Part II, 2001, Paper 3

# Part II, 2001, Paper 3

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B3.10

commentLet $C$ be the projective curve (over an algebraically closed field $k$ of characteristic zero) defined by the affine equation

$x^{5}+y^{5}=1$

Determine the points at infinity of $C$ and show that $C$ is smooth.

Determine the divisors of the rational functions $x, y \in k(C)$.

Show that $\omega=d x / y^{4}$ is a regular differential on $C$.

Compute the divisor of $\omega$. What is the genus of $C$ ?

B3.7

commentWrite down the Mayer-Vietoris sequence and describe all the maps involved.

Use the Mayer-Vietoris sequence to compute the homology of the $n$-sphere $S^{n}$ for all $n$.

A3.10

comment(i) Let $P$ be the problem

$\text { minimize } f(x) \quad \text { subject to } h(x)=b, \quad x \in X \text {. }$

Explain carefully what it means for the problem $P$ to be Strong Lagrangian.

Outline the main steps in a proof that a quadratic programming problem

$\operatorname{minimize} \frac{1}{2} x^{T} Q x+c^{T} x \quad \text { subject to } A x \geqslant b$

where $Q$ is a symmetric positive semi-definite matrix, is Strong Lagrangian.

[You should carefully state the results you need, but should not prove them.]

(ii) Consider the quadratic programming problem:

$\begin{aligned} \operatorname{minimize} & x_{1}^{2}+2 x_{1} x_{2}+2 x_{2}^{2}+x_{1}-4 x_{2} \\ \text { subject to } 3 x_{1}+2 x_{2} \leqslant 6, \quad x_{1}+x_{2} \geqslant 1 . \end{aligned}$

State necessary and sufficient conditions for $\left(x_{1}, x_{2}\right)$ to be optimal, and use the activeset algorithm (explaining your steps briefly) to solve the problem starting with initial condition $(2,0)$. Demonstrate that the solution you have found is optimal by showing that it satisfies the necessary and sufficient conditions stated previously.

B3.23

commentWrite down the commutation relations satisfied by the cartesian components of the total angular momentum operator $\mathbf{J}$.

In quantum mechanics an operator $\mathbf{V}$ is said to be a vector operator if, under rotations, its components transform in the same way as those of the coordinate operator r. Show from first principles that this implies that its cartesian components satisfy the commutation relations

$\left[J_{j}, V_{k}\right]=i \epsilon_{j k l} V_{l}$

Hence calculate the total angular momentum of the nonvanishing states $V_{j}|0\rangle$, where $|0\rangle$ is the vacuum state.

B3.13

commentConsider an $M / G / 1$ queue with arrival rate $\lambda$ and traffic intensity less

than 1. Prove that the moment-generating function of a typical busy period, $M_{B}(\theta)$, satisfies

$M_{B}(\theta)=M_{S}\left(\theta-\lambda+\lambda M_{B}(\theta)\right),$

where $M_{S}(\theta)$ is the moment-generating function of a typical service time.

If service times are exponentially distributed with parameter $\mu>\lambda$, show that

$M_{B}(\theta)=\frac{\lambda+\mu-\theta-\left\{(\lambda+\mu-\theta)^{2}-4 \lambda \mu\right\}^{1 / 2}}{2 \lambda}$

for all sufficiently small but positive values of $\theta$.

B3.17

commentIf $A=\left(\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right)$ show that $A^{n+2}=A^{n+1}+A^{n}$ for all $n \geqslant 0$. Show that $A^{5}$ has trace 11 and deduce that the subshift map defined by $A$ has just two cycles of exact period 5. What are they?

A3.6 B3.4

comment(i) Define a hyperbolic fixed point $x_{0}$ of a flow determined by a differential equation $\dot{x}=f(x)$ where $x \in R^{n}$ and $f$ is $C^{1}$ (i.e. differentiable). State the Hartman-Grobman Theorem for flow near a hyperbolic fixed point. For nonlinear flows in $R^{2}$ with a hyperbolic fixed point $x_{0}$, does the theorem necessarily allow us to distinguish, on the basis of the linearized flow near $x_{0}$ between (a) a stable focus and a stable node; and (b) a saddle and a stable node? Justify your answers briefly.

(ii) Show that the system:

$\begin{aligned} &\dot{x}=-(\mu+1)+(\mu-3) x-y+6 x^{2}+12 x y+5 y^{2}-2 x^{3}-6 x^{2} y-5 x y^{2} \\ &\dot{y}=2-2 x+(\mu-5) y+4 x y+6 y^{2}-2 x^{2} y-6 x y^{2}-5 y^{3} \end{aligned}$

has a fixed point $\left(x_{0}, 0\right)$ on the $x$-axis. Show that there is a bifurcation at $\mu=0$ and determine the stability of the fixed point for $\mu>0$ and for $\mu<0$.

Make a linear change of variables of the form $u=x-x_{0}+\alpha y, v=x-x_{0}+\beta y$, where $\alpha$ and $\beta$ are constants to be determined, to bring the system into the form:

$\begin{aligned} &\dot{u}=v+u\left[\mu-\left(u^{2}+v^{2}\right)\right] \\ &\dot{v}=-u+v\left[\mu-\left(u^{2}+v^{2}\right)\right] \end{aligned}$

and hence determine whether the periodic orbit produced in the bifurcation is stable or unstable, and whether it exists in $\mu<0$ or $\mu>0$.

A3.5 B3.3

comment(i) Develop the theory of electromagnetic waves starting from Maxwell equations in vacuum. You should relate the wave-speed $c$ to $\epsilon_{0}$ and $\mu_{0}$ and establish the existence of plane, plane-polarized waves in which $\mathbf{E}$ takes the form

$\mathbf{E}=\left(E_{0} \cos (k z-\omega t), 0,0\right) .$

You should give the form of the magnetic field $\mathbf{B}$ in this case.

(ii) Starting from Maxwell's equation, establish Poynting's theorem.

$-\mathbf{j} \cdot \mathbf{E}=\frac{\partial W}{\partial t}+\nabla \cdot \mathbf{S},$

where $W=\frac{\epsilon_{0}}{2} \mathbf{E}^{2}+\frac{1}{2 \mu_{0}} \mathbf{B}^{2}$ and $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \wedge \mathbf{B}$. Give physical interpretations of $W, S$ and the theorem.

Compute the averages over space and time of $W$ and $\mathbf{S}$ for the plane wave described in (i) and relate them. Comment on the result.

B3.24

commentA planar flow of an inviscid, incompressible fluid is everywhere in the $x$-direction and has velocity profile

$u=\left\{\begin{array}{cc} U & y>0, \\ 0 & y<0 . \end{array}\right.$

By examining linear perturbations to the vortex sheet at $y=0$ that have the form $e^{i k x-i \omega t}$, show that

$\omega=\frac{1}{2} k U(1 \pm i)$

and deduce the temporal stability of the sheet to disturbances of wave number $k$.

Use this result to determine also the spatial growth rate and propagation speed of disturbances of frequency $\omega$ introduced at a fixed spatial position.

A3.13 B3.21

comment(i) Write the Hamiltonian for the harmonic oscillator,

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

in terms of creation and annihilation operators, defined by

$a^{\dagger}=\left(\frac{m \omega}{2 \hbar}\right)^{\frac{1}{2}}\left(x-i \frac{p}{m \omega}\right), \quad a=\left(\frac{m \omega}{2 \hbar}\right)^{\frac{1}{2}}\left(x+i \frac{p}{m \omega}\right)$

Obtain an expression for $\left[a^{\dagger}, a\right]$ by using the usual commutation relation between $p$ and $x$. Deduce the quantized energy levels for this system.

(ii) Define the number operator, $N$, in terms of creation and annihilation operators, $a^{\dagger}$ and $a$. The normalized eigenvector of $N$ with eigenvalue $n$ is $|n\rangle$. Show that $n \geq 0$.

Determine $a|n\rangle$ and $a^{\dagger}|n\rangle$ in the basis defined by $\{|n\rangle\}$.

Show that

$a^{\dagger m} a^{m}|n\rangle=\left\{\begin{aligned} \frac{n !}{(n-m) !}|n\rangle, & m \leq n \\ 0, & m>n \end{aligned}\right.$

Verify the relation

$|0\rangle\langle 0|=\sum_{m=0} \frac{1}{m !}(-1)^{m} a^{\dagger m} a^{m}$

by considering the action of both sides of the equation on an arbitrary basis vector.

A3.3 B3.2

comment(i) Define the notion of a measurable function between measurable spaces. Show that a continuous function $\mathbb{R}^{2} \rightarrow \mathbb{R}$ is measurable with respect to the Borel $\sigma$-fields on $\mathbb{R}^{2}$ and $\mathbb{R}$.

By using this, or otherwise, show that, when $f, g: X \rightarrow \mathbb{R}$ are measurable with respect to some $\sigma$-field $\mathcal{F}$ on $X$ and the Borel $\sigma$-field on $\mathbb{R}$, then $f+g$ is also measurable.

(ii) State the Monotone Convergence Theorem for $[0, \infty]$-valued functions. Prove the Dominated Convergence Theorem.

[You may assume the Monotone Convergence Theorem but any other results about integration that you use will need to be stated carefully and proved.]

Let $X$ be the real Banach space of continuous real-valued functions on $[0,1]$ with the uniform norm. Fix $u \in X$ and define

$T: X \rightarrow \mathbb{R} ; \quad f \mapsto \int_{0}^{1} f(t) u(t) d t$

Show that $T$ is a bounded, linear map with norm

$\|T\|=\int_{0}^{1}|u(t)| d t$

Is it true, for every choice of $u$, that there is function $f \in X$ with $\|f\|=1$ and $\|T(f)\|=\|T\|$ ?

B3.6

commentLet $\mathbf{F}_{p}$ be the finite field with $p$ elements ( $p$ a prime), and let $k$ be a finite extension of $\mathbf{F}_{p}$. Define the Frobenius automorphism $\sigma: k \longrightarrow k$, verifying that it is an $\mathbf{F}_{p^{-}}$ automorphism of $k$.

Suppose $f=X^{p+1}+X^{p}+1 \in \mathbf{F}_{p}[X]$ and that $K$ is its splitting field over $\mathbf{F}_{p}$. Why are the zeros of $f$ distinct? If $\alpha$ is any zero of $f$ in $K$, show that $\sigma(\alpha)=-\frac{1}{\alpha+1}$. Prove that $f$ has at most two zeros in $\mathbf{F}_{p}$ and that $\sigma^{3}=i d$. Deduce that the Galois group of $f$ over $\mathbf{F}_{p}$ is a cyclic group of order three.

A3.7

comment(i) Give the definition of the surface area of a parametrized surface in $\mathbf{R}^{3}$ and show that it does not depend on the parametrization.

(ii) Let $\varphi(u)>0$ be a differentiable function of $u$. Consider the surface of revolution:

$\left(\begin{array}{l} u \\ v \end{array}\right) \mapsto f(u, v)=\left(\begin{array}{c} \varphi(u) \cos (v) \\ \varphi(u) \sin (v) \\ u \end{array}\right)$

Find a formula for each of the following: (a) The first fundamental form. (b) The unit normal. (c) The second fundamental form. (d) The Gaussian curvature.

A3.4

comment(i) Let $G$ be the cyclic subgroup of $G L_{2}(\mathbf{C})$ generated by the matrix $\left(\begin{array}{cc}1 & 2 \\ 0 & -1\end{array}\right)$, acting on the polynomial ring $\mathbf{C}[X, Y]$. Determine the ring of invariants $\mathbf{C}[X, Y]^{G}$.

(ii) Determine $\mathbf{C}[X, Y]^{G}$ when $G$ is the cyclic group generated by $\left(\begin{array}{cc}0 & -1 \\ 1 & -1\end{array}\right)$.

[Hint: consider the eigenvectors.]

B3.8

commentLet $T$ be a bounded linear operator on a Hilbert space $H$. Define what it means to say that $T$ is (i) compact, and (ii) Fredholm. What is the index, ind $(T)$, of a Fredholm operator $T$ ?

Let $S, T$ be bounded linear operators on $H$. Prove that $S$ and $T$ are Fredholm if and only if both $S T$ and $T S$ are Fredholm. Prove also that if $S$ is invertible and $T$ is Fredholm then $\operatorname{ind}(S T)=\operatorname{ind}(T S)=\operatorname{ind}(T)$.

Let $K$ be a compact linear operator on $H$. Prove that $I+K$ is Fredholm with index zero.

A3.8 B3.11

comment(i) Write down a set of axioms for the theory of dense linear order with a bottom element but no top element.

(ii) Prove that this theory has, up to isomorphism, precisely one countable model.

A3.1 B3.1

comment(i) Explain what is meant by the transition semigroup $\left\{P_{t}\right\}$ of a Markov chain $X$ in continuous time. If the state space is finite, show under assumptions to be stated clearly, that $P_{t}^{\prime}=G P_{t}$ for some matrix $G$. Show that a distribution $\pi$ satisfies $\pi G=0$ if and only if $\pi P_{t}=\pi$ for all $t \geqslant 0$, and explain the importance of such $\pi$.

(ii) Let $X$ be a continuous-time Markov chain on the state space $S=\{1,2\}$ with generator

$G=\left(\begin{array}{cc} -\beta & \beta \\ \gamma & -\gamma \end{array}\right), \quad \text { where } \beta, \gamma>0 .$

Show that the transition semigroup $P_{t}=\exp (t G)$ is given by

$(\beta+\gamma) P_{t}=\left(\begin{array}{cc} \gamma+\beta h(t) & \beta(1-h(t)) \\ \gamma(1-h(t)) & \beta+\gamma h(t) \end{array}\right),$

where $h(t)=e^{-t(\beta+\gamma)}$.

For $0<\alpha<1$, let

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

For a continuous-time chain $X$, let $M$ be a matrix with $(i, j)$ entry

$P(X(1)=j \mid X(0)=i)$, for $i, j \in S$. Show that there is a chain $X$ with $M=H(\alpha)$ if and only if $\alpha>\frac{1}{2}$.

A3.17

comment(i) The function $y(x)$ satisfies the differential equation

$y^{\prime \prime}+b y^{\prime}+c y=0, \quad 0<x<1,$

where $b$ and $c$ are constants, with boundary conditions $y(0)=0, y^{\prime}(0)=1$. By integrating this equation or otherwise, show that $y$ must also satisfy the integral equation

$y(x)=g(x)+\int_{0}^{1} K(x, t) y(t) d t$

and find the functions $g(x)$ and $K(x, t)$.

(ii) Solve the integral equation

$\varphi(x)=1+\lambda^{2} \int_{0}^{x}(x-t) \varphi(t) d t, \quad x>0, \quad \lambda \text { real }$

by finding an ordinary differential equation satisfied by $\varphi(x)$ together with boundary conditions.

Now solve the integral equation by the method of successive approximations and show that the solutions are the same.

B3.19

commentConsider the integral

$\int_{0}^{\infty} \frac{t^{z} \mathrm{e}^{-a t}}{1+t} d t$

where $t^{z}$ is the principal branch and $a$ is a positive constant. State the region of the complex $z$-plane in which the integral defines a holomorphic function.

Show how the analytic continuation of this function can be obtained by means of an alternative integral representation using the Hankel contour.

Hence show that the analytic continuation is holomorphic except for simple poles at $z=-1,-2, \ldots$, and that the residue at $z=-n$ is

$(-1)^{n-1} \sum_{r=0}^{n-1} \frac{a^{r}}{r !}$

Part II

A3.18

comment(i) The so-called breather solution of the sine-Gordon equation is

$\phi(x, t)=4 \tan ^{-1}\left(\frac{\left(1-\lambda^{2}\right)^{\frac{1}{2}}}{\lambda} \frac{\sin \lambda t}{\cosh \left(1-\lambda^{2}\right)^{\frac{1}{2}} x}\right), \quad 0<\lambda<1$

Describe qualitatively the behaviour of $\phi(x, t)$, for $\lambda \ll 1$, when $|x| \gg \ln (2 / \lambda)$, when $|x| \ll 1$, and when $\cosh x \approx \frac{1}{\lambda}|\sin \lambda t|$. Explain how this solution can be interpreted in terms of motion of a kink and an antikink. Estimate the greatest separation of the kink and antikink.

(ii) The field $\psi(x, t)$ obeys the nonlinear wave equation

$\frac{\partial^{2} \psi}{\partial t^{2}}-\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{d U}{d \psi}=0$

where the potential $U$ has the form

$U(\psi)=\frac{1}{2}\left(\psi-\psi^{3}\right)^{2} .$

Show that $\psi=0$ and $\psi=1$ are stable constant solutions.

Find a steady wave solution $\psi=f(x-v t)$ satisfying the boundary conditions $\psi \rightarrow 0$ as $x \rightarrow-\infty, \psi \rightarrow 1$ as $x \rightarrow \infty$. What constraint is there on the velocity $v ?$

A3.9

comment(i) State the law of quadratic reciprocity.

Prove that 5 is a quadratic residue modulo primes $p \equiv \pm 1 \quad(\bmod 10)$ and a quadratic non-residue modulo primes $p \equiv \pm 3 \quad(\bmod 10)$.

Determine whether 5 is a quadratic residue or non-residue modulo 119 and modulo $539 .$

(ii) Explain what is meant by the continued fraction of a real number $\theta$. Define the convergents to $\theta$ and write down the recurrence relations satisfied by their numerators and denominators.

Use the continued fraction method to find two solutions in positive integers $x, y$ of the equation $x^{2}-15 y^{2}=1$.

A3.19 B3.20

comment(i) The diffusion equation

$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$

is discretized by the finite-difference method

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

where $u_{m}^{n} \approx u(m \Delta x, n \Delta t), \mu=\Delta t /(\Delta x)^{2}$ and $\alpha$ is a constant. Derive the order of magnitude (as a power of $\Delta x$ ) of the local error for different choices of $\alpha$.

(ii) Investigate the stability of the above finite-difference method for different values of $\alpha$ by the Fourier technique.

B3.14

commentA file of $X \mathrm{Mb}$ is to be transmitted over a communications link. At each time $t$ the sender can choose a transmission rate, $u(t)$, within the range $[0,1]$ Mb per second. The charge for transmitting at rate $u(t)$ at time $t$ is $u(t) p(t)$. The function $p$ is fully known at time 0. If it takes a total time $T$ to transmit the file then there is a delay cost of $\gamma T^{2}$, $\gamma>0$. Thus $u$ and $T$ are to be chosen to minimize

$\int_{0}^{T} u(t) p(t) d t+\gamma T^{2}$

where $u(t) \in[0,1], d x(t) / d t=-u(t), x(0)=X$ and $x(T)=0$. Quoting and applying appropriate results of Pontryagin's maximum principle show that a property of the optimal policy is that there exists $p^{*}$ such that $u(t)=1$ if $p(t)<p^{*}$ and $u(t)=0$ if $p(t)>p^{*}$.

Show that the optimal $p^{*}$ and $T$ are related by $p^{*}=p(T)+2 \gamma T$.

Suppose $p(t)=t+1 / t$ and $X=1$. For what value of $\gamma$ is it optimal to transmit at a constant rate 1 between times $1 / 2$ and $3 / 2$ ?

B3.18

commentWrite down a formula for the solution $u=u(t, x)$, for $t>0$ and $x \in \mathbb{R}^{n}$, of the initial value problem for the heat equation:

$\frac{\partial u}{\partial t}-\Delta u=0 \quad u(0, x)=f(x)$

for $f$ a bounded continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. State (without proof) a theorem which ensures that this formula is the unique solution in some class of functions (which should be explicitly described).

By writing $u=e^{t} v$, or otherwise, solve the initial value problem

$\tag{†} \frac{\partial v}{\partial t}+v-\Delta v=0, \quad v(0, x)=g(x)$

for $g$ a bounded continuous function $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and give a class of functions in which your solution is the unique one.

Hence, or otherwise, prove that for all $t>0$ :

$\sup _{x \in \mathbb{R}^{n}} v(t, x) \leqslant \sup _{x \in \mathbb{R}^{n}} g(x)$

and deduce that the solutions $v_{1}(t, x)$ and $v_{2}(t, x)$ of $(†)$ corresponding to initial values $g_{1}(x)$ and $g_{2}(x)$ satisfy, for $t>0$,

$\sup _{x \in \mathbb{R}^{n}}\left|v_{1}(t, x)-v_{2}(t, x)\right| \leqslant \sup _{x \in \mathbb{R}^{n}}\left|g_{1}(x)-g_{2}(x)\right| .$

A3.2

comment(i) (a) Write down Hamilton's equations for a dynamical system. Under what condition is the Hamiltonian a constant of the motion? What is the condition for one of the momenta to be a constant of the motion?

(b) Explain the notion of an adiabatic invariant. Give an expression, in terms of Hamiltonian variables, for one such invariant.

(ii) A mass $m$ is attached to one end of a straight spring with potential energy $\frac{1}{2} k r^{2}$, where $k$ is a constant and $r$ is the length. The other end is fixed at a point $O$. Neglecting gravity, consider a general motion of the mass in a plane containing $O$. Show that the Hamiltonian is given by

$H=\frac{1}{2} \frac{p_{\theta}^{2}}{m r^{2}}+\frac{1}{2} \frac{p_{r}^{2}}{m}+\frac{1}{2} k r^{2},$

where $\theta$ is the angle made by the spring relative to a fixed direction, and $p_{\theta}, p_{r}$ are the generalised momenta. Show that $p_{\theta}$ and the energy $E$ are constants of the motion, using Hamilton's equations.

If the mass moves in a non-circular orbit, and the spring constant $k$ is slowly varied, the orbit gradually changes. Write down the appropriate adiabatic invariant $J\left(E, p_{\theta}, k, m\right)$. Show that $J$ is proportional to

$\sqrt{m k}\left(r_{+}-r_{-}\right)^{2},$

where

$r_{\pm}^{2}=\frac{E}{k} \pm \sqrt{\left(\frac{E}{k}\right)^{2}-\frac{p_{\theta}^{2}}{m k}}$

Consider an orbit for which $p_{\theta}$ is zero. Show that, as $k$ is slowly varied, the energy $E \propto k^{\alpha}$, for a constant $\alpha$ which should be found.

[You may assume without proof that

$\left.\int_{r_{-}}^{r_{+}} d r \sqrt{\left(1-\frac{r^{2}}{r_{+}^{2}}\right)\left(1-\frac{r_{-}^{2}}{r^{2}}\right)}=\frac{\pi}{4 r_{+}}\left(r_{+}-r_{-}\right)^{2} \cdot\right]$

A3.12 B3.15

comment(i) Explain what is meant by a uniformly most powerful unbiased test of a null hypothesis against an alternative.

Let $Y_{1}, \ldots, Y_{n}$ be independent, identically distributed $N\left(\mu, \sigma^{2}\right)$ random variables, with $\sigma^{2}$ known. Explain how to construct a uniformly most powerful unbiased size $\alpha$ test of the null hypothesis that $\mu=0$ against the alternative that $\mu \neq 0$.

(ii) Outline briefly the Bayesian approach to hypothesis testing based on Bayes factors.

Let the distribution of $Y_{1}, \ldots, Y_{n}$ be as in (i) above, and suppose we wish to test, as in (i), $\mu=0$ against the alternative $\mu \neq 0$. Suppose we assume a $N\left(0, \tau^{2}\right)$ prior for $\mu$ under the alternative. Find the form of the Bayes factor $B$, and show that, for fixed $n, B$ $\rightarrow \infty$ as $\tau \rightarrow \infty$.

B3.12

commentState and prove Birkhoff's almost-everywhere ergodic theorem.

[You need not prove convergence in $L_{p}$ and the maximal ergodic lemma may be assumed provided that it is clearly stated.]

Let $\Omega=[0,1), \mathcal{F}=\mathcal{B}([0,1))$ be the Borel $\sigma$-field and let $\mathbb{P}$ be Lebesgue measure on $(\Omega, \mathcal{F})$. Give an example of an ergodic measure-preserving map $\theta: \Omega \rightarrow \Omega$ (you need not prove it is ergodic).

Let $f(x)=x$ for $x \in[0,1)$. Find (at least for all $x$ outside a set of measure zero)

$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n}\left(f \circ \theta^{i-1}\right)(x) .$

Briefly justify your answer.

B3.5

commentLet $G=S U_{2}$, and $V_{n}$ be the vector space of homogeneous polynomials of degree $n$ in the variables $x$ and $y$.

(i) Define the action of $G$ on $V_{n}$, and prove that $V_{n}$ is an irreducible representation of $G$.

(ii) Decompose $V_{4} \otimes V_{3}$ into irreducible representations of $S U_{2}$. Briefly justify your answer.

(iii) $S U_{2}$ acts on the vector space $M_{3}(\mathbf{C})$ of complex $3 \times 3$ matrices via

$\rho\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \cdot X=\left(\begin{array}{lll} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{array}\right) X\left(\begin{array}{lll} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{array}\right)^{-1}, \quad X \in M_{3}(\mathbf{C}) .$

Decompose this representation into irreducible representations.

B3.9

commentLet $f: X \rightarrow Y$ be a nonconstant holomorphic map between compact connected Riemann surfaces. Define the valency of $f$ at a point, and the degree of $f$.

Define the genus of a compact connected Riemann surface $X$ (assuming the existence of a triangulation).

State the Riemann-Hurwitz theorem. Show that a holomorphic non-constant selfmap of a compact Riemann surface of genus $g>1$ is bijective, with holomorphic inverse. Verify that the Riemann surface in $\mathbb{C}^{2}$ described in the equation $w^{4}=z^{4}-1$ is non-singular, and describe its topological type.

[Note: The description can be in the form of a picture or in words. If you apply RiemannHurwitz, explain first how you compactify the surface.]

B3.22

commentA system consists of $N$ weakly interacting non-relativistic fermions, each of mass $m$, in a three-dimensional volume, $V$. Derive the Fermi-Dirac distribution

$n(\epsilon)=K V g \frac{\epsilon^{1 / 2}}{\exp ((\epsilon-\mu) / k T)+1}$

where $n(\epsilon) d \epsilon$ is the number of particles with energy in $(\epsilon, \epsilon+d \epsilon)$ and $K=2 \pi(2 m)^{3 / 2} / h^{3}$. Explain the physical significance of $g$.

Explain how the constant $\mu$ is determined by the number of particles $N$ and write down expressions for $N$ and the internal energy $E$ in terms of $n(\epsilon)$.

Show that, in the limit $\kappa \equiv e^{-\mu / k T} \gg 1$,

$N=\frac{V}{A \kappa}\left(1-\frac{1}{2 \sqrt{2} \kappa}+O\left(\frac{1}{\kappa^{2}}\right)\right)$

where $A=h^{3} / g(2 \pi m k T)^{3 / 2}$.

Show also that in this limit

$E=\frac{3}{2} N k T\left(1+\frac{1}{4 \sqrt{2} \kappa}+O\left(\frac{1}{\kappa^{2}}\right)\right) \text {. }$

Deduce that the condition $\kappa \gg 1$ implies that $A N / V \ll 1$. Discuss briefly whether or not this latter condition is satisfied in a white dwarf star and in a dilute electron gas at room temperature.

$\left[\right.$ Note that $\left.\int_{0}^{\infty} d u e^{-u^{2} a}=\frac{1}{2} \sqrt{\frac{\pi}{a}}\right]$.

A3.14

comment(i) A spherically symmetric star has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is distance from the star's centre. Stating without proof any theorems you may need, show that mechanical equilibrium implies the Newtonian pressure support equation

$P^{\prime}=-\frac{G m \rho}{r^{2}},$

where $m(r)$ is the mass within radius $r$ and $P^{\prime}=d P / d r$.

Write down an integral expression for the total gravitational potential energy, $E_{g r}$. Use this to derive the "virial theorem"

$E_{g r}=-3\langle P\rangle V$

when $\langle P\rangle$ is the average pressure.

(ii) Given that the total kinetic energy, $E_{k i n}$, of a spherically symmetric star is related to its average pressure by the formula

$E_{k i n}=\alpha\langle P\rangle V$

for constant $\alpha$, use the virial theorem (stated in part (i)) to determine the condition on $\alpha$ needed for gravitational binding. State the relation between pressure $P$ and "internal energy" $U$ for an ideal gas of non-relativistic particles. What is the corresponding relation for ultra-relativistic particles? Hence show that the formula $(*)$ applies in these cases, and determine the values of $\alpha$.

Why does your result imply a maximum mass for any star, whatever the source of its pressure? What is the maximum mass, approximately, for stars supported by

(a) thermal pressure,

(b) electron degeneracy pressure (White Dwarf),

(c) neutron degeneracy pressure (Neutron Star).

A White Dwarf can accrete matter from a companion star until its mass exceeds the Chandrasekar limit. Explain briefly the process by which it then evolves into a neutron star.

A3.11 B3.16

comment(i) Suppose that $Z$ is a random variable having the normal distribution with $\mathbb{E} Z=\beta$ and $\operatorname{Var} Z=\tau^{2}$.

For positive constants $a, c$ show that

$\mathbb{E}\left(a e^{Z}-c\right)_{+}=a e^{\left(\beta+\tau^{2} / 2\right)} \Phi\left(\frac{\log (a / c)+\beta}{\tau}+\tau\right)-c \Phi\left(\frac{\log (a / c)+\beta}{\tau}\right)$

where $\Phi$ is the standard normal distribution function.

In the context of the Black-Scholes model, derive the formula for the price at time 0 of a European call option on the stock at strike price $c$ and maturity time $t_{0}$ when the interest rate is $\rho$ and the volatility of the stock is $\sigma$.

(ii) Let $p$ denote the price of the call option in the Black-Scholes model specified in (i). Show that $\frac{\partial p}{\partial \rho}>0$ and sketch carefully the dependence of $p$ on the volatility $\sigma$ (when the other parameters in the model are held fixed).

Now suppose that it is observed that the interest rate lies in the range $0<\rho<\rho_{0}$ and when it changes it is linked to the volatility by the formula $\sigma=\ln \left(\rho_{0} / \rho\right)$. Consider a call option at strike price $c=S_{0}$, where $S_{0}$ is the stock price at time 0 . There is a small increase $\Delta \rho$ in the interest rate; will the price of the option increase or decrease (assuming that the stock price is unaffected)? Justify your answer carefully.

[You may assume that the function $\phi(x) / \Phi(x)$ is decreasing in $x,-\infty<x<\infty$, and increases to $+\infty$ as $x \rightarrow-\infty$, where $\Phi$ is the standard-normal distribution function and $\phi=\Phi^{\prime}$.]

A3.15

comment(i) The pions form an isospin triplet with $\pi^{+}=|1,1\rangle, \pi^{0}=|1,0\rangle$ and $\pi^{-}=|1,-1\rangle$, whilst the nucleons form an isospin doublet with $p=\left|\frac{1}{2}, \frac{1}{2}\right\rangle$ and $n=\left|\frac{1}{2},-\frac{1}{2}\right\rangle$. Consider the isospin representation of two-particle states spanned by the basis

$T=\left\{\left|\pi^{+} p\right\rangle,\left|\pi^{+} n\right\rangle,\left|\pi^{0} p\right\rangle,\left|\pi^{0} n\right\rangle,\left|\pi^{-} p\right\rangle,\left|\pi^{-} n\right\rangle\right\}$

State which irreducible representations are contained in this representation and explain why $\left|\pi^{+} p\right\rangle$ is an isospin eigenstate.

Using

$I_{-}|j, m\rangle=\sqrt{(j-m+1)(j+m)}|j, m-1\rangle,$

where $I_{-}$is the isospin ladder operator, write the isospin eigenstates in terms of the basis, $T$.

(ii) The Lie algebra $s u(2)$ of generators of $S U(2)$ is spanned by the operators $\left\{J_{+}, J_{-}, J_{3}\right\}$ satisfying the commutator algebra $\left[J_{+}, J_{-}\right]=2 J_{3}$ and $\left[J_{3}, J_{\pm}\right]=\pm J_{\pm}$. Let $\Psi_{j}$ be an eigenvector of $J_{3}: J_{3}\left(\Psi_{j}\right)=j \Psi_{j}$ such that $J_{+} \Psi_{j}=0$. The vector space $V_{j}=\operatorname{span}\left\{J_{-}^{n} \Psi_{j}: n \in \mathbb{N}_{0}\right\}$ together with the action of an arbitrary su(2) operator $A$ on $V_{j}$ defined by

$J_{-}\left(J_{-}^{n} \Psi_{j}\right)=J_{-}^{n+1} \Psi_{j}, \quad A\left(J_{-}^{n} \Psi_{j}\right)=\left[A, J_{-}\right]\left(J_{-}^{n-1} \Psi_{j}\right)+J_{-}\left(A\left(J_{-}^{n-1} \Psi_{j}\right)\right)$

forms a representation (not necessarily reducible) of $s u(2)$. Show that if $J_{-}^{n} \Psi_{j}$ is nontrivial then it is an eigenvector of $J_{3}$ and find its eigenvalue. Given that $\left[J_{+}, J_{-}^{n}\right]=$ $\alpha_{n} J_{-}^{n-1} J_{3}+\beta_{n} J_{-}^{n-1}$ show that $\alpha_{n}$ and $\beta_{n}$ satisfy

$\alpha_{n}=\alpha_{n-1}+2, \quad \beta_{n}=\beta_{n-1}-\alpha_{n-1}$

By solving these equations evaluate $\left[J_{+}, J_{-}^{n}\right]$. Show that $J_{+} J_{-}^{2 j+1} \Psi_{j}=0$. Hence show that $J_{-}^{2 j+1} \Psi_{j}$ is contained in a proper sub-representation of $V_{j}$.

A3.16

comment(i) Incompressible fluid of kinematic viscosity $\nu$ occupies a parallel-sided channel $0 \leqslant y \leqslant h_{0},-\infty<x<\infty$. The wall $y=0$ is moving parallel to itself, in the $x$ direction, with velocity $\operatorname{Re}\left\{U e^{i \omega t}\right\}$, where $t$ is time and $U, \omega$ are real constants. The fluid velocity $u(y, t)$ satisfies the equation

$u_{t}=\nu u_{y y}$

write down the boundary conditions satisfied by $u$.

Assuming that

$u=\operatorname{Re}\left\{a \sinh [b(1-\eta)] e^{i \omega t}\right\}$

where $\eta=y / h_{0}$, find the complex constants $a, b$. Calculate the velocity (in real, not complex, form) in the limit $h_{0}(\omega / \nu)^{1 / 2} \rightarrow 0$.

(ii) Incompressible fluid of viscosity $\mu$ fills the narrow gap between the rigid plane $y=0$, which moves parallel to itself in the $x$-direction with constant speed $U$, and the rigid wavy wall $y=h(x)$, which is at rest. The length-scale, $L$, over which $h$ varies is much larger than a typical value, $h_{0}$, of $h$.

Assume that inertia is negligible, and therefore that the governing equations for the velocity field $(u, v)$ and the pressure $p$ are

$u_{x}+v_{y}=0, p_{x}=\mu\left(u_{x x}+u_{y y}\right), p_{y}=\mu\left(v_{x x}+v_{y y}\right)$

Use scaling arguments to show that these equations reduce approximately to

$p_{x}=\mu u_{y y}, \quad p_{y}=0$

Hence calculate the velocity $u(x, y)$, the flow rate

$Q=\int_{0}^{h} u d y$

and the viscous shear stress exerted by the fluid on the plane wall,

$\tau=-\left.\mu u_{y}\right|_{y=0}$

in terms of $p_{x}, h, U$ and $\mu$.

Now assume that $h=h_{0}(1+\epsilon \sin k x)$, where $\epsilon \ll 1$ and $k h_{0} \ll 1$, and that $p$ is periodic in $x$ with wavelength $2 \pi / k$. Show that

$Q=\frac{h_{0} U}{2}\left(1-\frac{3}{2} \epsilon^{2}+O\left(\epsilon^{4}\right)\right)$

and calculate $\tau$ correct to $O\left(\epsilon^{2}\right)$. Does increasing the amplitude $\epsilon$ of the corrugation cause an increase or a decrease in the force required to move the plane $y=0$ at the chosen speed $U ?$

B3.25

commentConsider the equation

$\phi_{t t}+\alpha^{2} \phi_{x x x x}+\beta^{2} \phi=0,$

where $\alpha$ and $\beta$ are real constants. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Find the phase velocity $c(k)$ and the group velocity $c_{g}(k)$ and sketch graphs of these functions.

Multiplying equation $(*)$ by $\phi_{t}$, obtain an equation of the form

$\frac{\partial A}{\partial t}+\frac{\partial B}{\partial x}=0$

where $A$ and $B$ are expressions involving $\phi$ and its derivatives. Give a physical interpretation of this equation.

Evaluate the time-averaged energy $\langle E\rangle$ and energy flux $\langle I\rangle$ of a monochromatic wave $\phi=\cos (k x-w t)$, and show that

$\langle I\rangle=c_{g}\langle E\rangle .$