Part IB, 2007, Paper 1

# Part IB, 2007, Paper 1

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1.II.11H

commentDefine what it means for a function $f: \mathbb{R}^{a} \rightarrow \mathbb{R}^{b}$ to be differentiable at a point $p \in \mathbb{R}^{a}$ with derivative a linear map $\left.D f\right|_{p} .$

State the Chain Rule for differentiable maps $f: \mathbb{R}^{a} \rightarrow \mathbb{R}^{b}$ and $g: \mathbb{R}^{b} \rightarrow \mathbb{R}^{c}$. Prove the Chain Rule.

Let $\|x\|$ denote the standard Euclidean norm of $x \in \mathbb{R}^{a}$. Find the partial derivatives $\frac{\partial f}{\partial x_{i}}$ of the function $f(x)=\|x\|$ where they exist.

1.I.3F

commentFor the function

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

determine the Taylor series of $f$ around the point $z_{0}=1$, and give the largest $r$ for which this series converges in the disc $|z-1|<r$.

1.II.13F

commentBy integrating round the contour $C_{R}$, which is the boundary of the domain

$D_{R}=\left\{z=r e^{i \theta}: 0<r<R, \quad 0<\theta<\frac{\pi}{4}\right\}$

evaluate each of the integrals

$\int_{0}^{\infty} \sin x^{2} d x, \quad \int_{0}^{\infty} \cos x^{2} d x$

[You may use the relations $\int_{0}^{\infty} e^{-r^{2}} d r=\frac{\sqrt{\pi}}{2}$ and $\sin t \geq \frac{2}{\pi} t$ for $\left.0 \leq t \leq \frac{\pi}{2} \cdot\right]$

1.II.16E

commentA steady magnetic field $\mathbf{B}(\mathbf{x})$ is generated by a current distribution $\mathbf{j}(\mathbf{x})$ that vanishes outside a bounded region $V$. Use the divergence theorem to show that

$\int_{V} \mathbf{j} d V=0 \quad \text { and } \quad \int_{V} x_{i} j_{k} d V=-\int_{V} x_{k} j_{i} d V$

Define the magnetic vector potential $\mathbf{A}(\mathbf{x})$. Use Maxwell's equations to obtain a differential equation for $\mathbf{A}(\mathbf{x})$ in terms of $\mathbf{j}(\mathbf{x})$.

It may be shown that for an unbounded domain the equation for $\mathbf{A}(\mathbf{x})$ has solution

$\mathbf{A}(\mathbf{x})=\frac{\mu_{0}}{4 \pi} \int_{V} \frac{\mathbf{j}\left(\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d V^{\prime}$

Deduce that in general the leading order approximation for $\mathbf{A}(\mathbf{x})$ as $|\mathbf{x}| \rightarrow \infty$ is

$\mathbf{A}=\frac{\mu_{0}}{4 \pi} \frac{\mathbf{m} \times \mathbf{x}}{|\mathbf{x}|^{3}} \quad \text { where } \quad \mathbf{m}=\frac{1}{2} \int_{V} \mathbf{x}^{\prime} \times \mathbf{j}\left(\mathbf{x}^{\prime}\right) d V^{\prime}$

Find the corresponding far-field expression for $\mathbf{B}(\mathbf{x})$.

1.I.5D

commentA steady two-dimensional velocity field is given by

$\mathbf{u}(x, y)=(\alpha x-\beta y, \beta x-\alpha y), \quad \alpha>0, \quad \beta>0$

(i) Calculate the vorticity of the flow.

(ii) Verify that $\mathbf{u}$ is a possible flow field for an incompressible fluid, and calculate the stream function.

(iii) Show that the streamlines are bounded if and only if $\alpha<\beta$.

(iv) What are the streamlines in the case $\alpha=\beta ?$

1.II.17D

commentWrite down the Euler equation for the steady motion of an inviscid, incompressible fluid in a constant gravitational field. From this equation, derive (a) Bernoulli's equation and (b) the integral form of the momentum equation for a fixed control volume $V$ with surface $S$.

(i) A circular jet of water is projected vertically upwards with speed $U_{0}$ from a nozzle of cross-sectional area $A_{0}$ at height $z=0$. Calculate how the speed $U$ and crosssectional area $A$ of the jet vary with $z$, for $z \ll U_{0}^{2} / 2 g$.

(ii) A circular jet of speed $U$ and cross-sectional area $A$ impinges axisymmetrically on the vertex of a cone of semi-angle $\alpha$, spreading out to form an almost parallel-sided sheet on the surface. Choose a suitable control volume and, neglecting gravity, show that the force exerted by the jet on the cone is

$\rho A U^{2}(1-\cos \alpha)$

(iii) A cone of mass $M$ is supported, axisymmetrically and vertex down, by the jet of part (i), with its vertex at height $z=h$, where $h \ll U_{0}^{2} / 2 g$. Assuming that the result of part (ii) still holds, show that $h$ is given by

$\rho A_{0} U_{0}^{2}\left(1-\frac{2 g h}{U_{0}^{2}}\right)^{\frac{1}{2}}(1-\cos \alpha)=M g$

1.I.2A

commentState the Gauss-Bonnet theorem for spherical triangles, and deduce from it that for each convex polyhedron with $F$ faces, $E$ edges, and $V$ vertices, $F-E+V=2$.

1.II.10G

comment(i) State a structure theorem for finitely generated abelian groups.

(ii) If $K$ is a field and $f$ a polynomial of degree $n$ in one variable over $K$, what is the maximal number of zeroes of $f$ ? Justify your answer in terms of unique factorization in some polynomial ring, or otherwise.

(iii) Show that any finite subgroup of the multiplicative group of non-zero elements of a field is cyclic. Is this true if the subgroup is allowed to be infinite?

1.I.1G

commentSuppose that $\left\{e_{1}, \ldots, e_{3}\right\}$ is a basis of the complex vector space $\mathbb{C}^{3}$ and that $A: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$ is the linear operator defined by $A\left(e_{1}\right)=e_{2}, A\left(e_{2}\right)=e_{3}$, and $A\left(e_{3}\right)=e_{1}$.

By considering the action of $A$ on column vectors of the form $\left(1, \xi, \xi^{2}\right)^{T}$, where $\xi^{3}=1$, or otherwise, find the diagonalization of $A$ and its characteristic polynomial.

1.II.9G

commentState and prove Sylvester's law of inertia for a real quadratic form.

[You may assume that for each real symmetric matrix A there is an orthogonal matrix $U$, such that $U^{-1} A U$ is diagonal.]

Suppose that $V$ is a real vector space of even dimension $2 m$, that $Q$ is a non-singular quadratic form on $V$ and that $U$ is an $m$-dimensional subspace of $V$ on which $Q$ vanishes. What is the signature of $Q ?$

1.II.19C

commentConsider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ on states $\{0,1, \ldots, r\}$ with transition matrix $\left(P_{i j}\right)$, where $P_{0,0}=1=P_{r, r}$, so that 0 and $r$ are absorbing states. Let

$A=\left(X_{n}=0, \text { for some } n \geqslant 0\right) \text {, }$

be the event that the chain is absorbed in 0 . Assume that $h_{i}=\mathbb{P}\left(A \mid X_{0}=i\right)>0$ for $1 \leqslant i<r$.

Show carefully that, conditional on the event $A,\left(X_{n}\right)_{n \geqslant 0}$ is a Markov chain and determine its transition matrix.

Now consider the case where $P_{i, i+1}=\frac{1}{2}=P_{i, i-1}$, for $1 \leqslant i<r$. Suppose that $X_{0}=i, 1 \leqslant i<r$, and that the event $A$ occurs; calculate the expected number of transitions until the chain is first in the state 0 .

1.II.14D

commentDefine the Fourier transform $\tilde{f}(k)$ of a function $f(x)$ that tends to zero as $|x| \rightarrow \infty$, and state the inversion theorem. State and prove the convolution theorem.

Calculate the Fourier transforms of

Hence show that

$\int_{-\infty}^{\infty} \frac{\sin (b k) e^{i k x}}{k\left(a^{2}+k^{2}\right)} d k=\frac{\pi \sinh (a b)}{a^{2}} e^{-a x} \quad \text { for } \quad x>b$

and evaluate this integral for all other (real) values of $x$.

1.II.12A

commentLet $X$ and $Y$ be topological spaces. Define the product topology on $X \times Y$ and show that if $X$ and $Y$ are Hausdorff then so is $X \times Y$.

Show that the following statements are equivalent.

(i) $X$ is a Hausdorff space.

(ii) The diagonal $\Delta=\{(x, x): x \in X\}$ is a closed subset of $X \times X$, in the product topology.

(iii) For any topological space $Y$ and any continuous maps $f, g: Y \rightarrow X$, the set $\{y \in Y: f(y)=g(y)\}$ is closed in $Y$.

1.I.6F

commentSolve the least squares problem

$\left[\begin{array}{ll} 1 & 3 \\ 0 & 2 \\ 0 & 2 \\ 0 & 1 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{r} 4 \\ 1 \\ 4 \\ -1 \end{array}\right]$

using $Q R$ method with Householder transformation. (A solution using normal equations is not acceptable.)

1.I.8C

commentState and prove the max-flow min-cut theorem for network flows.

1.II.15B

commentThe relative motion of a neutron and proton is described by the Schrödinger equation for a single particle of mass $m$ under the influence of the central potential

$V(r)=\left\{\begin{array}{rc} -U & r<a \\ 0 & r>a \end{array}\right.$

where $U$ and $a$ are positive constants. Solve this equation for a spherically symmetric state of the deuteron, which is a bound state of a proton and neutron, giving the condition on $U$ for this state to exist.

[If $\psi$ is spherically symmetric then $\left.\nabla^{2} \psi=\frac{1}{r} \frac{d^{2}}{d r^{2}}(r \psi) .\right]$

1.I.4B

commentWrite down the position four-vector. Suppose this represents the position of a particle with rest mass $M$ and velocity v. Show that the four momentum of the particle is

$p_{a}=(M \gamma c, M \gamma \mathbf{v})$

where $\gamma=\left(1-|\mathbf{v}|^{2} / c^{2}\right)^{-1 / 2}$.

For a particle of zero rest mass show that

$p_{a}=(|\mathbf{p}|, \mathbf{p})$

where $\mathbf{p}$ is the three momentum.

1.I.7C

commentLet $X_{1}, \ldots, X_{n}$ be independent, identically distributed random variables from the $N\left(\mu, \sigma^{2}\right)$ distribution where $\mu$ and $\sigma^{2}$ are unknown. Use the generalized likelihood-ratio test to derive the form of a test of the hypothesis $H_{0}: \mu=\mu_{0}$ against $H_{1}: \mu \neq \mu_{0}$.

Explain carefully how the test should be implemented.

1.II.18C

commentLet $X_{1}, \ldots, X_{n}$ be independent, identically distributed random variables with

$\mathbb{P}\left(X_{i}=1\right)=\theta=1-\mathbb{P}\left(X_{i}=0\right)$

where $\theta$ is an unknown parameter, $0<\theta<1$, and $n \geqslant 2$. It is desired to estimate the quantity $\phi=\theta(1-\theta)=n \operatorname{Var}\left(\left(X_{1}+\cdots+X_{n}\right) / n\right)$.

(i) Find the maximum-likelihood estimate, $\hat{\phi}$, of $\phi$.

(ii) Show that $\hat{\phi}_{1}=X_{1}\left(1-X_{2}\right)$ is an unbiased estimate of $\phi$ and hence, or otherwise, obtain an unbiased estimate of $\phi$ which has smaller variance than $\hat{\phi}_{1}$ and which is a function of $\hat{\phi}$.

(iii) Now suppose that a Bayesian approach is adopted and that the prior distribution for $\theta, \pi(\theta)$, is taken to be the uniform distribution on $(0,1)$. Compute the Bayes point estimate of $\phi$ when the loss function is $L(\phi, a)=(\phi-a)^{2}$.

[You may use that fact that when $r, s$ are non-negative integers,

$\left.\int_{0}^{1} x^{r}(1-x)^{s} d x=r ! s ! /(r+s+1) !\right]$