Part IB, 2018, Paper 3

Analysis II

• Paper 3, Section I, $2 F$

  Analysis II | Part IB, 2018

  For a continuous function $f=\left(f_{1}, f_{2}, \ldots, f_{m}\right):[0,1] \rightarrow \mathbb{R}^{m}$, define

  $$\int_{0}^{1} f(t) d t=\left(\int_{0}^{1} f_{1}(t) d t, \int_{0}^{1} f_{2}(t) d t, \ldots, \int_{0}^{1} f_{m}(t) d t\right)$$

  Show that

  $$\left\|\int_{0}^{1} f(t) d t\right\|_{2} \leqslant \int_{0}^{1}\|f(t)\|_{2} d t$$

  for every continuous function $f:[0,1] \rightarrow \mathbb{R}^{m}$, where $\|\cdot\|_{2}$ denotes the Euclidean norm on $\mathbb{R}^{m}$.

  Find all continuous functions $f:[0,1] \rightarrow \mathbb{R}^{m}$ with the property that

  $$\left\|\int_{0}^{1} f(t) d t\right\|=\int_{0}^{1}\|f(t)\| d t$$

  regardless of the norm $\|\cdot\|$ on $\mathbb{R}^{m}$.

  [Hint: start by analysing the case when $\|\cdot\|$ is the Euclidean norm $\|\cdot\|_{2}$.]

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• Paper 3, Section II, F

  Analysis II | Part IB, 2018

  (a) Let $A \subset \mathbb{R}^{m}$ and let $f, f_{n}: A \rightarrow \mathbb{R}$ be functions for $n=1,2,3, \ldots$ What does it mean to say that the sequence $\left(f_{n}\right)$ converges uniformly to $f$ on $A$ ? What does it mean to say that $f$ is uniformly continuous?

  (b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a uniformly continuous function. Determine whether each of the following statements is true or false. Give reasons for your answers.

  (i) If $f_{n}(x)=f(x+1 / n)$ for each $n=1,2,3, \ldots$ and each $x \in \mathbb{R}$, then $f_{n} \rightarrow f$ uniformly on $\mathbb{R}$.

  (ii) If $g_{n}(x)=(f(x+1 / n))^{2}$ for each $n=1,2,3, \ldots$ and each $x \in \mathbb{R}$, then $g_{n} \rightarrow(f)^{2}$ uniformly on $\mathbb{R}$.

  (c) Let $A$ be a closed, bounded subset of $\mathbb{R}^{m}$. For each $n=1,2,3, \ldots$, let $g_{n}: A \rightarrow \mathbb{R}$ be a continuous function such that $\left(g_{n}(x)\right)$ is a decreasing sequence for each $x \in A$. If $\delta \in \mathbb{R}$ is such that for each $n$ there is $x_{n} \in A$ with $g_{n}\left(x_{n}\right) \geqslant \delta$, show that there is $x_{0} \in A$ such that $\lim _{n \rightarrow \infty} g_{n}\left(x_{0}\right) \geqslant \delta$.

  Deduce the following: If $f_{n}: A \rightarrow \mathbb{R}$ is a continuous function for each $n=1,2,3, \ldots$ such that $\left(f_{n}(x)\right)$ is a decreasing sequence for each $x \in A$, and if the pointwise limit of $\left(f_{n}\right)$ is a continuous function $f: A \rightarrow \mathbb{R}$, then $f_{n} \rightarrow f$ uniformly on $A$.

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Complex Analysis

• Paper 3, Section II, F

  Complex Analysis | Part IB, 2018

  Let $D=\{z \in \mathbb{C}:|z|<1\}$ and let $f: D \rightarrow \mathbb{C}$ be analytic.

  (a) If there is a point $a \in D$ such that $|f(z)| \leqslant|f(a)|$ for all $z \in D$, prove that $f$ is constant.

  (b) If $f(0)=0$ and $|f(z)| \leqslant 1$ for all $z \in D$, prove that $|f(z)| \leqslant|z|$ for all $z \in D$.

  (c) Show that there is a constant $C$ independent of $f$ such that if $f(0)=1$ and $f(z) \notin(-\infty, 0]$ for all $z \in D$ then $|f(z)| \leqslant C$ whenever $|z| \leqslant 1 / 2 .$

  [Hint: you may find it useful to consider the principal branch of the map $z \mapsto z^{1 / 2}$.]

  (d) Does the conclusion in (c) hold if we replace the hypothesis $f(z) \notin(-\infty, 0]$ for $z \in D$ with the hypothesis $f(z) \neq 0$ for $z \in D$, and keep all other hypotheses? Justify your answer.

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Complex Methods

• Paper 3, Section I, A

  Complex Methods | Part IB, 2018

  (a) Let $f(z)=\left(z^{2}-1\right)^{1 / 2}$. Define the branch cut of $f(z)$ as $[-1,1]$ such that

  $$f(x)=+\sqrt{x^{2}-1} \quad x>1$$

  Show that $f(z)$ is an odd function.

  (b) Let $g(z)=\left[(z-2)\left(z^{2}-1\right)\right]^{1 / 2}$.

  (i) Show that $z=\infty$ is a branch point of $g(z)$.

  (ii) Define the branch cuts of $g(z)$ as $[-1,1] \cup[2, \infty)$ such that

  $$g(x)=e^{\pi i / 2} \sqrt{|x-2|\left|x^{2}-1\right|} \quad x \in(1,2) .$$

  Find $g\left(0_{\pm}\right)$, where $0_{+}$denotes $z=0$ just above the branch cut, and $0_{-}$denotes $z=0$ just below the branch cut.

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Electromagnetism

• Paper 3, Section II, C

  Electromagnetism | Part IB, 2018

  Use Maxwell's equations to show that

  $$\frac{d}{d t} \int_{\Omega}\left(\frac{\epsilon_{0}}{2} \mathbf{E} \cdot \mathbf{E}+\frac{1}{2 \mu_{0}} \mathbf{B} \cdot \mathbf{B}\right) d V+\int_{\Omega} \mathbf{J} \cdot \mathbf{E} d V=-\frac{1}{\mu_{0}} \int_{\partial \Omega}(\mathbf{E} \times \mathbf{B}) \cdot \mathbf{n} d S$$

  where $\Omega \subset \mathbb{R}^{3}$ is a bounded region, $\partial \Omega$ its boundary and $\mathbf{n}$ its outward-pointing normal. Give an interpretation for each of the terms in this equation.

  A certain capacitor consists of two conducting, circular discs, each of large area $A$, separated by a small distance along their common axis. Initially, the plates carry charges $q_{0}$ and $-q_{0}$. At time $t=0$ the plates are connected by a resistive wire, causing the charge on the plates to decay slowly as $q(t)=q_{0} \mathrm{e}^{-\lambda t}$ for some constant $\lambda$. Construct the Poynting vector and show that energy flows radially out of the capacitor as it discharges.

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Fluid Dynamics

• Paper 3, Section II, D

  Fluid Dynamics | Part IB, 2018

  A soap bubble of radius $a(t)$ is attached to the end of a long, narrow straw of internal radius $\epsilon$ and length $L$, the other end of which is open to the atmosphere. The pressure difference between the inside and outside of the bubble is $2 \gamma / a$, where $\gamma$ is the surface tension of the soap bubble. At time $t=0, a=a_{0}$ and the air in the straw is at rest. Assume that the flow of air through the straw is irrotational and consider the pressure drop along the straw to show that subsequently

  $$a^{3} \ddot{a}+2 a^{2} \dot{a}^{2}=-\frac{\gamma \epsilon^{2}}{2 \rho L},$$

  where $\rho$ is the density of air.

  By multiplying the equation by $2 a \dot{a}$ and integrating, or otherwise, determine an implicit equation for $a(t)$ and show that the bubble disappears in a time

  $$t=\frac{\pi}{2} \frac{a_{0}^{2}}{\epsilon}\left(\frac{\rho L}{2 \gamma}\right)^{1 / 2}$$

  [Hint: The substitution $a=a_{0} \sin \theta$ can be used.]

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Geometry

• Paper 3, Section I, G

  Geometry | Part IB, 2018

  Consider a quadrilateral $A B C D$ in the hyperbolic plane whose sides are hyperbolic line segments. Suppose angles $A B C, B C D$ and $C D A$ are right-angles. Prove that $A D$ is longer than $B C$.

  [You may use without proof the distance formula in the upper-half-plane model

  $$\left.\rho\left(z_{1}, z_{2}\right)=2 \tanh ^{-1}\left|\frac{z_{1}-z_{2}}{z_{1}-\bar{z}_{2}}\right| \cdot\right]$$

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• Paper 3, Section II, G

  Geometry | Part IB, 2018

  Let $U$ be an open subset of the plane $\mathbb{R}^{2}$, and let $\sigma: U \rightarrow S$ be a smooth parametrization of a surface $S$. A coordinate curve is an arc either of the form

  $$\alpha_{v_{0}}(t)=\sigma\left(t, v_{0}\right)$$

  for some constant $v_{0}$ and $t \in\left[u_{1}, u_{2}\right]$, or of the form

  $$\beta_{u_{0}}(t)=\sigma\left(u_{0}, t\right)$$

  for some constant $u_{0}$ and $t \in\left[v_{1}, v_{2}\right]$. A coordinate rectangle is a rectangle in $S$ whose sides are coordinate curves.

  Prove that all coordinate rectangles in $S$ have opposite sides of the same length if and only if $\frac{\partial E}{\partial v}=\frac{\partial G}{\partial u}=0$ at all points of $S$, where $E$ and $G$ are the usual components of the first fundamental form, and $(u, v)$ are coordinates in $U$.

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Groups, Rings and Modules

• Paper 3, Section I, $1 G$

  Groups, Rings and Modules | Part IB, 2018

  (a) Find all integer solutions to $x^{2}+5 y^{2}=9$.

  (b) Find all the irreducibles in $\mathbb{Z}[\sqrt{-5}]$ of norm 9 .

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• Paper 3, Section II, G

  Groups, Rings and Modules | Part IB, 2018

  (a) State Gauss's Lemma.

  (b) State and prove Eisenstein's criterion for the irreducibility of a polynomial.

  (c) Determine whether or not the polynomial

  $$f(X)=2 X^{3}+19 X^{2}-54 X+3$$

  is irreducible over $\mathbb{Q}$.

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Linear Algebra

• Paper 3, Section II, E

  Linear Algebra | Part IB, 2018

  State and prove the Cayley-Hamilton Theorem.

  Let $A$ be an $n \times n$ complex matrix. Using division of polynomials, show that if $p(x)$ is a polynomial then there is another polynomial $r(x)$ of degree at most $(n-1)$ such that $p(\lambda)=r(\lambda)$ for each eigenvalue $\lambda$ of $A$ and such that $p(A)=r(A)$.

  Hence compute the $(1,1)$ entry of the matrix $A^{1000}$ when

  $$A=\left[\begin{array}{ccc} 2 & -1 & 0 \\ 1 & -1 & 1 \\ -1 & -1 & 1 \end{array}\right]$$

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Markov Chains

• Paper 3, Section I, H

  Markov Chains | Part IB, 2018

  The mathematics course at the University of Barchester is a three-year one. After the end-of-year examinations there are three possibilities:

  (i) failing and leaving (probability $p$ );

  (ii) taking that year again (probability $q$ );

  (iii) going on to the next year (or graduating, if the current year is the third one) (probability $r$ ).

  Thus there are five states for a student $\left(1^{\text {st }}\right.$ year, $2^{\text {nd }}$year, $3^{\text {rd }}$year, left without a degree, graduated).

  Write down the $5 \times 5$ transition matrix. Classify the states, assuming $p, q, r \in(0,1)$. Find the probability that a student will eventually graduate.

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Methods

• Paper 3, Section I, A

  Methods | Part IB, 2018

  (a) Determine the Green's function $G(x ; \xi)$ satisfying

  $$G^{\prime \prime}-4 G^{\prime}+4 G=\delta(x-\xi),$$

  with $G(0 ; \xi)=G(1 ; \xi)=0$. Here ' denotes differentiation with respect to $x$.

  (b) Using the Green's function, solve

  $$y^{\prime \prime}-4 y^{\prime}+4 y=e^{2 x}$$

  with $y(0)=y(1)=0$.

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• Paper 3, Section II, A

  Methods | Part IB, 2018

  Consider the Dirac delta function, $\delta(x)$, defined by the sampling property

  $$\int_{-\infty}^{\infty} f(x) \delta\left(x-x_{0}\right) d x=f\left(x_{0}\right)$$

  for any suitable function $f(x)$ and real constant $x_{0}$.

  (a) Show that $\delta(\alpha x)=|\alpha|^{-1} \delta(x)$ for any non-zero $\alpha \in \mathbb{R}$.

  (b) Show that $x \delta^{\prime}(x)=-\delta(x)$, where ${ }^{\prime}$ denotes differentiation with respect to $x$.

  (c) Calculate

  $$\int_{-\infty}^{\infty} f(x) \delta^{(m)}(x) d x$$

  where $\delta^{(m)}(x)$ is the $m^{\text {th }}$derivative of the delta function.

  (d) For

  $$\gamma_{n}(x)=\frac{1}{\pi} \frac{n}{(n x)^{2}+1}$$

  show that $\lim _{n \rightarrow \infty} \gamma_{n}(x)=\delta(x)$.

  (e) Find expressions in terms of the delta function and its derivatives for

  (i)

  $$\lim _{n \rightarrow \infty} n^{3} x e^{-x^{2} n^{2}}$$

  (ii)

  $$\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{0}^{n} \cos (k x) d k .$$

  (f) Hence deduce that

  $$\lim _{n \rightarrow \infty} \frac{1}{2 \pi} \int_{-n}^{n} e^{i k x} d k=\delta(x)$$

  [You may assume that

  $$\left.\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi} \quad \text { and } \quad \int_{-\infty}^{\infty} \frac{\sin y}{y} d y=\pi .\right]$$

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Metric and Topological Spaces

• Paper 3, Section I, $3 E$

  Metric and Topological Spaces | Part IB, 2018

  What does it mean to say that a topological space is connected? If $X$ is a topological space and $x \in X$, show that there is a connected subspace $K_{x}$ of $X$ so that if $S$ is any other connected subspace containing $x$ then $S \subseteq K_{x}$.

  Show that the sets $K_{x}$ partition $X$.

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Numerical Analysis

• Paper 3, Section II, D

  Numerical Analysis | Part IB, 2018

  Taylor's theorem for functions $f \in C^{k+1}[a, b]$ is given in the form

  $$f(x)=f(a)+(x-a) f^{\prime}(a)+\cdots+\frac{(x-a)^{k}}{k !} f^{(k)}(a)+R(x) .$$

  Use integration by parts to show that

  $$R(x)=\frac{1}{k !} \int_{a}^{x}(x-\theta)^{k} f^{(k+1)}(\theta) d \theta$$

  Let $\lambda_{k}$ be a linear functional on $C^{k+1}[a, b]$ such that $\lambda_{k}[p]=0$ for $p \in \mathbb{P}_{k}$. Show that

  $$\lambda_{k}[f]=\frac{1}{k !} \int_{a}^{b} K(\theta) f^{(k+1)}(\theta) d \theta$$

  where the Peano kernel function $K(\theta)=\lambda_{k}\left[(x-\theta)_{+}^{k}\right] . \quad[$ You may assume that the functional commutes with integration over a fixed interval.]

  The error in the mid-point rule for numerical quadrature on $[0,1]$ is given by

  $$e[f]=\int_{0}^{1} f(x) d x-f\left(\frac{1}{2}\right)$$

  Show that $e[p]=0$ if $p$ is a linear polynomial. Find the Peano kernel function corresponding to $e$ explicitly and verify the formula ( $\dagger$ ) in the case $f(x)=x^{2}$.

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Optimization

• Paper 3, Section II, 21H

  Optimization | Part IB, 2018

  State and prove the Lagrangian Sufficiency Theorem.

  The manufacturers, $A$ and $B$, of two competing soap powders must plan how to allocate their advertising resources ( $X$ and $Y$ pounds respectively) among $n$ distinct geographical regions. If $x_{i} \geqslant 0$ and $y_{i} \geqslant 0$ denote, respectively, the resources allocated to area $i$ by $A$ and $B$ then the number of packets sold by $A$ and $B$ in area $i$ are

  $$\frac{x_{i} u_{i}}{x_{i}+y_{i}}, \quad \frac{y_{i} u_{i}}{x_{i}+y_{i}}$$

  respectively, where $u_{i}$ is the total market in area $i$, and $u_{1}, u_{2}, \ldots, u_{n}$ are known constants. The difference between the amount sold by $A$ and $B$ is then

  $$\sum_{i=1}^{n} \frac{x_{i}-y_{i}}{x_{i}+y_{i}} u_{i}$$

  $A$ seeks to maximize this quantity, while $B$ seeks to minimize it.

  (i) If $A$ knows $B$ 's allocation, how should $A$ choose $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ ?

  (ii) Determine the best strategies for $A$ and $B$ if each assumes the other will know its strategy and react optimally.

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Quantum Mechanics

• Paper 3, Section I, B

  Quantum Mechanics | Part IB, 2018

  What is meant by the statement that an operator is Hermitian?

  Consider a particle of mass $m$ in a real potential $V(x)$ in one dimension. Show that the Hamiltonian of the system is Hermitian.

  Starting from the time-dependent Schrödinger equation, show that

  $$\frac{d}{d t}\langle\hat{x}\rangle=\frac{1}{m}\langle\hat{p}\rangle, \quad \frac{d}{d t}\langle\hat{p}\rangle=-\left\langle V^{\prime}(\hat{x})\right\rangle$$

  where $\hat{p}$ is the momentum operator and $\langle\hat{A}\rangle$ denotes the expectation value of the operator $\hat{A}$.

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• Paper 3, Section II, B

  Quantum Mechanics | Part IB, 2018

  What is the physical significance of the expectation value

  $$\langle Q\rangle=\int \psi^{*}(x) Q \psi(x) d x$$

  of an observable $Q$ in the normalised state $\psi(x)$ ? Let $P$ and $Q$ be two observables. By considering the norm of $(Q+i \lambda P) \psi$ for real values of $\lambda$, show that

  $$\left\langle Q^{2}\right\rangle\left\langle P^{2}\right\rangle \geqslant \frac{1}{4}|\langle[Q, P]\rangle|^{2} .$$

  Deduce the generalised uncertainty relation

  $$\Delta Q \Delta P \geqslant \frac{1}{2}|\langle[Q, P]\rangle|,$$

  where the uncertainty $\Delta Q$ in the state $\psi(x)$ is defined by

  $$(\Delta Q)^{2}=\left\langle(Q-\langle Q\rangle)^{2}\right\rangle$$

  A particle of mass $m$ moves in one dimension under the influence of the potential $\frac{1}{2} m \omega^{2} x^{2}$. By considering the commutator $[x, p]$, show that every energy eigenvalue $E$ satisfies

  $$E \geqslant \frac{1}{2} \hbar \omega$$

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Statistics

• Paper 3, Section II, H

  Statistics | Part IB, 2018

  A treatment is suggested for a particular illness. The results of treating a number of patients chosen at random from those in a hospital suffering from the illness are shown in the following table, in which the entries $a, b, c, d$ are numbers of patients.

  $\begin{array}{lcc} & \text { Recovery } & \text { Non-recovery } \\ \text { Untreated } & a & b \\ \text { Treated } & c & d\end{array}$

  Describe the use of Pearson's $\chi^{2}$ statistic in testing whether the treatment affects recovery, and outline a justification derived from the generalised likelihood ratio statistic. Show that

  $$\chi^{2}=\frac{(a d-b c)^{2}(a+b+c+d)}{(a+b)(c+d)(a+c)(b+d)}$$

  [Hint: You may find it helpful to observe that $a(a+b+c+d)-(a+b)(a+c)=a d-b c .]$

  Comment on the use of this statistical technique when

  $$a=50, \quad b=10, \quad c=15, \quad d=5 .$$

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Variational Principles

• Paper 3, Section I, B

  Variational Principles | Part IB, 2018

  For a particle of unit mass moving freely on a unit sphere, the Lagrangian in polar coordinates is

  $$L=\frac{1}{2} \dot{\theta}^{2}+\frac{1}{2} \sin ^{2} \theta \dot{\phi}^{2}$$

  Determine the equations of motion. Show that $l=\sin ^{2} \theta \dot{\phi}$ is a conserved quantity, and use this result to simplify the equation of motion for $\theta$. Deduce that

  $$h=\dot{\theta}^{2}+\frac{l^{2}}{\sin ^{2} \theta}$$

  is a conserved quantity. What is the interpretation of $h$ ?

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