{1}{n^{4}}=\frac{\pi^{4}}{96}$

1.I.8D

commentFrom the time-dependent Schrödinger equation for $\psi(x, t)$, derive the equation

$\frac{\partial \rho}{\partial t}+\frac{\partial j}{\partial x}=0$

for $\rho(x, t)=\psi^{*}(x, t) \psi(x, t)$ and some suitable $j(x, t)$.

Show that $\psi(x, t)=e^{i(k x-\omega t)}$ is a solution of the time-dependent Schrödinger equation with zero potential for suitable $\omega(k)$ and calculate $\rho$ and $j$. What is the interpretation of this solution?

1.II.19D

commentThe angular momentum operators are $\mathbf{L}=\left(L_{1}, L_{2}, L_{3}\right)$. Write down their commutation relations and show that $\left[L_{i}, \mathbf{L}^{2}\right]=0$. Let

$L_{\pm}=L_{1} \pm i L_{2},$

and show that

$\mathbf{L}^{2}=L_{-} L_{+}+L_{3}^{2}+\hbar L_{3} .$

Verify that $\mathbf{L} f(r)=0$, where $r^{2}=x_{i} x_{i}$, for any function $f$. Show that

$L_{3}\left(x_{1}+i x_{2}\right)^{n} f(r)=n \hbar\left(x_{1}+i x_{2}\right)^{n} f(r), \quad L_{+}\left(x_{1}+i x_{2}\right)^{n} f(r)=0$

for any integer $n$. Show that $\left(x_{1}+i x_{2}\right)^{n} f(r)$ is an eigenfunction of $\mathbf{L}^{2}$ and determine its eigenvalue. Why must $L_{-}\left(x_{1}+i x_{2}\right)^{n} f(r)$ be an eigenfunction of $\mathbf{L}^{2}$ ? What is its eigenvalue?

1.I.10H

commentUse the generalized likelihood-ratio test to derive Student's $t$-test for the equality of the means of two populations. You should explain carefully the assumptions underlying the test.

1.II.21H

commentState and prove the Rao-Blackwell Theorem.

Suppose that $X_{1}, X_{2}, \ldots, X_{n}$ are independent, identically-distributed random variables with distribution

$P\left(X_{1}=r\right)=p^{r-1}(1-p), \quad r=1,2, \ldots$

where $p, 0<p<1$, is an unknown parameter. Determine a one-dimensional sufficient statistic, $T$, for $p$.

By first finding a simple unbiased estimate for $p$, or otherwise, determine an unbiased estimate for $p$ which is a function of $T$.