Part IB, 2007, Paper 4

Analysis II

• 4.I.3H

  Analysis II | Part IB, 2007

  Define uniform convergence for a sequence $f_{1}, f_{2}, \ldots$ of real-valued functions on the interval $(0,1)$.

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

  (i) $f_{n}(x)=\log \left(x+\frac{1}{n}\right)$,

  (ii) $f_{n}(x)=\cos \left(\frac{x}{n}\right)$.

  Justify your answers.

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• 4.II.13H

  Analysis II | Part IB, 2007

  State and prove the Contraction Mapping Theorem.

  Find numbers $a$ and $b$, with $a<0<b$, such that the mapping $T: C[a, b] \rightarrow C[a, b]$ defined by

  $$T(f)(x)=1+\int_{0}^{x} 3 t f(t) d t$$

  is a contraction, in the sup norm on $C[a, b]$. Deduce that the differential equation

  $$\frac{d y}{d x}=3 x y, \quad \text { with } y=1 \text { when } x=0,$$

  has a unique solution in some interval containing 0 .

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Complex Analysis

• 4.I.4H

  Complex Analysis | Part IB, 2007

  State the argument principle.

  Show that if $f$ is an analytic function on an open set $U \subset \mathbb{C}$ which is one-to-one, then $f^{\prime}(z) \neq 0$ for all $z \in U$.

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Complex Methods

• 4.II.15F

  Complex Methods | Part IB, 2007

  (i) Use the definition of the Laplace transform of $f(t)$ :

  $$L\{f(t)\}=F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$

  to show that, for $f(t)=t^{n}$,

  $$L\{f(t)\}=F(s)=\frac{n !}{s^{n+1}}, \quad L\left\{e^{a t} f(t)\right\}=F(s-a)=\frac{n !}{(s-a)^{n+1}}$$

  (ii) Use contour integration to find the inverse Laplace transform of

  $$F(s)=\frac{1}{s^{2}(s+1)^{2}}$$

  (iii) Verify the result in (ii) by using the results in (i) and the convolution theorem.

  (iv) Use Laplace transforms to solve the differential equation

  $$f^{(i v)}(t)+2 f^{\prime \prime \prime}(t)+f^{\prime \prime}(t)=0$$

  subject to the initial conditions

  $$f(0)=f^{\prime}(0)=f^{\prime \prime}(0)=0, \quad f^{\prime \prime \prime}(0)=1$$

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Electromagnetism

• 4.I $7 \mathrm{E} \quad$

  Electromagnetism | Part IB, 2007

  Write down Faraday's law of electromagnetic induction for a moving circuit $C(t)$ in a magnetic field $\mathbf{B}(\mathbf{x}, t)$. Explain carefully the meaning of each term in the equation.

  A thin wire is bent into a circular loop of radius $a$. The loop lies in the $(x, z)$-plane at time $t=0$. It spins steadily with angular velocity $\Omega \mathbf{k}$, where $\Omega$ is a constant and $\mathbf{k}$ is a unit vector in the $z$-direction. A spatially uniform magnetic field $\mathbf{B}=B_{0}(\cos \omega t, \sin \omega t, 0)$ is applied, with $B_{0}$ and $\omega$ both constant. If the resistance of the wire is $R$, find the current in the wire at time $t$.

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Fluid Dynamics

• 4.II.18D

  Fluid Dynamics | Part IB, 2007

  Starting from Euler's equation for an inviscid, incompressible fluid in the absence of body forces,

  $$\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u} . \nabla) \mathbf{u}=-\frac{1}{\rho} \nabla p$$

  derive the equation for the vorticity $\boldsymbol{\omega}=\nabla_{\wedge} \mathbf{u}$.

  [You may assume that $\left.\nabla_{\wedge}\left(\mathbf{a}_{\wedge} \mathbf{b}\right)=\mathbf{a} \nabla \cdot \mathbf{b}-\mathbf{b} \nabla \cdot \mathbf{a}+(\mathbf{b} . \nabla) \mathbf{a}-(\mathbf{a} . \nabla) \mathbf{b} .\right]$

  Show that, in a two-dimensional flow, vortex lines keep their strength and move with the fluid.

  Show that a two-dimensional flow driven by a line vortex of circulation $\Gamma$ at distance $b$ from a rigid plane wall is the same as if the wall were replaced by another vortex of circulation $-\Gamma$ at the image point, distance $b$ from the wall on the other side. Deduce that the first vortex will move at speed $\Gamma / 4 \pi b$ parallel to the wall.

  A line vortex of circulation $\Gamma$ moves in a quarter-plane, bounded by rigid plane walls at $x=0, y>0$ and $y=0, x>0$. Show that the vortex follows a trajectory whose equation in plane polar coordinates is $r \sin 2 \theta=$ constant.

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Geometry

• 4.II.12A

  Geometry | Part IB, 2007

  Write down the Riemannian metric for the upper half-plane model $\mathbf{H}$ of the hyperbolic plane. Describe, without proof, the group of isometries of $\mathbf{H}$ and the hyperbolic lines (i.e. the geodesics) on $\mathbf{H}$.

  Show that for any two hyperbolic lines $\ell_{1}, \ell_{2}$, there is an isometry of $\mathbf{H}$ which maps $\ell_{1}$ onto $\ell_{2}$.

  Suppose that $g$ is a composition of two reflections in hyperbolic lines which are ultraparallel (i.e. do not meet either in the hyperbolic plane or at its boundary). Show that $g$ cannot be an element of finite order in the group of isometries of $\mathbf{H}$.

  [Existence of a common perpendicular to two ultraparallel hyperbolic lines may be assumed. You might like to choose carefully which hyperbolic line to consider as a common perpendicular.]

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Groups, Rings and Modules

• 4.I.2G

  Groups, Rings and Modules | Part IB, 2007

  If $p$ is a prime, how many abelian groups of order $p^{4}$ are there, up to isomorphism?

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• 4.II.11G

  Groups, Rings and Modules | Part IB, 2007

  A regular icosahedron has 20 faces, 12 vertices and 30 edges. The group $G$ of its rotations acts transitively on the set of faces, on the set of vertices and on the set of edges.

  (i) List the conjugacy classes in $G$ and give the size of each.

  (ii) Find the order of $G$ and list its normal subgroups.

  [A normal subgroup of $G$ is a union of conjugacy classes in $G$.]

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Linear Algebra

• 4.I.1G

  Linear Algebra | Part IB, 2007

  Suppose that $\alpha: V \rightarrow W$ is a linear map of finite-dimensional complex vector spaces. What is the dual map $\alpha^{*}$ of the dual vector spaces?

  Suppose that we choose bases of $V, W$ and take the corresponding dual bases of the dual vector spaces. What is the relation between the matrices that represent $\alpha$ and $\alpha^{*}$ with respect to these bases? Justify your answer.

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• 4.II.10G

  Linear Algebra | Part IB, 2007

  (i) State and prove the Cayley-Hamilton theorem for square complex matrices.

  (ii) A square matrix $A$ is of order $n$ for a strictly positive integer $n$ if $A^{n}=I$ and no smaller positive power of $A$ is equal to $I$.

  Determine the order of a complex $2 \times 2$ matrix $A$ of trace zero and determinant 1 .

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Markov Chains

• 4.I.9C

  Markov Chains | Part IB, 2007

  For a Markov chain with state space $S$, define what is meant by the following:

  (i) states $i, j \in S$ communicate;

  (ii) state $i \in S$ is recurrent.

  Prove that communication is an equivalence relation on $S$ and that if two states $i, j$ communicate and $i$ is recurrent then $j$ is recurrent.

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Methods

• 4.I.5B

  Methods | Part IB, 2007

  Show that the general solution of the wave equation

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

  where $c$ is a constant, is

  $$y=f(x+c t)+g(x-c t),$$

  where $f$ and $g$ are twice differentiable functions. Briefly discuss the physical interpretation of this solution.

  Calculate $y(x, t)$ subject to the initial conditions

  $$y(x, 0)=0 \quad \text { and } \quad \frac{\partial y}{\partial t}(x, 0)=\psi(x)$$

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• 4.II.16E

  Methods | Part IB, 2007

  Write down the Euler-Lagrange equation for extrema of the functional

  $$I=\int_{a}^{b} F\left(y, y^{\prime}\right) d x$$

  Show that a first integral of this equation is given by

  $$F-y^{\prime} \frac{\partial F}{\partial y^{\prime}}=C$$

  A road is built between two points $A$ and $B$ in the plane $z=0$ whose polar coordinates are $r=a, \theta=0$ and $r=a, \theta=\pi / 2$ respectively. Owing to congestion, the traffic speed at points along the road is $k r^{2}$ with $k$ a positive constant. If the equation describing the road is $r=r(\theta)$, obtain an integral expression for the total travel time $T$ from $A$ to $B$.

  [Arc length in polar coordinates is given by $d s^{2}=d r^{2}+r^{2} d \theta^{2}$.]

  Calculate $T$ for the circular road $r=a$.

  Find the equation for the road that minimises $T$ and determine this minimum value.

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Metric and Topological Spaces

• 4.II.14A

  Metric and Topological Spaces | Part IB, 2007

  (a) For a subset $A$ of a topological space $X$, define the closure cl $(A)$ of $A$. Let $f: X \rightarrow Y$ be a map to a topological space $Y$. Prove that $f$ is continuous if and only if $f(c l(A)) \subseteq c l(f(A))$, for each $A \subseteq X$.

  (b) Let $M$ be a metric space. A subset $S$ of $M$ is called dense in $M$ if the closure of $S$ is equal to $M$.

  Prove that if a metric space $M$ is compact then it has a countable subset which is dense in $M$.

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Numerical Analysis

• 4.I.8F

  Numerical Analysis | Part IB, 2007

  Given $f \in C^{3}[0,2]$, we approximate $f^{\prime}(0)$ by the linear combination

  $$\mu(f)=-\frac{3}{2} f(0)+2 f(1)-\frac{1}{2} f(2)$$

  Using the Peano kernel theorem, determine the least constant $c$ in the inequality

  $$\left|f^{\prime}(0)-\mu(f)\right| \leq c\left\|f^{\prime \prime \prime}\right\|_{\infty},$$

  and give an example of $f$ for which the inequality turns into equality.

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Optimization

• 4.II $20 \mathrm{C}$

  Optimization | Part IB, 2007

  Consider the linear programming problem

  $$\begin{aligned} \operatorname{minimize} \quad & 2 x_{1}-3 x_{2}-2 x_{3} \\ \text { subject to } \quad-& 2 x_{1}+2 x_{2}+4 x_{3} \leqslant 5 \\ & 4 x_{1}+2 x_{2}-5 x_{3} \leqslant 8 \\ & 5 x_{1}-4 x_{2}+\frac{1}{2} x_{3} \leqslant 5, \quad x_{i} \geqslant 0, \quad i=1,2,3 . \end{aligned}$$

  (i) After adding slack variables $z_{1}, z_{2}$ and $z_{3}$ and performing one iteration of the simplex algorithm, the following tableau is obtained.

  \begin{tabular}{c|rrrrrr|c} & $x_{1}$ & $x_{2}$ & $x_{3}$ & $z_{1}$ & $z_{2}$ & $z_{3}$ & \ \hline$x_{2}$ & $-1$ & 1 & 2 & $1 / 2$ & 0 & 0 & $5 / 2$ \ $z_{2}$ & 6 & 0 & $-9$ & $-1$ & 1 & 0 & 3 \ $z_{3}$ & 1 & 0 & $17 / 2$ & 2 & 0 & 1 & 15 \ \hline Payoff & $-1$ & 0 & 4 & $3 / 2$ & 0 & 0 & $15 / 2$ \end{tabular}

  Complete the solution of the problem.

  (ii) Now suppose that the problem is amended so that the objective function becomes

  $$2 x_{1}-3 x_{2}-5 x_{3}$$

  Find the solution of this new problem.

  (iii) Formulate the dual of the problem in (ii) and identify the optimal solution to the dual.

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Quantum Mechanics

• 4.I.6B

  Quantum Mechanics | Part IB, 2007

  A particle moving in one space dimension with wave-function $\Psi(x, t)$ obeys the time-dependent Schrödinger equation. Write down the probability density, $\rho$, and current density, $j$, in terms of the wave-function and show that they obey the equation

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

  The wave-function is

  $$\Psi(x, t)=\left(e^{i k x}+R e^{-i k x}\right) e^{-i E t / \hbar},$$

  where $E=\hbar^{2} k^{2} / 2 m$ and $R$ is a constant, which may be complex. Evaluate $j$.

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Special Relativity

• 4.II.17B

  Special Relativity | Part IB, 2007

  (a) A moving $\pi^{0}$ particle of rest-mass $m_{\pi}$ decays into two photons of zero rest-mass,

  $$\pi^{0} \rightarrow \gamma+\gamma$$

  Show that

  $$\sin \frac{\theta}{2}=\frac{m_{\pi} c^{2}}{2 \sqrt{E_{1} E_{2}}}$$

  where $\theta$ is the angle between the three-momenta of the two photons and $E_{1}, E_{2}$ are their energies.

  (b) The $\pi^{-}$particle of rest-mass $m_{\pi}$ decays into an electron of rest-mass $m_{e}$ and a neutrino of zero rest mass,

  $$\pi^{-} \rightarrow e^{-}+\nu .$$

  Show that $v$, the speed of the electron in the rest frame of the $\pi^{-}$, is

  $$v=c\left[\frac{1-\left(m_{e} / m_{\pi}\right)^{2}}{1+\left(m_{e} / m_{\pi}\right)^{2}}\right]$$

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Statistics

• 4.II.19C

  Statistics | Part IB, 2007

  Consider the linear regression model

  $$Y_{i}=\alpha+\beta x_{i}+\epsilon_{i}, \quad 1 \leqslant i \leqslant n$$

  where $\epsilon_{1}, \ldots, \epsilon_{n}$ are independent, identically distributed $N\left(0, \sigma^{2}\right), x_{1}, \ldots, x_{n}$ are known real numbers with $\sum_{i=1}^{n} x_{i}=0$ and $\alpha, \beta$ and $\sigma^{2}$ are unknown.

  (i) Find the least-squares estimates $\widehat{\alpha}$and $\widehat{\beta}$ of $\alpha$ and $\beta$, respectively, and explain why in this case they are the same as the maximum-likelihood estimates.

  (ii) Determine the maximum-likelihood estimate $\widehat{\sigma}^{2}$ of $\sigma^{2}$ and find a multiple of it which is an unbiased estimate of $\sigma^{2}$.

  (iii) Determine the joint distribution of $\widehat{\alpha}, \widehat{\beta}$ and $\widehat{\sigma}^{2}$.

  (iv) Explain carefully how you would test the hypothesis $H_{0}: \alpha=\alpha_{0}$ against the alternative $H_{1}: \alpha \neq \alpha_{0}$.

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