Part II, 2012

Algebraic Geometry

• Paper 1, Section II, I

  Algebraic Geometry | Part II, 2012

  (a) Let $X$ be an affine variety, $k[X]$ its ring of functions, and let $p \in X$. Assume $k$ is algebraically closed. Define the tangent space $T_{p} X$ at $p$. Prove the following assertions.

  (i) A morphism of affine varieties $f: X \rightarrow Y$ induces a linear map

  $$d f: T_{p} X \rightarrow T_{f(p)} Y$$

  (ii) If $g \in k[X]$ and $U:=\{x \in X \mid g(x) \neq 0\}$, then $U$ has the natural structure of an affine variety, and the natural morphism of $U$ into $X$ induces an isomorphism $T_{p} U \rightarrow T_{p} X$ for all $p \in U$.

  (iii) For all $s \geqslant 0$, the subset $\left\{x \in X \mid \operatorname{dim} T_{x} X \geqslant s\right\}$ is a Zariski-closed subvariety of $X$.

  (b) Show that the set of nilpotent $2 \times 2$ matrices

  $$X=\left\{x \in \operatorname{Mat}_{2}(k) \mid x^{2}=0\right\}$$

  may be realised as an affine surface in $\mathbf{A}^{3}$, and determine its tangent space at all points $x \in X$.

  Define what it means for two varieties $Y_{1}$ and $Y_{2}$ to be birationally equivalent, and show that the variety $X$ of nilpotent $2 \times 2$ matrices is birationally equivalent to $\mathbf{A}^{2}$.

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• Paper 2, Section II, I

  Algebraic Geometry | Part II, 2012

  Let $k$ be a field, $J$ an ideal of $k\left[x_{1}, \ldots, x_{n}\right]$, and let $R=k\left[x_{1}, \ldots, x_{n}\right] / J$. Define the radical $\sqrt{J}$ of $J$ and show that it is also an ideal.

  The Nullstellensatz says that if $J$ is a maximal ideal, then the inclusion $k \subseteq R$ is an algebraic extension of fields. Suppose from now on that $k$ is algebraically closed. Assuming the above statement of the Nullstellensatz, prove the following.

  (i) If $J$ is a maximal ideal, then $J=\left(x_{1}-a_{1}, \ldots, x_{n}-a_{n}\right)$, for some $\left(a_{1}, \ldots, a_{n}\right) \in k^{n}$.

  (ii) If $J \neq k\left[x_{1}, \ldots, x_{n}\right]$, then $Z(J) \neq \emptyset$, where

  $$Z(J)=\left\{a \in k^{n} \mid f(a)=0 \text { for all } f \in J\right\}$$

  (iii) For $V$ an affine subvariety of $k^{n}$, we set

  $$I(V)=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid f(a)=0 \text { for all } a \in V\right\}$$

  Prove that $J=I(V)$ for some affine subvariety $V \subseteq k^{n}$, if and only if $J=\sqrt{J}$.

  [Hint. Given $f \in J$, you may wish to consider the ideal in $k\left[x_{1}, \ldots, x_{n}, y\right]$ generated $b y J$ and $y f-1$.]

  (iv) If $A$ is a finitely generated algebra over $k$, and $A$ does not contain nilpotent elements, then there is an affine variety $V \subseteq k^{n}$, for some $n$, with $A=k\left[x_{1}, \ldots, x_{n}\right] / I(V)$.

  Assuming $\operatorname{char}(k) \neq 2$, find $\sqrt{J}$ when $J$ is the ideal $\left(x(x-y)^{2}, y(x+y)^{2}\right)$ in $k[x, y]$.

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• Paper 3, Section II, I

  Algebraic Geometry | Part II, 2012

  Let $X \subset \mathbf{P}^{2}(\mathbf{C})$ be the projective closure of the affine curve $y^{3}=x^{4}+1$. Let $\omega$ denote the differential $d x / y^{2}$. Show that $X$ is smooth, and compute $v_{p}(\omega)$ for all $p \in X$.

  Calculate the genus of $X$.

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• Paper 4, Section II, I

  Algebraic Geometry | Part II, 2012

  Let $X$ be a smooth projective curve of genus 2, defined over the complex numbers. Show that there is a morphism $f: X \rightarrow \mathbf{P}^{1}$ which is a double cover, ramified at six points.

  Explain briefly why $X$ cannot be embedded into $\mathbf{P}^{2}$.

  For any positive integer $n$, show that there is a smooth affine plane curve which is a double cover of $\mathbf{A}^{1}$ ramified at $n$ points.

  [State clearly any theorems that you use.]

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Algebraic Topology

• Paper 1, Section II, G

  Algebraic Topology | Part II, 2012

  Define the notions of covering projection and of locally path-connected space. Show that a locally path-connected space is path-connected if it is connected.

  Suppose $f: Y \rightarrow X$ and $g: Z \rightarrow X$ are continuous maps, the space $Y$ is connected and locally path-connected and that $g$ is a covering projection. Suppose also that we are given base-points $x_{0}, y_{0}, z_{0}$ satisfying $f\left(y_{0}\right)=x_{0}=g\left(z_{0}\right)$. Show that there is a continuous $\tilde{f}: Y \rightarrow Z$ satisfying $\tilde{f}\left(y_{0}\right)=z_{0}$ and $g \tilde{f}=f$ if and only if the image of $f_{*}: \Pi_{1}\left(Y, y_{0}\right) \rightarrow \Pi_{1}\left(X, x_{0}\right)$ is contained in that of $g_{*}: \Pi_{1}\left(Z, z_{0}\right) \rightarrow \Pi_{1}\left(X, x_{0}\right)$. [You may assume the path-lifting and homotopy-lifting properties of covering projections.]

  Now suppose $X$ is locally path-connected, and both $f: Y \rightarrow X$ and $g: Z \rightarrow X$ are covering projections with connected domains. Show that $Y$ and $Z$ are homeomorphic as spaces over $X$ if and only if the images of their fundamental groups under $f_{*}$ and $g_{*}$ are conjugate subgroups of $\Pi_{1}\left(X, x_{0}\right)$.

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• Paper 2, Section II, G

  Algebraic Topology | Part II, 2012

  State the Seifert-Van Kampen Theorem. Deduce that if $f: S^{1} \rightarrow X$ is a continuous map, where $X$ is path-connected, and $Y=X \cup_{f} B^{2}$ is the space obtained by adjoining a disc to $X$ via $f$, then $\Pi_{1}(Y)$ is isomorphic to the quotient of $\Pi_{1}(X)$ by the smallest normal subgroup containing the image of $f_{*}: \Pi_{1}\left(S^{1}\right) \rightarrow \Pi_{1}(X)$.

  State the classification theorem for connected triangulable 2-manifolds. Use the result of the previous paragraph to obtain a presentation of $\Pi_{1}\left(M_{g}\right)$, where $M_{g}$ denotes the compact orientable 2 -manifold of genus $g>0$.

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• Paper 3, Section II, G

  Algebraic Topology | Part II, 2012

  State the Mayer-Vietoris Theorem for a simplicial complex $K$ expressed as the union of two subcomplexes $L$ and $M$. Explain briefly how the connecting homomorphism $\delta_{*}: H_{n}(K) \rightarrow H_{n-1}(L \cap M)$, which appears in the theorem, is defined. [You should include a proof that $\delta_{*}$ is well-defined, but need not verify that it is a homomorphism.]

  Now suppose that $|K| \cong S^{3}$, that $|L|$ is a solid torus $S^{1} \times B^{2}$, and that $|L \cap M|$ is the boundary torus of $|L|$. Show that $\delta_{*}: H_{3}(K) \rightarrow H_{2}(L \cap M)$ is an isomorphism, and hence calculate the homology groups of $M$. [You may assume that a generator of $H_{3}(K)$ may be represented by a 3 -cycle which is the sum of all the 3 -simplices of $K$, with 'matching' orientations.]

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• Paper 4, Section II, G

  Algebraic Topology | Part II, 2012

  State and prove the Lefschetz fixed-point theorem. Hence show that the $n$-sphere $S^{n}$ does not admit a topological group structure for any even $n>0$. [The existence and basic properties of simplicial homology with rational coefficients may be assumed.]

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Applications of Quantum Mechanics

• Paper 1, Section II, E

  Applications of Quantum Mechanics | Part II, 2012

  Give an account of the variational principle for establishing an upper bound on the ground-state energy $E_{0}$ of a particle moving in a potential $V(x)$ in one dimension.

  A particle of unit mass moves in the potential

  $$V(x)= \begin{cases}\infty & x \leqslant 0 \\ \lambda x & x>0\end{cases}$$

  with $\lambda$ a positive constant. Explain why it is important that any trial wavefunction used to derive an upper bound on $E_{0}$ should be chosen to vanish for $x \leqslant 0$.

  Use the trial wavefunction

  $$\psi(x)= \begin{cases}0 & x \leqslant 0 \\ x e^{-a x} & x>0\end{cases}$$

  where $a$ is a positive real parameter, to establish an upper bound $E_{0} \leqslant E(a, \lambda)$ for the energy of the ground state, and hence derive the lowest upper bound on $E_{0}$ as a function of $\lambda$.

  Explain why the variational method cannot be used in this case to derive an upper bound for the energy of the first excited state.

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• Paper 2, Section II, E

  Applications of Quantum Mechanics | Part II, 2012

  A solution of the $S$-wave Schrödinger equation at large distances for a particle of mass $m$ with momentum $\hbar k$ and energy $E=\hbar^{2} k^{2} / 2 m$, has the form

  $$\psi_{0}(\boldsymbol{r}) \sim \frac{A}{r}[\sin k r+g(k) \cos k r] .$$

  Define the phase shift $\delta_{0}$ and verify that $\tan \delta_{0}(k)=g(k)$.

  Write down a formula for the cross-section $\sigma$, for a particle of momentum $\hbar k$ scattering on a radially symmetric potential of finite range, as a function of the phase shifts $\delta_{l}$ for the partial waves with quantum number $l$.

  (i) Suppose that $g(k)=-k / K$ for $K>0$. Show that there is a bound state of energy $E_{B}=-\hbar^{2} K^{2} / 2 m$. Neglecting the contribution from partial waves with $l>0$ show that the cross section is

  $$\sigma=\frac{4 \pi}{K^{2}+k^{2}} .$$

  (ii) Suppose now that $g(k)=\gamma /\left(K_{0}-k\right)$ with $K_{0}>0, \gamma>0$ and $\gamma \ll K_{0}$. Neglecting the contribution from partial waves with $l>0$, derive an expression for the cross section $\sigma$, and show that it has a local maximum when $E \approx \hbar^{2} K_{0}^{2} / 2 \mathrm{~m}$. Discuss the interpretation of this phenomenon in terms of resonant behaviour and derive an expression for the decay width of the resonant state.

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• Paper 3, Section II, E

  Applications of Quantum Mechanics | Part II, 2012

  A simple model of a crystal consists of a 1D linear array of sites at positions $x=n a$, for all integer $n$ and separation $a$, each occupied by a similar atom. The potential due to the atom at the origin is $U(x)$, which is symmetric: $U(-x)=U(x)$. The Hamiltonian, $H_{0}$, for the atom at the $n$-th site in isolation has electron eigenfunction $\psi_{n}(x)$ with energy $E_{0}$. Write down $H_{0}$ and state the relationship between $\psi_{n}(x)$ and $\psi_{0}(x)$.

  The Hamiltonian $H$ for an electron moving in the crystal is $H=H_{0}+V(x)$. Give an expression for $V(x)$.

  In the tight-binding approximation for this model the $\psi_{n}$ are assumed to be orthonormal, $\left(\psi_{n}, \psi_{m}\right)=\delta_{n m}$, and the only non-zero matrix elements of $H_{0}$ and $V$ are

  $$\left(\psi_{n}, H_{0} \psi_{n}\right)=E_{0}, \quad\left(\psi_{n}, V \psi_{n}\right)=\alpha, \quad\left(\psi_{n}, V \psi_{n \pm 1}\right)=-A$$

  where $A>0$. By considering the trial wavefunction $\Psi(x, t)=\sum_{n} c_{n}(t) \psi_{n}(x)$, show that the time-dependent Schrödinger equation governing the amplitudes $c_{n}(t)$ is

  $$i \hbar \dot{c}_{n}=\left(E_{0}+\alpha\right) c_{n}-A\left(c_{n+1}+c_{n-1}\right)$$

  By examining a solution of the form

  $$c_{n}=e^{i(k n a-E t / \hbar)}$$

  show that $E$, the energy of the electron in the crystal, lies in a band given by

  $$E=E_{0}+\alpha-2 A \cos k a$$

  Using the fact that $\psi_{0}(x)$ is a parity eigenstate show that

  $$\left(\psi_{n}, x \psi_{n}\right)=n a .$$

  The electron in this model is now subject to an electric field $\mathcal{E}$ in the direction of increasing $x$, so that $V(x)$ is replaced by $V(x)-e \mathcal{E} x$, where $-e$ is the charge on the electron. Assuming that $\left(\psi_{n}, x \psi_{m}\right)=0, n \neq m$, write down the new form of the time-dependent Schrödinger equation for the probability amplitudes $c_{n}$. Verify that it has solutions of the form

  $$c_{n}=\exp \left[-\frac{i}{\hbar} \int_{0}^{t} \epsilon\left(t^{\prime}\right) d t^{\prime}+i\left(k+\frac{e \mathcal{E} t}{\hbar}\right) n a\right]$$

  where

  $$\epsilon(t)=E_{0}+\alpha-2 A \cos \left[\left(k+\frac{e \mathcal{E} t}{\hbar}\right) a\right]$$

  Use this result to show that the dynamical behaviour of an electron near the bottom of an energy band is the same as that for a free particle in the presence of an electric field with an effective mass $m^{*}=\hbar^{2} /\left(2 A a^{2}\right)$.

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• Paper 4, Section II, $33 E$

  Applications of Quantum Mechanics | Part II, 2012

  Consider a one-dimensional crystal lattice of lattice spacing $a$ with the $n$-th atom having position $r_{n}=n a+x_{n}$ and momentum $p_{n}$, for $n=0,1, \ldots, N-1$. The atoms interact with their nearest neighbours with a harmonic force and the classical Hamiltonian is

  $$H=\sum_{n} \frac{p_{n}^{2}}{2 m}+\frac{1}{2} \lambda\left(x_{n}-x_{n-1}\right)^{2}$$

  where we impose periodic boundary conditions: $x_{N}=x_{0}$. Show that the normal mode frequencies for the classical harmonic vibrations of the system are given by

  $$\omega_{l}=2 \sqrt{\frac{\lambda}{m}}\left|\sin \left(\frac{k_{l} a}{2}\right)\right|,$$

  where $k_{l}=2 \pi l / N a$, with $l$ integer and (for $N$ even, which you may assume) $-N / 2<l \leqslant$ $N / 2$. What is the velocity of sound in this crystal?

  Show how the system may be quantized to give the quantum operator

  $$x_{n}(t)=X_{0}(t)+\sum_{l \neq 0} \sqrt{\frac{\hbar}{2 N m \omega_{l}}}\left[a_{l} e^{-i\left(\omega_{l} t-k_{l} n a\right)}+a_{l}^{\dagger} e^{i\left(\omega_{l} t-k_{l} n a\right)}\right],$$

  where $a_{l}^{\dagger}$ and $a_{l}$ are creation and annihilation operators, respectively, whose commutation relations should be stated. Briefly describe the spectrum of energy eigenstates for this system, stating the definition of the ground state $|0\rangle$ and giving the expression for the energy eigenvalue of any eigenstate.

  The Debye-Waller factor $e^{-W(Q)}$ associated with Bragg scattering from this crystal is defined by the matrix element

  $$e^{-W(Q)}=\left\langle 0\left|e^{i Q x_{0}(0)}\right| 0\right\rangle$$

  In the case where $\left\langle 0\left|X_{0}\right| 0\right\rangle=0$, calculate $W(Q)$.

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Applied Probability

• Paper 1, Section II, $27 \mathrm{~K}$

  Applied Probability | Part II, 2012

  (a) Give the definition of a Poisson process $\left(N_{t}, t \geqslant 0\right)$ with rate $\lambda$, using its transition rates. Show that for each $t \geqslant 0$, the distribution of $N_{t}$ is Poisson with a parameter to be specified.

  Let $J_{0}=0$ and let $J_{1}, J_{2}, \ldots$ denote the jump times of $\left(N_{t}, t \geqslant 0\right)$. What is the distribution of $\left(J_{n+1}-J_{n}, n \geqslant 0\right) ?$ (You do not need to justify your answer.)

  (b) Let $n \geqslant 1$. Compute the joint probability density function of $\left(J_{1}, J_{2}, \ldots, J_{n}\right)$ given $\left\{N_{t}=n\right\}$. Deduce that, given $\left\{N_{t}=n\right\},\left(J_{1}, \ldots, J_{n}\right)$ has the same distribution as the nondecreasing rearrangement of $n$ independent uniform random variables on $[0, t]$.

  (c) Starting from time 0, passengers arrive on platform $9 \mathrm{~B}$ at King's Cross station, with constant rate $\lambda>0$, in order to catch a train due to depart at time $t>0$. Using the above results, or otherwise, find the expected total time waited by all passengers (the sum of all passengers' waiting times).

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• Paper 2, Section II, K

  Applied Probability | Part II, 2012

  (a) A colony of bacteria evolves as follows. Let $X$ be a random variable with values in the positive integers. Each bacterium splits into $X$ copies of itself after an exponentially distributed time of parameter $\lambda>0$. Each of the $X$ daughters then splits in the same way but independently of everything else. This process keeps going forever. Let $Z_{t}$ denote the number of bacteria at time $t$. Specify the $Q$-matrix of the Markov chain $Z=\left(Z_{t}, t \geqslant 0\right)$. [It will be helpful to introduce $p_{n}=\mathbb{P}(X=n)$, and you may assume for simplicity that $\left.p_{0}=p_{1}=0 .\right]$

  (b) Using the Kolmogorov forward equation, or otherwise, show that if $u(t)=$ $\mathbb{E}\left(Z_{t} \mid Z_{0}=1\right)$, then $u^{\prime}(t)=\alpha u(t)$ for some $\alpha$ to be explicitly determined in terms of $X$. Assuming that $\mathbb{E}(X)<\infty$, deduce the value of $u(t)$ for all $t \geqslant 0$, and show that $Z$ does not explode. [You may differentiate series term by term and exchange the order of summation without justification.]

  (c) We now assume that $X=2$ with probability 1 . Fix $0<q<1$ and let $\phi(t)=\mathbb{E}\left(q^{Z_{t}} \mid Z_{0}=1\right)$. Show that $\phi$ satisfies

  $$\phi(t)=q e^{-\lambda t}+\int_{0}^{t} \lambda e^{-\lambda s} \phi(t-s)^{2} d s$$

  By making the change of variables $u=t-s$, show that $d \phi / d t=\lambda \phi(\phi-1)$. Deduce that for all $n \geqslant 1, \mathbb{P}\left(Z_{t}=n \mid Z_{0}=1\right)=\beta^{n-1}(1-\beta)$ where $\beta=1-e^{-\lambda t}$.

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• Paper 3, Section II, K

  Applied Probability | Part II, 2012

  We consider a system of two queues in tandem, as follows. Customers arrive in the first queue at rate $\lambda$. Each arriving customer is immediately served by one of infinitely many servers at rate $\mu_{1}$. Immediately after service, customers join a single-server second queue which operates on a first-come, first-served basis, and has a service rate $\mu_{2}$. After service in this second queue, each customer returns to the first queue with probability $0<1-p<1$, and otherwise leaves the system forever. A schematic representation is given below:

  

  (a) Let $M_{t}$ and $N_{t}$ denote the number of customers at time $t$ in queues number 1 and 2 respectively, including those currently in service at time $t$. Give the transition rates of the Markov chain $\left(M_{t}, N_{t}\right)_{t \geqslant 0}$.

  (b) Write down an equation satisfied by any invariant measure $\pi$ for this Markov chain. Let $\alpha>0$ and $\beta \in(0,1)$. Define a measure $\pi$ by

  $$\pi(m, n):=e^{-\alpha} \frac{\alpha^{m}}{m !} \beta^{n}(1-\beta), \quad m, n \in\{0,1, \ldots\}$$

  Show that it is possible to find $\alpha>0, \beta \in(0,1)$ so that $\pi$ is an invariant measure of $\left(M_{t}, N_{t}\right)_{t \geqslant 0}$, if and only if $\lambda<\mu_{2} p$. Give the values of $\alpha$ and $\beta$ in this case.

  (c) Assume now that $\lambda p>\mu_{2}$. Show that the number of customers is not positive recurrent.

  [Hint. One way to solve the problem is as follows. Assume it is positive recurrent. Observe that $M_{t}$ is greater than a $M / M / \infty$ queue with arrival rate $\lambda$. Deduce that $N_{t}$ is greater than a $M / M / 1$ queue with arrival rate $\lambda p$ and service rate $\mu_{2}$. You may use without proof the fact that the departure process from the first queue is then, at equilibrium, a Poisson process with rate $\lambda$, and you may use without proof properties of thinned Poisson processes.]

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• Paper 4, Section II, $26 K$

  Applied Probability | Part II, 2012

  (a) Define the Moran model and Kingman's $n$-coalescent. State and prove a theorem which describes the relationship between them. [You may use without proof a construction of the Moran model for all $-\infty<t<\infty$.]

  (b) Let $\theta>0$. Suppose that a population of $N \geqslant 2$ individuals evolves according to the rules of the Moran model. Assume also that each individual in the population undergoes a mutation at constant rate $u=\theta /(N-1)$. Each time a mutation occurs, we assume that the allelic type of the corresponding individual changes to an entirely new type, never seen before in the population. Let $p(\theta)$ be the homozygosity probability, i.e., the probability that two individuals sampled without replacement from the population have the same genetic type. Give an expression for $p(\theta)$.

  (c) Let $q(\theta)$ denote the probability that a sample of size $n$ consists of one allelic type (monomorphic population). Show that $q(\theta)=\mathbb{E}\left(\exp \left\{-(\theta / 2) L_{n}\right\}\right)$, where $L_{n}$ denotes the sum of all the branch lengths in the genealogical tree of the sample - that is, $L_{n}=\sum_{i=2}^{n} i\left(\tau_{i}-\tau_{i-1}\right)$, where $\tau_{i}$ is the first time that the genealogical tree of the sample has $i$ lineages. Deduce that

  $$q(\theta)=\frac{(n-1) !}{\prod_{i=1}^{n-1}(\theta+i)}$$

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Asymptotic Methods

• Paper 1, Section II, B

  Asymptotic Methods | Part II, 2012

  What precisely is meant by the statement that

  $$f(x) \sim \sum_{n=0}^{\infty} d_{n} x^{n}$$

  as $x \rightarrow 0 ?$

  Consider the Stieltjes integral

  $$I(x)=\int_{1}^{\infty} \frac{\rho(t)}{1+x t} d t$$

  where $\rho(t)$ is bounded and decays rapidly as $t \rightarrow \infty$, and $x>0$. Find an asymptotic series for $I(x)$ of the form $(*)$, as $x \rightarrow 0$, and prove that it has the asymptotic property.

  In the case that $\rho(t)=e^{-t}$, show that the coefficients $d_{n}$ satisfy the recurrence relation

  $$d_{n}=(-1)^{n} \frac{1}{e}-n d_{n-1} \quad(n \geqslant 1)$$

  and that $d_{0}=\frac{1}{e}$. Hence find the first three terms in the asymptotic series.

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• Paper 3, Section II, B

  Asymptotic Methods | Part II, 2012

  Find the two leading terms in the asymptotic expansion of the Laplace integral

  $$I(x)=\int_{0}^{1} f(t) e^{x t^{4}} d t$$

  as $x \rightarrow \infty$, where $f(t)$ is smooth and positive on $[0,1]$.

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• Paper 4, Section II, B

  Asymptotic Methods | Part II, 2012

  The stationary Schrödinger equation in one dimension has the form

  $$\epsilon^{2} \frac{d^{2} \psi}{d x^{2}}=-(E-V(x)) \psi$$

  where $\epsilon$ can be assumed to be small. Using the Liouville-Green method, show that two approximate solutions in a region where $V(x)<E$ are

  $$\psi(x) \sim \frac{1}{(E-V(x))^{1 / 4}} \exp \left\{\pm \frac{i}{\epsilon} \int_{c}^{x}\left(E-V\left(x^{\prime}\right)\right)^{1 / 2} d x^{\prime}\right\}$$

  where $c$ is suitably chosen.

  Without deriving connection formulae in detail, describe how one obtains the condition

  $$\frac{1}{\epsilon} \int_{a}^{b}\left(E-V\left(x^{\prime}\right)\right)^{1 / 2} d x^{\prime}=\left(n+\frac{1}{2}\right) \pi$$

  for the approximate energies $E$ of bound states in a smooth potential well. State the appropriate values of $a, b$ and $n$.

  Estimate the range of $n$ for which $(*)$ gives a good approximation to the true bound state energies in the cases

  (i) $V(x)=|x|$,

  (ii) $V(x)=x^{2}+\lambda x^{6}$ with $\lambda$ small and positive,

  (iii) $V(x)=x^{2}-\lambda x^{6}$ with $\lambda$ small and positive.

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Classical Dynamics

• Paper 1, Section I, A

  Classical Dynamics | Part II, 2012

  Consider a heavy symmetric top of mass $M$, pinned at point $P$, which is a distance $l$ from the centre of mass.

  (a) Working in the body frame $\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right)$ (where $\mathbf{e}_{3}$ is the symmetry axis of the top) define the Euler angles $(\psi, \theta, \phi)$ and show that the components of the angular velocity can be expressed in terms of the Euler angles as

  $$\boldsymbol{\omega}=(\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi, \dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi, \dot{\psi}+\dot{\phi} \cos \theta)$$

  (b) Write down the Lagrangian of the top in terms of the Euler angles and the principal moments of inertia $I_{1}, I_{3}$.

  (c) Find the three constants of motion.

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• Paper 2, Section I, A

  Classical Dynamics | Part II, 2012

  (a) The action for a system with a generalized coordinate $q$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  State the Principle of Least Action and state the Euler-Lagrange equation.

  (b) Consider a light rigid circular wire of radius $a$ and centre $O$. The wire lies in a vertical plane, which rotates about the vertical axis through $O$. At time $t$ the plane containing the wire makes an angle $\phi(t)$ with a fixed vertical plane. A bead of mass $m$ is threaded onto the wire. The bead slides without friction along the wire, and its location is denoted by $A$. The angle between the line $O A$ and the downward vertical is $\theta(t)$.

  Show that the Lagrangian of this system is

  $$\frac{m a^{2}}{2} \dot{\theta}^{2}+\frac{m a^{2}}{2} \dot{\phi}^{2} \sin ^{2} \theta+m g a \cos \theta .$$

  Calculate two independent constants of the motion, and explain their physical significance.

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• Paper 2, Section II, A

  Classical Dynamics | Part II, 2012

  Consider a rigid body with principal moments of inertia $I_{1}, I_{2}, I_{3}$.

  (a) Derive Euler's equations of torque-free motion

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}, \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$$

  with components of the angular velocity $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ given in the body frame.

  (b) Show that rotation about the second principal axis is unstable if $\left(I_{2}-I_{3}\right)\left(I_{1}-I_{2}\right)>0$.

  (c) The principal moments of inertia of a uniform cylinder of radius $R$, height $h$ and mass $M$ about its centre of mass are

  $$I_{1}=I_{2}=\frac{M R^{2}}{4}+\frac{M h^{2}}{12} \quad ; \quad I_{3}=\frac{M R^{2}}{2} .$$

  The cylinder has two identical cylindrical holes of radius $r$ drilled along its length. The axes of symmetry of the holes are at a distance $a$ from the axis of symmetry of the cylinder such that $r<R / 2$ and $r<a<R-r$. All three axes lie in a single plane. Compute the principal moments of inertia of the body.

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• Paper 3, Section I, A

  Classical Dynamics | Part II, 2012

  The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

  $$L=\frac{m \dot{\mathbf{r}}^{2}}{2}-q(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$$

  (a) Show that the Euler-Lagrange equation is invariant under the gauge transformation

  $$\phi \rightarrow \phi-\frac{\partial \Lambda}{\partial t}, \quad \mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda$$

  for an arbitrary function $\Lambda(\mathbf{r}, t)$.

  (b) Derive the equations of motion in terms of the electric and magnetic fields $\mathbf{E}(\mathbf{r}, t)$ and $\mathbf{B}(\mathbf{r}, t)$.

  [Recall that $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t}$.]

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• Paper 4, Section I, A

  Classical Dynamics | Part II, 2012

  Consider a one-dimensional dynamical system with generalized coordinate and momentum $(q, p)$.

  (a) Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t)$ and $g(q, p, t)$.

  (b) Find the Poisson brackets $\{q, q\},\{p, p\}$ and $\{q, p\}$.

  (c) Assuming Hamilton's equations of motion prove that

  $$\frac{d f}{d t}=\{f, H\}+\frac{\partial f}{\partial t}$$

  (d) State the condition for a transformation $(q, p) \rightarrow(Q, P)$ to be canonical in terms of the Poisson brackets found in (b). Use this to determine whether or not the following transformations are canonical:

  (i) $Q=\sin q, P=\frac{p-a}{\cos q}$,

  (ii) $Q=\cos q, P=\frac{p-a}{\sin q}$,

  where $a$ is constant.

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• Paper 4, Section II, A

  Classical Dynamics | Part II, 2012

  A homogenous thin rod of mass $M$ and length $l$ is constrained to rotate in a horizontal plane about its centre $O$. A bead of mass $m$ is set to slide along the rod without friction. The bead is attracted to $O$ by a force resulting in a potential $k x^{2} / 2$, where $x$ is the distance from $O$.

  (a) Identify suitable generalized coordinates and write down the Lagrangian of the system.

  (b) Identify all conserved quantities.

  (c) Derive the equations of motion and show that one of them can be written as

  $$m \ddot{x}=-\frac{\partial V_{\mathrm{eff}}(x)}{\partial x}$$

  where the form of the effective potential $V_{\text {eff }}(x)$ should be found explicitly.

  (d) Sketch the effective potential. Find and characterize all points of equilibrium.

  (e) Find the frequencies of small oscillations around the stable equilibria.

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Coding and Cryptography

• Paper 1, Section I, G

  Coding and Cryptography | Part II, 2012

  Let $\mathcal{A}$ and $\mathcal{B}$ be alphabets of sizes $m$ and a respectively. What does it mean to say that $c: \mathcal{A} \rightarrow \mathcal{B}^{*}$ is a decodable code? State Kraft's inequality.

  Suppose that a source emits letters from the alphabet $\mathcal{A}=\{1,2, \ldots, m\}$, each letter $j$ occurring with (known) probability $p_{j}>0$. Let $S$ be the codeword-length random variable for a decodable code $c: \mathcal{A} \rightarrow \mathcal{B}^{*}$, where $|\mathcal{B}|=a$. It is desired to find a decodable code that minimizes the expected value of $a^{S}$. Establish the lower bound $\mathbb{E}\left(a^{S}\right) \geqslant\left(\sum_{j=1}^{m} \sqrt{p_{j}}\right)^{2}$, and characterise when equality occurs. [Hint. You may use without proof the Cauchy-Schwarz inequality, that (for positive $x_{i}, y_{i}$ )

  $$\sum_{i=1}^{m} x_{i} y_{i} \leqslant\left(\sum_{i=1}^{m} x_{i}^{2}\right)^{1 / 2}\left(\sum_{i=1}^{m} y_{i}^{2}\right)^{1 / 2}$$

  with equality if and only if $x_{i}=\lambda y_{i}$ for all i.]

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• Paper 1, Section II, 12G

  Coding and Cryptography | Part II, 2012

  Define a cyclic binary code of length $n$.

  Show how codewords can be identified with polynomials in such a way that cyclic binary codes correspond to ideals in the polynomial ring with a suitably chosen multiplication rule.

  Prove that any cyclic binary code $C$ has a unique generator, that is, a polynomial $c(X)$ of minimum degree, such that the code consists of the multiples of this polynomial. Prove that the rank of the code equals $n-\operatorname{deg} c(X)$, and show that $c(X)$ divides $X^{n}-1$.

  Show that the repetition and parity check codes are cyclic, and determine their generators.

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• Paper 2, Section I, G

  Coding and Cryptography | Part II, 2012

  What is a (binary) linear code? What does it mean to say that a linear code has length $n$ and minimum weight $d$ ? When is a linear code perfect? Show that, if $n=2^{r}-1$, there exists a perfect linear code of length $n$ and minimum weight 3 .

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• Paper 2, Section II, G

  Coding and Cryptography | Part II, 2012

  What does it mean to say that $f: \mathbb{F}_{2}^{d} \rightarrow \mathbb{F}_{2}^{d}$ is a linear feedback shift register? Let $\left(x_{n}\right)_{n \geqslant 0}$ be a stream produced by such a register. Show that there exist $N, M$ with $N+M \leqslant 2^{d}-1$ such that $x_{r+N}=x_{r}$ for all $r \geqslant M$.

  Describe and justify the Berlekamp-Massey method for 'breaking' a cipher stream arising from a linear feedback register of unknown length.

  Let $x_{n}, y_{n}, z_{n}$ be three streams produced by linear feedback registers. Set

  $$\begin{aligned} &k_{n}=x_{n} \quad \text { if } y_{n}=z_{n} \\ &k_{n}=y_{n} \quad \text { if } y_{n} \neq z_{n} \end{aligned}$$

  Show that $k_{n}$ is also a stream produced by a linear feedback register. Sketch proofs of any theorems you use.

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• Paper 3, Section $I$, G

  Coding and Cryptography | Part II, 2012

  Describe the RSA system with public key $(N, e)$ and private key $d$. Give a simple example of how the system is vulnerable to a homomorphism attack. Explain how a signature system prevents such an attack.

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• Paper 4, Section I, G

  Coding and Cryptography | Part II, 2012

  Describe the BB84 protocol for quantum key exchange.

  Suppose we attempt to implement the BB84 protocol but cannot send single photons. Instead we send $K$ photons at a time all with the same polarization. An enemy can separate one of these photons from the other $K-1$. Explain briefly how the enemy can intercept the key exchange without our knowledge.

  Show that an enemy can find our common key if $K=3$. Can she do so when $K=2$ (with suitable equipment)?

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Cosmology

• Paper 1, Section I, E

  Cosmology | Part II, 2012

  The number density of photons in equilibrium at temperature $T$ is given by

  $$n=\frac{8 \pi}{(h c)^{3}} \int_{0}^{\infty} \frac{\nu^{2} d \nu}{e^{\beta h \nu}-1}$$

  where $\beta=1 /\left(k_{B} T\right)\left(k_{B}\right.$ is Boltzmann's constant). Show that $n \propto T^{3}$. Show further that $\epsilon \propto T^{4}$, where $\epsilon$ is the photon energy density.

  Write down the Friedmann equation for the scale factor $a(t)$ of a flat homogeneous and isotropic universe. State the relation between $a$ and the mass density $\rho$ for a radiation-dominated universe and hence deduce the time-dependence of $a$. How does the temperature $T$ depend on time?

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• Paper 1, Section II, E

  Cosmology | Part II, 2012

  The Friedmann equation for the scale factor $a(t)$ of a homogeneous and isotropic universe of mass density $\rho$ is

  $$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi}{3} G \rho-\frac{k c^{2}}{a^{2}}$$

  where $\dot{a}=d a / d t$. Explain how the value of the constant $k$ affects the late-time $(t \rightarrow \infty)$ behaviour of $a$.

  Explain briefly why $\rho \propto 1 / a^{3}$ in a matter-dominated (zero-pressure) universe. By considering the scale factor $a$ of a closed universe as a function of conformal time $\tau$, defined by $d \tau=a^{-1} d t$, show that

  $$a(\tau)=\frac{\Omega_{0}}{2\left(\Omega_{0}-1\right)}[1-\cos (\sqrt{k} c \tau)]$$

  where $\Omega_{0}$ is the present $\left(\tau=\tau_{0}\right)$ density parameter, with $a\left(\tau_{0}\right)=1$. Use this result to show that

  $$t(\tau)=\frac{\Omega_{0}}{2 H_{0}\left(\Omega_{0}-1\right)^{3 / 2}}[\sqrt{k} c \tau-\sin (\sqrt{k} c \tau)],$$

  where $H_{0}$ is the present Hubble parameter. Find the time $t_{B C}$ at which this model universe ends in a "big crunch".

  Given that $\sqrt{k} c \tau_{0} \ll 1$, obtain an expression for the present age of the universe in terms of $H_{0}$ and $\Omega_{0}$, according to this model. How does it compare with the age of a flat universe?

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• Paper 2, Section I, E

  Cosmology | Part II, 2012

  The Friedmann equation for the scale factor $a(t)$ of a homogeneous and isotropic universe of mass density $\rho$ is

  $$H^{2}=\frac{8 \pi G \rho}{3}-\frac{k c^{2}}{a^{2}}, \quad\left(H=\frac{\dot{a}}{a}\right)$$

  where $\dot{a}=d a / d t$ and $k$ is a constant. The mass conservation equation for a fluid of mass density $\rho$ and pressure $P$ is

  $$\dot{\rho}=-3\left(\rho+P / c^{2}\right) H$$

  Conformal time $\tau$ is defined by $d \tau=a^{-1} d t$. Show that

  $$\mathcal{H}=a H, \quad\left(\mathcal{H}=\frac{a^{\prime}}{a}\right)$$

  where $a^{\prime}=d a / d \tau$. Hence show that the acceleration equation can be written as

  $$\mathcal{H}^{\prime}=-\frac{4 \pi}{3} G\left(\rho+3 P / c^{2}\right) a^{2}$$

  Define the density parameter $\Omega_{m}$ and show that in a matter-dominated era, in which $P=0$, it satisfies the equation

  $$\Omega_{m}^{\prime}=\mathcal{H} \Omega_{m}\left(\Omega_{m}-1\right)$$

  Use this result to briefly explain the "flatness problem" of cosmology.

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• Paper 3, Section I, E

  Cosmology | Part II, 2012

  For an ideal Fermi gas in equilibrium at temperature $T$ and chemical potential $\mu$, the average occupation number of the $k$ th energy state, with energy $E_{k}$, is

  $$\bar{n}_{k}=\frac{1}{e^{\left(E_{k}-\mu\right) / k_{B} T}+1} .$$

  Discuss the limit $T \rightarrow 0$. What is the Fermi energy $\epsilon_{F} ?$ How is it related to the Fermi momentum $p_{F}$ ? Explain why the density of states with momentum between $p$ and $p+d p$ is proportional to $p^{2} d p$ and use this fact to deduce that the fermion number density at zero temperature takes the form

  $$n \propto p_{F}^{3} .$$

  Consider an ideal Fermi gas that, at zero temperature, is either (i) non-relativistic or (ii) ultra-relativistic. In each case show that the fermion energy density $\epsilon$ takes the form

  $$\epsilon \propto n^{\gamma}$$

  for some constant $\gamma$ which you should compute.

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• Paper 3, Section II, E

  Cosmology | Part II, 2012

  In a flat expanding universe with scale factor $a(t)$, average mass density $\bar{\rho}$ and average pressure $\bar{P} \ll \bar{\rho} c^{2}$, the fractional density perturbations $\delta_{k}(t)$ at co-moving wavenumber $k$ satisfy the equation

  $$\ddot{\delta}_{k}=-2\left(\frac{\dot{a}}{a}\right) \dot{\delta}_{k}+4 \pi G \bar{\rho} \delta_{k}-\frac{c_{s}^{2} k^{2}}{a^{2}} \delta_{k}$$

  Discuss briefly the meaning of each term on the right hand side of this equation. What is the Jeans length $\lambda_{J}$, and what is its significance? How is it related to the Jeans mass?

  How does the equation $(*)$ simplify at $\lambda \gg \lambda_{J}$ in a flat universe? Use your result to show that density perturbations can grow. For a growing density perturbation, how does $\dot{\delta} / \delta$ compare to the inverse Hubble time?

  Explain qualitatively why structure only forms after decoupling, and why cold dark matter is needed for structure formation.

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• Paper 4, Section I, E

  Cosmology | Part II, 2012

  The number density of a species $\star$ of non-relativistic particles of mass $m$, in equilibrium at temperature $T$ and chemical potential $\mu$, is

  $$n_{\star}=g_{\star}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} e^{\left(\mu-m c^{2}\right) / k T},$$

  where $g_{\star}$ is the spin degeneracy. During primordial nucleosynthesis, deuterium, $D$, forms through the nuclear reaction

  $$p+n \leftrightarrow D,$$

  where $p$ and $n$ are non-relativistic protons and neutrons. Write down the relationship between the chemical potentials in equilibrium.

  Using the fact that $g_{D}=4$, and explaining the approximations you make, show that

  $$\frac{n_{D}}{n_{n} n_{p}} \approx\left(\frac{h^{2}}{\pi m_{p} k T}\right)^{3 / 2} \exp \left(\frac{B_{D}}{k T}\right)$$

  where $B_{D}$ is the deuterium binding energy, i.e. $B_{D}=\left(m_{n}+m_{p}-m_{D}\right) c^{2}$.

  Let $X_{\star}=n_{\star} / n_{B}$ where $n_{B}$ is the baryon number density of the universe. Using the fact that $n_{\gamma} \propto T^{3}$, show that

  $$\frac{X_{D}}{X_{n} X_{p}} \propto T^{3 / 2} \eta \exp \left(\frac{B_{D}}{k T}\right)$$

  where $\eta$ is the baryon asymmetry parameter

  $$\eta=\frac{n_{B}}{n_{\gamma}}$$

  Briefly explain why primordial deuterium does not form until temperatures well below $k T \sim B_{D}$.

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Differential Geometry

• Paper 1, Section II, I

  Differential Geometry | Part II, 2012

  Define the geodesic curvature $k_{g}$ of a regular curve in an oriented surface $S \subset \mathbb{R}^{3}$. When is $k_{g}=0$ along a curve?

  Explain briefly what is meant by the Euler characteristic $\chi$ of a compact surface $S \subset \mathbb{R}^{3}$. State the global Gauss-Bonnet theorem with boundary terms.

  Let $S$ be a surface with positive Gaussian curvature that is diffeomorphic to the sphere $S^{2}$ and let $\gamma_{1}, \gamma_{2}$ be two disjoint simple closed curves in $S$. Can both $\gamma_{1}$ and $\gamma_{2}$ be geodesics? Can both $\gamma_{1}$ and $\gamma_{2}$ have constant geodesic curvature? Justify your answers.

  [You may assume that the complement of a simple closed curve in $S^{2}$ consists of two open connected regions.]

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• Paper 2, Section II, I

  Differential Geometry | Part II, 2012

  Define the Gauss map $N$ for an oriented surface $S \subset \mathbb{R}^{3}$. Show that at each $p \in S$ the derivative of the Gauss map

  $$d N_{p}: T_{p} S \rightarrow T_{N(p)} S^{2}=T_{p} S$$

  is self-adjoint. Define the principal curvatures $k_{1}, k_{2}$ of $S$.

  Now suppose that $S$ is compact (and without boundary). By considering the square of the distance to the origin, or otherwise, prove that $S$ has a point $p$ with $k_{1}(p) k_{2}(p)>0$.

  [You may assume that the intersection of $S$ with a plane through the normal direction at $p \in S$ contains a regular curve through $p$.]

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• Paper 3, Section II, I

  Differential Geometry | Part II, 2012

  For a surface $S \subset \mathbb{R}^{3}$, define what is meant by the exponential mapping exp $p$ at $p \in S$, geodesic polar coordinates $(r, \theta)$ and geodesic circles.

  Let $E, F, G$ be the coefficients of the first fundamental form in geodesic polar coordinates $(r, \theta)$. Prove that $\lim _{r \rightarrow 0} \sqrt{G}(r, \theta)=0$ and $\lim _{r \rightarrow 0}(\sqrt{G})_{r}(r, \theta)=1$. Give an expression for the Gaussian curvature $K$ in terms of $G$.

  Prove that the Gaussian curvature at a point $p \in S$ satisfies

  $$K(p)=\lim _{r \rightarrow 0} \frac{12\left(\pi r^{2}-A_{p}(r)\right)}{\pi r^{4}}$$

  where $A_{p}(r)$ is the area of the region bounded by the geodesic circle of radius $r$ centred at $p$.

  [You may assume that $E=1, F=0$ and $d\left(\exp _{p}\right)_{0}$ is an isometry. Taylor's theorem with any form of the remainder may be assumed if accurately stated.]

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• Paper 4, Section II, I

  Differential Geometry | Part II, 2012

  For manifolds $X, Y \subset \mathbb{R}^{n}$, define the terms tangent space to $X$ at a point $x \in X$ and derivative $d f_{x}$ of a smooth map $f: X \rightarrow Y$. State the Inverse Function Theorem for smooth maps between manifolds without boundary.

  Now let $X$ be a submanifold of $Y$ and $f: X \rightarrow Y$ the inclusion map. By considering the map $f^{-1}: f(X) \rightarrow X$, or otherwise, show that $d f_{x}$ is injective for each $x \in X$.

  Show further that there exist local coordinates around $x$ and around $y=f(x)$ such that $f$ is given in these coordinates by

  $$\left(x_{1}, \ldots, x_{l}\right) \in \mathbb{R}^{l} \mapsto\left(x_{1}, \ldots, x_{l}, 0, \ldots, 0\right) \in \mathbb{R}^{k},$$

  where $l=\operatorname{dim} X$ and $k=\operatorname{dim} Y$. [You may assume that any open ball in $\mathbb{R}^{l}$ is diffeomorphic to $\mathbb{R}^{l}$.]

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Dynamical Systems

• Paper 1, Section I, $7 \mathrm{D}$

  Dynamical Systems | Part II, 2012

  State the Poincaré-Bendixson theorem.

  A model of a chemical process obeys the second-order system

  $$\dot{x}=1-x(1+a)+x^{2} y, \quad \dot{y}=a x-x^{2} y$$

  where $a>0$. Show that there is a unique fixed point at $(x, y)=(1, a)$ and that it is unstable if $a>2$. Show that trajectories enter the region bounded by the lines $x=1 / q$, $y=0, y=a q$ and $x+y=1+a q$, provided $q>(1+a)$. Deduce that there is a periodic orbit when $a>2$.

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• Paper 2, Section I, D

  Dynamical Systems | Part II, 2012

  Consider the dynamical system

  $$\dot{x}=\mu x+x^{3}-a x y, \quad \dot{y}=\mu-x^{2}-y,$$

  where $a$ is a constant.

  (a) Show that there is a bifurcation from the fixed point $(0, \mu)$ at $\mu=0$.

  (b) Find the extended centre manifold at leading non-trivial order in $x$. Hence find the type of bifurcation, paying particular attention to the special values $a=1$ and $a=-1$. [Hint. At leading order, the extended centre manifold is of the form $y=\mu+\alpha x^{2}+\beta \mu x^{2}+\gamma x^{4}$, where $\alpha, \beta, \gamma$ are constants to be determined.]

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• Paper 3, Section I, D

  Dynamical Systems | Part II, 2012

  State without proof Lyapunov's first theorem, carefully defining all the terms that you use.

  Consider the dynamical system

  $$\begin{aligned} &\dot{x}=-2 x+y-x y+3 y^{2}-x y^{2}+x^{3} \\ &\dot{y}=-2 y-x-y^{2}-3 x y+2 x^{2} y \end{aligned}$$

  By choosing a Lyapunov function $V(x, y)=x^{2}+y^{2}$, prove that the origin is asymptotically stable.

  By factorising the expression for $\dot{V}$, or otherwise, show that the basin of attraction of the origin includes the set $V<7 / 4$.

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• Paper 3, Section II, D

  Dynamical Systems | Part II, 2012

  Consider the dynamical system

  $$\ddot{x}-(a-b x) \dot{x}+x-x^{2}=0, \quad a, b>0 .$$

  (a) Show that the fixed point at the origin is an unstable node or focus, and that the fixed point at $x=1$ is a saddle point.

  (b) By considering the phase plane $(x, \dot{x})$, or otherwise, show graphically that the maximum value of $x$ for any periodic orbit is less than one.

  (c) By writing the system in terms of the variables $x$ and $z=\dot{x}-\left(a x-b x^{2} / 2\right)$, or otherwise, show that for any periodic orbit $\mathcal{C}$

  $$\oint_{\mathcal{C}}\left(x-x^{2}\right)\left(2 a x-b x^{2}\right) d t=0$$

  Deduce that if $a / b>1 / 2$ there are no periodic orbits.

  (d) If $a=b=0$ the system (1) is Hamiltonian and has homoclinic orbit

  $$X(t)=\frac{1}{2}\left(3 \tanh ^{2}\left(\frac{t}{2}\right)-1\right)$$

  which approaches $X=1$ as $t \rightarrow \pm \infty$. Now suppose that $a, b$ are very small and that we seek the value of $a / b$ corresponding to a periodic orbit very close to $X(t)$. By using equation (3) in equation (2), find an approximation to the largest value of $a / b$ for a periodic orbit when $a, b$ are very small.

  [Hint. You may use the fact that $\left.(1-X)=\frac{3}{2} \operatorname{sech}^{2}\left(\frac{t}{2}\right)=3 \frac{d}{d t}\left(\tanh \left(\frac{t}{2}\right)\right)\right]$