Part IB, 2006, Paper 4

Analysis II

• 4.I.3F

  Analysis II | Part IB, 2006

  Let $V$ be the vector space of all sequences $\left(x_{1}, x_{2}, \ldots\right)$ of real numbers such that $x_{i}$ converges to zero. Show that the function

  $$\left|\left(x_{1}, x_{2}, \ldots\right)\right|=\max _{i \geqslant 1}\left|x_{i}\right|$$

  defines a norm on $V$.

  Is the sequence

  $$(1,0,0,0, \ldots),(0,1,0,0, \ldots), \ldots$$

  convergent in $V ?$ Justify your answer.

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• 4.II.13F

  Analysis II | Part IB, 2006

  State precisely the contraction mapping theorem.

  An ancient way to approximate the square root of a positive number $a$ is to start with a guess $x>0$ and then hope that the average of $x$ and $a / x$ gives a better guess. We can then repeat the procedure using the new guess. Justify this procedure as follows. First, show that all the guesses after the first one are greater than or equal to $\sqrt{a}$. Then apply the properties of contraction mappings to the interval $[\sqrt{a}, \infty)$ to show that the procedure always converges to $\sqrt{a}$.

  Once the above procedure is close enough to $\sqrt{a}$, estimate how many more steps of the procedure are needed to get one more decimal digit of accuracy in computing $\sqrt{a}$.

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Complex Analysis

• 4.I.4H

  Complex Analysis | Part IB, 2006

  State the principle of isolated zeros for an analytic function on a domain in $\mathbf{C}$.

  Suppose $f$ is an analytic function on $\mathbf{C} \backslash\{0\}$, which is real-valued at the points $1 / n$, for $n=1,2, \ldots$, and does not have an essential singularity at the origin. Prove that $f(z)=\overline{f(\bar{z})}$ for all $z \in \mathbf{C} \backslash\{0\}$.

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Complex Methods

• 4.II.15D

  Complex Methods | Part IB, 2006

  Denote by $f * g$ the convolution of two functions, and by $\widehat{f}$ the Fourier transform, i.e.,

  $$[f * g](x)=\int_{-\infty}^{\infty} f(t) g(x-t) d t, \quad \widehat{f}(\lambda)=\int_{-\infty}^{\infty} f(x) e^{-i \lambda x} d x$$

  (a) Show that, for suitable functions $f$ and $g$, the Fourier transform $\widehat{F}$of the convolution $F=f * g$ is given by $\widehat{F}=\widehat{f} \cdot \widehat{g}$.

  (b) Let

  $$f_{1}(x)= \begin{cases}1 & |x| \leqslant 1 / 2 \\ 0 & \text { otherwise }\end{cases}$$

  and let $f_{2}=f_{1} * f_{1}$ be the convolution of $f_{1}$ with itself. Find the Fourier transforms of $f_{1}$ and $f_{2}$, and, by applying Parseval's theorem, determine the value of the integral

  $$\int_{-\infty}^{\infty}\left(\frac{\sin y}{y}\right)^{4} d y$$

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Electromagnetism

• 4.I.7G

  Electromagnetism | Part IB, 2006

  Starting from Maxwell's equations, deduce Faraday's law of induction

  $$\frac{d \Phi}{d t}=-\varepsilon$$

  for a moving circuit $C$, where $\Phi$ is the flux of $\mathbf{B}$ through the circuit and where the EMF $\varepsilon$ is defined to be

  $$\varepsilon=\oint_{C}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}$$

  with $\mathbf{v}(\mathbf{r})$ denoting the velocity of a point $\mathbf{r}$ of $C$.

  [Hint: consider the closed surface consisting of the surface $S(t)$ bounded by $C(t)$, the surface $S(t+\delta t)$ bounded by $C(t+\delta t)$ and the surface $S^{\prime}$ stretching from $C(t)$ to $C(t+\delta t)$. Show that the flux of $\mathbf{B}$ through $S^{\prime}$ is $-\oint_{C} \mathbf{B} \cdot(\mathbf{v} \times d \mathbf{r}) \delta t$.]

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Fluid Dynamics

• 4.II.18A

  Fluid Dynamics | Part IB, 2006

  A rectangular tank has a horizontal base and vertical sides. Viewed from above, the cross-section of the tank is a square of side $a$. At rest, the depth of water in the tank is $h$. Suppose that the free-surface is disturbed in such a way that the flow in the water is irrotational. Take the pressure at the free surface as atmospheric. Starting from the appropriate non-linear expressions, obtain free-surface boundary conditions for the velocity potential appropriate for small-amplitude disturbances of the surface.

  Show that the governing equations and boundary conditions admit small-amplitude normal mode solutions for which the free-surface elevation above its equilibrium level is everywhere proportional to $e^{i \omega t}$, and find the frequencies, $\omega$, of such modes.

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Geometry

• 4.II.12H

  Geometry | Part IB, 2006

  Describe the hyperbolic lines in both the disc and upper half-plane models of the hyperbolic plane. Given a hyperbolic line $l$ and a point $P \notin l$, we define

  $$d(P, l):=\inf _{Q \in l} \rho(P, Q)$$

  where $\rho$ denotes the hyperbolic distance. Show that $d(P, l)=\rho\left(P, Q^{\prime}\right)$, where $Q^{\prime}$ is the unique point of $l$ for which the hyperbolic line segment $P Q^{\prime}$ is perpendicular to $l$.

  Suppose now that $L_{1}$ is the positive imaginary axis in the upper half-plane model of the hyperbolic plane, and $L_{2}$ is the semicircle with centre $a>0$ on the real line, and radius $r$, where $0<r<a$. For any $P \in L_{2}$, show that the hyperbolic line through $P$ which is perpendicular to $L_{1}$ is a semicircle centred on the origin of radius $\leqslant a+r$, and prove that

  $$d\left(P, L_{1}\right) \geqslant \frac{a-r}{a+r} .$$

  For arbitrary hyperbolic lines $L_{1}, L_{2}$ in the hyperbolic plane, we define

  $$d\left(L_{1}, L_{2}\right):=\inf _{P \in L_{1}, Q \in L_{2}} \rho(P, Q)$$

  If $L_{1}$ and $L_{2}$ are ultraparallel (i.e. hyperbolic lines which do not meet, either inside the hyperbolic plane or at its boundary), prove that $d\left(L_{1}, L_{2}\right)$ is strictly positive.

  [The equivalence of the disc and upper half-plane models of the hyperbolic plane, and standard facts about the metric and isometries of these models, may be quoted without proof.]

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Groups, Rings and Modules

• 4.I $. 2 \mathrm{E} \quad$

  Groups, Rings and Modules | Part IB, 2006

  How many elements does the ring $\mathbb{Z}[X] /\left(3, X^{2}+X+1\right)$ have?

  Is this ring an integral domain?

  Briefly justify your answers.

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• 4.II.11E

  Groups, Rings and Modules | Part IB, 2006

  (a) Suppose that $R$ is a commutative ring, $M$ an $R$-module generated by $m_{1}, \ldots, m_{n}$ and $\phi \in \operatorname{End}_{R}(M)$. Show that, if $A=\left(a_{i j}\right)$ is an $n \times n$ matrix with entries in $R$ that represents $\phi$ with respect to this generating set, then in the sub-ring $R[\phi]$ of $\operatorname{End}_{R}(M)$ we have $\operatorname{det}\left(a_{i j}-\phi \delta_{i j}\right)=0 .$

  [Hint: $A$ is a matrix such that $\phi\left(m_{i}\right)=\sum a_{i j} m_{j}$ with $a_{i j} \in R$. Consider the matrix $C=\left(a_{i j}-\phi \delta_{i j}\right)$ with entries in $R[\phi]$ and use the fact that for any $n \times n$ matrix $N$ over any commutative ring, there is a matrix $N^{\prime}$ such that $N^{\prime} N=(\operatorname{det} N) 1_{n}$.]

  (b) Suppose that $k$ is a field, $V$ a finite-dimensional $k$-vector space and that $\phi \in \operatorname{End}_{k}(V)$. Show that if $A$ is the matrix of $\phi$ with respect to some basis of $V$ then $\phi$ satisfies the characteristic equation $\operatorname{det}(A-\lambda 1)=0$ of $A$.

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Linear Algebra

• 4.I.1H

  Linear Algebra | Part IB, 2006

  Suppose $V$ is a vector space over a field $k$. A finite set of vectors is said to be a basis for $V$ if it is both linearly independent and spanning. Prove that any two finite bases for $V$ have the same number of elements.

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• 4.II.10E

  Linear Algebra | Part IB, 2006

  Suppose that $\alpha$ is an orthogonal endomorphism of the finite-dimensional real inner product space $V$. Suppose that $V$ is decomposed as a direct sum of mutually orthogonal $\alpha$-invariant subspaces. How small can these subspaces be made, and how does $\alpha$ act on them? Justify your answer.

  Describe the possible matrices for $\alpha$ with respect to a suitably chosen orthonormal basis of $V$ when $\operatorname{dim} V=3$.

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Markov Chains

• 4.I.9C

  Markov Chains | Part IB, 2006

  A game of chance is played as follows. At each turn the player tosses a coin, which lands heads or tails with equal probability $1 / 2$. The outcome determines a score for that turn, which depends also on the cumulative score so far. Write $S_{n}$ for the cumulative score after $n$ turns. In particular $S_{0}=0$. When $S_{n}$ is odd, a head scores 1 but a tail scores 0 . When $S_{n}$ is a multiple of 4 , a head scores 4 and a tail scores 1 . When $S_{n}$ is even but is not a multiple of 4 , a head scores 2 and a tail scores 1 . By considering a suitable four-state Markov chain, determine the long run proportion of turns for which $S_{n}$ is a multiple of 4 . State clearly any general theorems to which you appeal.

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Methods

• 4.I.5G

  Methods | Part IB, 2006

  A finite-valued function $f(r, \theta, \phi)$, where $r, \theta, \phi$ are spherical polar coordinates, satisfies Laplace's equation in the regions $r<1$ and $r>1$, and $f \rightarrow 0$ as $r \rightarrow \infty$. At $r=1, f$ is continuous and its derivative with respect to $r$ is discontinuous by $A \sin ^{2} \theta$, where $A$ is a constant. Write down the general axisymmetric solution for $f$ in the two regions and use the boundary conditions to find $f$.

  $$\left[\text { Hint }: \quad P_{2}(\cos \theta)=\frac{1}{2}\left(3 \cos ^{2} \theta-1\right) \cdot\right]$$

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• 4.II.16B

  Methods | Part IB, 2006

  The integral

  $$I=\int_{a}^{b} F\left(y(x), y^{\prime}(x)\right) d x,$$

  where $F$ is some functional, is defined for the class of functions $y(x)$ for which $y(a)=y_{0}$, with the value $y(b)$ at the upper endpoint unconstrained. Suppose that $y(x)$ extremises the integral among the functions in this class. By considering perturbed paths of the form $y(x)+\epsilon \eta(x)$, with $\epsilon \ll 1$, show that

  $$\frac{d}{d x}\left(\frac{\partial F}{\partial y^{\prime}}\right)-\frac{\partial F}{\partial y}=0$$

  and that

  $$\left.\frac{\partial F}{\partial y^{\prime}}\right|_{x=b}=0 .$$

  Show further that

  $$F-y^{\prime} \frac{\partial F}{\partial y^{\prime}}=k$$

  for some constant $k$.

  A bead slides along a frictionless wire under gravity. The wire lies in a vertical plane with coordinates $(x, y)$ and connects the point $A$ with coordinates $(0,0)$ to the point $B$ with coordinates $\left(x_{0}, y\left(x_{0}\right)\right)$, where $x_{0}$ is given and $y\left(x_{0}\right)$ can take any value less than zero. The bead is released from rest at $A$ and slides to $B$ in a time $T$. For a prescribed $x_{0}$ find both the shape of the wire, and the value of $y\left(x_{0}\right)$, for which $T$ is as small as possible.

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Metric and Topological Spaces

• 4.II.14F

  Metric and Topological Spaces | Part IB, 2006

  (a) Show that every compact subset of a Hausdorff topological space is closed.

  (b) Let $X$ be a compact metric space. For $F$ a closed subset of $X$ and $p$ any point of $X$, show that there is a point $q$ in $F$ with

  $$d(p, q)=\inf _{q^{\prime} \in F} d\left(p, q^{\prime}\right)$$

  Suppose that for every $x$ and $y$ in $X$ there is a point $m$ in $X$ with $d(x, m)=(1 / 2) d(x, y)$ and $d(y, m)=(1 / 2) d(x, y)$. Show that $X$ is connected.

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Numerical Analysis

• 4.I.8D

  Numerical Analysis | Part IB, 2006

  (a) Given the data

  \begin{tabular}{c|r|r|r|r} $x_{i}$ & $-1$ & 0 & 1 & 3 \ \hline$f\left(x_{i}\right)$ & $-7$ & $-3$ & $-3$ & 9 \end{tabular}

  find the interpolating cubic polynomial $p \in \mathcal{P}_{3}$ in the Newton form, and transform it to the power form.

  (b) We add to the data one more value $f\left(x_{i}\right)$ at $x_{i}=2$. Find the power form of the interpolating quartic polynomial $q \in \mathcal{P}_{4}$ to the extended data

  \begin{tabular}{c|c|r|r|r|r} $x_{i}$ & $-1$ & 0 & 1 & 2 & 3 \ \hline$f\left(x_{i}\right)$ & $-7$ & $-3$ & $-3$ & $-7$ & 9 \end{tabular}$.$

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Optimization

• 4.II.20C

  Optimization | Part IB, 2006

  Use a suitable version of the simplex algorithm to solve the following linear programming problem:

  $\begin{array}{lrrrrr}\operatorname{maximize} & 50 x_{1} & - & 30 x_{2}+x_{3} & \\ \text { subject to } & x_{1}+ & x_{2}+x_{3} & \leq 30 \\ & 2 x_{1} & - & x_{2} & & \leq 35 \\ & x_{1} & + & 2 x_{2} & -x_{3} & \geq 40 \\ \text { and } & & & x_{1}, x_{2}, x_{3} & \geq 0\end{array}$

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Quantum Mechanics

• 4.I.6B

  Quantum Mechanics | Part IB, 2006

  (a) Define the probability density $\rho(\mathbf{x}, t)$ and the probability current $\mathbf{J}(\mathbf{x}, t)$ for a quantum mechanical wave function $\psi(\mathbf{x}, t)$, where the three dimensional vector $\mathbf{x}$ defines spatial coordinates.

  Given that the potential $V(\mathbf{x})$ is real, show that

  $$\boldsymbol{\nabla} \cdot \mathbf{J}+\frac{\partial \rho}{\partial t}=0$$

  (b) Write down the standard integral expressions for the expectation value $\langle A\rangle_{\psi}$ and the uncertainty $\Delta_{\psi} A$ of a quantum mechanical observable $A$ in a state with wavefunction $\psi(\mathbf{x})$. Give an expression for $\Delta_{\psi} A$ in terms of $\left\langle A^{2}\right\rangle_{\psi}$ and $\langle A\rangle_{\psi}$, and justify your answer.

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Special Relativity

• 4.II.17B

  Special Relativity | Part IB, 2006

  A javelin of length 4 metres is thrown at a speed of $\frac{12}{13} c$ horizontally and lengthwise through a barn of length 3 metres, which is open at both ends. (Here $c$ denotes the speed of light.)

  (a) What is the length of the javelin in the rest frame of the barn?

  (b) What is the length of the barn in the rest frame of the javelin?

  (c) Define the rest frame coordinates of the barn and of the javelin such that the point where the trailing end of the javelin enters the barn is the origin in both frames. Draw a space-time diagram in the rest frame coordinates $(c t, x)$ of the barn, showing the world lines of both ends of the javelin and of the front and back of the barn. Draw a second space-time diagram in the rest frame coordinates $\left(c t^{\prime}, x^{\prime}\right)$ of the javelin, again showing the world lines of both ends of the javelin and of the front and back of the barn.

  (d) Clearly mark the space-time events corresponding to (A) the trailing end of the javelin entering the barn, and (B) the leading end of the javelin exiting the back of the barn. Give the corresponding $(c t, x)$ and $\left(c t^{\prime}, x^{\prime}\right)$ coordinates for (B).

  Are the events (A) and (B) space-like, null, or time-like separated?

  As the javelin is longer than the barn in one frame and shorter than the barn in another, it might be argued that the javelin is contained entirely within the barn for a period according to an observer in one frame, but not according to an observer in another. Explain how this apparent inconsistency is resolved.

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Statistics

• 4.II.19C

  Statistics | Part IB, 2006

  Two series of experiments are performed, the first resulting in observations $X_{1}, \ldots, X_{m}$, the second resulting in observations $Y_{1}, \ldots, Y_{n}$. We assume that all observations are independent and normally distributed, with unknown means $\mu_{X}$ in the first series and $\mu_{Y}$ in the second series. We assume further that the variances of the observations are unknown but are all equal.

  Write down the distributions of the sample mean $\bar{X}=m^{-1} \sum_{i=1}^{m} X_{i}$ and sum of squares $S_{X X}=\sum_{i=1}^{m}\left(X_{i}-\bar{X}\right)^{2}$.

  Hence obtain a statistic $T(X, Y)$ to test the hypothesis $H_{0}: \mu_{X}=\mu_{Y}$ against $H_{1}: \mu_{X}>\mu_{Y}$ and derive its distribution under $H_{0}$. Explain how you would carry out a test of size $\alpha=1 / 100$.

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