Part IB, 2013, Paper 3

Analysis II

• Paper 3, Section I, $2 F$

  Analysis II | Part IB, 2013

  For each of the following sequences of functions on $[0,1]$, indexed by $n=1,2, \ldots$, determine whether or not the sequence has a pointwise limit, and if so, determine whether or not the convergence to the pointwise limit is uniform.

  1. $f_{n}(x)=1 /\left(1+n^{2} x^{2}\right)$

  2. $g_{n}(x)=n x(1-x)^{n}$

  3. $h_{n}(x)=\sqrt{n} x(1-x)^{n}$

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• Paper 3, Section II, F

  Analysis II | Part IB, 2013

  For each of the following statements, provide a proof or justify a counterexample.

  1. The norms $\|x\|_{1}=\sum_{i=1}^{n}\left|x_{i}\right|$ and $\|x\|_{\infty}=\max _{1 \leqslant i \leqslant n}\left|x_{i}\right|$ on $\mathbb{R}^{n}$ are Lipschitz equivalent.

  2. The norms $\|x\|_{1}=\sum_{i=1}^{\infty}\left|x_{i}\right|$ and $\|x\|_{\infty}=\max _{i}\left|x_{i}\right|$ on the vector space of sequences $\left(x_{i}\right)_{i \geqslant 1}$ with $\sum\left|x_{i}\right|<\infty$ are Lipschitz equivalent.

  3. Given a linear function $\phi: V \rightarrow W$ between normed real vector spaces, there is some $N$ for which $\|\phi(x)\| \leqslant N$ for every $x \in V$ with $\|x\| \leqslant 1$.

  4. Given a linear function $\phi: V \rightarrow W$ between normed real vector spaces for which there is some $N$ for which $\|\phi(x)\| \leqslant N$ for every $x \in V$ with $\|x\| \leqslant 1$, then $\phi$ is continuous.

  5. The uniform norm $\|f\|=\sup _{x \in \mathbb{R}}|f(x)|$ is complete on the vector space of continuous real-valued functions $f$ on $\mathbb{R}$ for which $f(x)=0$ for $|x|$ sufficiently large.

  6. The uniform norm $\|f\|=\sup _{x \in \mathbb{R}}|f(x)|$ is complete on the vector space of continuous real-valued functions $f$ on $\mathbb{R}$ which are bounded.

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Complex Analysis

• Paper 3, Section II, E

  Complex Analysis | Part IB, 2013

  Let $D=\{z \in \mathbb{C}|| z \mid<1\}$ be the open unit disk, and let $C$ be its boundary (the unit circle), with the anticlockwise orientation. Suppose $\phi: C \rightarrow \mathbb{C}$ is continuous. Stating clearly any theorems you use, show that

  $$g_{\phi}(w)=\frac{1}{2 \pi i} \int_{C} \frac{\phi(z)}{z-w} d z$$

  is an analytic function of $w$ for $w \in D$.

  Now suppose $\phi$ is the restriction of a holomorphic function $F$ defined on some annulus $1-\epsilon<|z|<1+\epsilon$. Show that $g_{\phi}(w)$ is the restriction of a holomorphic function defined on the open disc $|w|<1+\epsilon$.

  Let $f_{\phi}:[0,2 \pi] \rightarrow \mathbb{C}$ be defined by $f_{\phi}(\theta)=\phi\left(e^{i \theta}\right)$. Express the coefficients in the power series expansion of $g_{\phi}$ centered at 0 in terms of $f_{\phi}$.

  Let $n \in \mathbb{Z}$. What is $g_{\phi}$ in the following cases?

  1. $\phi(z)=z^{n}$.

  2. $\phi(z)=\bar{z}^{n}$.

  3. $\phi(z)=(\operatorname{Re} z)^{2}$.

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Complex Methods

• Paper 3, Section I, D

  Complex Methods | Part IB, 2013

  Let $y(t)=0$ for $t<0$, and let $\lim _{t \rightarrow 0^{+}} y(t)=y_{0}$.

  (i) Find the Laplace transforms of $H(t)$ and $t H(t)$, where $H(t)$ is the Heaviside step function.

  (ii) Given that the Laplace transform of $y(t)$ is $\widehat{y}(s)$, find expressions for the Laplace transforms of $\dot{y}(t)$ and $y(t-1)$.

  (iii) Use Laplace transforms to solve the equation

  $$\dot{y}(t)-y(t-1)=H(t)-(t-1) H(t-1)$$

  in the case $y_{0}=0$.

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Electromagnetism

• Paper 3, Section II, D

  Electromagnetism | Part IB, 2013

  Three sides of a closed rectangular circuit $C$ are fixed and one is moving. The circuit lies in the plane $z=0$ and the sides are $x=0, y=0, x=a(t), y=b$, where $a(t)$ is a given function of time. A magnetic field $\mathbf{B}=\left(0,0, \frac{\partial f}{\partial x}\right)$ is applied, where $f(x, t)$ is a given function of $x$ and $t$ only. Find the magnetic flux $\Phi$ of $\mathbf{B}$ through the surface $S$ bounded by $C$.

  Find an electric field $\mathbf{E}_{\mathbf{0}}$ that satisfies the Maxwell equation

  $$\boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$

  and then write down the most general solution $\mathbf{E}$ in terms of $\mathbf{E}_{0}$ and an undetermined scalar function independent of $f$.

  Verify that

  $$\oint_{C}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}=-\frac{d \Phi}{d t},$$

  where $\mathbf{v}$ is the velocity of the relevant side of $C$. Interpret the left hand side of this equation.

  If a unit current flows round $C$, what is the rate of work required to maintain the motion of the moving side of the rectangle? You should ignore any electromagnetic fields produced by the current.

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Fluid Dynamics

• Paper 3, Section II, A

  Fluid Dynamics | Part IB, 2013

  A layer of incompressible fluid of density $\rho$ and viscosity $\mu$ flows steadily down a plane inclined at an angle $\theta$ to the horizontal. The layer is of uniform thickness $h$ measured perpendicular to the plane and the viscosity of the overlying air can be neglected. Using coordinates $x$ parallel to the plane (in steepest downwards direction) and $y$ normal to the plane, write down the equations of motion and the boundary conditions on the plane and on the free top surface. Determine the pressure and velocity fields and show that the volume flux down the plane is

  $$\frac{\rho g h^{3} \sin \theta}{3 \mu}$$

  Consider now the case where a second layer of fluid, of uniform thickness $\alpha h$, viscosity $\beta \mu$ and density $\rho$, flows steadily on top of the first layer. Explain why one of the appropriate boundary conditions between the two fluids is

  $$\mu \frac{\partial}{\partial y} u\left(h_{-}\right)=\beta \mu \frac{\partial}{\partial y} u\left(h_{+}\right),$$

  where $u$ is the component of velocity in the $x$ direction and $h_{-}$and $h_{+}$refer to just below and just above the boundary respectively. Determine the velocity field in each layer.

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Geometry

• Paper 3, Section I, F

  Geometry | Part IB, 2013

  Let $S$ be a surface with Riemannian metric having first fundamental form $d u^{2}+G(u, v) d v^{2}$. State a formula for the Gauss curvature $K$ of $S$.

  Suppose that $S$ is flat, so $K$ vanishes identically, and that $u=0$ is a geodesic on $S$ when parametrised by arc-length. Using the geodesic equations, or otherwise, prove that $G(u, v) \equiv 1$, i.e. $S$ is locally isometric to a plane.

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• Paper 3, Section II, F

  Geometry | Part IB, 2013

  Show that the set of all straight lines in $\mathbb{R}^{2}$ admits the structure of an abstract smooth surface $S$. Show that $S$ is an open Möbius band (i.e. the Möbius band without its boundary circle), and deduce that $S$ admits a Riemannian metric with vanishing Gauss curvature.

  Show that there is no metric $d: S \times S \rightarrow \mathbb{R}_{\geqslant 0}$, in the sense of metric spaces, which

  1. induces the locally Euclidean topology on $S$ constructed above;

  2. is invariant under the natural action on $S$ of the group of translations of $\mathbb{R}^{2}$.

  Show that the set of great circles on the two-dimensional sphere admits the structure of a smooth surface $S^{\prime}$. Is $S^{\prime}$ homeomorphic to $S$ ? Does $S^{\prime}$ admit a Riemannian metric with vanishing Gauss curvature? Briefly justify your answers.

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Groups, Rings and Modules

• Paper 3, Section I, $1 G$

  Groups, Rings and Modules | Part IB, 2013

  Define the notion of a free module over a ring. When $R$ is a PID, show that every ideal of $R$ is free as an $R$-module.

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• Paper 3, Section II, G

  Groups, Rings and Modules | Part IB, 2013

  Let $R=\mathbb{C}[X, Y]$ be the polynomial ring in two variables over the complex numbers, and consider the principal ideal $I=\left(X^{3}-Y^{2}\right)$ of $R$.

  (i) Using the fact that $R$ is a UFD, show that $I$ is a prime ideal of $R$. [Hint: Elements in $\mathbb{C}[X, Y]$ are polynomials in $Y$ with coefficients in $\mathbb{C}[X] .]$

  (ii) Show that $I$ is not a maximal ideal of $R$, and that it is contained in infinitely many distinct proper ideals in $R$.

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Linear Algebra

• Paper 3, Section II, E

  Linear Algebra | Part IB, 2013

  Let $V$ and $W$ be finite dimensional real vector spaces and let $T: V \rightarrow W$ be a linear map. Define the dual space $V^{*}$ and the dual map $T^{*}$. Show that there is an isomorphism $\iota: V \rightarrow\left(V^{*}\right)^{*}$ which is canonical, in the sense that $\iota \circ S=\left(S^{*}\right)^{*} \circ \iota$ for any automorphism $S$ of $V$.

  Now let $W$ be an inner product space. Use the inner product to show that there is an injective map from im $T$ to $\operatorname{im} T^{*}$. Deduce that the row rank of a matrix is equal to its column rank.

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Markov Chains

• Paper 3, Section I, H

  Markov Chains | Part IB, 2013

  Prove that if a distribution $\pi$ is in detailed balance with a transition matrix $P$ then it is an invariant distribution for $P$.

  Consider the following model with 2 urns. At each time, $t=0,1, \ldots$ one of the following happens:

  • with probability $\beta$ a ball is chosen at random and moved to the other urn (but nothing happens if both urns are empty);

  • with probability $\gamma$ a ball is chosen at random and removed (but nothing happens if both urns are empty);

  • with probability $\alpha$ a new ball is added to a randomly chosen urn,

  where $\alpha+\beta+\gamma=1$ and $\alpha<\gamma$. State $(i, j)$ denotes that urns 1,2 contain $i$ and $j$ balls respectively. Prove that there is an invariant measure

  $$\lambda_{i, j}=\frac{(i+j) !}{i ! j !}(\alpha / 2 \gamma)^{i+j}$$

  Find the proportion of time for which there are $n$ balls in the system.

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Methods

• Paper 3, Section I, C

  Methods | Part IB, 2013

  The solution to the Dirichlet problem on the half-space $D=\{\mathbf{x}=(x, y, z): z>0\}$ :

  $$\nabla^{2} u(\mathbf{x})=0, \quad \mathbf{x} \in D, \quad u(\mathbf{x}) \rightarrow 0 \quad \text { as } \quad|\mathbf{x}| \rightarrow \infty, \quad u(x, y, 0)=h(x, y)$$

  is given by the formula

  $$u\left(\mathbf{x}_{0}\right)=u\left(x_{0}, y_{0}, z_{0}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h(x, y) \frac{\partial}{\partial n} G\left(\mathbf{x}, \mathbf{x}_{0}\right) d x d y$$

  where $n$ is the outward normal to $\partial D$.

  State the boundary conditions on $G$ and explain how $G$ is related to $G_{3}$, where

  $$G_{3}\left(\mathbf{x}, \mathbf{x}_{0}\right)=-\frac{1}{4 \pi} \frac{1}{\left|\mathbf{x}-\mathbf{x}_{0}\right|}$$

  is the fundamental solution to the Laplace equation in three dimensions.

  Using the method of images find an explicit expression for the function $\frac{\partial}{\partial n} G\left(\mathbf{x}, \mathbf{x}_{0}\right)$ in the formula.

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• Paper 3, Section II, C

  Methods | Part IB, 2013

  The Laplace equation in plane polar coordinates has the form

  $$\nabla^{2} \phi=\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right] \phi(r, \theta)=0 .$$

  Using separation of variables, derive the general solution to the equation that is singlevalued in the domain $1<r<2$.

  For

  $$f(\theta)=\sum_{n=1}^{\infty} A_{n} \sin n \theta$$

  solve the Laplace equation in the annulus with the boundary conditions:

  $$\nabla^{2} \phi=0, \quad 1<r<2, \quad \phi(r, \theta)= \begin{cases}f(\theta), & r=1 \\ f(\theta)+1, & r=2\end{cases}$$

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Metric and Topological Spaces

• Paper 3, Section I, G

  Metric and Topological Spaces | Part IB, 2013

  Let $X$ be a metric space with the metric $d: X \times X \rightarrow \mathbb{R}$.

  (i) Show that if $X$ is compact as a topological space, then $X$ is complete.

  (ii) Show that the completeness of $X$ is not a topological property, i.e. give an example of two metrics $d, d^{\prime}$ on a set $X$, such that the associated topologies are the same, but $(X, d)$ is complete and $\left(X, d^{\prime}\right)$ is not.

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Numerical Analysis

• Paper 3, Section II, C

  Numerical Analysis | Part IB, 2013

  $$f^{\prime}(0) \approx a_{0} f(0)+a_{1} f(1)+a_{2} f(2)=: \lambda(f)$$

  be a formula of numerical differentiation which is exact on polynomials of degree 2 , and let

  $$e(f)=f^{\prime}(0)-\lambda(f)$$

  be its error.

  Find the values of the coefficients $a_{0}, a_{1}, a_{2}$.

  Using the Peano kernel theorem, find the least constant $c$ such that, for all functions $f \in C^{3}[0,2]$, we have

  $$|e(f)| \leqslant c\left\|f^{\prime \prime \prime}\right\|_{\infty} .$$

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Optimization

• Paper 3, Section II, H

  Optimization | Part IB, 2013

  Use the two phase method to find all optimal solutions to the problem

  $$\begin{aligned} \operatorname{maximize} 2 x_{1}+3 x_{2}+x_{3} \\ \text { subject to } x_{1}+x_{2}+x_{3} & \leqslant 40 \\ 2 x_{1}+x_{2}-x_{3} & \geqslant 10 \\ -x_{2}+x_{3} & \geqslant 10 \\ x_{1}, x_{2}, x_{3} & \geqslant 0 \end{aligned}$$

  Suppose that the values $(40,10,10)$ are perturbed to $(40,10,10)+\left(\epsilon_{1}, \epsilon_{2}, \epsilon_{3}\right)$. Find an expression for the change in the optimal value, which is valid for all sufficiently small values of $\epsilon_{1}, \epsilon_{2}, \epsilon_{3}$.

  Suppose that $\left(\epsilon_{1}, \epsilon_{2}, \epsilon_{3}\right)=(\theta,-2 \theta, 0)$. For what values of $\theta$ is your expression valid?

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Quantum Mechanics

• Paper 3, Section I, B

  Quantum Mechanics | Part IB, 2013

  If $\alpha, \beta$ and $\gamma$ are linear operators, establish the identity

  $$[\alpha \beta, \gamma]=\alpha[\beta, \gamma]+[\alpha, \gamma] \beta$$

  In what follows, the operators $x$ and $p$ are Hermitian and represent position and momentum of a quantum mechanical particle in one-dimension. Show that

  $$\left[x^{n}, p\right]=i \hbar n x^{n-1}$$

  and

  $$\left[x, p^{m}\right]=i \hbar m p^{m-1}$$

  where $m, n \in \mathbb{Z}^{+}$. Assuming $\left[x^{n}, p^{m}\right] \neq 0$, show that the operators $x^{n}$ and $p^{m}$ are Hermitian but their product is not. Determine whether $x^{n} p^{m}+p^{m} x^{n}$ is Hermitian.

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• Paper 3, Section II, B

  Quantum Mechanics | Part IB, 2013

  Obtain, with the aid of the time-dependent Schrödinger equation, the conservation equation

  $$\frac{\partial}{\partial t} \rho(\mathbf{x}, t)+\nabla \cdot \mathbf{j}(\mathbf{x}, t)=0$$

  where $\rho(\mathbf{x}, t)$ is the probability density and $\mathbf{j}(\mathbf{x}, t)$ is the probability current. What have you assumed about the potential energy of the system?

  Show that if the potential $U(\mathbf{x}, t)$ is complex the conservation equation becomes

  $$\frac{\partial}{\partial t} \rho(\mathbf{x}, t)+\nabla \cdot \mathbf{j}(\mathbf{x}, t)=\frac{2}{\hbar} \rho(\mathbf{x}, t) \operatorname{Im} U(\mathbf{x}, t)$$

  Take the potential to be time-independent. Show, with the aid of the divergence theorem, that

  $$\frac{d}{d t} \int_{\mathbb{R}^{3}} \rho(\mathbf{x}, t) d V=\frac{2}{\hbar} \int_{\mathbb{R}^{3}} \rho(\mathbf{x}, t) \operatorname{Im} U(\mathbf{x}) d V$$

  Assuming the wavefunction $\psi(\mathbf{x}, 0)$ is normalised to unity, show that if $\rho(\mathbf{x}, t)$ is expanded about $t=0$ so that $\rho(\mathbf{x}, t)=\rho_{0}(\mathbf{x})+t \rho_{1}(\mathbf{x})+\cdots$, then

  $$\int_{\mathbb{R}^{3}} \rho(\mathbf{x}, t) d V=1+\frac{2 t}{\hbar} \int_{\mathbb{R}^{3}} \rho_{0}(\mathbf{x}) \operatorname{Im} U(\mathbf{x}) d V+\cdots$$

  As time increases, how does the quantity on the left of this equation behave if $\operatorname{Im} U(\mathbf{x})<0$ ?

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Statistics

• Paper 3, Section II, H

  Statistics | Part IB, 2013

  Suppose $x_{1}$ is a single observation from a distribution with density $f$ over $[0,1]$. It is desired to test $H_{0}: f(x)=1$ against $H_{1}: f(x)=2 x$.

  Let $\delta:[0,1] \rightarrow\{0,1\}$ define a test by $\delta\left(x_{1}\right)=i \Longleftrightarrow$ 'accept $H_{i}$ '. Let $\alpha_{i}(\delta)=P\left(\delta\left(x_{1}\right)=1-i \mid H_{i}\right)$. State the Neyman-Pearson lemma using this notation.

  Let $\delta$ be the best test of size $0.10$. Find $\delta$ and $\alpha_{1}(\delta)$.

  Consider now $\delta:[0,1] \rightarrow\{0,1, \star\}$ where $\delta\left(x_{1}\right)=\star$ means 'declare the test to be inconclusive'. Let $\gamma_{i}(\delta)=P\left(\delta(x)=\star \mid H_{i}\right)$. Given prior probabilities $\pi_{0}$ for $H_{0}$ and $\pi_{1}=1-\pi_{0}$ for $H_{1}$, and some $w_{0}, w_{1}$, let

  $$\operatorname{cost}(\delta)=\pi_{0}\left(w_{0} \alpha_{0}(\delta)+\gamma_{0}(\delta)\right)+\pi_{1}\left(w_{1} \alpha_{1}(\delta)+\gamma_{1}(\delta)\right)$$

  Let $\delta^{*}\left(x_{1}\right)=i \Longleftrightarrow x_{1} \in A_{i}$, where $A_{0}=[0,0.5), A_{\star}=[0.5,0.6), A_{1}=[0.6,1]$. Prove that for each value of $\pi_{0} \in(0,1)$ there exist $w_{0}, w_{1}$ (depending on $\left.\pi_{0}\right)$ such that $\operatorname{cost}\left(\delta^{*}\right)=\min _{\delta} \operatorname{cost}(\delta) .\left[\right.$ Hint $\left.: w_{0}=1+2(0.6)\left(\pi_{1} / \pi_{0}\right) .\right]$

  Hence prove that if $\delta$ is any test for which

  $$\alpha_{i}(\delta) \leqslant \alpha_{i}\left(\delta^{*}\right), \quad i=0,1$$

  then $\gamma_{0}(\delta) \geqslant \gamma_{0}\left(\delta^{*}\right)$ and $\gamma_{1}(\delta) \geqslant \gamma_{1}\left(\delta^{*}\right)$.

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Variational Principles

• Paper 3, Section I, A

  Variational Principles | Part IB, 2013

  A cylindrical drinking cup has thin curved sides with density $\rho$ per unit area, and a disk-shaped base with density $k \rho$ per unit area. The cup has capacity to hold a fixed volume $V$ of liquid. Use the method of Lagrange multipliers to find the minimum mass of the cup.

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