Part II, 2006, Paper 4

Algebraic Topology

• 4.II.21H

  Algebraic Topology | Part II, 2006

  Fix a point $p$ in the torus $S^{1} \times S^{1}$. Let $G$ be the group of homeomorphisms $f$ from the torus $S^{1} \times S^{1}$ to itself such that $f(p)=p$. Determine a non-trivial homomorphism $\phi$ from $G$ to the group $\operatorname{GL}(2, \mathbb{Z})$.

  [The group $\mathrm{GL}(2, \mathbb{Z})$ consists of $2 \times 2$ matrices with integer coefficients that have an inverse which also has integer coefficients.]

  Establish whether each $f$ in the kernel of $\phi$ is homotopic to the identity map.

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Applications of Quantum Mechanics

• 4.II.33D

  Applications of Quantum Mechanics | Part II, 2006

  For the one-dimensional potential

  $$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{n} \delta(x-n a)$$

  solve the Schrödinger equation for negative energy and obtain an equation that determines possible energy bands. Show that the results agree with the tight-binding model in appropriate limits.

  [It may be useful to note that $\left.V(x)=-\frac{\hbar^{2} \lambda}{m a} \sum_{n} e^{2 \pi i n x / a} .\right]$

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Applied Probability

• 4.II.26J

  Applied Probability | Part II, 2006

  (a) Let $\left(N_{t}\right)_{t \geqslant 0}$ be a Poisson process of rate $\lambda>0$. Let $p$ be a number between 0 and 1 and suppose that each jump in $\left(N_{t}\right)$ is counted as type one with probability $p$ and type two with probability $1-p$, independently for different jumps and independently of the Poisson process. Let $M_{t}^{(1)}$ be the number of type-one jumps and $M_{t}^{(2)}=N_{t}-M_{t}^{(1)}$ the number of type-two jumps by time $t$. What can you say about the pair of processes $\left(M_{t}^{(1)}\right)_{t \geqslant 0}$ and $\left(M_{t}^{(2)}\right)_{t \geqslant 0}$ ? What if we fix probabilities $p_{1}, \ldots, p_{m}$ with $p_{1}+\ldots+p_{m}=1$ and consider $m$ types instead of two?

  (b) A person collects coupons one at a time, at jump times of a Poisson process $\left(N_{t}\right)_{t \geqslant 0}$ of rate $\lambda$. There are $m$ types of coupons, and each time a coupon of type $j$ is obtained with probability $p_{j}$, independently of the previously collected coupons and independently of the Poisson process. Let $T$ be the first time when a complete set of coupon types is collected. Show that

  $$\mathbb{P}(T<t)=\prod_{j=1}^{m}\left(1-e^{-p_{j} \lambda t}\right)$$

  Let $L=N_{T}$ be the total number of coupons collected by the time the complete set of coupon types is obtained. Show that $\lambda \mathbb{E} T=\mathbb{E} L$. Hence, or otherwise, deduce that $\mathbb{E} L$ does not depend on $\lambda$.

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Asymptotic Methods

• 4.II $. 31 \mathrm{~B}$

  Asymptotic Methods | Part II, 2006

  (a) Outline the Liouville-Green approximation to solutions $w(z)$ of the ordinary differential equation

  $$\frac{d^{2} w}{d z^{2}}=f(z) w$$

  in a neighbourhood of infinity, in the case that, near infinity, $f(z)$ has the convergent series expansion

  $$f(z)=\sum_{s=0}^{\infty} \frac{f_{s}}{z^{s}}$$

  with $f_{0} \neq 0$.

  In the case

  $$f(z)=1+\frac{1}{z}+\frac{2}{z^{2}},$$

  explain why you expect a basis of two asymptotic solutions $w_{1}(z), w_{2}(z)$, with

  $$\begin{aligned} &w_{1}(z) \sim z^{\frac{1}{2}} e^{z}\left(1+\frac{a_{1}}{z}+\frac{a_{2}}{z^{2}}+\cdots\right), \\ &w_{2}(z) \sim z^{-\frac{1}{2}} e^{-z}\left(1+\frac{b_{1}}{z}+\frac{b_{2}}{z^{2}}+\cdots\right), \end{aligned}$$

  as $z \rightarrow+\infty$, and show that $a_{1}=-\frac{9}{8}$.

  (b) Determine, at leading order in the large positive real parameter $\lambda$, an approximation to the solution $u(x)$ of the eigenvalue problem:

  $$u^{\prime \prime}(x)+\lambda^{2} g(x) u(x)=0 ; \quad u(0)=u(1)=0$$

  where $g(x)$ is greater than a positive constant for $x \in[0,1]$.

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Classical Dynamics

• 4.I.9C

  Classical Dynamics | Part II, 2006

  Calculate the principal moments of inertia for a uniform cylinder, of mass $M$, radius $R$ and height $2 h$, about its centre of mass. For what height-to-radius ratio does the cylinder spin like a sphere?

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Coding and Cryptography

• 4.I.4G

  Coding and Cryptography | Part II, 2006

  A binary erasure channel with erasure probability $p$ is a discrete memoryless channel with channel matrix

  $$\left(\begin{array}{ccc} 1-p & p & 0 \\ 0 & p & 1-p \end{array}\right)$$

  State Shannon's second coding theorem, and use it to compute the capacity of this channel.

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Cosmology

• 4.I.10D

  Cosmology | Part II, 2006

  The number density of fermions of mass $m$ at equilibrium in the early universe with temperature $T$, is given by the integral

  $$n=\frac{4 \pi}{h^{3}} \int_{0}^{\infty} \frac{p^{2} d p}{\exp [(\mathcal{E}(p)-\mu) / k T]+1}$$

  where $\mathcal{E}(p)=c \sqrt{p^{2}+m^{2} c^{2}}$, and $\mu$ is the chemical potential. Assuming that the fermions remain in equilibrium when they become non-relativistic $\left(k T, \mu \ll m c^{2}\right)$, show that the number density can be expressed as

  $$n=\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} \exp \left[\left(\mu-m c^{2}\right) / k T\right]$$

  [Hint: You may assume $\left.\int_{0}^{\infty} d x e^{-\sigma^{2} x^{2}}=\sqrt{\pi} /(2 \sigma), \quad(\sigma>0) .\right]$

  Suppose that the fermions decouple at a temperature given by $k T=m c^{2} / \alpha$ where $\alpha \gg 1$. Assume also that $\mu=0$. By comparing with the photon number density at $n_{\gamma}=16 \pi \zeta(3)(k T / h c)^{3}$, where $\zeta(3)=\sum_{n=1}^{\infty} n^{-3}=1.202 \ldots$, show that the ratio of number densities at decoupling is given by

  $$\frac{n}{n_{\gamma}}=\frac{\sqrt{2 \pi}}{8 \zeta(3)} \alpha^{3 / 2} e^{-\alpha}$$

  Now assume that $\alpha \approx 20$, (which implies $n / n_{\gamma} \approx 5 \times 10^{-8}$ ), and that the fermion mass $m=m_{p} / 20$, where $m_{p}$ is the proton mass. Explain clearly why this new fermion would be a good candidate for solving the dark matter problem of the standard cosmology.

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• 4.II.15D

  Cosmology | Part II, 2006

  The perturbed motion of cold dark matter particles (pressure-free, $P=0$ ) in an expanding universe can be parametrized by the trajectories

  $$\mathbf{r}(\mathbf{q}, t)=a(t)[\mathbf{q}+\boldsymbol{\psi}(\mathbf{q}, t)]$$

  where $a(t)$ is the scale factor of the universe, $\mathbf{q}$ is the unperturbed comoving trajectory and $\boldsymbol{\psi}$ is the comoving displacement. The particle equation of motion is $\ddot{\mathbf{r}}=-\nabla \Phi$, where the Newtonian potential satisfies the Poisson equation $\nabla^{2} \Phi=4 \pi G \rho$ with mass density $\rho(\mathbf{r}, t)$.

  (a) Discuss how matter conservation in a small volume $d^{3} \mathbf{r}$ ensures that the perturbed density $\rho(\mathbf{r}, t)$ and the unperturbed background density $\bar{\rho}(t)$ are related by

  $$\rho(\mathbf{r}, t) d^{3} \mathbf{r}=\bar{\rho}(t) a^{3}(t) d^{3} \mathbf{q}$$

  By changing co-ordinates with the Jacobian

  $$\left|\partial r_{i} / \partial q_{j}\right|^{-1}=\left|a \delta_{i j}+a \partial \psi_{i} / \partial q_{j}\right|^{-1} \approx a^{-3}\left(1-\nabla_{q} \cdot \psi\right),$$

  show that the fractional density perturbation $\delta(\mathbf{q}, t)$ can be written to leading order as

  $$\delta \equiv \frac{\rho-\bar{\rho}}{\bar{\rho}}=-\nabla_{q} \cdot \psi,$$

  where $\nabla_{q} \cdot \boldsymbol{\psi}=\sum_{i} \partial \psi_{i} / \partial q_{i}$.

  Use this result to integrate the Poisson equation once. Hence, express the particle equation of motion in terms of the comoving displacement as

  $$\ddot{\boldsymbol{\psi}}+2 \frac{\dot{a}}{a} \dot{\boldsymbol{\psi}}-4 \pi G \bar{\rho} \boldsymbol{\psi}=0$$

  Infer that the density perturbation evolution equation is

  $$\ddot{\delta}+2 \frac{\dot{a}}{a} \dot{\delta}-4 \pi G \bar{\rho} \delta=0$$

  [Hint: You may assume that the integral of $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ is $\nabla \Phi=-4 \pi G \bar{\rho} \mathbf{r} / 3$. Note also that the Raychaudhuri equation (for $P=0$ ) is $\ddot{a} / a=-4 \pi G \bar{\rho} / 3 .$.]

  (b) Find the general solution of equation $(*)$ in a flat $(k=0)$ universe dominated by cold dark matter $(P=0)$. Discuss the effect of late-time $\Lambda$ or dark energy domination on the growth of density perturbations.

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Differential Geometry

• 4.II.24H

  Differential Geometry | Part II, 2006

  (a) Let $S \subset \mathbb{R}^{3}$ be an oriented surface and let $\lambda$ be a real number. Given a point $p \in S$ and a vector $v \in T_{p} S$ with unit norm, show that there exist $\varepsilon>0$ and a unique curve $\gamma:(-\varepsilon, \varepsilon) \rightarrow S$ parametrized by arc-length and with constant geodesic curvature $\lambda$ such that $\gamma(0)=p$ and $\dot{\gamma}(0)=v$.

  [You may use the theorem on existence and uniqueness of solutions of ordinary differential equations.]

  (b) Let $S$ be an oriented surface with positive Gaussian curvature and diffeomorphic to $S^{2}$. Show that two simple closed geodesics in $S$ must intersect. Is it true that two smooth simple closed curves in $S$ with constant geodesic curvature $\lambda \neq 0$ must intersect?

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Dynamical Systems

• 4.I.7E

  Dynamical Systems | Part II, 2006

  Consider the logistic map $F(x)=\mu x(1-x)$ for $0 \leqslant x \leqslant 1,0 \leqslant \mu \leqslant 4$. Show that there is a period-doubling bifurcation of the nontrivial fixed point at $\mu=3$. Show further that the bifurcating 2 -cycle $\left(x_{1}, x_{2}\right)$ is given by the roots of

  $$\mu^{2} x^{2}-\mu(\mu+1) x+\mu+1=0 .$$

  Show that there is a second period-doubling bifurcation at $\mu=1+\sqrt{6}$.

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Electrodynamics

• 4.II $. 35 \mathrm{E} \quad$

  Electrodynamics | Part II, 2006

  The retarded scalar potential produced by a charge distribution $\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)$ is

  $$\varphi(t, \mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \int d^{3} x^{\prime} \frac{\rho\left(t-R, \mathbf{x}^{\prime}\right)}{R},$$

  where $R=|\mathbf{R}|$ and $\mathbf{R}=\mathbf{x}-\mathbf{x}^{\prime}$. By use of an appropriate delta function rewrite the integral as an integral over both $d^{3} x^{\prime}$ and $d t^{\prime}$ involving $\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)$.

  Now specialize to a point charge $q$ moving on a path $\mathbf{x}^{\prime}=\mathbf{x}_{0}\left(t^{\prime}\right)$ so that we may set

  $$\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)=q \delta^{(3)}\left(\mathbf{x}^{\prime}-\mathbf{x}_{0}\left(t^{\prime}\right)\right) .$$

  By performing the volume integral first obtain the Liénard-Wiechert potential

  $$\varphi(t, \mathbf{x})=\frac{q}{4 \pi \epsilon_{0}} \frac{1}{\left(R^{*}-\mathbf{v} \cdot \mathbf{R}^{*}\right)},$$

  where $\mathbf{R}^{*}$ and $\mathbf{v}$ should be specified.

  Obtain the corresponding result for the magnetic potential.

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Fluid Dynamics II

• 4.II.37B

  Fluid Dynamics II | Part II, 2006

  A line force of magnitude $F$ is applied in the positive $x$-direction to an unbounded fluid, generating a thin two-dimensional jet along the positive $x$-axis. The fluid is at rest at $y=\pm \infty$ and there is negligible motion in $x<0$. Write down the pressure gradient within the boundary layer. Deduce that the function $M(x)$ defined by

  $$M(x)=\int_{-\infty}^{\infty} \rho u^{2}(x, y) d y$$

  is independent of $x$ for $x>0$. Interpret this result, and explain why $M=F$. Use scaling arguments to deduce that there is a similarity solution having stream function

  $$\psi=(F \nu x / \rho)^{1 / 3} f(\eta) \quad \text { where } \quad \eta=y\left(F / \rho \nu^{2} x^{2}\right)^{1 / 3}$$

  Hence show that $f$ satisfies

  $$3 f^{\prime \prime \prime}+f^{\prime 2}+f f^{\prime \prime}=0 .$$

  Show that a solution of $(*)$ is

  $$f(\eta)=A \tanh (A \eta / 6)$$

  where $A$ is a constant to be determined by requiring that $M$ is independent of $x$. Find the volume flux, $Q(x)$, in the jet. Briefly indicate why $Q(x)$ increases as $x$ increases.

  [Hint: You may use $\left.\int_{-\infty}^{\infty} \operatorname{sech}^{4}(x) d x=4 / 3 .\right]$

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Further Complex Methods

• 4.I.8E

  Further Complex Methods | Part II, 2006

  By means of the change of variable $u=r s, v=r(1-s)$ in a suitable double integral, or otherwise, show that for $\operatorname{Re} z>0$

  $$\left[\Gamma\left(\frac{1}{2} z\right)\right]^{2}=B\left(\frac{1}{2} z, \frac{1}{2} z\right) \Gamma(z)$$

  Deduce that, if $\Gamma(z)=0$ for some $z$ with $\operatorname{Re} z>0$, then $\Gamma\left(z / 2^{m}\right)=0$ for any positive integer $m$.

  Prove that $\Gamma(z) \neq 0$ for any $z$.

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• 4.II.14E

  Further Complex Methods | Part II, 2006

  Let $I=\int_{0}^{1}\left[x\left(1-x^{2}\right)\right]^{1 / 3} d x$

  (a) Express $I$ in terms of an integral of the form $\oint\left(z^{3}-z\right)^{1 / 3} d z$, where the path of integration is a large circle. You should explain carefully which branch of $\left(z^{3}-z\right)^{1 / 3}$ you choose, by using polar co-ordinates with respect to the branch points. Hence show that $I=\frac{1}{6} \pi \operatorname{cosec} \frac{1}{3} \pi$.

  (b) Give an alternative method of evaluating $I$, by making a suitable change of variable and expressing $I$ in terms of a beta function.

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Galois Theory

• 4.II.18H

  Galois Theory | Part II, 2006

  Let $K$ be a field of characteristic different from 2 .

  Show that if $L / K$ is an extension of degree 2 , then $L=K(x)$ for some $x \in L$ such that $x^{2}=a \in K$. Show also that if $L^{\prime}=K(y)$ with $0 \neq y^{2}=b \in K$ then $L$ and $L^{\prime}$ are isomorphic (as extensions of $K$ ) if and only $b / a$ is a square in $K$.

  Now suppose that $F=K\left(x_{1}, \ldots, x_{n}\right)$ where $0 \neq x_{i}^{2}=a_{i} \in K$. Show that $F / K$ is a Galois extension, with Galois group isomorphic to $(\mathbb{Z} / 2 \mathbb{Z})^{m}$ for some $m \leqslant n$. By considering the subgroups of $\operatorname{Gal}(F / K)$, show that if $K \subset L \subset F$ and $[L: K]=2$, then $L=K(y)$ where $y=\prod_{i \in I} x_{i}$ for some subset $I \subset\{1, \ldots, n\}$.

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General Relativity

• 4.II.36A

  General Relativity | Part II, 2006

  What are local inertial co-ordinates? What is their physical significance and how are they related to the equivalence principle?

  If $V_{a}$ are the components of a covariant vector field, show that

  $$\partial_{a} V_{b}-\partial_{b} V_{a}$$

  are the components of an anti-symmetric second rank covariant tensor field.

  If $K^{a}$ are the components of a contravariant vector field and $g_{a b}$ the components of a metric tensor, let

  $$Q_{a b}=K^{c} \partial_{c} g_{a b}+g_{a c} \partial_{b} K^{c}+g_{c b} \partial_{a} K^{c}$$

  Show that

  $$Q_{a b}=2 \nabla_{(a} K_{b)},$$

  where $K_{a}=g_{a b} K^{b}$, and $\nabla_{a}$ is the Levi-Civita covariant derivative operator of the metric $g_{a b}$.

  In a particular co-ordinate system $\left(x^{1}, x^{2}, x^{3}, x^{4}\right)$, it is given that $K^{a}=(0,0,0,1)$, $Q_{a b}=0$. Deduce that, in this co-ordinate system, the metric tensor $g_{a b}$ is independent of the co-ordinate $x^{4}$. Hence show that

  $$\nabla_{a} K_{b}=\frac{1}{2}\left(\partial_{a} K_{b}-\partial_{b} K_{a}\right)$$

  and that

  $$E=-K_{a} \frac{d x^{a}}{d \lambda}$$

  is constant along every geodesic $x^{a}(\lambda)$ in every co-ordinate system.

  What further conditions must one impose on $K^{a}$ and $d x^{a} / d \lambda$ to ensure that the metric is stationary and that $E$ is proportional to the energy of a particle moving along the geodesic?

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Geometry and Groups

• 4.I.3F

  Geometry and Groups | Part II, 2006

  What is a crystallographic group in the Euclidean plane? Prove that, if $G$ is crystallographic and $g$ is a nontrivial rotation in $G$, then $g$ has order $2,3,4$, or 6 .

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• 4.II.12F

  Geometry and Groups | Part II, 2006

  Let $G$ be a discrete subgroup of $\operatorname{PSL}_{2}(\mathbb{C})$. Show that $G$ is countable. Let $G=\left\{g_{1}, g_{2}, \ldots\right\}$ be some enumeration of the elements of $G$. Show that for any point $p$ in hyperbolic 3-space $\mathbb{H}^{3}$, the distance $d_{h y p}\left(p, g_{n}(p)\right)$ tends to infinity. Deduce that a subgroup $G$ of $\mathrm{PSL}_{2}(\mathbb{C})$ is discrete if and only if it acts properly discontinuously on $\mathbb{H}^{3}$.

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Graph Theory

• 4.II.17F

  Graph Theory | Part II, 2006

  What is meant by a graph $G$ of order $n$ being strongly regular with parameters $(d, a, b) ?$ Show that, if such a graph $G$ exists and $b>0$, then

  $$\frac{1}{2}\left\{n-1+\frac{(n-1)(b-a)-2 d}{\sqrt{(a-b)^{2}+4(d-b)}}\right\}$$

  is an integer.

  Let $G$ be a graph containing no triangles, in which every pair of non-adjacent vertices has exactly three common neighbours. Show that $G$ must be $d$-regular and $|G|=1+d(d+2) / 3$ for some $d \in\{1,3,21\}$. Show that such a graph exists for $d=3$.

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Linear Analysis

• 4.II.22G

  Linear Analysis | Part II, 2006

  Let $H$ be a complex Hilbert space. Define what it means for a linear operator $T: H \rightarrow H$ to be self-adjoint. State a version of the spectral theorem for compact selfadjoint operators on a Hilbert space. Give an example of a Hilbert space $H$ and a compact self-adjoint operator on $H$ with infinite dimensional range. Define the notions spectrum, point spectrum, and resolvent set, and describe these in the case of the operator you wrote down. Justify your answers.

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Logic and Set Theory

• 4.II.16H

  Logic and Set Theory | Part II, 2006

  Explain carefully what is meant by a well-founded relation on a set. State the recursion theorem, and use it to prove that a binary relation $r$ on a set $a$ is well-founded if and only if there exists a function $f$ from $a$ to some ordinal $\alpha$ such that $(x, y) \in r$ implies $f(x)<f(y)$.

  Deduce, using the axiom of choice, that any well-founded relation on a set may be extended to a well-ordering.

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Mathematical Biology

• 4.I.6B

  Mathematical Biology | Part II, 2006

  A nonlinear model of insect dispersal with exponential death rate takes the form (for insect population $n(x, t)$ )

  $$\tag{*} \frac{\partial n}{\partial t}=-\mu n+\frac{\partial}{\partial x}\left(n \frac{\partial n}{\partial x}\right)$$

  At time $t=0$ the total insect population is $Q$, and all the insects are at the origin. A solution is sought in the form

  $$\tag{†} n=\frac{e^{-\mu t}}{\lambda(t)} f(\eta) ; \quad \eta=\frac{x}{\lambda(t)}, \quad \lambda(0)=0$$

  (a) Verify that $\int_{-\infty}^{\infty} f d \eta=Q$, provided $f$ decays sufficiently rapidly as $|x| \rightarrow \infty$.

  (b) Show, by substituting the form of $n$ given in equation $(†)$ into equation $(*)$, that $(*)$ is satisfied, for nonzero $f$, when

  $$\frac{d \lambda}{d t}=\lambda^{-2} e^{-\mu t} \quad \text { and } \quad \frac{d f}{d \eta}=-\eta$$

  Hence find the complete solution and show that the insect population is always confined to a finite region that never exceeds the range

  $$|x| \leqslant\left(\frac{9 Q}{2 \mu}\right)^{1 / 3}$$

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Number Fields

• 4.II.20G

  Number Fields | Part II, 2006

  Let $\zeta=e^{2 \pi i / 5}$ and let $K=\mathbb{Q}(\zeta)$. Show that the discriminant of $K$ is 125 . Hence prove that the ideals in $K$ are all principal.

  Verify that $\left(1-\zeta^{n}\right) /(1-\zeta)$ is a unit in $K$ for each integer $n$ with $1 \leqslant n \leqslant 4$. Deduce that $5 /(1-\zeta)^{4}$ is a unit in $K$. Hence show that the ideal $[1-\zeta]$ is prime and totally ramified in $K$. Indicate briefly why there are no other ramified prime ideals in $K$.

  [It can be assumed that $\zeta, \zeta^{2}, \zeta^{3}, \zeta^{4}$ is an integral basis for $K$ and that the Minkowski constant for $K$ is $3 /\left(2 \pi^{2}\right)$.]

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Number Theory

• 4.I.1H

  Number Theory | Part II, 2006

  Let $x$ be a real number greater than or equal to 2 , and define

  $$P(x)=\prod_{p \leqslant x}\left(1-\frac{1}{p}\right)$$

  where the product is taken over all primes $p$ which are less than or equal to $x$. Prove that $P(x) \rightarrow 0$ as $x \rightarrow \infty$, and deduce that $\sum_{p} \frac{1}{p}$ diverges when the summation is taken over all primes $p$.

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• 4.II.11H

  Number Theory | Part II, 2006

  Define the notion of a Fermat, Euler, and strong pseudo-prime to the base $b$, where $b$ is an integer greater than $1 .$

  Let $N$ be an odd integer greater than 1. Prove that:

  (a) If $N$ is a prime number, then $N$ is a strong pseudo-prime for every base $b$ with $(b, N)=1$.

  (b) If there exists a base $b_{1}$ with $1<b_{1}<N$ and $\left(b_{1}, N\right)=1$ for which $N$ is not a pseudo-prime, then in fact $N$ is not a pseudo-prime for at least half of all bases $b$ with $1<b<N$ and $(b, N)=1$.

  Prove that 341 is a Fermat pseudo-prime, but not an Euler pseudo-prime, to the base $2 .$

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Numerical Analysis

• 4.II.39C

  Numerical Analysis | Part II, 2006

  The difference equation

  $$u_{m}^{n+1}=u_{m}^{n}+\frac{3}{2} \mu\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right)-\frac{1}{2} \mu\left(u_{m-1}^{n-1}-2 u_{m}^{n-1}+u_{m+1}^{n-1}\right),$$

  where $\mu=\Delta t /(\Delta x)^{2}$, is used to approximate a solution of the diffusion equation $u_{t}=u_{x x}$.

  (a) Prove that, as $\Delta t \rightarrow 0$ with constant $\mu$, the local error of the method is $\mathcal{O}(\Delta t)^{2}$.

  (b) Applying the Fourier stability test, show that the method is stable if and only if $\mu \leqslant \frac{1}{4}$.

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Optimization and Control

• 4.II.29I

  Optimization and Control | Part II, 2006

  An investor has a (possibly negative) bank balance $x(t)$ at time $t$. For given positive $x(0), T, \mu, A$ and $r$, he wishes to choose his spending rate $u(t) \geqslant 0$ so as to maximize

  $$\Phi(u ; \mu) \equiv \int_{0}^{T} e^{-\beta t} \log u(t) d t+\mu e^{-\beta T} x(T),$$

  where $d x(t) / d t=A+r x(t)-u(t)$. Find the investor's optimal choice of control $u(t)=u_{*}(t ; \mu)$.

  Let $x_{*}(t ; \mu)$ denote the optimally-controlled bank balance. By considering next how $x_{*}(T ; \mu)$ depends on $\mu$, show that there is a unique positive $\mu_{*}$ such that $x_{*}\left(T ; \mu_{*}\right)=0$. If the original problem is modified by setting $\mu=0$, but requiring that $x(T) \geqslant 0$, show that the optimal control for this modified problem is $u(t)=u_{*}\left(t ; \mu_{*}\right)$.

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Partial Differential Equations

• 4.II.30A

  Partial Differential Equations | Part II, 2006

  (a) State the Fourier inversion theorem for Schwartz functions $\mathcal{S}(\mathbb{R})$ on the real line. Define the Fourier transform of a tempered distribution and compute the Fourier transform of the distribution defined by the function $F(x)=\frac{1}{2}$ for $-t \leqslant x \leqslant+t$ and $F(x)=0$ otherwise. (Here $t$ is any positive number.)

  Use the Fourier transform in the $x$ variable to deduce a formula for the solution to the one dimensional wave equation

  $$u_{t t}-u_{x x}=0, \quad \text { with initial data } \quad u(0, x)=0, \quad u_{t}(0, x)=g(x),$$

  for $g$ a Schwartz function. Explain what is meant by "finite propagation speed" and briefly explain why the formula you have derived is in fact valid for arbitrary smooth $g \in C^{\infty}(\mathbb{R})$.

  (b) State a theorem on the representation of a smooth $2 \pi$-periodic function $g$ as a Fourier series

  $$g(x)=\sum_{\alpha \in \mathbb{Z}} \hat{g}(\alpha) e^{i \alpha x}$$

  and derive a representation for solutions to $(*)$ as Fourier series in $x$.

  (c) Verify that the formulae obtained in (a) and (b) agree for the case of smooth $2 \pi$ periodic $g$.

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Principles of Quantum Mechanics

• 4.II.32A

  Principles of Quantum Mechanics | Part II, 2006

  Define the Heisenberg picture of quantum mechanics in relation to the Schrödinger picture and explain how these formulations give rise to identical physical predictions. Derive an equation of motion for an operator in the Heisenberg picture, assuming the operator is independent of time in the Schrödinger picture.

  State clearly the form of the unitary operator corresponding to a rotation through an angle $\theta$ about an axis $\mathbf{n}$ (a unit vector) for a general quantum system. Verify your statement for the case in which the system is a single particle by considering the effect of an infinitesimal rotation on the particle's position $\hat{\mathbf{x}}$ and on its spin $\mathbf{S}$.

  Show that if the Hamiltonian for a particle is of the form

  $$H=\frac{1}{2 m} \hat{\mathbf{p}}^{2}+U\left(\hat{\mathbf{x}}^{2}\right) \hat{\mathbf{x}} \cdot \mathbf{S}$$

  then all components of the total angular momentum are independent of time in the Heisenberg picture. Is the same true for either orbital or spin angular momentum?

  [You may quote commutation relations involving components of $\hat{\mathbf{x}}, \hat{\mathbf{p}}, \mathbf{L}$ and $\mathbf{S}$.]

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Principles of Statistics

• 4.II.27J

  Principles of Statistics | Part II, 2006

  (a) State the strong law of large numbers. State the central limit theorem.

  (b) Assuming whatever regularity conditions you require, show that if $\hat{\theta}_{n} \equiv \hat{\theta}_{n}\left(X_{1}, \ldots, X_{n}\right)$ is the maximum-likelihood estimator of the unknown parameter $\theta$ based on an independent identically distributed sample of size $n$, then under $P_{\theta}$

  $$\sqrt{n}\left(\hat{\theta}_{n}-\theta\right) \rightarrow N\left(0, J(\theta)^{-1}\right) \quad \text { in distribution }$$

  as $n \rightarrow \infty$, where $J(\theta)$ is a matrix which you should identify. A rigorous derivation is not required.

  (c) Suppose that $X_{1}, X_{2}, \ldots$ are independent binomial $\operatorname{Bin}(1, \theta)$ random variables. It is required to test $H_{0}: \theta=\frac{1}{2}$ against the alternative $H_{1}: \theta \in(0,1)$. Show that the construction of a likelihood-ratio test leads us to the statistic

  $$T_{n}=2 n\left\{\hat{\theta}_{n} \log \hat{\theta}_{n}+\left(1-\hat{\theta}_{n}\right) \log \left(1-\hat{\theta}_{n}\right)+\log 2\right\}$$

  where $\hat{\theta}_{n} \equiv n^{-1} \sum_{k=1}^{n} X_{k}$. Stating clearly any result to which you appeal, for large $n$, what approximately is the distribution of $T_{n}$ under $H_{0}$ ? Writing $\hat{\theta}_{n}=\frac{1}{2}+Z_{n}$, and assuming that $Z_{n}$ is small, show that

  $$T_{n} \simeq 4 n Z_{n}^{2}$$

  Using this and the central limit theorem, briefly justify the approximate distribution of $T_{n}$ given by asymptotic maximum-likelihood theory. What could you say if the assumption that $Z_{n}$ is small failed?

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Probability and Measure

• 4.II $. 25 \mathrm{~J} \quad$

  Probability and Measure | Part II, 2006

  Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and $f: \Omega \rightarrow \mathbb{R}$ a measurable function.

  (a) Explain what is meant by saying that $f$ is integrable, and how the integral $\int_{\Omega} f d \mu$ is defined, starting with integrals of $\mathcal{A}$-simple functions.

  [Your answer should consist of clear definitions, including the ones for $\mathcal{A}$-simple functions and their integrals.]

  (b) For $f: \Omega \rightarrow[0, \infty)$ give a specific sequence $\left(g_{n}\right)_{n \in \mathbb{N}}$ of $\mathcal{A}$-simple functions such that $0 \leqslant g_{n} \leqslant f$ and $g_{n}(x) \rightarrow f(x)$ for all $x \in \Omega$. Justify your answer.

  (c) Suppose that that $\mu(\Omega)<\infty$ and let $f_{1}, f_{2}, \ldots: \Omega \rightarrow \mathbb{R}$ be measurable functions such that $f_{n}(x) \rightarrow 0$ for all $x \in \Omega$. Prove that, if

  $$\lim _{c \rightarrow \infty} \sup _{n \in \mathbb{N}} \int_{\left|f_{n}\right|>c}\left|f_{n}\right| d \mu=0$$

  then $\int_{\Omega} f_{n} d \mu \rightarrow 0$.

  Give an example with $\mu(\Omega)<\infty$ such that $f_{n}(x) \rightarrow 0$ for all $x \in \Omega$, but $\int_{\Omega} f_{n} d \mu \nrightarrow 0$, and justify your answer.

  (d) State and prove Fatou's Lemma for a sequence of non-negative measurable functions.

  [Standard results on measurability and integration may be used without proof.]

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Representation Theory

• 4.II.19F

  Representation Theory | Part II, 2006

  In this question, all vector spaces will be complex.

  (a) Let $A$ be a finite abelian group.

  (i) Show directly from the definitions that any irreducible representation must be one-dimensional.

  (ii) Show that $A$ has a faithful one-dimensional representation if and only if $A$ is cyclic.

  (b) Now let $G$ be an arbitrary finite group and suppose that the centre of $G$ is nontrivial. Write $Z=\{z \in G \mid z g=g z \quad \forall g \in G\}$ for this centre.

  (i) Let $W$ be an irreducible representation of $G$. Show that $\operatorname{Res}_{Z}^{G} W=\operatorname{dim} W \cdot \chi$, where $\chi$ is an irreducible representation of $Z$.

  (ii) Show that every irreducible representation of $Z$ occurs in this way.

  (iii) Suppose that $Z$ is not a cyclic group. Show that there does not exist an irreducible representation $W$ of $G$ such that every irreducible representation $V$ occurs as a summand of $W^{\otimes n}$ for some $n$.

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Riemann Surfaces

• 4.II.23F

  Riemann Surfaces | Part II, 2006

  Define what is meant by a divisor on a compact Riemann surface, the degree of a divisor, and a linear equivalence between divisors. For a divisor $D$, define $\ell(D)$ and show that if a divisor $D^{\prime}$ is linearly equivalent to $D$ then $\ell(D)=\ell\left(D^{\prime}\right)$. Determine, without using the Riemann-Roch theorem, the value $\ell(P)$ in the case when $P$ is a point on the Riemann sphere $S^{2}$.

  [You may use without proof any results about holomorphic maps on $S^{2}$ provided that these are accurately stated.]

  State the Riemann-Roch theorem for a compact connected Riemann surface $C$. (You are not required to give a definition of a canonical divisor.) Show, by considering an appropriate divisor, that if $C$ has genus $g$ then $C$ admits a non-constant meromorphic function (that is a holomorphic map $C \rightarrow S^{2}$ ) of degree at most $g+1$.

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Statistical Modelling

• 4.I.5I

  Statistical Modelling | Part II, 2006

  The table below summarises the yearly numbers of named storms in the Atlantic basin over the period 1944-2004, and also gives an index of average July ocean temperature in the northern hemisphere over the same period. To save space, only the data for the first four and last four years are shown.

  $\begin{array}{rrr}\text { Year } & \text { Storms } & \text { Temp } \\ 1944 & 11 & 0.165 \\ 1945 & 11 & 0.080 \\ 1946 & 6 & 0.000 \\ 1947 & 9 & -0.024 \\ \vdots & \vdots & \vdots \\ 2001 & 15 & 0.592 \\ 2002 & 12 & 0.627 \\ 2003 & 16 & 0.608 \\ 2004 & 15 & 0.546\end{array}$

  Explain and interpret the $\mathrm{R}$ commands and (slightly abbreviated) output below.

  

  In 2005 , the ocean temperature index was 0.743. Explain how you would predict the number of named storms for that year.

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• 4.II.13I

  Statistical Modelling | Part II, 2006

  Consider a linear model for $Y=\left(Y_{1}, \ldots, Y_{n}\right)^{T}$ given by

  $$Y=X \beta+\epsilon,$$

  where $X$ is a known $n \times p$ matrix of full rank $p<n$, where $\beta$ is an unknown vector and $\epsilon \sim N_{n}\left(0, \sigma^{2} I\right)$. Derive an expression for the maximum likelihood estimator $\hat{\beta}$ of $\beta$, and write down its distribution.

  Find also the maximum likelihood estimator $\hat{\sigma}^{2}$ of $\sigma^{2}$, and derive its distribution.

  [You may use Cochran's theorem, provided that it is stated carefully. You may also assume that the matrix $P=X\left(X^{T} X\right)^{-1} X^{T}$ has rank $p$, and that $I-P$ has rank $n-p$.]

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Statistical Physics

• 4.II.34D

  Statistical Physics | Part II, 2006

  Two examples of phenomenological temperature measurements are (i) the mark reached along the length of a liquid-in-glass thermometer; and (ii) the wavelength of the brightest colour of electromagnetic radiation emitted by a hot body (used, for example, to measure the surface temperature of a star).

  Give the definition of temperature in statistical physics, and explain how the analysis of ideal gases and black body radiation is used to calibrate and improve phenomenological temperature measurements like those mentioned above. You should give brief derivations of any key results that you use.

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Stochastic Financial Models

• 4.II.28I

  Stochastic Financial Models | Part II, 2006

  State the definitions of a martingale and a stopping time.

  State and prove the optional sampling theorem.

  If $\left(M_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale, under what conditions is it true that $M_{n}$ converges with probability 1 as $n \rightarrow \infty$ ? Show by an example that some condition is necessary.

  A market consists of $K>1$ agents, each of whom is either optimistic or pessimistic. At each time $n=0,1, \ldots$, one of the agents is selected at random, and chooses to talk to one of the other agents (again selected at random), whose type he then adopts. If choices in different periods are independent, show that the proportion of pessimists is a martingale. What can you say about the limiting behaviour of the proportion of pessimists as time $n$ tends to infinity?

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Topics in Analysis

• 4.I.2G

  Topics in Analysis | Part II, 2006

  (a) State the Baire category theorem, in its closed-sets version.

  (b) For every $n \in \mathbb{N}$ let $f_{n}$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$, and let $g(x)=1$ when $x$ is rational and 0 otherwise. For each $N \in \mathbb{N}$, let

  $$F_{N}=\left\{x \in \mathbb{R}: \forall n \geqslant N \quad f_{n}(x) \leqslant \frac{1}{3} \quad \text { or } \quad f_{n}(x) \geqslant \frac{2}{3}\right\} .$$

  By applying the Baire category theorem, prove that the functions $f_{n}$ cannot converge pointwise to $g$. (That is, it is not the case that $f_{n}(x) \rightarrow g(x)$ for every $x \in \mathbb{R}$.)

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Waves

• 4.II $. 38 \mathrm{C} \quad$

  Waves | Part II, 2006

  Obtain an expression for the compressive energy $W(\rho)$ per unit volume for adiabatic motion of a perfect gas, for which the pressure $p$ is given in terms of the density $\rho$ by a relation of the form

  $$\tag{*} p=p_{0}\left(\rho / \rho_{0}\right)^{\gamma},$$

  where $p_{0}, \rho_{0}$ and $\gamma$ are positive constants.

  For one-dimensional motion with speed $u$ write down expressions for the mass flux and the momentum flux. Deduce from the energy flux $u\left(p+W+\frac{1}{2} \rho u^{2}\right)$ together with the mass flux that if the motion is steady then

  $$\tag{†} \frac{\gamma}{\gamma-1} \frac{p}{\rho}+\frac{1}{2} u^{2}=\text { constant }$$

  A one-dimensional shock wave propagates at constant speed along a tube containing the gas. Ahead of the shock the gas is at rest with pressure $p_{0}$ and density $\rho_{0}$. Behind the shock the pressure is maintained at the constant value $(1+\beta) p_{0}$ with $\beta>0$. Determine the density $\rho_{1}$ behind the shock, assuming that $(†)$ holds throughout the flow.

  For small $\beta$ show that the changes in pressure and density across the shock satisfy the adiabatic relation $(*)$ approximately, correect to order $\beta^{2}$.

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