Part IB 2008

• 1.I.1E

  Linear Algebra

  Part IB, 2008

  Let $A$ be an $n \times n$ matrix over $\mathbb{C}$. What does it mean to say that $\lambda$ is an eigenvalue of $A$ ? Show that $A$ has at least one eigenvalue. For each of the following statements, provide a proof or a counterexample as appropriate.

  (i) If $A$ is Hermitian, all eigenvalues of $A$ are real.

  (ii) If all eigenvalues of $A$ are real, $A$ is Hermitian.

  (iii) If all entries of $A$ are real and positive, all eigenvalues of $A$ have positive real part.

  (iv) If $A$ and $B$ have the same trace and determinant then they have the same eigenvalues.

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• 1.II.9E

  Linear Algebra

  Part IB, 2008

  Let $A$ be an $m \times n$ matrix of real numbers. Define the row rank and column rank of $A$ and show that they are equal.

  Show that if a matrix $A^{\prime}$ is obtained from $A$ by elementary row and column operations then $\operatorname{rank}\left(A^{\prime}\right)=\operatorname{rank}(A)$.

  Let $P, Q$ and $R$ be $n \times n$ matrices. Show that the $2 n \times 2 n$ matrices $\left(\begin{array}{cc}P Q & 0 \\ Q & Q R\end{array}\right)$ and $\left(\begin{array}{cc}0 & P Q R \\ Q & 0\end{array}\right)$ have the same rank.

  Hence, or otherwise, prove that

  $$\operatorname{rank}(P Q)+\operatorname{rank}(Q R) \leqslant \operatorname{rank}(Q)+\operatorname{rank}(P Q R)$$

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• 3.I.1G

  Groups, Rings and Modules

  Part IB, 2008

  Let $G$ be the abelian group generated by elements $a, b, c, d$ subject to the relations

  $$4 a-2 b+2 c+12 d=0, \quad-2 b+2 c=0, \quad 2 b+2 c=0, \quad 8 a+4 c+24 d=0$$

  Express $G$ as a product of cyclic groups, and find the number of elements of $G$ of order 2 .

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• $3 . \mathrm{II} . 11 \mathrm{G}$

  Groups, Rings and Modules

  Part IB, 2008

  What is a Euclidean domain? Show that a Euclidean domain is a principal ideal domain.

  Show that $\mathbb{Z}[\sqrt{-7}]$ is not a Euclidean domain (for any choice of norm), but that the ring

  $$\mathbb{Z}\left[\frac{1+\sqrt{-7}}{2}\right]$$

  is Euclidean for the norm function $N(z)=z \bar{z}$.

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• 4.I.2G

  Groups, Rings and Modules

  Part IB, 2008

  Let $n \geq 2$ be an integer. Show that the polynomial $\left(X^{n}-1\right) /(X-1)$ is irreducible over $\mathbb{Z}$ if and only if $n$ is prime.

  [You may use Eisenstein's criterion without proof.]

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• 4.II.11G

  Groups, Rings and Modules

  Part IB, 2008

  Let $R$ be a ring and $M$ an $R$-module. What does it mean to say that $M$ is a free $R$-module? Show that $M$ is free if there exists a submodule $N \subseteq M$ such that both $N$ and $M / N$ are free.

  Let $M$ and $M^{\prime}$ be $R$-modules, and $N \subseteq M, N^{\prime} \subseteq M^{\prime}$ submodules. Suppose that $N \cong N^{\prime}$ and $M / N \cong M^{\prime} / N^{\prime}$. Determine (by proof or counterexample) which of the following statements holds:

  (1) If $N$ is free then $M \cong M^{\prime}$.

  (2) If $M / N$ is free then $M \cong M^{\prime}$.

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• 1.I.2G

  Geometry

  Part IB, 2008

  Show that any element of $S O(3, \mathbb{R})$ is a rotation, and that it can be written as the product of two reflections.

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• 2.II.12G

  Geometry

  Part IB, 2008

  Show that the area of a spherical triangle with angles $\alpha, \beta, \gamma$ is $\alpha+\beta+\gamma-\pi$. Hence derive the formula for the area of a convex spherical $n$-gon.

  Deduce Euler's formula $F-E+V=2$ for a decomposition of a sphere into $F$ convex polygons with a total of $E$ edges and $V$ vertices.

  A sphere is decomposed into convex polygons, comprising $m$ quadrilaterals, $n$ pentagons and $p$ hexagons, in such a way that at each vertex precisely three edges meet. Show that there are at most 7 possibilities for the pair $(m, n)$, and that at least 3 of these do occur.

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• 3.I.2G

  Geometry

  Part IB, 2008

  A smooth surface in $\mathbb{R}^{3}$ has parametrization

  $$\sigma(u, v)=\left(u-\frac{u^{3}}{3}+u v^{2}, v-\frac{v^{3}}{3}+u^{2} v, u^{2}-v^{2}\right) .$$

  Show that a unit normal vector at the point $\sigma(u, v)$ is

  $$\left(\frac{-2 u}{1+u^{2}+v^{2}}, \frac{2 v}{1+u^{2}+v^{2}}, \frac{1-u^{2}-v^{2}}{1+u^{2}+v^{2}}\right)$$

  and that the curvature is $\frac{-4}{\left(1+u^{2}+v^{2}\right)^{4}}$.

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• 3.II.12G

  Geometry

  Part IB, 2008

  Let $D$ be the unit disc model of the hyperbolic plane, with metric

  $$\frac{4|d \zeta|^{2}}{\left(1-|\zeta|^{2}\right)^{2}}$$

  (i) Show that the group of Möbius transformations mapping $D$ to itself is the group of transformations

  $$\zeta \mapsto \omega \frac{\zeta-\lambda}{\bar{\lambda} \zeta-1},$$

  where $|\lambda|<1$ and $|\omega|=1$.

  (ii) Assuming that the transformations in (i) are isometries of $D$, show that any hyperbolic circle in $D$ is a Euclidean circle.

  (iii) Let $P$ and $Q$ be points on the unit circle with $\angle P O Q=2 \alpha$. Show that the hyperbolic distance from $O$ to the hyperbolic line $P Q$ is given by

  $$2 \tanh ^{-1}\left(\frac{1-\sin \alpha}{\cos \alpha}\right)$$

  (iv) Deduce that if $a>2 \tanh ^{-1}(2-\sqrt{3})$ then no hyperbolic open disc of radius $a$ is contained in a hyperbolic triangle.

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• 4.II.12G

  Geometry

  Part IB, 2008

  Let $\gamma:[a, b] \rightarrow S$ be a curve on a smoothly embedded surface $S \subset \mathbf{R}^{3}$. Define the energy of $\gamma$. Show that if $\gamma$ is a stationary point for the energy for proper variations of $\gamma$, then $\gamma$ satisfies the geodesic equations

  $$\begin{aligned} \frac{d}{d t}\left(E \dot{\gamma}_{1}+F \dot{\gamma}_{2}\right) &=\frac{1}{2}\left(E_{u} \dot{\gamma}_{1}^{2}+2 F_{u} \dot{\gamma}_{1} \dot{\gamma}_{2}+G_{u} \dot{\gamma}_{2}^{2}\right) \\ \frac{d}{d t}\left(F \dot{\gamma}_{1}+G \dot{\gamma}_{2}\right) &=\frac{1}{2}\left(E_{v} \dot{\gamma}_{1}^{2}+2 F_{v} \dot{\gamma}_{1} \dot{\gamma}_{2}+G_{v} \dot{\gamma}_{2}^{2}\right) \end{aligned}$$

  where $\gamma=\left(\gamma_{1}, \gamma_{2}\right)$ in terms of a smooth parametrization $(u, v)$ for $S$, with first fundamental form $E d u^{2}+2 F d u d v+G d v^{2}$.

  Now suppose that for every $c, d$ the curves $u=c, v=d$ are geodesics.

  (i) Show that $(F / \sqrt{G})_{v}=(\sqrt{G})_{u}$ and $(F / \sqrt{E})_{u}=(\sqrt{E})_{v}$.

  (ii) Suppose moreover that the angle between the curves $u=c, v=d$ is independent of $c$ and $d$. Show that $E_{v}=0=G_{u}$.

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• 1.II.11F

  Analysis II

  Part IB, 2008

  State and prove the Contraction Mapping Theorem.

  Let $(X, d)$ be a nonempty complete metric space and $f: X \rightarrow X$ a mapping such that, for some $k>0$, the $k$ th iterate $f^{k}$ of $f$ (that is, $f$ composed with itself $k$ times) is a contraction mapping. Show that $f$ has a unique fixed point.

  Now let $X$ be the space of all continuous real-valued functions on $[0,1]$, equipped with the uniform norm $\|h\|_{\infty}=\sup \{|h(t)|: t \in[0,1]\}$, and let $\phi: \mathbb{R} \times[0,1] \rightarrow \mathbb{R}$ be a continuous function satisfying the Lipschitz condition

  $$|\phi(x, t)-\phi(y, t)| \leqslant M|x-y|$$

  for all $t \in[0,1]$ and all $x, y \in \mathbb{R}$, where $M$ is a constant. Let $F: X \rightarrow X$ be defined by

  $$F(h)(t)=g(t)+\int_{0}^{t} \phi(h(s), s) d s$$

  where $g$ is a fixed continuous function on $[0,1]$. Show by induction on $n$ that

  $$\left|F^{n}(h)(t)-F^{n}(k)(t)\right| \leqslant \frac{M^{n} t^{n}}{n !}\|h-k\|_{\infty}$$

  for all $h, k \in X$ and all $t \in[0,1]$. Deduce that the integral equation

  $$f(t)=g(t)+\int_{0}^{t} \phi(f(s), s) d s$$

  has a unique continuous solution $f$ on $[0,1]$.

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• 2.I.1E

  Linear Algebra

  Part IB, 2008

  Suppose that $V$ and $W$ are finite-dimensional vector spaces over $\mathbb{R}$. What does it mean to say that $\psi: V \rightarrow W$ is a linear map? State the rank-nullity formula. Using it, or otherwise, prove that a linear map $\psi: V \rightarrow V$ is surjective if, and only if, it is injective.

  Suppose that $\psi: V \rightarrow V$ is a linear map which has a right inverse, that is to say there is a linear map $\phi: V \rightarrow V$ such that $\psi \phi=\mathrm{id}_{V}$, the identity map. Show that $\phi \psi=\mathrm{id}_{V}$.

  Suppose that $A$ and $B$ are two $n \times n$ matrices over $\mathbb{R}$ such that $A B=I$. Prove that $B A=I$.

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• 2.I.3F

  Analysis II

  Part IB, 2008

  Explain what is meant by the statement that a sequence $\left(f_{n}\right)$ of functions defined on an interval $[a, b]$ converges uniformly to a function $f$. If $\left(f_{n}\right)$ converges uniformly to $f$, and each $f_{n}$ is continuous on $[a, b]$, prove that $f$ is continuous on $[a, b]$.

  Now suppose additionally that $\left(x_{n}\right)$ is a sequence of points of $[a, b]$ converging to a limit $x$. Prove that $f_{n}\left(x_{n}\right) \rightarrow f(x)$.

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• 2.II.13F

  Analysis II

  Part IB, 2008

  Let $\left(u_{n}(x): n=0,1,2, \ldots\right)$ be a sequence of real-valued functions defined on a subset $E$ of $\mathbb{R}$. Suppose that for all $n$ and all $x \in E$ we have $\left|u_{n}(x)\right| \leqslant M_{n}$, where $\sum_{n=0}^{\infty} M_{n}$ converges. Prove that $\sum_{n=0}^{\infty} u_{n}(x)$ converges uniformly on $E$.

  Now let $E=\mathbb{R} \backslash \mathbb{Z}$, and consider the series $\sum_{n=0}^{\infty} u_{n}(x)$, where $u_{0}(x)=1 / x^{2}$ and

  $$u_{n}(x)=1 /(x-n)^{2}+1 /(x+n)^{2}$$

  for $n>0$. Show that the series converges uniformly on $E_{R}=\{x \in E:|x|<R\}$ for any real number $R$. Deduce that $f(x)=\sum_{n=0}^{\infty} u_{n}(x)$ is a continuous function on $E$. Does the series converge uniformly on $E$ ? Justify your answer.

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• $3 . \mathrm{I} . 3 \mathrm{~F} \quad$

  Analysis II

  Part IB, 2008

  Explain what it means for a function $f(x, y)$ of two variables to be differentiable at a point $\left(x_{0}, y_{0}\right)$. If $f$ is differentiable at $\left(x_{0}, y_{0}\right)$, show that for any $\alpha$ the function $g_{\alpha}$ defined by

  $$g_{\alpha}(t)=f\left(x_{0}+t \cos \alpha, y_{0}+t \sin \alpha\right)$$

  is differentiable at $t=0$, and find its derivative in terms of the partial derivatives of $f$ at $\left(x_{0}, y_{0}\right)$.

  Consider the function $f$ defined by

  

  Is $f$ differentiable at $(0,0)$ ? Justify your answer.

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• 3.II.13F

  Analysis II

  Part IB, 2008

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function, and $\left(x_{0}, y_{0}\right)$ a point of $\mathbb{R}^{2}$. Prove that if the partial derivatives of $f$ exist in some open disc around $\left(x_{0}, y_{0}\right)$ and are continuous at $\left(x_{0}, y_{0}\right)$, then $f$ is differentiable at $\left(x_{0}, y_{0}\right)$.

  Now let $X$ denote the vector space of all $(n \times n)$ real matrices, and let $f: X \rightarrow \mathbb{R}$ be the function assigning to each matrix its determinant. Show that $f$ is differentiable at the identity matrix $I$, and that $\left.D f\right|_{I}$ is the linear map $H \mapsto \operatorname{tr} H$. Deduce that $f$ is differentiable at any invertible matrix $A$, and that $\left.D f\right|_{A}$ is the linear map $H \mapsto \operatorname{det} A \operatorname{tr}\left(A^{-1} H\right) .$

  Show also that if $K$ is a matrix with $\|K\|<1$, then $(I+K)$ is invertible. Deduce that $f$ is twice differentiable at $I$, and find $\left.D^{2} f\right|_{I}$ as a bilinear map $X \times X \rightarrow \mathbb{R}$.

  [You may assume that the norm $\|-\|$ on $X$ is complete, and that it satisfies the inequality $\|A B\| \leqslant\|A\| \cdot\|B\|$ for any two matrices $A$ and $B .]$

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• 4.I.3F

  Analysis II

  Part IB, 2008

  Let $X$ be the vector space of all continuous real-valued functions on the unit interval $[0,1]$. Show that the functions

  $$\|f\|_{1}=\int_{0}^{1}|f(t)| d t \quad \text { and } \quad\|f\|_{\infty}=\sup \{|f(t)|: 0 \leqslant t \leqslant 1\}$$

  both define norms on $X$.

  Consider the sequence $\left(f_{n}\right)$ defined by $f_{n}(t)=n t^{n}(1-t)$. Does $\left(f_{n}\right)$ converge in the norm $\|-\|_{1}$ ? Does it converge in the norm $\|-\|_{\infty}$ ? Justify your answers.

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• 4.II.13F

  Analysis II

  Part IB, 2008

  Explain what it means for two norms on a real vector space to be Lipschitz equivalent. Show that if two norms are Lipschitz equivalent, then one is complete if and only if the other is.

  Let $\|-\|$ be an arbitrary norm on the finite-dimensional space $\mathbb{R}^{n}$, and let $\|-\|_{2}$ denote the standard (Euclidean) norm. Show that for every $\mathbf{x} \in \mathbb{R}^{n}$ with $\|\mathbf{x}\|_{2}=1$, we have

  $$\|\mathbf{x}\| \leqslant\left\|\mathbf{e}_{1}\right\|+\left\|\mathbf{e}_{2}\right\|+\cdots+\left\|\mathbf{e}_{n}\right\|$$

  where $\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}\right)$ is the standard basis for $\mathbb{R}^{n}$, and deduce that the function $\|-\|$ is continuous with respect to $\|-\|_{2}$. Hence show that there exists a constant $m>0$ such that $\|\mathbf{x}\| \geqslant m$ for all $\mathbf{x}$ with $\|\mathbf{x}\|_{2}=1$, and deduce that $\|-\|$ and $\|-\|_{2}$ are Lipschitz equivalent.

  [You may assume the Bolzano-Weierstrass Theorem.]

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• 1.II.12F

  Metric and Topological Spaces

  Part IB, 2008

  Write down the definition of a topology on a set $X$.

  For each of the following families $\mathcal{T}$ of subsets of $\mathbb{Z}$, determine whether $\mathcal{T}$ is a topology on $\mathbb{Z}$. In the cases where the answer is 'yes', determine also whether $(\mathbb{Z}, \mathcal{T})$ is a Hausdorff space and whether it is compact.

  (a) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : either $U$ is finite or $0 \in U\}$.

  (b) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : either $\mathbb{Z} \backslash U$ is finite or $0 \notin U\}$.

  (c) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : there exists $k>0$ such that, for all $n, n \in U \Leftrightarrow n+k \in U\}$.

  (d) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : for all $n \in U$, there exists $k>0$ such that $\{n+k m: m \in \mathbb{Z}\} \subseteq U\}$.

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• 2.I.4F

  Metric and Topological Spaces

  Part IB, 2008

  Stating carefully any results on compactness which you use, show that if $X$ is a compact space, $Y$ is a Hausdorff space and $f: X \rightarrow Y$ is bijective and continuous, then $f$ is a homeomorphism.

  Hence or otherwise show that the unit circle $S=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$ is homeomorphic to the quotient space $[0,1] / \sim$, where $\sim$ is the equivalence relation defined by

  $$x \sim y \Leftrightarrow \text { either } x=y \text { or }\{x, y\}=\{0,1\} .$$

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• $3 . \mathrm{I} . 4 \mathrm{~F} \quad$

  Metric and Topological Spaces

  Part IB, 2008

  Explain what it means for a topological space to be connected.

  Are the following subspaces of the unit square $[0,1] \times[0,1]$ connected? Justify your answers.

  (a) $\{(x, y): x \neq 0, y \neq 0$, and $x / y \in \mathbb{Q}\}$.

  (b) $\{(x, y):(x=0)$ or $(x \neq 0$ and $y \in \mathbb{Q})\}$.

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• 4.II.14F

  Metric and Topological Spaces

  Part IB, 2008

  Explain what is meant by a base for a topology. Illustrate your definition by describing bases for the topology induced by a metric on a set, and for the product topology on the cartesian product of two topological spaces.

  A topological space $(X, \mathcal{T})$ is said to be separable if there is a countable subset $C \subseteq X$ which is dense, i.e. such that $C \cap U \neq \emptyset$ for every nonempty $U \in \mathcal{T}$. Show that a product of two separable spaces is separable. Show also that a metric space is separable if and only if its topology has a countable base, and deduce that every subspace of a separable metric space is separable.

  Now let $X=\mathbb{R}$ with the topology $\mathcal{T}$ having as a base the set of all half-open intervals

  $$[a, b)=\{x \in \mathbb{R}: a \leqslant x<b\}$$

  with $a<b$. Show that $X$ is separable, but that the subspace $Y=\{(x,-x): x \in \mathbb{R}\}$ of $X \times X$ is not separable.

  [You may assume standard results on countability.]

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• 2.II.10E

  Linear Algebra

  Part IB, 2008

  Define the determinant $\operatorname{det}(A)$ of an $n \times n$ square matrix $A$ over the complex numbers. If $A$ and $B$ are two such matrices, show that $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$.

  Write $p_{M}(\lambda)=\operatorname{det}(M-\lambda I)$ for the characteristic polynomial of a matrix $M$. Let $A, B, C$ be $n \times n$ matrices and suppose that $C$ is nonsingular. Show that $p_{B C}=p_{C B}$. Taking $C=A+t I$ for appropriate values of $t$, or otherwise, deduce that $p_{B A}=p_{A B}$.

  Show that if $p_{A}=p_{B}$ then $\operatorname{tr}(A)=\operatorname{tr}(B)$. Which of the following statements is true for all $n \times n$ matrices $A, B, C$ ? Justify your answers.

  (i) $p_{A B C}=p_{A C B}$;

  (ii) $p_{A B C}=p_{B C A}$.

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• 1.I.3C

  Complex Analysis or Complex Methods

  Part IB, 2008

  Given that $f(z)$ is an analytic function, show that the mapping $w=f(z)$

  (a) preserves angles between smooth curves intersecting at $z$ if $f^{\prime}(z) \neq 0$;

  (b) has Jacobian given by $\left|f^{\prime}(z)\right|^{2}$.

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• 1.II.13C

  Complex Analysis or Complex Methods

  Part IB, 2008

  By a suitable choice of contour show the following:

  (a)

  $$\int_{0}^{\infty} \frac{x^{1 / n}}{1+x^{2}} d x=\frac{\pi}{2 \cos (\pi / 2 n)}$$

  where $n>1$,

  (b)

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

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• 2.II.14C

  Complex Analysis or Complex Methods

  Part IB, 2008

  Let $f(z)=1 /\left(e^{z}-1\right)$. Find the first three terms in the Laurent expansion for $f(z)$ valid for $0<|z|<2 \pi$.

  Now let $n$ be a positive integer, and define

  $$\begin{aligned} &f_{1}(z)=\frac{1}{z}+\sum_{r=1}^{n} \frac{2 z}{z^{2}+4 \pi^{2} r^{2}} \\ &f_{2}(z)=f(z)-f_{1}(z) \end{aligned}$$

  Show that the singularities of $f_{2}$ in $\{z:|z|<2(n+1) \pi\}$ are all removable. By expanding $f_{1}$ as a Laurent series valid for $|z|>2 n \pi$, and $f_{2}$ as a Taylor series valid for $|z|<2(n+1) \pi$, find the coefficients of $z^{j}$ for $-1 \leq j \leq 1$ in the Laurent series for $f$ valid for $2 n \pi<|z|<2(n+1) \pi$.

  By estimating an appropriate integral around the contour $|z|=(2 n+1) \pi$, show that

  $$\sum_{r=1}^{\infty} \frac{1}{r^{2}}=\frac{\pi^{2}}{6}$$

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• 3.II.14E

  Complex Analysis

  Part IB, 2008

  State and prove Rouché's theorem, and use it to count the number of zeros of $3 z^{9}+8 z^{6}+z^{5}+2 z^{3}+1$ inside the annulus $\{z: 1<|z|<2\}$.

  Let $\left(p_{n}\right)_{n=1}^{\infty}$ be a sequence of polynomials of degree at most $d$ with the property that $p_{n}(z)$ converges uniformly on compact subsets of $\mathbb{C}$ as $n \rightarrow \infty$. Prove that there is a polynomial $p$ of degree at most $d$ such that $p_{n} \rightarrow p$ uniformly on compact subsets of $\mathbb{C}$. [If you use any results about uniform convergence of analytic functions, you should prove them.]

  Suppose that $p$ has $d$ distinct roots $z_{1}, \ldots, z_{d}$. Using Rouché's theorem, or otherwise, show that for each $i$ there is a sequence $\left(z_{i, n}\right)_{n=1}^{\infty}$ such that $p_{n}\left(z_{i, n}\right)=0$ and $z_{i, n} \rightarrow z_{i}$ as $n \rightarrow \infty$.

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• 4.I.4E

  Complex Analysis

  Part IB, 2008

  Suppose that $f$ and $g$ are two functions which are analytic on the whole complex plane $\mathbb{C}$. Suppose that there is a sequence of distinct points $z_{1}, z_{2}, \ldots$ with $\left|z_{i}\right| \leqslant 1$ such that $f\left(z_{i}\right)=g\left(z_{i}\right)$. Show that $f(z)=g(z)$ for all $z \in \mathbb{C}$. [You may assume any results on Taylor expansions you need, provided they are clearly stated.]

  What happens if the assumption that $\left|z_{i}\right| \leqslant 1$ is dropped?

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• 3.I.5C

  Complex Methods

  Part IB, 2008

  Using the contour integration formula for the inversion of Laplace transforms find the inverse Laplace transforms of the following functions: (a) $\frac{s}{s^{2}+a^{2}} \quad(a$ real and non-zero $)$, (b) $\frac{1}{\sqrt{s}}$.

  [You may use the fact that $\int_{-\infty}^{\infty} e^{-b x^{2}} d x=\sqrt{\pi / b}$.]

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• 4.II.15C

  Complex Methods

  Part IB, 2008

  Let $H$ be the domain $\mathbb{C}-\{x+i y: x \leq 0, y=0\}$ (i.e., $\mathbb{C}$ cut along the negative $x$-axis). Show, by a suitable choice of branch, that the mapping

  $$z \mapsto w=-i \log z$$

  maps $H$ onto the strip $S=\{z=x+i y,-\pi<x<\pi\}$.

  How would a different choice of branch change the result?

  Let $G$ be the domain $\{z \in \mathbb{C}:|z|<1,|z+i|>\sqrt{2}\}$. Find an analytic transformation that maps $G$ to $S$, where $S$ is the strip defined above.

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• 1.II.14D

  Methods

  Part IB, 2008

  Write down the Euler-Lagrange equation for the variational problem for $y(x)$ that extremizes the integral $I$ defined as

  $$I=\int_{x_{1}}^{x_{2}} f\left(x, y, y^{\prime}\right) d x$$

  with boundary conditions $y\left(x_{1}\right)=y_{1}, y\left(x_{2}\right)=y_{2}$, where $y_{1}$ and $y_{2}$ are positive constants such that $y_{2}>y_{1}$, with $x_{2}>x_{1}$. Find a first integral of the equation when $f$ is independent of $y$, i.e. $f=f\left(x, y^{\prime}\right)$.

  A light ray moves in the $(x, y)$ plane from $\left(x_{1}, y_{1}\right)$ to $\left(x_{2}, y_{2}\right)$ with speed $c(x)$ taking a time $T$. Show that the equation of the path that makes $T$ an extremum satisfies

  $$\frac{d y}{d x}=\frac{c(x)}{\sqrt{k^{2}-c^{2}(x)}}$$

  where $k$ is a constant and write down an integral relating $k, x_{1}, x_{2}, y_{1}$ and $y_{2}$.

  When $c(x)=a x$ where $a$ is a constant and $k=a x_{2}$, show that the path is given by

  $$\left(y_{2}-y\right)^{2}=x_{2}^{2}-x^{2} .$$

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• $2 . \mathrm{I} . 5 \mathrm{D} \quad$

  Methods

  Part IB, 2008

  Describe briefly the method of Lagrange multipliers for finding the stationary values of a function $f(x, y)$ subject to a constraint $g(x, y)=0$.

  Use the method to find the largest possible volume of a circular cylinder that has surface area $A$ (including both ends).

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• 2.II.15D

  Methods

  Part IB, 2008

  (a) Legendre's equation may be written in the form

  $$\frac{d}{d x}\left(\left(1-x^{2}\right) \frac{d y}{d x}\right)+\lambda y=0$$

  Show that there is a series solution for $y$ of the form

  $$y=\sum_{k=0}^{\infty} a_{k} x^{k},$$

  where the $a_{k}$ satisfy the recurrence relation

  $$\frac{a_{k+2}}{a_{k}}=-\frac{(\lambda-k(k+1))}{(k+1)(k+2)} .$$

  Hence deduce that there are solutions for $y(x)=P_{n}(x)$ that are polynomials of degree $n$, provided that $\lambda=n(n+1)$. Given that $a_{0}$ is then chosen so that $P_{n}(1)=1$, find the explicit form for $P_{2}(x)$.

  (b) Laplace's equation for $\Phi(r, \theta)$ in spherical polar coordinates $(r, \theta, \phi)$ may be written in the axisymmetric case as

  $$\frac{\partial^{2} \Phi}{\partial r^{2}}+\frac{2}{r} \frac{\partial \Phi}{\partial r}+\frac{1}{r^{2}} \frac{\partial}{\partial x}\left(\left(1-x^{2}\right) \frac{\partial \Phi}{\partial x}\right)=0$$

  where $x=\cos \theta$.

  Write down without proof the general form of the solution obtained by the method of separation of variables. Use it to find the form of $\Phi$ exterior to the sphere $r=a$ that satisfies the boundary conditions, $\Phi(a, x)=1+x^{2}$, and $\lim _{r \rightarrow \infty} \Phi(r, x)=0$.

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• 3.II.10E

  Linear Algebra

  Part IB, 2008

  Let $k=\mathbb{R}$ or $\mathbb{C}$. What is meant by a quadratic form $q: k^{n} \rightarrow k$ ? Show that there is a basis $\left\{v_{1}, \ldots, v_{n}\right\}$ for $k^{n}$ such that, writing $x=x_{1} v_{1}+\ldots+x_{n} v_{n}$, we have $q(x)=a_{1} x_{1}^{2}+\ldots+a_{n} x_{n}^{2}$ for some scalars $a_{1}, \ldots, a_{n} \in\{-1,0,1\} .$

  Suppose that $k=\mathbb{R}$. Define the rank and signature of $q$ and compute these quantities for the form $q: \mathbb{R}^{3} \rightarrow \mathbb{R}$ given by $q(x)=-3 x_{1}^{2}+x_{2}^{2}+2 x_{1} x_{2}-2 x_{1} x_{3}+2 x_{2} x_{3}$.

  Suppose now that $k=\mathbb{C}$ and that $q_{1}, \ldots, q_{d}: \mathbb{C}^{n} \rightarrow \mathbb{C}$ are quadratic forms. If $n \geqslant 2^{d}$, show that there is some nonzero $x \in \mathbb{C}^{n}$ such that $q_{1}(x)=\ldots=q_{d}(x)=0$.

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• 3.I.6D

  Methods

  Part IB, 2008

  Let $\mathcal{L}$ be the operator

  $$\mathcal{L} y=\frac{d^{2} y}{d x^{2}}-k^{2} y$$

  on functions $y(x)$ satisfying $\lim _{x \rightarrow-\infty} \quad y(x)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$.

  Given that the Green's function $G(x ; \xi)$ for $\mathcal{L}$ satisfies

  $$\mathcal{L} G=\delta(x-\xi)$$

  show that a solution of

  $$\mathcal{L} y=S(x)$$

  for a given function $S(x)$, is given by

  $$y(x)=\int_{-\infty}^{\infty} G(x ; \xi) S(\xi) d \xi$$

  Indicate why this solution is unique.

  Show further that the Green's function is given by

  $$G(x ; \xi)=-\frac{1}{2|k|} \exp (-|k||x-\xi|)$$

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• 3.II.15D

  Methods

  Part IB, 2008

  Let $\lambda_{1}<\lambda_{2}<\ldots \lambda_{n} \ldots$ and $y_{1}(x), y_{2}(x), \ldots y_{n}(x) \ldots$ be the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville system

  $$\mathcal{L} y_{n}=\lambda_{n} w(x) y_{n},$$

  where

  $$\mathcal{L} y \equiv \frac{d}{d x}\left(-p(x) \frac{d y}{d x}\right)+q(x) y,$$

  with $p(x)>0$ and $w(x)>0$. The boundary conditions on $y$ are that $y(0)=y(1)=0$.

  Show that two distinct eigenfunctions are orthogonal in the sense that

  $$\int_{0}^{1} w y_{n} y_{m} d x=\delta_{n m} \int_{0}^{1} w y_{n}^{2} d x .$$

  Show also that if $y$ has the form

  $$y=\sum_{n=1}^{\infty} a_{n} y_{n},$$

  with $a_{n}$ being independent of $x$, then

  $$\frac{\int_{0}^{1} y \mathcal{L} y d x}{\int_{0}^{1} w y^{2} d x} \geq \lambda_{1}$$

  Assuming that the eigenfunctions are complete, deduce that a solution of the diffusion equation,

  $$\frac{\partial y}{\partial t}=-\frac{1}{w} \mathcal{L} y$$

  that satisfies the boundary conditions given above is such that

  $$\frac{1}{2} \frac{d}{d t}\left(\int_{0}^{1} w y^{2} d x\right) \leq-\lambda_{1} \int_{0}^{1} w y^{2} d x .$$

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• 4.I.5A

  Methods

  Part IB, 2008

  Find the half-range Fourier cosine series for $f(x)=x^{2}, 0<x<1$. Hence show that

  $$\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6} .$$

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• 4.II.16A

  Methods

  Part IB, 2008

  Assume $F(x)$ satisfies

  $$\int_{-\infty}^{\infty}|F(x)| d x<\infty$$

  and that the series

  $$g(\tau)=\sum_{n=-\infty}^{\infty} F(2 n \pi+\tau)$$

  converges uniformly in $[0 \leqslant \tau \leqslant 2 \pi]$.

  If $\tilde{F}$ is the Fourier transform of $F$, prove that

  $$g(\tau)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} \tilde{F}(n) e^{i n \tau}$$

  [Hint: prove that $g$ is periodic and express its Fourier expansion coefficients in terms of $\tilde{F}]$.

  In the case that $F(x)=e^{-|x|}$, evaluate the sum

  $$\sum_{n=-\infty}^{\infty} \frac{1}{1+n^{2}} .$$

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• 1.II.15A

  Quantum Mechanics

  Part IB, 2008

  The radial wavefunction $g(r)$ for the hydrogen atom satisfies the equation

  $$-\frac{\hbar^{2}}{2 m r^{2}} \frac{d}{d r}\left(r^{2} \frac{d g(r)}{d r}\right)-\frac{e^{2} g(r)}{4 \pi \epsilon_{0} r}+\hbar^{2} \frac{\ell(\ell+1)}{2 m r^{2}} g(r)=E g(r)$$

  With reference to the general form for the time-independent Schrödinger equation, explain the origin of each term. What are the allowed values of $\ell$ ?

  The lowest-energy bound-state solution of $(*)$, for given $\ell$, has the form $r^{\alpha} e^{-\beta r}$. Find $\alpha$ and $\beta$ and the corresponding energy $E$ in terms of $\ell$.

  A hydrogen atom makes a transition between two such states corresponding to $\ell+1$ and $\ell$. What is the frequency of the emitted photon?

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• 2.II.16A

  Quantum Mechanics

  Part IB, 2008

  Give the physical interpretation of the expression

  $$\langle A\rangle_{\psi}=\int \psi(x)^{*} \hat{A} \psi(x) d x$$

  for an observable $A$, where $\hat{A}$ is a Hermitian operator and $\psi$ is normalised. By considering the norm of the state $(A+i \lambda B) \psi$ for two observables $A$ and $B$, and real values of $\lambda$, show that

  $$\left\langle A^{2}\right\rangle_{\psi}\left\langle B^{2}\right\rangle_{\psi} \geqslant \frac{1}{4}\left|\langle[A, B]\rangle_{\psi}\right|^{2} .$$

  Deduce the uncertainty relation

  $$\Delta A \Delta B \geqslant \frac{1}{2}\left|\langle[A, B]\rangle_{\psi}\right|,$$

  where $\Delta A$ is the uncertainty of $A$.

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

  $$\langle H\rangle_{\psi} \geqslant \frac{1}{2} \hbar \omega .$$

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• 3.I.7A

  Quantum Mechanics

  Part IB, 2008

  Write down a formula for the orbital angular momentum operator $\hat{\mathbf{L}}$. Show that its components satisfy

  $$\left[L_{i}, L_{j}\right]=i \hbar \epsilon_{i j k} L_{k} .$$

  If $L_{3} \psi=0$, show that $\left(L_{1} \pm i L_{2}\right) \psi$ are also eigenvectors of $L_{3}$, and find their eigenvalues.

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• 3.II.16A

  Quantum Mechanics

  Part IB, 2008

  What is the probability current for a particle of mass $m$, wavefunction $\psi$, moving in one dimension?

  A particle of energy $E$ is incident from $x<0$ on a barrier given by

  $$V(x)=\left\{\begin{array}{cc} 0 & x \leqslant 0 \\ V_{1} & 0<x<a \\ V_{0} & x \geqslant a \end{array}\right.$$

  where $V_{1}>V_{0}>0$. What are the conditions satisfied by $\psi$ at $x=0$ and $x=a$ ? Write down the form taken by the wavefunction in the regions $x \leqslant 0$ and $x \geqslant a$ distinguishing between the cases $E>V_{0}$ and $E<V_{0}$. For both cases, use your expressions for $\psi$ to calculate the probability currents in these two regions.

  Define the reflection and transmission coefficients, $R$ and $T$. Using current conservation, show that the expressions you have derived satisfy $R+T=1$. Show that $T=0$ if $0<E<V_{0}$.

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• 4.I.6A

  Quantum Mechanics

  Part IB, 2008

  What is meant by a stationary state? What form does the wavefunction take in such a state? A particle has wavefunction $\psi(x, t)$, such that

  $$\psi(x, 0)=\sqrt{\frac{1}{2}}\left(\chi_{1}(x)+\chi_{2}(x)\right)$$

  where $\chi_{1}$ and $\chi_{2}$ are normalised eigenstates of the Hamiltonian with energies $E_{1}$ and $E_{2}$. Write down $\psi(x, t)$ at time $t$. Show that the expectation value of $A$ at time $t$ is

  $$\langle A\rangle_{\psi}=\frac{1}{2} \int_{-\infty}^{\infty}\left(\chi_{1}^{*} \hat{A} \chi_{1}+\chi_{2}^{*} \hat{A} \chi_{2}\right) d x+\operatorname{Re}\left(e^{i\left(E_{1}-E_{2}\right) t / \hbar} \int_{-\infty}^{\infty} \chi_{1}^{*} \hat{A} \chi_{2} d x\right)$$

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• 1.II.16B

  Electromagnetism

  Part IB, 2008

  Suppose that the current density $\mathbf{J}(\mathbf{r})$ is constant in time but the charge density $\rho(\mathbf{r}, t)$ is not.

  (i) Show that $\rho$ is a linear function of time:

  $$\rho(\mathbf{r}, t)=\rho(\mathbf{r}, 0)+\dot{\rho}(\mathbf{r}, 0) t$$

  where $\dot{\rho}(\mathbf{r}, 0)$ is the time derivative of $\rho$ at time $t=0$.

  (ii) The magnetic induction due to a current density $\mathbf{J}(\mathbf{r})$ can be written as

  $$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} d V^{\prime}$$

  Show that this can also be written as

  $$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \nabla \times \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d V^{\prime}$$

  (iii) Assuming that $\mathbf{J}$ vanishes at infinity, show that the curl of the magnetic field in (1) can be written as

  $$\nabla \times \mathbf{B}(\mathbf{r})=\mu_{0} \mathbf{J}(\mathbf{r})+\frac{\mu_{0}}{4 \pi} \nabla \int \frac{\nabla^{\prime} \cdot \mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d V^{\prime}$$

  [You may find useful the identities $\nabla \times(\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^{2} \mathbf{A}$ and also $\left.\nabla^{2}\left(1 /\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)=-4 \pi \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) .\right]$

  (iv) Show that the second term on the right hand side of (2) can be expressed in terms of the time derivative of the electric field in such a way that $\mathbf{B}$ itself obeys Ampère's law with Maxwell's displacement current term, i.e. $\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\mu_{0} \epsilon_{0} \partial \mathbf{E} / \partial t$.

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• 4.I.1E

  Linear Algebra

  Part IB, 2008

  Describe (without proof) what it means to put an $n \times n$ matrix of complex numbers into Jordan normal form. Explain (without proof) the sense in which the Jordan normal form is unique.

  Put the following matrix in Jordan normal form:

  $$\left(\begin{array}{ccc} -7 & 3 & -5 \\ 7 & -1 & 5 \\ 17 & -6 & 12 \end{array}\right)$$

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• 2.I.6B

  Electromagnetism

  Part IB, 2008

  Given the electric potential of a dipole

  $$\phi(r, \theta)=\frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}},$$

  where $p$ is the magnitude of the dipole moment, calculate the corresponding electric field and show that it can be written as

  $$\mathbf{E}(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}\left[3\left(\mathbf{p} \cdot \hat{\mathbf{e}}_{r}\right) \hat{\mathbf{e}}_{r}-\mathbf{p}\right]$$

  where $\hat{\mathbf{e}}_{r}$ is the unit vector in the radial direction.

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• 2.II.17B

  Electromagnetism

  Part IB, 2008

  Two perfectly conducting rails are placed on the $x y$-plane, one coincident with the $x$-axis, starting at $(0,0)$, the other parallel to the first rail a distance $\ell$ apart, starting at $(0, \ell)$. A resistor $R$ is connected across the rails between $(0,0)$ and $(0, \ell)$, and a uniform magnetic field $\mathbf{B}=B \hat{\mathbf{e}}_{z}$, where $\hat{\mathbf{e}}_{z}$ is the unit vector along the $z$-axis and $B>0$, fills the entire region of space. A metal bar of negligible resistance and mass $m$ slides without friction on the two rails, lying perpendicular to both of them in such a way that it closes the circuit formed by the rails and the resistor. The bar moves with speed $v$ to the right such that the area of the loop becomes larger with time.

  (i) Calculate the current in the resistor and indicate its direction of flow in a diagram of the system.

  (ii) Show that the magnetic force on the bar is

  $$\mathbf{F}=-\frac{B^{2} \ell^{2} v}{R} \hat{\mathbf{e}}_{x}$$

  (iii) Assume that the bar starts moving with initial speed $v_{0}$ at time $t=0$, and is then left to slide freely. Using your result from part (ii) and Newton's laws show that its velocity at the time $t$ is

  $$v(t)=v_{0} e^{-\left(B^{2} \ell^{2} / m R\right) t} .$$

  (iv) By calculating the total energy delivered to the resistor, verify that energy is conserved.

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• 3.II.17B

  Electromagnetism

  Part IB, 2008

  (i) From Maxwell's equations in vacuum,

  $$\begin{array}{ll} \nabla \cdot \mathbf{E}=0 & \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0 & \nabla \times \mathbf{B}=\mu_{0} \epsilon_{0} \frac{\partial \mathbf{E}}{\partial t} \end{array}$$

  obtain the wave equation for the electric field E. [You may find the following identity useful: $\left.\nabla \times(\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^{2} \mathbf{A} .\right]$

  (ii) If the electric and magnetic fields of a monochromatic plane wave in vacuum are

  $$\mathbf{E}(z, t)=\mathbf{E}_{0} \mathrm{e}^{i(k z-\omega t)} \text { and } \mathbf{B}(z, t)=\mathbf{B}_{0} \mathrm{e}^{i(k z-\omega t)}$$

  show that the corresponding electromagnetic waves are transverse (that is, both fields have no component in the direction of propagation).

  (iii) Use Faraday's law for these fields to show that

  $$\mathbf{B}_{0}=\frac{k}{\omega}\left(\hat{\mathbf{e}}_{z} \times \mathbf{E}_{0}\right)$$

  (iv) Explain with symmetry arguments how these results generalise to

  $$\mathbf{E}(\mathbf{r}, t)=E_{0} \mathrm{e}^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)} \hat{\mathbf{n}} \quad \text { and } \quad \mathbf{B}(\mathbf{r}, t)=\frac{1}{c} E_{0} \mathrm{e}^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}(\hat{\mathbf{k}} \times \hat{\mathbf{n}})$$

  where $\hat{\mathbf{n}}$ is the polarisation vector, i.e., the unit vector perpendicular to the direction of motion and along the direction of the electric field, and $\hat{\mathbf{k}}$ is the unit vector in the direction of propagation of the wave.

  (v) Using Maxwell's equations in vacuum prove that:

  $$\oint_{\mathcal{A}}\left(1 / \mu_{0}\right)(\mathbf{E} \times \mathbf{B}) \cdot d \mathcal{A}=-\frac{\partial}{\partial t} \int_{\mathcal{V}}\left(\frac{\epsilon_{0} E^{2}}{2}+\frac{B^{2}}{2 \mu_{0}}\right) d V$$

  where $\mathcal{V}$ is the closed volume and $\mathcal{A}$ is the bounding surface. Comment on the differing time dependencies of the left-hand-side of (1) for the case of (a) linearly-polarized and (b) circularly-polarized monochromatic plane waves.

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• 4.I.7B

  Electromagnetism

  Part IB, 2008

  The energy stored in a static electric field $\mathbf{E}$ is

  $$U=\frac{1}{2} \int \rho \phi d V,$$

  where $\phi$ is the associated electric potential, $\mathbf{E}=-\nabla \phi$, and $\rho$ is the volume charge density.

  (i) Assuming that the energy is calculated over all space and that $\mathbf{E}$ vanishes at infinity, show that the energy can be written as

  $$U=\frac{\epsilon_{0}}{2} \int|\mathbf{E}|^{2} d V$$

  (ii) Find the electric field produced by a spherical shell with total charge $Q$ and radius $R$, assuming it to vanish inside the shell. Find the energy stored in the electric field.

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• 1.I.4C

  Special Relativity

  Part IB, 2008

  In an inertial frame $S$ a photon of energy $E$ is observed to travel at an angle $\theta$ relative to the $x$-axis. The inertial frame $S^{\prime}$ moves relative to $S$ at velocity $v$ in the $x$ direction and the $x^{\prime}$-axis of $S^{\prime}$ is taken parallel to the $x$-axis of $S$. Observed in $S^{\prime}$, the photon has energy $E^{\prime}$ and travels at an angle $\theta^{\prime}$ relative to the $x^{\prime}$-axis. Show that

  $$E^{\prime}=\frac{E(1-\beta \cos \theta)}{\sqrt{1-\beta^{2}}}, \quad \cos \theta^{\prime}=\frac{\cos \theta-\beta}{1-\beta \cos \theta},$$

  where $\beta=v / c$.

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• 2.I.7C

  Special Relativity

  Part IB, 2008

  A photon of energy $E$ collides with a particle of rest mass $m$, which is at rest. The final state consists of a photon and a particle of rest mass $M, M>m$. Show that the minimum value of $E$ for which it is possible for this reaction to take place is

  $$E_{\min }=\frac{M^{2}-m^{2}}{2 m} c^{2}$$

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• 4.II.17C

  Special Relativity

  Part IB, 2008

  Write down the formulae for the one-dimensional Lorentz transformation $(x, t) \rightarrow$ $\left(x^{\prime}, t^{\prime}\right)$ for frames moving with relative velocity $v$ along the $x$-direction. Derive the relativistic formula for the addition of velocities $v$ and $u$.

  A train, of proper length $L$, travels past a station at velocity $v>0$. The origin of the inertial frame $S$, with coordinates $(x, t)$, in which the train is stationary, is located at the mid-point of the train. The origin of the inertial frame $S^{\prime}$, with coordinates $\left(x^{\prime}, t^{\prime}\right)$, in which the station is stationary, is located at the mid-point of the platform. Coordinates are chosen such that when the origins coincide then $t=t^{\prime}=0$.

  Observers A and B, stationary in $S$, are located, respectively, at the front and rear of the train. Observer C, stationary in $S^{\prime}$, is located at the origin of $S^{\prime}$. At $t^{\prime}=0$, C sends two signals, which both travel at speed $u$, where $v<u \leq c$, one directed towards $\mathrm{A}$ and the other towards $\mathrm{B}$, who receive the signals at respective times $t_{A}$ and $t_{B}$. $\mathrm{C}$ observes these events to occur, respectively, at times $t_{A}^{\prime}$ and $t_{B}^{\prime}$. At $t^{\prime}=0, \mathrm{C}$ also observes that the two ends of the platform coincide with the positions of $A$ and $B$.

  (a) Draw two space-time diagrams, one for $S$ and the other for $S^{\prime}$, showing the trajectories of the observers and the events that take place.

  (b) What is the length of the platform in terms of $L$ ? Briefly illustrate your answer by reference to the space-time diagrams.

  (c) Calculate the time differences $t_{B}-t_{A}$ and $t_{B}^{\prime}-t_{A}^{\prime}$.

  (d) Setting $u=c$, use this example to discuss briefly the fact that two events observed to be simultaneous in one frame need not be observed to be simultaneous in another.

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• 1.I.5B

  Fluid Dynamics

  Part IB, 2008

  Verify that the two-dimensional flow given in Cartesian coordinates by

  $$\mathbf{u}=\left(\mathrm{e}^{y} \sinh x,-\mathrm{e}^{y} \cosh x\right)$$

  satisfies $\nabla \cdot \mathbf{u}=0$. Find the stream function $\psi(x, y)$. Sketch the streamlines.

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• 1.II.17B

  Fluid Dynamics

  Part IB, 2008

  Two incompressible fluids flow in infinite horizontal streams, the plane of contact being $z=0$, with $z$ positive upwards. The flow is given by

  $$\mathbf{U}(\mathbf{r})= \begin{cases}U_{2} \hat{\mathbf{e}}_{x}, & z>0 \\ U_{1} \hat{\mathbf{e}}_{x}, & z<0\end{cases}$$

  where $\hat{\mathbf{e}}_{x}$ is the unit vector in the positive $x$ direction. The upper fluid has density $\rho_{2}$ and pressure $p_{0}-g \rho_{2} z$, the lower has density $\rho_{1}$ and pressure $p_{0}-g \rho_{1} z$, where $p_{0}$ is a constant and $g$ is the acceleration due to gravity.

  (i) Consider a perturbation to the flat surface $z=0$ of the form

  $$z \equiv \zeta(x, y, t)=\zeta_{0} e^{i(k x+\ell y)+s t} .$$

  State the kinematic boundary conditions on the velocity potentials $\phi_{i}$ that hold on the interface in the two domains, and show by linearising in $\zeta$ that they reduce to

  $$\frac{\partial \phi_{i}}{\partial z}=\frac{\partial \zeta}{\partial t}+U_{i} \frac{\partial \zeta}{\partial x} \quad(z=0, i=1,2) .$$

  (ii) State the dynamic boundary condition on the perturbed interface, and show by linearising in $\zeta$ that it reduces to

  $$\rho_{1}\left(U_{1} \frac{\partial \phi_{1}}{\partial x}+\frac{\partial \phi_{1}}{\partial t}+g \zeta\right)=\rho_{2}\left(U_{2} \frac{\partial \phi_{2}}{\partial x}+\frac{\partial \phi_{2}}{\partial t}+g \zeta\right) \quad(z=0)$$

  (iii) Use the velocity potentials

  $$\phi_{1}=U_{1} x+A_{1} e^{q z} e^{i(k x+\ell y)+s t}, \quad \phi_{2}=U_{2} x+A_{2} e^{-q z} e^{i(k x+\ell y)+s t},$$

  where $q=\sqrt{k^{2}+\ell^{2}}$, and the conditions in (i) and (ii) to perform a stability analysis. Show that the relation between $s, k$ and $\ell$ is

  $$s=-i k \frac{\rho_{1} U_{1}+\rho_{2} U_{2}}{\rho_{1}+\rho_{2}} \pm\left[\frac{k^{2} \rho_{1} \rho_{2}\left(U_{1}-U_{2}\right)^{2}}{\left(\rho_{1}+\rho_{2}\right)^{2}}-\frac{q g\left(\rho_{1}-\rho_{2}\right)}{\rho_{1}+\rho_{2}}\right]^{1 / 2} .$$

  Find the criterion for instability.

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• 2.I.8B

  Fluid Dynamics

  Part IB, 2008

  (i) Show that for a two-dimensional incompressible flow $(u(x, y), v(x, y), 0)$, the vorticity is given by $\boldsymbol{\omega} \equiv \omega_{z} \hat{\mathbf{e}}_{z}=\left(0,0,-\nabla^{2} \psi\right)$ where $\psi$ is the stream function.

  (ii) Express the $z$-component of the vorticity equation

  $$\frac{\partial \boldsymbol{\omega}}{\partial t}+(\mathbf{u} \cdot \nabla) \boldsymbol{\omega}=(\boldsymbol{\omega} \cdot \nabla) \mathbf{u}$$

  in terms of the stream function $\psi$.

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• 4.II.10E

  Linear Algebra

  Part IB, 2008

  What is meant by a Hermitian matrix? Show that if $A$ is Hermitian then all its eigenvalues are real and that there is an orthonormal basis for $\mathbb{C}^{n}$ consisting of eigenvectors of $A$.

  A Hermitian matrix is said to be positive definite if $\langle A x, x\rangle>0$ for all $x \neq 0$. We write $A>0$ in this case. Show that $A$ is positive definite if, and only if, all of its eigenvalues are positive. Show that if $A>0$ then $A$ has a unique positive definite square root $\sqrt{A}$.

  Let $A, B$ be two positive definite Hermitian matrices with $A-B>0$. Writing $C=\sqrt{A}$ and $X=\sqrt{A}-\sqrt{B}$, show that $C X+X C>0$. By considering eigenvalues of $X$, or otherwise, show that $X>0$.

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• 3.II.18B

  Fluid Dynamics

  Part IB, 2008

  An ideal liquid contained within a closed circular cylinder of radius $a$ rotates about the axis of the cylinder (assume this axis to be in the vertical $z$-direction).

  (i) Prove that the equation of continuity and the boundary conditions are satisfied by the velocity $\mathbf{v}=\boldsymbol{\Omega} \times \mathbf{r}$, where $\boldsymbol{\Omega}=\Omega \hat{\mathbf{e}}_{z}$ is the angular velocity, with $\hat{\mathbf{e}}_{z}$ the unit vector in the $z$-direction, which depends only on time, and $\mathbf{r}$ is the position vector measured from a point on the axis of rotation.

  (ii) Calculate the angular momentum $\mathbf{M}=\rho \int(\mathbf{r} \times \mathbf{v}) d V$ per unit length of the cylinder.

  (iii) Suppose the the liquid starts from rest and flows under the action of an external force per unit mass $\mathbf{f}=(\alpha x+\beta y, \gamma x+\delta y, 0)$. By taking the curl of the Euler equation, prove that

  $$\frac{d \Omega}{d t}=\frac{1}{2}(\gamma-\beta)$$

  (iv) Find the pressure.

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• 4.II.18B

  Fluid Dynamics

  Part IB, 2008

  (i) Starting from Euler's equation for an incompressible fluid show that for potential flow with $\mathbf{u}=\nabla \phi$,

  $$\frac{\partial \phi}{\partial t}+\frac{1}{2} u^{2}+\chi=f(t)$$

  where $u=|\mathbf{u}|, \chi=p / \rho+V$, the body force per unit mass is $-\nabla V$ and $f(t)$ is an arbitrary function of time.

  (ii) Hence show that, for the steady flow of a liquid of density $\rho$ through a pipe of varying cross-section that is subject to a pressure difference $\Delta p=p_{1}-p_{2}$ between its two ends, the mass flow through the pipe per unit time is given by

  $$m \equiv \frac{d M}{d t}=S_{1} S_{2} \sqrt{\frac{2 \rho \Delta p}{S_{1}^{2}-S_{2}^{2}}}$$

  where $S_{1}$ and $S_{2}$ are the cross-sectional areas of the two ends.

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• 1.I.6D

  Numerical Analysis

  Part IB, 2008

  Show that if $A=L D L^{T}$, where $L \in \mathbb{R}^{m \times m}$ is a lower triangular matrix with all elements on the main diagonal being unity and $D \in \mathbb{R}^{m \times m}$ is a diagonal matrix with positive elements, then $A$ is positive definite. Find $L$ and the corresponding $D$ when

  $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 3 \end{array}\right]$$

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• 2.II.18D

  Numerical Analysis

  Part IB, 2008

  (a) A Householder transformation (reflection) is given by

  $$H=I-\frac{2 u u^{T}}{\|u\|^{2}},$$

  where $H \in \mathbb{R}^{m \times m}, u \in \mathbb{R}^{m}$, and $I$ is the $m \times m$ unit matrix and $u$ is a non-zero vector which has norm $\|u\|=\left(\sum_{i=1}^{m} u_{i}^{2}\right)^{1 / 2}$. Show that $H$ is orthogonal.

  (b) Suppose that $A \in \mathbb{R}^{m \times n}, x \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$ with $n<m$. Show that if $x$ minimises $\|A x-b\|^{2}$ then it also minimises $\|Q A x-Q b\|^{2}$, where $Q$ is an arbitrary $m \times m$ orthogonal matrix.

  (c) Using Householder reflection, find the $x$ that minimises $\|A x-b\|^{2}$ when

  $$A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 4 \\ 0 & 2 \\ 0 & 4 \end{array}\right] \quad b=\left[\begin{array}{r} 1 \\ 1 \\ 2 \\ -1 \end{array}\right]$$

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• 3.II.19D

  Numerical Analysis

  Part IB, 2008

  Starting from the Taylor formula for $f(x) \in C^{k+1}[a, b]$ with an integral remainder term, show that the error of an approximant $L(f)$ can be written in the form (Peano kernel theorem)

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

  when $L(f)$, which is identically zero if $f(x)$ is a polynomial of degree $k$, satisfies conditions that you should specify. Give an expression for $K(\theta)$.

  Hence determine the minimum value of $c$ in the inequality

  $$|L(f)| \leq c\left\|f^{\prime \prime \prime}\right\|_{\infty}$$

  when

  $$L(f)=f^{\prime}(1)-\frac{1}{2}(f(2)-f(0)) \text { for } f(x) \in C^{3}[0,2]$$

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• 4.I.8D

  Numerical Analysis

  Part IB, 2008

  Show that the Chebyshev polynomials, $T_{n}(x)=\cos \left(n \cos ^{-1} x\right), n=0,1,2, \ldots$ obey the orthogonality relation

  $$\int_{-1}^{1} \frac{T_{n}(x) T_{m}(x)}{\sqrt{1-x^{2}}} d x=\frac{\pi}{2} \delta_{n, m}\left(1+\delta_{n, 0}\right)$$

  State briefly how an optimal choice of the parameters $a_{k}, x_{k}, k=1,2 \ldots n$ is made in the Gaussian quadrature formula

  $$\int_{-1}^{1} \frac{f(x)}{\sqrt{1-x^{2}}} d x \sim \sum_{k=1}^{n} a_{k} f\left(x_{k}\right)$$

  Find these parameters for the case $n=3$.

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• $1 . \mathrm{I} . 7 \mathrm{H} \quad$

  Statistics

  Part IB, 2008

  A Bayesian statistician observes a random sample $X_{1}, \ldots, X_{n}$ drawn from a $N\left(\mu, \tau^{-1}\right)$ distribution. He has a prior density for the unknown parameters $\mu, \tau$ of the form

  $$\pi_{0}(\mu, \tau) \propto \tau^{\alpha_{0}-1} \exp \left(-\frac{1}{2} K_{0} \tau\left(\mu-\mu_{0}\right)^{2}-\beta_{0} \tau\right) \sqrt{\tau},$$

  where $\alpha_{0}, \beta_{0}, \mu_{0}$ and $K_{0}$ are constants which he chooses. Show that after observing $X_{1}, \ldots, X_{n}$ his posterior density $\pi_{n}(\mu, \tau)$ is again of the form

  $$\pi_{n}(\mu, \tau) \propto \tau^{\alpha_{n}-1} \exp \left(-\frac{1}{2} K_{n} \tau\left(\mu-\mu_{n}\right)^{2}-\beta_{n} \tau\right) \sqrt{\tau}$$

  where you should find explicitly the form of $\alpha_{n}, \beta_{n}, \mu_{n}$ and $K_{n}$.

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• 1.II.18H

  Statistics

  Part IB, 2008

  Suppose that $X_{1}, \ldots, X_{n}$ is a sample of size $n$ with common $N\left(\mu_{X}, 1\right)$ distribution, and $Y_{1}, \ldots, Y_{n}$ is an independent sample of size $n$ from a $N\left(\mu_{Y}, 1\right)$ distribution.

  (i) Find (with careful justification) the form of the size- $\alpha$ likelihood-ratio test of the null hypothesis $H_{0}: \mu_{Y}=0$ against alternative $H_{1}:\left(\mu_{X}, \mu_{Y}\right)$ unrestricted.

  (ii) Find the form of the size- $\alpha$ likelihood-ratio test of the hypothesis

  $$H_{0}: \mu_{X} \geqslant A, \mu_{Y}=0$$

  against $H_{1}:\left(\mu_{X}, \mu_{Y}\right)$ unrestricted, where $A$ is a given constant.

  Compare the critical regions you obtain in (i) and (ii) and comment briefly.

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• 2.II.19H

  Statistics

  Part IB, 2008

  Suppose that the joint distribution of random variables $X, Y$ taking values in $\mathbb{Z}^{+}=\{0,1,2, \ldots\}$ is given by the joint probability generating function

  $$\varphi(s, t) \equiv E\left[s^{X} t^{Y}\right]=\frac{1-\alpha-\beta}{1-\alpha s-\beta t}$$

  where the unknown parameters $\alpha$ and $\beta$ are positive, and satisfy the inequality $\alpha+\beta<1$. Find $E(X)$. Prove that the probability mass function of $(X, Y)$ is

  $$f(x, y \mid \alpha, \beta)=(1-\alpha-\beta)\left(\begin{array}{c} x+y \\ x \end{array}\right) \alpha^{x} \beta^{y} \quad\left(x, y \in \mathbb{Z}^{+}\right)$$

  and prove that the maximum-likelihood estimators of $\alpha$ and $\beta$ based on a sample of size $n$ drawn from the distribution are

  $$\hat{\alpha}=\frac{\bar{X}}{1+\bar{X}+\bar{Y}}, \quad \hat{\beta}=\frac{\bar{Y}}{1+\bar{X}+\bar{Y}},$$

  where $\bar{X}$ (respectively, $\bar{Y}$ ) is the sample mean of $X_{1}, \ldots, X_{n}$ (respectively, $Y_{1}, \ldots, Y_{n}$ ).

  By considering $\hat{\alpha}+\hat{\beta}$ or otherwise, prove that the maximum-likelihood estimator is biased. Stating clearly any results to which you appeal, prove that as $n \rightarrow \infty, \hat{\alpha} \rightarrow \alpha$, making clear the sense in which this convergence happens.

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• $3 . \mathrm{I} . 8 \mathrm{H} \quad$

  Statistics

  Part IB, 2008

  If $X_{1}, \ldots, X_{n}$ is a sample from a density $f(\cdot \mid \theta)$ with $\theta$ unknown, what is a $95 \%$ confidence set for $\theta$ ?

  In the case where the $X_{i}$ are independent $N\left(\mu, \sigma^{2}\right)$ random variables with $\sigma^{2}$ known, $\mu$ unknown, find (in terms of $\sigma^{2}$ ) how large the size $n$ of the sample must be in order for there to exist a $95 \%$ confidence interval for $\mu$ of length no more than some given $\varepsilon>0$.

  [Hint: If $Z \sim N(0,1)$ then $P(Z>1.960)=0.025 .]$

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• 1.II.10G

  Groups, Rings and Modules

  Part IB, 2008

  (i) Show that $A_{4}$ is not simple.

  (ii) Show that the group Rot $(D)$ of rotational symmetries of a regular dodecahedron is a simple group of order 60 .

  (iii) Show that $\operatorname{Rot}(D)$ is isomorphic to $A_{5}$.

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• 4.II.19H

  Statistics

  Part IB, 2008

  (i) Consider the linear model

  $$Y_{i}=\alpha+\beta x_{i}+\varepsilon_{i}$$

  where observations $Y_{i}, i=1, \ldots, n$, depend on known explanatory variables $x_{i}$, $i=1, \ldots, n$, and independent $N\left(0, \sigma^{2}\right)$ random variables $\varepsilon_{i}, i=1, \ldots, n$.

  Derive the maximum-likelihood estimators of $\alpha, \beta$ and $\sigma^{2}$.

  Stating clearly any results you require about the distribution of the maximum-likelihood estimators of $\alpha, \beta$ and $\sigma^{2}$, explain how to construct a test of the hypothesis that $\alpha=0$ against an unrestricted alternative.

  (ii) A simple ballistic theory predicts that the range of a gun fired at angle of elevation $\theta$ should be given by the formula

  $$Y=\frac{V^{2}}{g} \sin 2 \theta$$

  where $V$ is the muzzle velocity, and $g$ is the gravitational acceleration. Shells are fired at 9 different elevations, and the ranges observed are as follows:

  $\begin{array}{cccccccccc}\theta \text { (degrees) } & 5 & 15 & 25 & 35 & 45 & 55 & 65 & 75 & 85 \\ \sin 2 \theta & 0.1736 & 0.5 & 0.7660 & 0.9397 & 1 & 0.9397 & 0.7660 & 0.5 & 0.1736 \\ Y(\mathrm{~m}) & 4322 & 11898 & 17485 & 20664 & 21296 & 19491 & 15572 & 10027 & 3458\end{array}$

  The model

  $$Y_{i}=\alpha+\beta \sin 2 \theta_{i}+\varepsilon_{i}$$

  is proposed. Using the theory of part (i) above, find expressions for the maximumlikelihood estimators of $\alpha$ and $\beta$.

  The $t$-test of the null hypothesis that $\alpha=0$ against an unrestricted alternative does not reject the null hypothesis. Would you be willing to accept the model $(*)$ ? Briefly explain your answer.

  [You may need the following summary statistics of the data. If $x_{i}=\sin 2 \theta_{i}$, then $\bar{x} \equiv n^{-1} \sum x_{i}=0.63986, \bar{Y}=13802, S_{x x} \equiv \sum\left(x_{i}-\bar{x}\right)^{2}=0.81517, S_{x y}=\sum Y_{i}\left(x_{i}-\bar{x}\right)=$ 17186. ]

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• 1.I.8H

  Optimization

  Part IB, 2008

  State the Lagrangian Sufficiency Theorem for the maximization over $x$ of $f(x)$ subject to the constraint $g(x)=b$.

  For each $p>0$, solve

  $$\max \sum_{i=1}^{d} x_{i}^{p} \quad \text { subject to } \sum_{i=1}^{d} x_{i}=1, \quad x_{i} \geqslant 0 .$$

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• 2.I.9H

  Optimization

  Part IB, 2008

  Goods from three warehouses have to be delivered to five shops, the cost of transporting one unit of good from warehouse $i$ to shop $j$ being $c_{i j}$, where

  $$C=\left(\begin{array}{ccccc} 2 & 3 & 6 & 6 & 4 \\ 7 & 6 & 1 & 1 & 5 \\ 3 & 6 & 6 & 2 & 1 \end{array}\right)$$

  The requirements of the five shops are respectively $9,6,12,5$ and 10 units of the good, and each warehouse holds a stock of 15 units. Find a minimal-cost allocation of goods from warehouses to shops and its associated cost.

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• 3.II.20H

  Optimization

  Part IB, 2008

  Use the simplex algorithm to solve the problem

  $$\max x_{1}+2 x_{2}-6 x_{3}$$

  subject to $x_{1}, x_{2} \geqslant 0,\left|x_{3}\right| \leqslant 5$, and

  $$\begin{array}{r} x_{1}+x_{2}+x_{3} \leqslant 7 \\ 2 x_{2}+x_{3} \geqslant 1 \end{array}$$

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• 4.II.20H

  Optimization

  Part IB, 2008

  (i) Suppose that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$, and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ are continuously differentiable. Suppose that the problem

  $\max f(x)$ subject to $g(x)=b$

  is solved by a unique $\bar{x}=\bar{x}(b)$ for each $b \in \mathbb{R}^{m}$, and that there exists a unique $\lambda(b) \in \mathbb{R}^{m}$ such that

  $$\varphi(b) \equiv f(\bar{x}(b))=\sup _{x}\left\{f(x)+\lambda(b)^{T}(b-g(x))\right\}$$

  Assuming that $\bar{x}$ and $\lambda$ are continuously differentiable, prove that

  $$\frac{\partial \varphi}{\partial b_{i}}(b)=\lambda_{i}(b)$$

  (ii) The output of a firm is a function of the capital $K$ deployed, and the amount $L$ of labour employed, given by

  $$f(K, L)=K^{\alpha} L^{\beta}$$

  where $\alpha, \beta \in(0,1)$. The firm's manager has to optimize the output subject to the budget constraint

  $$K+w L=b,$$

  where $w>0$ is the wage rate and $b>0$ is the available budget. By casting the problem in Lagrangian form, find the optimal solution and verify the relation $(*)$.

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• 1.II.19H

  Markov Chains

  Part IB, 2008

  The village green is ringed by a fence with $N$ fenceposts, labelled $0,1, \ldots, N-1$. The village idiot is given a pot of paint and a brush, and started at post 0 with instructions to paint all the posts. He paints post 0 , and then chooses one of the two nearest neighbours, 1 or $N-1$, with equal probability, moving to the chosen post and painting it. After painting a post, he chooses with equal probability one of the two nearest neighbours, moves there and paints it (regardless of whether it is already painted). Find the distribution of the last post unpainted.

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• 2.II.20H

  Markov Chains

  Part IB, 2008

  A Markov chain with state-space $I=\mathbb{Z}^{+}$has non-zero transition probabilities $p_{00}=q_{0}$ and

  $$p_{i, i+1}=p_{i}, \quad p_{i+1, i}=q_{i+1} \quad(i \in I) .$$

  Prove that this chain is recurrent if and only if

  $$\sum_{n \geqslant 1} \prod_{r=1}^{n} \frac{q_{r}}{p_{r}}=\infty$$

  Prove that this chain is positive-recurrent if and only if

  $$\sum_{n \geqslant 1} \prod_{r=1}^{n} \frac{p_{r-1}}{q_{r}}<\infty$$

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• 3.I.9H

  Markov Chains

  Part IB, 2008

  What does it mean to say that a Markov chain is recurrent?

  Stating clearly any general results to which you appeal, prove that the symmetric simple random walk on $\mathbb{Z}$ is recurrent.

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• 4.I.9H

  Markov Chains

  Part IB, 2008

  A Markov chain on the state-space $I=\{1,2,3,4,5,6,7\}$ has transition matrix

  $$P=\left(\begin{array}{ccccccc} 0 & 1 / 2 & 1 / 4 & 0 & 1 / 4 & 0 & 0 \\ 1 / 3 & 0 & 1 / 2 & 0 & 0 & 1 / 6 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 / 2 & 0 & 1 / 2 \end{array}\right)$$

  Classify the chain into its communicating classes, deciding for each what the period is, and whether the class is recurrent.

  For each $i, j \in I$ say whether the $\operatorname{limit}^{-1} \lim _{n \rightarrow \infty} p_{i j}^{(n)}$ exists, and evaluate the limit when it does exist.

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• 2.I.2G

  Groups, Rings and Modules

  Part IB, 2008

  What does it means to say that a complex number $\alpha$ is algebraic over $\mathbb{Q}$ ? Define the minimal polynomial of $\alpha$.

  Suppose that $\alpha$ satisfies a nonconstant polynomial $f \in \mathbb{Z}[X]$ which is irreducible over $\mathbb{Z}$. Show that there is an isomorphism $\mathbb{Z}[X] /(f) \cong \mathbb{Z}[\alpha]$.

  [You may assume standard results about unique factorisation, including Gauss's lemma.]

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• 2.II.11G

  Groups, Rings and Modules

  Part IB, 2008

  Let $F$ be a field. Prove that every ideal of the ring $F\left[X_{1}, \ldots, X_{n}\right]$ is finitely generated.

  Consider the set

  $$R=\left\{p(X, Y)=\sum c_{i j} X^{i} Y^{j} \in F[X, Y] \mid c_{0 j}=c_{j 0}=0 \text { whenever } j>0\right\}$$

  Show that $R$ is a subring of $F[X, Y]$ which is not Noetherian.

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