Preliminary Exam - Spring 1995

Problem 1   For each positive integer $n$, define $f_n : \mathbb{R}\to \mathbb{R}$ by $f_n(x)=\cos nx$. Prove that the sequence of functions $\{f_n\}$ has no uniformly convergent subsequence.

Problem 2   Let $A$ be the 3$\times$3 matrix

$$\left( \begin{array}{rrr} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{array} \right)$$

Determine all real numbers a for which the limit $${\lim_{n\to\infty}a^nA^n}$$ exists and is nonzero (as a matrix).

Problem 3   Let n be a positive integer and $0<\theta<\pi$. Prove that

$${1\over2\pi i}\int_{\vert z\vert=2}{z^n\over1-2z\cos\theta+z^2}dz= {\sin n\theta \over\sin\theta}$$

where the circle $\vert z\vert=2$ is oriented counterclockwise.

Problem 4  

Let $\mbox{\bf {F}}$ be a finite field of cardinality $p^n$, with p prime and $n>0$, and let $G$ be the group of invertible 2$\times$2 matrices with coefficients in $\mbox{\bf {F}}$.

1. Prove that $G$ has order $(p^{2n}-1)(p^{2n}-p^n)$.
2. Show that any p-Sylow subgroup of $G$ is isomorphic to the additive group of $\mbox{\bf {F}}$.

Problem 5  

Let $f:\mathbb{R}\to \mathbb{R}$ be a bounded continuously differentiable function. Show that every solution of $y'(x)=f \left( y(x) \right)$ is monotone.

Problem 6   Suppose that $R$ is a subring of a commutative ring $S$ and that $R$ is of finite index $n$ in $S$. Let $m$ be an integer that is relatively prime to $n$. Prove that the natural map $R/mR\to S/mS$ is a ring isomorphism.

Problem 7  

Let $f$, $g\colon[0,1]\to[0,\infty)$ be continuous functions satisfying

$$\sup_{0\le x\le1}f(x)=\sup_{0\le x\le1}g(x).$$

Prove that there exists $t\in[0,1]$ with $f(t)^2+3f(t)=g(t)^2+3g(t)$.

Problem 8   Suppose that $W\subset V$ are finite-dimensional vector spaces over a field, and let $L\colon V\to V$ be a linear transformation with $L(V)\subset W$. Denote the restriction of $L$ to $W$ by $L_W$. Prove that $\det(1-tL)=\det(1-tL_W)$.

Problem 9  

Let $P(x)$ be a polynomial with real coefficients and with leading coefficient $1$. Suppose that $P(0)=-1$ and that $P(x)$ has no complex zeros inside the unit circle. Prove that $P(1)=0$.

Problem 10   Let $f_n\colon[0,1]\to[0,\infty)$ be a continuous function, for
$n=1, 2, \ldots$. Suppose that one has

$$(*) \;\;\;\;\; f_1(x)\ge f_2(x)\ge f_3(x)\ge\cdots\;\; for \; all \; x \in [0,1].$$

Let $f(x)=\lim_{n\to\infty}f_n(x)$ and $M=\sup_{0\le x\le1}f(x)$.

1. Prove that there exists $t\in[0,1]$ with $f(t)=M$.
2. Show by example that the conclusion of Part 1 need not hold if instead of $(*)$ we merely know that for each $x\in[0,1]$ there exists $n_x$ such that for all $n\ge n_x$ one has $f_n(x)\ge f_{n+1}(x)$.

Problem 11   Let $n$ be a positive integer, and let $S\subset \mathbb{R}^n$ a finite subset with $0\in S$. Suppose that $\varphi : S \to S$ is a map satisfying

$$\begin{eqnarray*} \varphi(0) & = & 0,\\ d(\varphi(s),\varphi(t)) & = & d(s,t)\qquad for \; all\;\quad s, t\in S, \end{eqnarray*}$$

where $d(\;,\;)$ denotes Euclidean distance. Prove that there is a linear map
$f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ whose restriction to $S$ is $\varphi$.

Problem 12   Let $n$ be a positive integer. Compute

$$\int_0^{2\pi}{1-\cos n\theta \over1-\cos\theta}d\theta\,.$$

Problem 13  

Let $n$ be an odd positive integer, and denote by $S_n$ the group of all permutations of $\{1,2,\ldots,n\}$. Suppose that $G$ is a subgroup of $S_n$ of $2$-power order. Prove that there exists $i\in\{1,2,\ldots,n\}$ such that for all $\sigma\in G$, one has $\sigma(i)=i$.

Problem 14  

Let $y : \mathbb{R}\to \mathbb{R}$ be a three times differentiable function satisfying the differential equation $y'''-y=0$. Suppose that $\lim_{x\to\infty}y(x)=0$. Find real numbers $a$, $b$, $c$, and $d$, not all zero, such that $ay(0)+y'(0)+cy''(0)=d$.

Problem 15   Let $\mbox{\bf {F}}$ be a finite field, and suppose that the subfield of $\mbox{\bf {F}}$ generated by $\{x^3 \;\vert\;x\in \mbox{\bf {F}}\}$ is different from $\mbox{\bf {F}}$. Show that $\mbox{\bf {F}}$ has cardinality $4$.

Problem 16  

Let $K$ be a nonempty compact set in a metric space with distance function $d$. Suppose that $\varphi\colon K\to K$ satisfies

$$d(\varphi(x),\varphi(y))<d(x,y)$$

for all $x\ne y$ in $K$. Show there exists precisely one point $x\in K$ such that $x=\varphi(x)$.

Problem 17   Let $V$ be a finite-dimensional vector space over a field $\mbox{\bf {F}}$, and let $L : V \to V$ be a linear transformation. Suppose that the characteristic polynomial $\chi$ of $L$ is written as $\chi=\chi_1\chi_2$, where $\chi_1$ and $\chi_2$ are two relatively prime polynomials with coefficients in $\mbox{\bf {F}}$. Show that $V$ can be written as the direct sum of two subspaces $V_1$ and $V_2$ with the property that $\chi_i(L)V_i=0$ for $i=1,2$.

Problem 18   Prove that there is no one-to-one conformal map of the punctured disc $G=\{z\in \mathbb{C}\;\vert\; 0<\vert z\vert<1\}$ onto the annulus $A=\{z\in \mathbb{C}\;\vert\;1<\vert z\vert<2\}$.