Part IB, 2010, Paper 1

# Part IB, 2010, Paper 1

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Paper 1, Section II, G

commentState and prove the contraction mapping theorem. Demonstrate its use by showing that the differential equation $f^{\prime}(x)=f\left(x^{2}\right)$, with boundary condition $f(0)=1$, has a unique solution on $[0,1)$, with one-sided derivative $f^{\prime}(0)=1$ at zero.

Paper 1, Section I, A

comment(a) Write down the definition of the complex derivative of the function $f(z)$ of a single complex variable.

(b) Derive the Cauchy-Riemann equations for the real and imaginary parts $u(x, y)$ and $v(x, y)$ of $f(z)$, where $z=x+i y$ and

$f(z)=u(x, y)+i v(x, y)$

(c) State necessary and sufficient conditions on $u(x, y)$ and $v(x, y)$ for the function $f(z)$ to be complex differentiable.

Paper 1, Section II, A

commentCalculate the following real integrals by using contour integration. Justify your steps carefully.

(a)

$I_{1}=\int_{0}^{\infty} \frac{x \sin x}{x^{2}+a^{2}} d x, \quad a>0$

(b)

$I_{2}=\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x$

Paper 1, Section II, C

commentA capacitor consists of three perfectly conducting coaxial cylinders of radii $a, b$ and $c$ where $0<a<b<c$, and length $L$ where $L \gg c$ so that end effects may be ignored. The inner and outer cylinders are maintained at zero potential, while the middle cylinder is held at potential $V$. Assuming its cylindrical symmetry, compute the electrostatic potential within the capacitor, the charge per unit length on the middle cylinder, the capacitance and the electrostatic energy, both per unit length.

Next assume that the radii $a$ and $c$ are fixed, as is the potential $V$, while the radius $b$ is allowed to vary. Show that the energy achieves a minimum when $b$ is the geometric mean of $a$ and $c$.

Paper 1, Section I, B

commentA planar solenoidal velocity field has the velocity potential

$\phi(x, y, t)=x e^{-t}+y e^{t}$

Find and sketch (i) the streamlines at $t=0$; (ii) the pathline that passes through the origin at $t=0$; (iii) the locus at $t=0$ of points that pass through the origin at earlier times (streakline).

Paper 1, Section II, B

commentStarting with the Euler equations for an inviscid incompressible fluid, derive Bernoulli's theorem for unsteady irrotational flow.

Inviscid fluid of density $\rho$ is contained within a U-shaped tube with the arms vertical, of height $h$ and with the same (unit) cross-section. The ends of the tube are closed. In the equilibrium state the pressures in the two arms are $p_{1}$ and $p_{2}$ and the heights of the fluid columns are $\ell_{1}, \ell_{2}$.

The fluid in arm 1 is displaced upwards by a distance $\xi$ (and in the other arm downward by the same amount). In the subsequent evolution the pressure above each column may be taken as inversely proportional to the length of tube above the fluid surface. Using Bernoulli's theorem, show that $\xi(t)$ obeys the equation

$\rho\left(\ell_{1}+\ell_{2}\right) \ddot{\xi}+\frac{p_{1} \xi}{h-\ell_{1}-\xi}+\frac{p_{2} \xi}{h-\ell_{2}+\xi}+2 \rho g \xi=0$

Now consider the special case $\ell_{1}=\ell_{2}=\ell_{0}, p_{1}=p_{2}=p_{0}$. Construct a first integral of this equation and hence give an expression for the total kinetic energy $\rho \ell_{0} \dot{\xi}^{2}$ of the flow in terms of $\xi$ and the maximum displacement $\xi_{\max }$.

Paper 1, Section I, F

comment(i) Define the notion of curvature for surfaces embedded in $\mathbb{R}^{3}$.

(ii) Prove that the unit sphere in $\mathbb{R}^{3}$ has curvature $+1$ at all points.

Paper 1, Section II, H

commentProve that the kernel of a group homomorphism $f: G \rightarrow H$ is a normal subgroup of the group $G$.

Show that the dihedral group $D_{8}$ of order 8 has a non-normal subgroup of order 2. Conclude that, for a group $G$, a normal subgroup of a normal subgroup of $G$ is not necessarily a normal subgroup of $G$.

Paper 1, Section I, F

commentSuppose that $V$ is the complex vector space of polynomials of degree at most $n-1$ in the variable $z$. Find the Jordan normal form for each of the linear transformations $\frac{d}{d z}$ and $z \frac{d}{d z}$ acting on $V$.

Paper 1, Section II, F

commentLet $V$ denote the vector space of $n \times n$ real matrices.

(1) Show that if $\psi(A, B)=\operatorname{tr}\left(A B^{T}\right)$, then $\psi$ is a positive-definite symmetric bilinear form on $V$.

(2) Show that if $q(A)=\operatorname{tr}\left(A^{2}\right)$, then $q$ is a quadratic form on $V$. Find its rank and signature.

[Hint: Consider symmetric and skew-symmetric matrices.]

Paper 1, Section II, E

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain.

(a) What does it mean to say that a state $i$ is positive recurrent? How is this property related to the equilibrium probability $\pi_{i}$ ? You do not need to give a full proof, but you should carefully state any theorems you use.

(b) What is a communicating class? Prove that if states $i$ and $j$ are in the same communicating class and $i$ is positive recurrent then $j$ is positive recurrent also.

A frog is in a pond with an infinite number of lily pads, numbered $1,2,3, \ldots$ She hops from pad to pad in the following manner: if she happens to be on pad $i$ at a given time, she hops to one of pads $(1,2, \ldots, i, i+1)$ with equal probability.

(c) Find the equilibrium distribution of the corresponding Markov chain.

(d) Now suppose the frog starts on pad $k$ and stops when she returns to it. Show that the expected number of times the frog hops is $e(k-1)$ ! where $e=2.718 \ldots$ What is the expected number of times she will visit the lily pad $k+1$ ?

Paper 1, Section II, A

comment(a) A function $f(t)$ is periodic with period $2 \pi$ and has continuous derivatives up to and including the $k$ th derivative. Show by integrating by parts that the Fourier coefficients of $f(t)$

$\begin{aligned} &a_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} f(t) \cos n t d t \\ &b_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} f(t) \sin n t d t \end{aligned}$

decay at least as fast as $1 / n^{k}$ as $n \rightarrow \infty$

(b) Calculate the Fourier series of $f(t)=|\sin t|$ on $[0,2 \pi]$.

(c) Comment on the decay rate of your Fourier series.

Paper 1, Section II, H

commentLet $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be continuous maps of topological spaces with $f \circ g=\mathrm{id}_{Y}$.

(1) Suppose that (i) $Y$ is path-connected, and (ii) for every $y \in Y$, its inverse image $f^{-1}(y)$ is path-connected. Prove that $X$ is path-connected.

(2) Prove the same statement when "path-connected" is everywhere replaced by "connected".

Paper 1, Section I, C

commentObtain the Cholesky decompositions of

$H_{3}=\left(\begin{array}{ccc} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \end{array}\right), \quad H_{4}=\left(\begin{array}{cccc} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \lambda \end{array}\right) .$

What is the minimum value of $\lambda$ for $H_{4}$ to be positive definite? Verify that if $\lambda=\frac{1}{7}$ then $H_{4}$ is positive definite.

Paper 1, Section II, 18C

commentLet

$\langle f, g\rangle=\int_{-\infty}^{\infty} e^{-x^{2}} f(x) g(x) d x$

be an inner product. The Hermite polynomials $H_{n}(x), n=0,1,2, \ldots$ are polynomials in $x$ of degree $n$ with leading term $2^{n} x^{n}$ which are orthogonal with respect to the inner product, with

$\left\langle H_{m}, H_{n}\right\rangle= \begin{cases}\gamma_{m}>0 & \text { if } m=n \\ 0 & \text { otherwise }\end{cases}$

and $H_{0}(x)=1$. Find a three-term recurrence relation which is satisfied by $H_{n}(x)$ and $\gamma_{n}$ for $n=1,2,3$. [You may assume without proof that

$\left.\langle 1,1\rangle=\sqrt{\pi}, \quad\langle x, x\rangle=\frac{1}{2} \sqrt{\pi}, \quad\left\langle x^{2}, x^{2}\right\rangle=\frac{3}{4} \sqrt{\pi}, \quad\left\langle x^{3}, x^{3}\right\rangle=\frac{15}{8} \sqrt{\pi} .\right]$

Next let $x_{0}, x_{1}, \ldots, x_{k}$ be the $k+1$ distinct zeros of $H_{k+1}(x)$ and for $i, j=0,1, \ldots, k$ define the Lagrangian polynomials

$L_{i}(x)=\prod_{j \neq i} \frac{x-x_{j}}{x_{i}-x_{j}}$

associated with these points. Prove that $\left\langle L_{i}, L_{j}\right\rangle=0$ if $i \neq j$.

Paper 1, Section I, 8E

commentWhat is the maximal flow problem in a network?

Explain the Ford-Fulkerson algorithm. Why must this algorithm terminate if the initial flow is set to zero and all arc capacities are rational numbers?

Paper 1, Section II, 15D

commentA particle of unit mass moves in one dimension in a potential

$V=\frac{1}{2} \omega^{2} x^{2}$

Show that the stationary solutions can be written in the form

$\psi_{n}(x)=f_{n}(x) \exp \left(-\alpha x^{2}\right)$

You should give the value of $\alpha$ and derive any restrictions on $f_{n}(x)$. Hence determine the possible energy eigenvalues $E_{n}$.

The particle has a wave function $\psi(x, t)$ which is even in $x$ at $t=0$. Write down the general form for $\psi(x, 0)$, using the fact that $f_{n}(x)$ is an even function of $x$ only if $n$ is even. Hence write down $\psi(x, t)$ and show that its probability density is periodic in time with period $\pi / \omega$.

Paper 1, Section I, E

commentSuppose $X_{1}, \ldots, X_{n}$ are independent $N\left(0, \sigma^{2}\right)$ random variables, where $\sigma^{2}$ is an unknown parameter. Explain carefully how to construct the uniformly most powerful test of size $\alpha$ for the hypothesis $H_{0}: \sigma^{2}=1$ versus the alternative $H_{1}: \sigma^{2}>1$.

Paper 1, Section II, E

commentConsider the the linear regression model

$Y_{i}=\beta x_{i}+\epsilon_{i},$

where the numbers $x_{1}, \ldots, x_{n}$ are known, the independent random variables $\epsilon_{1}, \ldots, \epsilon_{n}$ have the $N\left(0, \sigma^{2}\right)$ distribution, and the parameters $\beta$ and $\sigma^{2}$ are unknown. Find the maximum likelihood estimator for $\beta$.

State and prove the Gauss-Markov theorem in the context of this model.

Write down the distribution of an arbitrary linear estimator for $\beta$. Hence show that there exists a linear, unbiased estimator $\widehat{\beta}$ for $\beta$ such that

$\mathbb{E}_{\beta, \sigma^{2}}\left[(\widehat{\beta}-\beta)^{4}\right] \leqslant \mathbb{E}_{\beta, \sigma^{2}}\left[(\widetilde{\beta}-\beta)^{4}\right]$

for all linear, unbiased estimators $\widetilde{\beta}$.

[Hint: If $Z \sim N\left(a, b^{2}\right)$ then $\left.\mathbb{E}\left[(Z-a)^{4}\right]=3 b^{4} .\right]$

Paper 1, Section I, D

comment(a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex and strictly convex.

(b) State a necessary and sufficient first-order condition for strict convexity of $f \in C^{1}\left(\mathbb{R}^{n}\right)$, and give, with proof, an example of a function which is strictly convex but with second derivative which is not everywhere strictly positive.