Part IB, 2007, Paper 2

# Part IB, 2007, Paper 2

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2.II.13H

commentShow that the limit of a uniformly convergent sequence of real valued continuous functions on $[0,1]$ is continuous on $[0,1]$.

Let $f_{n}$ be a sequence of continuous functions on $[0,1]$ which converge point-wise to a continuous function. Suppose also that the integrals $\int_{0}^{1} f_{n}(x) d x$ converge to $\int_{0}^{1} f(x) d x$. Must the functions $f_{n}$ converge uniformly to $f ?$ Prove or give a counterexample.

Let $f_{n}$ be a sequence of continuous functions on $[0,1]$ which converge point-wise to a function $f$. Suppose that $f$ is integrable and that the integrals $\int_{0}^{1} f_{n}(x) d x$ converge to $\int_{0}^{1} f(x) d x$. Is the limit $f$ necessarily continuous? Prove or give a counterexample.

2.II.14F

commentLet $\Omega$ be the half-strip in the complex plane,

$\Omega=\left\{z=x+i y \in \mathbb{C}:-\frac{\pi}{2}<x<\frac{\pi}{2}, \quad y>0\right\}$

Find a conformal mapping that maps $\Omega$ onto the unit disc.

2.II.17E

commentIf $S$ is a fixed surface enclosing a volume $V$, use Maxwell's equations to show that

$\frac{d}{d t} \int_{V}\left(\frac{1}{2} \epsilon_{0} E^{2}+\frac{1}{2 \mu_{0}} B^{2}\right) d V+\int_{S} \mathbf{P} \cdot \mathbf{n} d S=-\int_{V} \mathbf{j} \cdot \mathbf{E} d V$

where $\mathbf{P}=(\mathbf{E} \times \mathbf{B}) / \mu_{0}$. Give a physical interpretation of each term in this equation.

Show that Maxwell's equations for a vacuum permit plane wave solutions with $\mathbf{E}=E_{0}(0,1,0) \cos (k x-\omega t)$ with $E_{0}, k$ and $\omega$ constants, and determine the relationship between $k$ and $\omega$.

Find also the corresponding $\mathbf{B}(\mathbf{x}, t)$ and hence the time average $<\mathbf{P}>$. What does $<\mathbf{P}>$ represent in this case?

2.I.8D

commentAn incompressible, inviscid fluid occupies the region beneath the free surface $y=\eta(x, t)$ and moves with a velocity field given by the velocity potential $\phi(x, y, t)$; gravity acts in the $-y$ direction. Derive the kinematic and dynamic boundary conditions that must be satisfied by $\phi$ on $y=\eta(x, t)$.

[You may assume Bernoulli's integral of the equation of motion:

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

In the absence of waves, the fluid has uniform velocity $U$ in the $x$ direction. Derive the linearised form of the above boundary conditions for small amplitude waves, and verify that they and Laplace's equation are satisfied by the velocity potential

$\phi=U x+\operatorname{Re}\left\{b e^{k y} e^{i(k x-\omega t)}\right\}$

where $|k b| \ll U$, with a corresponding expression for $\eta$, as long as

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

What are the propagation speeds of waves with a given wave-number $k ?$

2.II.12A

comment(i) The spherical circle with centre $P \in S^{2}$ and radius $r, 0<r<\pi$, is the set of all points on the unit sphere $S^{2}$ at spherical distance $r$ from $P$. Find the circumference of a spherical circle with spherical radius $r$. Compare, for small $r$, with the formula for a Euclidean circle and comment on the result.

(ii) The cross ratio of four distinct points $z_{i}$ in $\mathbf{C}$ is

$\frac{\left(z_{4}-z_{1}\right)\left(z_{2}-z_{3}\right)}{\left(z_{4}-z_{3}\right)\left(z_{2}-z_{1}\right)} .$

Show that the cross-ratio is a real number if and only if $z_{1}, z_{2}, z_{3}, z_{4}$ lie on a circle or a line.

[You may assume that Möbius transformations preserve the cross-ratio.]

2.I.2G

commentDefine the term Euclidean domain.

Show that the ring of integers $\mathbb{Z}$ is a Euclidean domain.

2.II.11G

comment(i) Give an example of a Noetherian ring and of a ring that is not Noetherian. Justify your answers.

(ii) State and prove Hilbert's basis theorem.

2.I.1G

commentSuppose that $S, T$ are endomorphisms of the 3-dimensional complex vector space $\mathbb{C}^{3}$ and that the eigenvalues of each of them are $1,2,3$. What are their characteristic and minimal polynomials? Are they conjugate?

2.II.10G

commentSuppose that $P$ is the complex vector space of complex polynomials in one variable, $z$.

(i) Show that the form $\langle$, $\rangle$ defined by

$\langle f, g\rangle=\frac{1}{2 \pi} \int_{0}^{2 \pi} f\left(e^{i \theta}\right) \cdot \overline{g\left(e^{i \theta}\right)} d \theta$

is a positive definite Hermitian form on $P$.

(ii) Find an orthonormal basis of $P$ for this form, in terms of the powers of $z$.

(iii) Generalize this construction to complex vector spaces of complex polynomials in any finite number of variables.

2.II.20C

commentConsider a Markov chain with state space $S=\{0,1,2, \ldots\}$ and transition matrix given by

$P_{i, j}= \begin{cases}q p^{j-i+1} & \text { for } i \geqslant 1 \text { and } j \geqslant i-1 \\ q p^{j} & \text { for } i=0 \text { and } j \geqslant 0\end{cases}$

and $P_{i, j}=0$ otherwise, where $0<p=1-q<1$.

For each value of $p, 0<p<1$, determine whether the chain is transient, null recurrent or positive recurrent, and in the last case find the invariant distribution.

2.II.15D

commentLet $y_{0}(x)$ be a non-zero solution of the Sturm-Liouville equation

$L\left(y_{0} ; \lambda_{0}\right) \equiv \frac{d}{d x}\left(p(x) \frac{d y_{0}}{d x}\right)+\left(q(x)+\lambda_{0} w(x)\right) y_{0}=0$

with boundary conditions $y_{0}(0)=y_{0}(1)=0$. Show that, if $y(x)$ and $f(x)$ are related by

$L\left(y ; \lambda_{0}\right)=f$

with $y(x)$ satisfying the same boundary conditions as $y_{0}(x)$, then

$\int_{0}^{1} y_{0} f d x=0$

Suppose that $y_{0}$ is normalised so that

$\int_{0}^{1} w y_{0}^{2} d x=1$

and consider the problem

$L(y ; \lambda)=y^{3} ; \quad y(0)=y(1)=0$

By choosing $f$ appropriately in $(*)$ deduce that, if

$\lambda-\lambda_{0}=\epsilon^{2} \mu[\mu=O(1), \epsilon \ll 1], \quad \text { and } \quad y(x)=\epsilon y_{0}(x)+\epsilon^{2} y_{1}(x)$

then

$\mu=\int_{0}^{1} y_{0}^{4} d x+O(\epsilon)$

2.I.4A

commentAre the following statements true or false? Give a proof or a counterexample as appropriate.

(i) If $f: X \rightarrow Y$ is a continuous map of topological spaces and $S \subseteq X$ is compact then $f(S)$ is compact.

(ii) If $f: X \rightarrow Y$ is a continuous map of topological spaces and $K \subseteq Y$ is compact then $\left.f^{-1}(K)=\{x \in X: f(x) \in K\}\right\}$ is compact.

(iii) If a metric space $M$ is complete and a metric space $T$ is homeomorphic to $M$ then $T$ is complete.

2.II.18F

commentFor a symmetric, positive definite matrix $A$ with the spectral radius $\rho(A)$, the linear system $A x=b$ is solved by the iterative procedure

$x^{(k+1)}=x^{(k)}-\tau\left(A x^{(k)}-b\right), \quad k \geq 0$

where $\tau$ is a real parameter. Find the range of $\tau$ that guarantees convergence of $x^{(k)}$ to the exact solution for any choice of $x^{(0)}$.

2.I.9C

commentConsider the game with payoff matrix

$\left(\begin{array}{lll} 2 & 5 & 4 \\ 3 & 2 & 2 \\ 2 & 1 & 3 \end{array}\right)$

where the $(i, j)$ entry is the payoff to the row player if the row player chooses row $i$ and the column player chooses column $j$.

Find the value of the game and the optimal strategies for each player.

2.II.16B

commentWrite down the angular momentum operators $L_{1}, L_{2}, L_{3}$ in terms of the position and momentum operators, $\mathbf{x}$ and $\mathbf{p}$, and the commutation relations satisfied by $\mathbf{x}$ and $\mathbf{p}$.

Verify the commutation relations

$\left[L_{i}, L_{j}\right]=i \hbar \epsilon_{i j k} L_{k}$

Further, show that

$\left[L_{i}, p_{j}\right]=i \hbar \epsilon_{i j k} p_{k}$

A wave-function $\Psi_{0}(r)$ is spherically symmetric. Verify that

$\mathbf{L} \Psi_{0}(r)=0$

Consider the vector function $\boldsymbol{\Phi}=\nabla \Psi_{0}(r)$. Show that $\Phi_{3}$ and $\Phi_{1} \pm i \Phi_{2}$ are eigenfunctions of $L_{3}$ with eigenvalues $0, \pm \hbar$ respectively.

2.I.7B

commentA particle in inertial frame $S$ has coordinates $(t, x)$, whilst the coordinates are $\left(t^{\prime}, x^{\prime}\right)$ in frame $S^{\prime}$, which moves with relative velocity $v$ in the $x$ direction. What is the relationship between the coordinates of $S$ and $S^{\prime}$ ?

Frame $S^{\prime \prime}$, with cooordinates $\left(t^{\prime \prime}, x^{\prime \prime}\right)$, moves with velocity $u$ with respect to $S^{\prime}$ and velocity $V$ with respect to $S$. Derive the relativistic formula for $V$ in terms of $u$ and $v$. Show how the Newtonian limit is recovered.

2.II.19C

commentState and prove the Neyman-Pearson lemma.

Suppose that $X$ is a random variable drawn from the probability density function

$f(x \mid \theta)=\frac{1}{2}|x|^{\theta-1} e^{-|x|} / \Gamma(\theta), \quad-\infty<x<\infty$

where $\Gamma(\theta)=\int_{0}^{\infty} y^{\theta-1} e^{-y} d y$ and $\theta \geqslant 1$ is unknown. Find the most powerful test of size $\alpha$, $0<\alpha<1$, of the hypothesis $H_{0}: \theta=1$ against the alternative $H_{1}: \theta=2$. Express the power of the test as a function of $\alpha$.

Is your test uniformly most powerful for testing $H_{0}: \theta=1$ against $H_{1}: \theta>1 ?$ Explain your answer carefully.