Part IB, 2016, Paper 2

# Part IB, 2016, Paper 2

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Paper 2, Section I, G

comment(a) What does it mean to say that the function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is differentiable at the point $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$ ? Show from your definition that if $f$ is differentiable at $x$, then $f$ is continuous at $x$.

(b) Suppose that there are functions $g_{j}: \mathbb{R} \rightarrow \mathbb{R}^{m}(1 \leqslant j \leqslant n)$ such that for every $x=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$,

$f(x)=\sum_{j=1}^{n} g_{j}\left(x_{j}\right) .$

Show that $f$ is differentiable at $x$ if and only if each $g_{j}$ is differentiable at $x_{j}$.

(c) Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by

$f(x, y)=|x|^{3 / 2}+|y|^{1 / 2}$

Determine at which points $(x, y) \in \mathbb{R}^{2}$ the function $f$ is differentiable.

Paper 2, Section II, G

comment(a) What is a norm on a real vector space?

(b) Let $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$ be the space of linear maps from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$. Show that

$\|A\|=\sup _{0 \neq x \in \mathbb{R}^{m}} \frac{\|A x\|}{\|x\|}, \quad A \in L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right),$

defines a norm on $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$, and that if $B \in L\left(\mathbb{R}^{\ell}, \mathbb{R}^{m}\right)$ then $\|A B\| \leqslant\|A\|\|B\|$.

(c) Let $M_{n}$ be the space of $n \times n$ real matrices, identified with $L\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right)$ in the usual way. Let $U \subset M_{n}$ be the subset

$U=\left\{X \in M_{n} \mid I-X \text { is invertible }\right\}$

Show that $U$ is an open subset of $M_{n}$ which contains the set $V=\left\{X \in M_{n} \mid\|X\|<1\right\}$.

(d) Let $f: U \rightarrow M_{n}$ be the map $f(X)=(I-X)^{-1}$. Show carefully that the series $\sum_{k=0}^{\infty} X^{k}$ converges on $V$ to $f(X)$. Hence or otherwise, show that $f$ is twice differentiable at 0 , and compute its first and second derivatives there.

Paper 2, Section II, A

commentLet $a=N+1 / 2$ for a positive integer $N$. Let $C_{N}$ be the anticlockwise contour defined by the square with its four vertices at $a \pm i a$ and $-a \pm i a$. Let

$I_{N}=\oint_{C_{N}} \frac{d z}{z^{2} \sin (\pi z)}$

Show that $1 / \sin (\pi z)$ is uniformly bounded on the contours $C_{N}$ as $N \rightarrow \infty$, and hence that $I_{N} \rightarrow 0$ as $N \rightarrow \infty$.

Using this result, establish that

$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}=\frac{\pi^{2}}{12}$

Paper 2, Section I, $6 \mathrm{D}$

comment(a) Derive the integral form of Ampère's law from the differential form of Maxwell's equations with a time-independent magnetic field, $\rho=0$ and $\mathbf{E}=\mathbf{0}$.

(b) Consider two perfectly-conducting concentric thin cylindrical shells of infinite length with axes along the $z$-axis and radii $a$ and $b(a<b)$. Current $I$ flows in the positive $z$-direction in each shell. Use Ampère's law to calculate the magnetic field in the three regions: (i) $r<a$, (ii) $a<r<b$ and (iii) $r>b$, where $r=\sqrt{x^{2}+y^{2}}$.

(c) If current $I$ now flows in the positive $z$-direction in the inner shell and in the negative $z$-direction in the outer shell, calculate the magnetic field in the same three regions.

Paper 2, Section II, D

comment(a) State the covariant form of Maxwell's equations and define all the quantities that appear in these expressions.

(b) Show that $\mathbf{E} \cdot \mathbf{B}$ is a Lorentz scalar (invariant under Lorentz transformations) and find another Lorentz scalar involving $\mathbf{E}$ and $\mathbf{B}$.

(c) In some inertial frame $S$ the electric and magnetic fields are respectively $\mathbf{E}=\left(0, E_{y}, E_{z}\right)$ and $\mathbf{B}=\left(0, B_{y}, B_{z}\right)$. Find the electric and magnetic fields, $\mathbf{E}^{\prime}=\left(0, E_{y}^{\prime}, E_{z}^{\prime}\right)$ and $\mathbf{B}^{\prime}=\left(0, B_{y}^{\prime}, B_{z}^{\prime}\right)$, in another inertial frame $S^{\prime}$ that is related to $S$ by the Lorentz transformation,

$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma v / c & 0 & 0 \\ -\gamma v / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

where $v$ is the velocity of $S^{\prime}$ in $S$ and $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$.

(d) Suppose that $\mathbf{E}=E_{0}(0,1,0)$ and $\mathbf{B}=\frac{E_{0}}{c}(0, \cos \theta, \sin \theta)$ where $0 \leqslant \theta \leqslant \pi / 2$, and $E_{0}$ is a real constant. An observer is moving in $S$ with velocity $v$ parallel to the $x$-axis. What must $v$ be for the electric and magnetic fields to appear to the observer to be parallel? Comment on the case $\theta=\pi / 2$.

Paper 2, Section I, C

commentA steady, two-dimensional unidirectional flow of a fluid with dynamic viscosity $\mu$ is set up between two plates at $y=0$ and $y=h$. The plate at $y=0$ is stationary while the plate at $y=h$ moves with constant speed $U \mathbf{e}_{x}$. The fluid is also subject to a constant pressure gradient $-G \mathbf{e}_{x}$. You may assume that the fluid velocity $\mathbf{u}$ has the form $\mathbf{u}=u(y) \mathbf{e}_{x}$.

(a) State the equation satisfied by $u(y)$ and its boundary conditions.

(b) Calculate $u(y)$.

(c) Show that the value of $U$ may be chosen to lead to zero viscous shear stress acting on the bottom plate and calculate the resulting flow rate.

Paper 2, Section II, F

comment(a) Let $A B C$ be a hyperbolic triangle, with the angle at $A$ at least $\pi / 2$. Show that the side $B C$ has maximal length amongst the three sides of $A B C$.

[You may use the hyperbolic cosine formula without proof. This states that if $a, b$ and $c$ are the lengths of $B C, A C$, and $A B$ respectively, and $\alpha, \beta$ and $\gamma$ are the angles of the triangle at $A, B$ and $C$ respectively, then

$\cosh a=\cosh b \cosh c-\sinh b \sinh c \cos \alpha .]$

(b) Given points $z_{1}, z_{2}$ in the hyperbolic plane, let $w$ be any point on the hyperbolic line segment joining $z_{1}$ to $z_{2}$, and let $w^{\prime}$ be any point not on the hyperbolic line passing through $z_{1}, z_{2}, w$. Show that

$\rho\left(w^{\prime}, w\right) \leqslant \max \left\{\rho\left(w^{\prime}, z_{1}\right), \rho\left(w^{\prime}, z_{2}\right)\right\}$

where $\rho$ denotes hyperbolic distance.

(c) The diameter of a hyperbolic triangle $\Delta$ is defined to be

$\sup \{\rho(P, Q) \mid P, Q \in \Delta\}$

Show that the diameter of $\Delta$ is equal to the length of its longest side.

Paper 2, Section I, E

commentLet $R$ be an integral domain.

Define what is meant by the field of fractions $F$ of $R$. [You do not need to prove the existence of $F$.]

Suppose that $\phi: R \rightarrow K$ is an injective ring homomorphism from $R$ to a field $K$. Show that $\phi$ extends to an injective ring homomorphism $\Phi: F \rightarrow K$.

Give an example of $R$ and a ring homomorphism $\psi: R \rightarrow S$ from $R$ to a ring $S$ such that $\psi$ does not extend to a ring homomorphism $F \rightarrow S$.

Paper 2, Section II, E

comment(a) State Sylow's theorems and give the proof of the second theorem which concerns conjugate subgroups.

(b) Show that there is no simple group of order 351 .

(c) Let $k$ be the finite field $\mathbb{Z} /(31)$ and let $G L_{2}(k)$ be the multiplicative group of invertible $2 \times 2$ matrices over $k$. Show that every Sylow 3-subgroup of $G L_{2}(k)$ is abelian.

Paper 2, Section I, F

commentFind a linear change of coordinates such that the quadratic form

$2 x^{2}+8 x y-6 x z+y^{2}-4 y z+2 z^{2}$

takes the form

$\alpha x^{2}+\beta y^{2}+\gamma z^{2}$

for real numbers $\alpha, \beta$ and $\gamma$.

Paper 2, Section II, F

commentLet $M_{n, n}$ denote the vector space over a field $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Given $B \in M_{n, n}$, consider the two linear transformations $R_{B}, L_{B}: M_{n, n} \rightarrow$ $M_{n, n}$ defined by

$L_{B}(A)=B A, \quad R_{B}(A)=A B$

(a) Show that $\operatorname{det} L_{B}=(\operatorname{det} B)^{n}$.

[For parts (b) and (c), you may assume the analogous result $\operatorname{det} R_{B}=(\operatorname{det} B)^{n}$ without proof.]

(b) Now let $F=\mathbb{C}$. For $B \in M_{n, n}$, write $B^{*}$ for the conjugate transpose of $B$, i.e., $B^{*}:=\bar{B}^{T}$. For $B \in M_{n, n}$, define the linear transformation $M_{B}: M_{n, n} \rightarrow M_{n, n}$ by

$M_{B}(A)=B A B^{*}$

Show that $\operatorname{det} M_{B}=|\operatorname{det} B|^{2 n}$.

(c) Again let $F=\mathbb{C}$. Let $W \subseteq M_{n, n}$ be the set of Hermitian matrices. [Note that $W$ is not a vector space over $\mathbb{C}$ but only over $\mathbb{R} .]$ For $B \in M_{n, n}$ and $A \in W$, define $T_{B}(A)=B A B^{*}$. Show that $T_{B}$ is an $\mathbb{R}$-linear operator on $W$, and show that as such,

$\operatorname{det} T_{B}=|\operatorname{det} B|^{2 n}$

Paper 2, Section II, H

comment(a) Prove that every open communicating class of a Markov chain is transient. Prove that every finite transient communicating class is open. Give an example of a Markov chain with an infinite transient closed communicating class.

(b) Consider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ with state space $\{a, b, c, d\}$ and transition probabilities given by the matrix

$P=\left(\begin{array}{cccc} 1 / 3 & 0 & 1 / 3 & 1 / 3 \\ 0 & 1 / 4 & 0 & 3 / 4 \\ 1 / 2 & 1 / 2 & 0 & 0 \\ 0 & 2 / 3 & 0 & 1 / 3 \end{array}\right)$

(i) Compute $\mathbb{P}\left(X_{n}=b \mid X_{0}=d\right)$ for a fixed $n \geqslant 0$.

(ii) Compute $\mathbb{P}\left(X_{n}=c\right.$ for some $\left.n \geqslant 1 \mid X_{0}=a\right)$.

(iii) Show that $P^{n}$ converges as $n \rightarrow \infty$, and determine the limit.

[Results from lectures can be used without proof if stated carefully.]

Paper 2, Section I, A

commentUse the method of characteristics to find $u(x, y)$ in the first quadrant $x \geqslant 0, y \geqslant 0$, where $u(x, y)$ satisfies

$\frac{\partial u}{\partial x}-2 x \frac{\partial u}{\partial y}=\cos x$

with boundary data $u(x, 0)=\cos x$.

Paper 2, Section II, A

commentConsider a bar of length $\pi$ with free ends, subject to longitudinal vibrations. You may assume that the longitudinal displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$ :

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

for $x \in(0, \pi)$ and $t>0$ with boundary conditions:

$\frac{\partial y}{\partial x}(0, t)=\frac{\partial y}{\partial x}(\pi, t)=0$

for $t>0$. The bar is initially at rest so that

$\frac{\partial y}{\partial t}(x, 0)=0$

for $x \in(0, \pi)$, with a spatially varying initial longitudinal displacement given by

$y(x, 0)=b x$

for $x \in(0, \pi)$, where $b$ is a real constant.

(a) Using separation of variables, show that

$y(x, t)=\frac{b \pi}{2}-\frac{4 b}{\pi} \sum_{n=1}^{\infty} \frac{\cos [(2 n-1) x] \cos [(2 n-1) c t]}{(2 n-1)^{2}}$

(b) Determine a periodic function $P(x)$ such that this solution may be expressed as

$y(x, t)=\frac{1}{2}[P(x+c t)+P(x-c t)]$

Paper 2, Section I, E

commentConsider $\mathbb{R}$ and $\mathbb{Q}$ with their usual topologies.

(a) Show that compact subsets of a Hausdorff topological space are closed. Show that compact subsets of $\mathbb{R}$ are closed and bounded.

(b) Show that there exists a complete metric space $(X, d)$ admitting a surjective continuous map $f: X \rightarrow \mathbb{Q}$.

Paper 2, Section II, D

comment(a) Define a Givens rotation $\Omega^{[p, q]} \in \mathbb{R}^{m \times m}$ and show that it is an orthogonal matrix.

(b) Define a QR factorization of a matrix $A \in \mathbb{R}^{m \times n}$ with $m \geqslant n$. Explain how Givens rotations can be used to find $Q \in \mathbb{R}^{m \times m}$ and $R \in \mathbb{R}^{m \times n}$.

(c) Let

$\mathbf{A}=\left[\begin{array}{ccc} 3 & 1 & 1 \\ 0 & 4 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 3 / 4 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} 98 / 25 \\ 25 \\ 25 \\ 0 \end{array}\right]$

(i) Find a QR factorization of $A$ using Givens rotations.

(ii) Hence find the vector $\mathbf{x}^{*} \in \mathbb{R}^{3}$ which minimises $\|A \mathbf{x}-\mathbf{b}\|$, where $\|\cdot\|$ is the Euclidean norm. What is $\left\|\mathrm{A} \mathbf{x}^{*}-\mathbf{b}\right\|$ ?

Paper 2, Section I, H

commentUse the simplex algorithm to find the optimal solution to the linear program:

$\operatorname{maximise} 3 x+5 y \text { subject to } \begin{aligned} 8 x+3 y+10 z & \leqslant 9, \quad x, y, z \geqslant 0 \\ 5 x+2 y+4 z & \leqslant 8 \\ 2 x+y+3 z & \leqslant 2 \end{aligned}$

Write down the dual problem and find its solution.

Paper 2, Section II, B

commentThe one dimensional quantum harmonic oscillator has Hamiltonian

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

where $m$ and $\omega$ are real positive constants and $\hat{x}$ and $\hat{p}$ are the standard position and momentum operators satisfying the commutation relation $[\hat{x}, \hat{p}]=i \hbar$. Consider the operators

$\hat{A}=\hat{p}-i m \omega \hat{x} \quad \text { and } \quad \hat{B}=\hat{p}+i m \omega \hat{x} .$

(a) Show that

$\hat{B} \hat{A}=2 m\left(\hat{H}-\frac{1}{2} \hbar \omega\right) \quad \text { and } \quad \hat{A} \hat{B}=2 m\left(\hat{H}+\frac{1}{2} \hbar \omega\right) .$

(b) Suppose that $\phi$ is an eigenfunction of $\hat{H}$ with eigenvalue $E$. Show that $\hat{A} \phi$ is then also an eigenfunction of $\hat{H}$ and that its corresponding eigenvalue is $E-\hbar \omega$.

(c) Show that for any normalisable wavefunctions $\chi$ and $\psi$,

$\int_{-\infty}^{\infty} \chi^{*}(\hat{A} \psi) d x=\int_{-\infty}^{\infty}(\hat{B} \chi)^{*} \psi d x$

[You may assume that the operators $\hat{x}$ and $\hat{p}$ are Hermitian.]

(d) With $\phi$ as in (b), obtain an expression for the norm of $\hat{A} \phi$ in terms of $E$ and the norm of $\phi$. [The squared norm of any wavefunction $\psi$ is $\int_{-\infty}^{\infty}|\psi|^{2} d x$.]

(e) Show that all eigenvalues of $\hat{H}$ are non-negative.

(f) Using the above results, deduce that each eigenvalue $E$ of $\hat{H}$ must be of the form $E=\left(n+\frac{1}{2}\right) \hbar \omega$ for some non-negative integer $n$.

Paper 2, Section I, H

commentThe efficacy of a new medicine was tested as follows. Fifty patients were given the medicine, and another fifty patients were given a placebo. A week later, the number of patients who got better, stayed the same, or got worse was recorded, as summarised in this table:

\begin{tabular}{|l|c|c|} \hline & medicine & placebo \ better & 28 & 22 \ same & 4 & 16 \ worse & 18 & 12 \ \hline \end{tabular}

Conduct a Pearson chi-squared test of size $1 \%$ of the hypothesis that the medicine and the placebo have the same effect.

[Hint: You may find the following values relevant:

$\left.\begin{array}{lcccccc}\text { Distribution } & \chi_{1}^{2} & \chi_{2}^{2} & \chi_{3}^{2} & \chi_{4}^{2} & \chi_{5}^{2} & \chi_{6}^{2} \\ 99 \% \text { percentile } & 6.63 & 9.21 & 11.34 & 13.3 & 15.09 & 16.81 .\end{array}\right]$

Paper 2, Section II, C

commentA flexible wire filament is described by the curve $(x, y(x), z(x))$ in cartesian coordinates for $0 \leqslant x \leqslant L$. The filament is assumed to be almost straight and thus we assume $\left|y^{\prime}\right| \ll 1$ and $\left|z^{\prime}\right| \ll 1$ everywhere.

(a) Show that the total length of the filament is approximately $L+\Delta$ where

$\Delta=\frac{1}{2} \int_{0}^{L}\left[\left(y^{\prime}\right)^{2}+\left(z^{\prime}\right)^{2}\right] d x$

(b) Under a uniform external axial force, $F>0$, the filament adopts the shape which minimises the total energy, $\mathcal{E}=E_{B}-F \Delta$, where $E_{B}$ is the bending energy given by

$E_{B}[y, z]=\frac{1}{2} \int_{0}^{L}\left[A(x)\left(y^{\prime \prime}\right)^{2}+B(x)\left(z^{\prime \prime}\right)^{2}\right] d x$

and where $A(x)$ and $B(x)$ are $x$-dependent bending rigidities (both known and strictly positive). The filament satisfies the boundary conditions

$y(0)=y^{\prime}(0)=z(0)=z^{\prime}(0)=0, \quad y(L)=y^{\prime}(L)=z(L)=z^{\prime}(L)=0$

Derive the Euler-Lagrange equations for $y(x)$ and $z(x)$.

(c) In the case where $A=B=1$ and $L=1$, show that below a critical force, $F_{c}$, which should be determined, the only energy-minimising solution for the filament is straight $(y=z=0)$, but that a new nonzero solution is admissible at $F=F_{c}$.