Part IB, 2007, Paper 3

# Part IB, 2007, Paper 3

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3.II.14H

commentSay that a function on the complex plane $\mathbb{C}$ is periodic if $f(z+1)=f(z)$ and $f(z+i)=f(z)$ for all $z$. If $f$ is a periodic analytic function, show that $f$ is constant.

If $f$ is a meromorphic periodic function, show that the number of zeros of $f$ in the square $[0,1) \times[0,1)$ is equal to the number of poles, both counted with multiplicities.

Define

$f(z)=\frac{1}{z^{2}}+\sum_{w}\left[\frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right]$

where the sum runs over all $w=a+b i$ with $a$ and $b$ integers, not both 0 . Show that this series converges to a meromorphic periodic function on the complex plane.

3.II.17E

commentA capacitor consists of three long concentric cylinders of radii $a, \lambda a$ and $2 a$ respectively, where $1<\lambda<2$. The inner and outer cylinders are earthed (i.e. held at zero potential); the cylinder of radius $\lambda a$ is raised to a potential $V$. Find the electrostatic potential in the regions between the cylinders and deduce the capacitance, $C(\lambda)$ per unit length, of the system.

For $\lambda=1+\delta$ with $0<\delta \ll 1$ find $C(\lambda)$ correct to leading order in $\delta$ and comment on your result.

Find also the value of $\lambda$ at which $C(\lambda)$ has an extremum. Is the extremum a maximum or a minimum? Justify your answer.

3.I $2 \mathrm{~A} \quad$

commentLet $l$ be a line in the Euclidean plane $\mathbf{R}^{2}$ and $P$ a point on $l$. Denote by $\rho$ the reflection in $l$ and by $\tau$ the rotation through an angle $\alpha$ about $P$. Describe, in terms of $l, P$, and $\alpha$, a line fixed by the composition $\tau \rho$ and show that $\tau \rho$ is a reflection.

3.II.12A

commentFor a parameterized smooth embedded surface $\sigma: V \rightarrow U \subset \mathbf{R}^{3}$, where $V$ is an open domain in $\mathbf{R}^{2}$, define the first fundamental form, the second fundamental form, and the Gaussian curvature $K$. State the Gauss-Bonnet formula for a compact embedded surface $S \subset \mathbf{R}^{3}$ having Euler number $e(S)$.

Let $S$ denote a surface defined by rotating a curve

$\eta(u)=(r+a \sin u, 0, b \cos u) \quad 0 \leq u \leq 2 \pi$

about the $z$-axis. Here $a, b, r$ are positive constants, such that $a^{2}+b^{2}=1$ and $a<r$. By considering a smooth parameterization, find the first fundamental form and the second fundamental form of $S$.

3.I.1G

commentWhat are the orders of the groups $G L_{2}\left(\mathbb{F}_{p}\right)$ and $S L_{2}\left(\mathbb{F}_{p}\right)$ where $\mathbb{F}_{p}$ is the field of $p$ elements?

3.II.11G

comment(i) State the Sylow theorems for Sylow $p$-subgroups of a finite group.

(ii) Write down one Sylow 3-subgroup of the symmetric group $S_{5}$ on 5 letters. Calculate the number of Sylow 3-subgroups of $S_{5}$.

3.II.10G

comment(i) Define the terms row-rank, column-rank and rank of a matrix, and state a relation between them.

(ii) Fix positive integers $m, n, p$ with $m, n \geqslant p$. Suppose that $A$ is an $m \times p$ matrix and $B$ a $p \times n$ matrix. State and prove the best possible upper bound on the rank of the product $A B$.

3.I.9C

commentConsider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ with state space $S=\{0,1\}$ and transition matrix

$P=\left(\begin{array}{cc} \alpha & 1-\alpha \\ 1-\beta & \beta \end{array}\right)$

where $0<\alpha<1$ and $0<\beta<1$.

Calculate $\mathbb{P}\left(X_{n}=0 \mid X_{0}=0\right)$ for each $n \geqslant 0$.

3.II.15E

commentLegendre's equation may be written

$\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+n(n+1) y=0 \quad \text { with } \quad y(1)=1$

Show that if $n$ is a positive integer, this equation has a solution $y=P_{n}(x)$ that is a polynomial of degree $n$. Find $P_{0}, P_{1}$ and $P_{2}$ explicitly.

Write down a general separable solution of Laplace's equation, $\nabla^{2} \phi=0$, in spherical polar coordinates $(r, \theta)$. (A derivation of this result is not required.)

Hence or otherwise find $\phi$ when

$\nabla^{2} \phi=0, \quad a<r<b$

with $\phi=\sin ^{2} \theta$ both when $r=a$ and when $r=b$.

3.I.4A

comment(a) Let $X$ be a connected topological space such that each point $x$ of $X$ has a neighbourhood homeomorphic to $\mathbb{R}^{n}$. Prove that $X$ is path-connected.

(b) Let $\tau$ denote the topology on $\mathbb{N}=\{1,2, \ldots\}$, such that the open sets are $\mathbb{N}$, the empty set, and all the sets $\{1,2, \ldots, n\}$, for $n \in \mathbb{N}$. Prove that any continuous map from the topological space $(\mathbb{N}, \tau)$ to the Euclidean $\mathbb{R}$ is constant.

3.II.19F

commentProve that the monic polynomials $Q_{n}, n \geq 0$, orthogonal with respect to a given weight function $w(x)>0$ on $[a, b]$, satisfy the three-term recurrence relation

$Q_{n+1}(x)=\left(x-a_{n}\right) Q_{n}(x)-b_{n} Q_{n-1}(x), \quad n \geq 0$

where $Q_{-1}(x) \equiv 0, Q_{0}(x) \equiv 1$. Express the values $a_{n}$ and $b_{n}$ in terms of $Q_{n}$ and $Q_{n-1}$ and show that $b_{n}>0$.

3.I.7B

commentThe quantum mechanical harmonic oscillator has Hamiltonian

$H=\frac{1}{2 m} p^{2}+\frac{1}{2} m \omega^{2} x^{2}$

and is in a stationary state of energy $<H>=E$. Show that

$E \geqslant \frac{1}{2 m}(\Delta p)^{2}+\frac{1}{2} m \omega^{2}(\Delta x)^{2},$

where $(\Delta p)^{2}=\left\langle p^{2}\right\rangle-\langle p\rangle^{2}$ and $(\Delta x)^{2}=\left\langle x^{2}\right\rangle-\langle x\rangle^{2}$. Use the Heisenberg Uncertainty Principle to show that

$E \geqslant \frac{1}{2} \hbar \omega$

3.II.16B

commentA quantum system has a complete set of orthonormal eigenstates, $\psi_{n}(x)$, with nondegenerate energy eigenvalues, $E_{n}$, where $n=1,2,3 \ldots$ Write down the wave-function, $\Psi(x, t), t \geqslant 0$ in terms of the eigenstates.

A linear operator acts on the system such that

$\begin{aligned} &A \psi_{1}=2 \psi_{1}-\psi_{2} \\ &A \psi_{2}=2 \psi_{2}-\psi_{1} \\ &A \psi_{n}=0, n \geqslant 3 \end{aligned}$

Find the eigenvalues of $A$ and obtain a complete set of normalised eigenfunctions, $\phi_{n}$, of $A$ in terms of the $\psi_{n}$.

At time $t=0$ a measurement is made and it is found that the observable corresponding to $A$ has value 3. After time $t, A$ is measured again. What is the probability that the value is found to be 1 ?

3.I.8C

commentLight bulbs are sold in packets of 3 but some of the bulbs are defective. A sample of 256 packets yields the following figures for the number of defectives in a packet:

\begin{tabular}{l|cccc} No. of defectives & 0 & 1 & 2 & 3 \ \hline No. of packets & 116 & 94 & 40 & 6 \end{tabular}

Test the hypothesis that each bulb has a constant (but unknown) probability $\theta$ of being defective independently of all other bulbs.

[Hint: You may wish to use some of the following percentage points:

$\left.\begin{array}{c|ccccccccc}\text { Distribution } & \chi_{1}^{2} & \chi_{2}^{2} & \chi_{3}^{2} & \chi_{4}^{2} & t_{1} & t_{2} & t_{3} & t_{4} \\ \hline 90 \% \text { percentile } & 2 \cdot 71 & 4 \cdot 61 & 6.25 & 7 \cdot 78 & 3 \cdot 08 & 1.89 & 1 \cdot 64 & 1.53 \\ 95 \% \text { percentile } & 3.84 & 5.99 & 7 \cdot 81 & 9 \cdot 49 & 6 \cdot 31 & 2.92 & 2 \cdot 35 & 2 \cdot 13\end{array}\right]$