Part IB, 2008, Paper 2

# Part IB, 2008, Paper 2

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2.I.3F

commentExplain what is meant by the statement that a sequence $\left(f_{n}\right)$ of functions defined on an interval $[a, b]$ converges uniformly to a function $f$. If $\left(f_{n}\right)$ converges uniformly to $f$, and each $f_{n}$ is continuous on $[a, b]$, prove that $f$ is continuous on $[a, b]$.

Now suppose additionally that $\left(x_{n}\right)$ is a sequence of points of $[a, b]$ converging to a limit $x$. Prove that $f_{n}\left(x_{n}\right) \rightarrow f(x)$.

2.II.13F

commentLet $\left(u_{n}(x): n=0,1,2, \ldots\right)$ be a sequence of real-valued functions defined on a subset $E$ of $\mathbb{R}$. Suppose that for all $n$ and all $x \in E$ we have $\left|u_{n}(x)\right| \leqslant M_{n}$, where $\sum_{n=0}^{\infty} M_{n}$ converges. Prove that $\sum_{n=0}^{\infty} u_{n}(x)$ converges uniformly on $E$.

Now let $E=\mathbb{R} \backslash \mathbb{Z}$, and consider the series $\sum_{n=0}^{\infty} u_{n}(x)$, where $u_{0}(x)=1 / x^{2}$ and

$u_{n}(x)=1 /(x-n)^{2}+1 /(x+n)^{2}$

for $n>0$. Show that the series converges uniformly on $E_{R}=\{x \in E:|x|<R\}$ for any real number $R$. Deduce that $f(x)=\sum_{n=0}^{\infty} u_{n}(x)$ is a continuous function on $E$. Does the series converge uniformly on $E$ ? Justify your answer.

2.II.14C

commentLet $f(z)=1 /\left(e^{z}-1\right)$. Find the first three terms in the Laurent expansion for $f(z)$ valid for $0<|z|<2 \pi$.

Now let $n$ be a positive integer, and define

$\begin{aligned} &f_{1}(z)=\frac{1}{z}+\sum_{r=1}^{n} \frac{2 z}{z^{2}+4 \pi^{2} r^{2}} \\ &f_{2}(z)=f(z)-f_{1}(z) \end{aligned}$

Show that the singularities of $f_{2}$ in $\{z:|z|<2(n+1) \pi\}$ are all removable. By expanding $f_{1}$ as a Laurent series valid for $|z|>2 n \pi$, and $f_{2}$ as a Taylor series valid for $|z|<2(n+1) \pi$, find the coefficients of $z^{j}$ for $-1 \leq j \leq 1$ in the Laurent series for $f$ valid for $2 n \pi<|z|<2(n+1) \pi$.

By estimating an appropriate integral around the contour $|z|=(2 n+1) \pi$, show that

$\sum_{r=1}^{\infty} \frac{1}{r^{2}}=\frac{\pi^{2}}{6}$

2.I.6B

commentGiven the electric potential of a dipole

$\phi(r, \theta)=\frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}},$

where $p$ is the magnitude of the dipole moment, calculate the corresponding electric field and show that it can be written as

$\mathbf{E}(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}\left[3\left(\mathbf{p} \cdot \hat{\mathbf{e}}_{r}\right) \hat{\mathbf{e}}_{r}-\mathbf{p}\right]$

where $\hat{\mathbf{e}}_{r}$ is the unit vector in the radial direction.

2.II.17B

commentTwo perfectly conducting rails are placed on the $x y$-plane, one coincident with the $x$-axis, starting at $(0,0)$, the other parallel to the first rail a distance $\ell$ apart, starting at $(0, \ell)$. A resistor $R$ is connected across the rails between $(0,0)$ and $(0, \ell)$, and a uniform magnetic field $\mathbf{B}=B \hat{\mathbf{e}}_{z}$, where $\hat{\mathbf{e}}_{z}$ is the unit vector along the $z$-axis and $B>0$, fills the entire region of space. A metal bar of negligible resistance and mass $m$ slides without friction on the two rails, lying perpendicular to both of them in such a way that it closes the circuit formed by the rails and the resistor. The bar moves with speed $v$ to the right such that the area of the loop becomes larger with time.

(i) Calculate the current in the resistor and indicate its direction of flow in a diagram of the system.

(ii) Show that the magnetic force on the bar is

$\mathbf{F}=-\frac{B^{2} \ell^{2} v}{R} \hat{\mathbf{e}}_{x}$

(iii) Assume that the bar starts moving with initial speed $v_{0}$ at time $t=0$, and is then left to slide freely. Using your result from part (ii) and Newton's laws show that its velocity at the time $t$ is

$v(t)=v_{0} e^{-\left(B^{2} \ell^{2} / m R\right) t} .$

(iv) By calculating the total energy delivered to the resistor, verify that energy is conserved.

2.I.8B

comment(i) Show that for a two-dimensional incompressible flow $(u(x, y), v(x, y), 0)$, the vorticity is given by $\boldsymbol{\omega} \equiv \omega_{z} \hat{\mathbf{e}}_{z}=\left(0,0,-\nabla^{2} \psi\right)$ where $\psi$ is the stream function.

(ii) Express the $z$-component of the vorticity equation

$\frac{\partial \boldsymbol{\omega}}{\partial t}+(\mathbf{u} \cdot \nabla) \boldsymbol{\omega}=(\boldsymbol{\omega} \cdot \nabla) \mathbf{u}$

in terms of the stream function $\psi$.

2.II.12G

commentShow that the area of a spherical triangle with angles $\alpha, \beta, \gamma$ is $\alpha+\beta+\gamma-\pi$. Hence derive the formula for the area of a convex spherical $n$-gon.

Deduce Euler's formula $F-E+V=2$ for a decomposition of a sphere into $F$ convex polygons with a total of $E$ edges and $V$ vertices.

A sphere is decomposed into convex polygons, comprising $m$ quadrilaterals, $n$ pentagons and $p$ hexagons, in such a way that at each vertex precisely three edges meet. Show that there are at most 7 possibilities for the pair $(m, n)$, and that at least 3 of these do occur.

2.I.2G

commentWhat does it means to say that a complex number $\alpha$ is algebraic over $\mathbb{Q}$ ? Define the minimal polynomial of $\alpha$.

Suppose that $\alpha$ satisfies a nonconstant polynomial $f \in \mathbb{Z}[X]$ which is irreducible over $\mathbb{Z}$. Show that there is an isomorphism $\mathbb{Z}[X] /(f) \cong \mathbb{Z}[\alpha]$.

[You may assume standard results about unique factorisation, including Gauss's lemma.]

2.II.11G

commentLet $F$ be a field. Prove that every ideal of the ring $F\left[X_{1}, \ldots, X_{n}\right]$ is finitely generated.

Consider the set

$R=\left\{p(X, Y)=\sum c_{i j} X^{i} Y^{j} \in F[X, Y] \mid c_{0 j}=c_{j 0}=0 \text { whenever } j>0\right\}$

Show that $R$ is a subring of $F[X, Y]$ which is not Noetherian.

2.I.1E

commentSuppose that $V$ and $W$ are finite-dimensional vector spaces over $\mathbb{R}$. What does it mean to say that $\psi: V \rightarrow W$ is a linear map? State the rank-nullity formula. Using it, or otherwise, prove that a linear map $\psi: V \rightarrow V$ is surjective if, and only if, it is injective.

Suppose that $\psi: V \rightarrow V$ is a linear map which has a right inverse, that is to say there is a linear map $\phi: V \rightarrow V$ such that $\psi \phi=\mathrm{id}_{V}$, the identity map. Show that $\phi \psi=\mathrm{id}_{V}$.

Suppose that $A$ and $B$ are two $n \times n$ matrices over $\mathbb{R}$ such that $A B=I$. Prove that $B A=I$.

2.II.10E

commentDefine the determinant $\operatorname{det}(A)$ of an $n \times n$ square matrix $A$ over the complex numbers. If $A$ and $B$ are two such matrices, show that $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$.

Write $p_{M}(\lambda)=\operatorname{det}(M-\lambda I)$ for the characteristic polynomial of a matrix $M$. Let $A, B, C$ be $n \times n$ matrices and suppose that $C$ is nonsingular. Show that $p_{B C}=p_{C B}$. Taking $C=A+t I$ for appropriate values of $t$, or otherwise, deduce that $p_{B A}=p_{A B}$.

Show that if $p_{A}=p_{B}$ then $\operatorname{tr}(A)=\operatorname{tr}(B)$. Which of the following statements is true for all $n \times n$ matrices $A, B, C$ ? Justify your answers.

(i) $p_{A B C}=p_{A C B}$;

(ii) $p_{A B C}=p_{B C A}$.

2.II.20H

commentA Markov chain with state-space $I=\mathbb{Z}^{+}$has non-zero transition probabilities $p_{00}=q_{0}$ and

$p_{i, i+1}=p_{i}, \quad p_{i+1, i}=q_{i+1} \quad(i \in I) .$

Prove that this chain is recurrent if and only if

$\sum_{n \geqslant 1} \prod_{r=1}^{n} \frac{q_{r}}{p_{r}}=\infty$

Prove that this chain is positive-recurrent if and only if

$\sum_{n \geqslant 1} \prod_{r=1}^{n} \frac{p_{r-1}}{q_{r}}<\infty$

2.II.15D

comment(a) Legendre's equation may be written in the form

$\frac{d}{d x}\left(\left(1-x^{2}\right) \frac{d y}{d x}\right)+\lambda y=0$

Show that there is a series solution for $y$ of the form

$y=\sum_{k=0}^{\infty} a_{k} x^{k},$

where the $a_{k}$ satisfy the recurrence relation

$\frac{a_{k+2}}{a_{k}}=-\frac{(\lambda-k(k+1))}{(k+1)(k+2)} .$

Hence deduce that there are solutions for $y(x)=P_{n}(x)$ that are polynomials of degree $n$, provided that $\lambda=n(n+1)$. Given that $a_{0}$ is then chosen so that $P_{n}(1)=1$, find the explicit form for $P_{2}(x)$.

(b) Laplace's equation for $\Phi(r, \theta)$ in spherical polar coordinates $(r, \theta, \phi)$ may be written in the axisymmetric case as

$\frac{\partial^{2} \Phi}{\partial r^{2}}+\frac{2}{r} \frac{\partial \Phi}{\partial r}+\frac{1}{r^{2}} \frac{\partial}{\partial x}\left(\left(1-x^{2}\right) \frac{\partial \Phi}{\partial x}\right)=0$

where $x=\cos \theta$.

Write down without proof the general form of the solution obtained by the method of separation of variables. Use it to find the form of $\Phi$ exterior to the sphere $r=a$ that satisfies the boundary conditions, $\Phi(a, x)=1+x^{2}$, and $\lim _{r \rightarrow \infty} \Phi(r, x)=0$.

2.I.4F

commentStating carefully any results on compactness which you use, show that if $X$ is a compact space, $Y$ is a Hausdorff space and $f: X \rightarrow Y$ is bijective and continuous, then $f$ is a homeomorphism.

Hence or otherwise show that the unit circle $S=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$ is homeomorphic to the quotient space $[0,1] / \sim$, where $\sim$ is the equivalence relation defined by

$x \sim y \Leftrightarrow \text { either } x=y \text { or }\{x, y\}=\{0,1\} .$

2.II.18D

comment(a) A Householder transformation (reflection) is given by

$H=I-\frac{2 u u^{T}}{\|u\|^{2}},$

where $H \in \mathbb{R}^{m \times m}, u \in \mathbb{R}^{m}$, and $I$ is the $m \times m$ unit matrix and $u$ is a non-zero vector which has norm $\|u\|=\left(\sum_{i=1}^{m} u_{i}^{2}\right)^{1 / 2}$. Show that $H$ is orthogonal.

(b) Suppose that $A \in \mathbb{R}^{m \times n}, x \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$ with $n<m$. Show that if $x$ minimises $\|A x-b\|^{2}$ then it also minimises $\|Q A x-Q b\|^{2}$, where $Q$ is an arbitrary $m \times m$ orthogonal matrix.

(c) Using Householder reflection, find the $x$ that minimises $\|A x-b\|^{2}$ when

$A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 4 \\ 0 & 2 \\ 0 & 4 \end{array}\right] \quad b=\left[\begin{array}{r} 1 \\ 1 \\ 2 \\ -1 \end{array}\right]$

2.I.9H

commentGoods from three warehouses have to be delivered to five shops, the cost of transporting one unit of good from warehouse $i$ to shop $j$ being $c_{i j}$, where

$C=\left(\begin{array}{ccccc} 2 & 3 & 6 & 6 & 4 \\ 7 & 6 & 1 & 1 & 5 \\ 3 & 6 & 6 & 2 & 1 \end{array}\right)$

The requirements of the five shops are respectively $9,6,12,5$ and 10 units of the good, and each warehouse holds a stock of 15 units. Find a minimal-cost allocation of goods from warehouses to shops and its associated cost.

2.II.16A

commentGive the physical interpretation of the expression

$\langle A\rangle_{\psi}=\int \psi(x)^{*} \hat{A} \psi(x) d x$

for an observable $A$, where $\hat{A}$ is a Hermitian operator and $\psi$ is normalised. By considering the norm of the state $(A+i \lambda B) \psi$ for two observables $A$ and $B$, and real values of $\lambda$, show that

$\left\langle A^{2}\right\rangle_{\psi}\left\langle B^{2}\right\rangle_{\psi} \geqslant \frac{1}{4}\left|\langle[A, B]\rangle_{\psi}\right|^{2} .$

Deduce the uncertainty relation

$\Delta A \Delta B \geqslant \frac{1}{2}\left|\langle[A, B]\rangle_{\psi}\right|,$

where $\Delta A$ is the uncertainty of $A$.

A particle of mass $m$ moves in one dimension under the influence of potential $\frac{1}{2} m \omega^{2} x^{2}$. By considering the commutator $[x, p]$, show that the expectation value of the Hamiltonian satisfies

$\langle H\rangle_{\psi} \geqslant \frac{1}{2} \hbar \omega .$

2.I.7C

commentA photon of energy $E$ collides with a particle of rest mass $m$, which is at rest. The final state consists of a photon and a particle of rest mass $M, M>m$. Show that the minimum value of $E$ for which it is possible for this reaction to take place is

$E_{\min }=\frac{M^{2}-m^{2}}{2 m} c^{2}$

2.II.19H

commentSuppose that the joint distribution of random variables $X, Y$ taking values in $\mathbb{Z}^{+}=\{0,1,2, \ldots\}$ is given by the joint probability generating function

$\varphi(s, t) \equiv E\left[s^{X} t^{Y}\right]=\frac{1-\alpha-\beta}{1-\alpha s-\beta t}$

where the unknown parameters $\alpha$ and $\beta$ are positive, and satisfy the inequality $\alpha+\beta<1$. Find $E(X)$. Prove that the probability mass function of $(X, Y)$ is

$f(x, y \mid \alpha, \beta)=(1-\alpha-\beta)\left(\begin{array}{c} x+y \\ x \end{array}\right) \alpha^{x} \beta^{y} \quad\left(x, y \in \mathbb{Z}^{+}\right)$

and prove that the maximum-likelihood estimators of $\alpha$ and $\beta$ based on a sample of size $n$ drawn from the distribution are

$\hat{\alpha}=\frac{\bar{X}}{1+\bar{X}+\bar{Y}}, \quad \hat{\beta}=\frac{\bar{Y}}{1+\bar{X}+\bar{Y}},$

where $\bar{X}$ (respectively, $\bar{Y}$ ) is the sample mean of $X_{1}, \ldots, X_{n}$ (respectively, $Y_{1}, \ldots, Y_{n}$ ).

By considering $\hat{\alpha}+\hat{\beta}$ or otherwise, prove that the maximum-likelihood estimator is biased. Stating clearly any results to which you appeal, prove that as $n \rightarrow \infty, \hat{\alpha} \rightarrow \alpha$, making clear the sense in which this convergence happens.