Spring97

Problem 1 For which values of the exponents $\alpha ,\,\beta$ does the following series converge?

$\begin{displaymath}\sum_{n=3}^{\infty}\frac{1}{n^{\alpha}(\log n)^{\beta}}\,\cdot\end{displaymath}$

Problem 2 Let $M$ be a metric space with metric $d$ . Let $C$ be a nonempty closed subset of $M$ . Define $% latex2html id marker 677 $f:M\to \mbox{$\mathbb{R}^{}$}$$ by

$\begin{displaymath}f(x)=\inf \{d(x,y)\,\vert\,y\in C\}.\end{displaymath}$

Show that $f$ is continuous, and that $f(x)=0$ if and only if $x\in C.$

Problem 3 Suppose that $f(x)$ is continuous, nonnegative for $x\geq 0$ , with $\int_{0}^{\infty}f(x)dx<\infty.$ Prove that

$\begin{displaymath}\lim_{n\to\infty}\int_{0}^{n}\frac{xf(x)}{n}dx=0\,.\end{displaymath}$

Problem 4 Let $f$ and $g$ be two entire functions such that, for all
$% latex2html id marker 724 $z\in\mbox{$\mathbb{C}\,^{}$}$$ , $\Re f(z)\leq k \Re g(z)$ for some real constant $k$ (independent of $z$ ). Show that there are constants $a,\,b$ such that

$\begin{displaymath}f(z) = ag(z)+b\, .\end{displaymath}$

Problem 5 Prove that

$\begin{displaymath}\int_{-\infty}^{\infty}\frac{e^{-(t-i\gamma )^2/2}}{\sqrt{2\pi}}dt\end{displaymath}$

is independent of the real parameter $\gamma$ .

Problem 6 Suppose that $X$ is a topological space and $V$ is a finite-dimensional subspace of the vector space of continuous real valued functions on $X$ . Prove that there exist a basis $(f_1,\dots , f_n)$ for $V$ and points $x_1,\dots , x_n$ in $X$ such that $f_i(x_j)=\delta_{ij}$ .

Problem 7 Suppose that $A$ and $B$ are endomorphisms of a finite-dimensional vector space $V$ over a field $% latex2html id marker 791 $\mbox{\bf {K}}$$ . Prove or disprove the following statements:

Every eigenvector of $AB$ is also an eigenvector of $BA$ .
Every eigenvalue of $AB$ is also an eigenvalue of $BA$ .

Problem 9 Let $R$ be the ring of $n\times n$ matrices over a field. Suppose $S$ is a ring and $h:R\to S$ is a homomorphism. Show that $h$ is either injective or zero.

Problem 10 Let $% latex2html id marker 846 $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$$ be a bounded function (i.e., there is a constant $M$ such that $\vert f(x)\vert\leq M$ for all $% latex2html id marker 849 $x\in\mbox{$\mathbb{R}^{}$}$$ ). Suppose the graph of $f$ is a closed subset of $% latex2html id marker 851 $\mbox{$\mathbb{R}^{2}$}$$ . Prove that $f$ is continuous.

Problem 11 Suppose that $f''(x)=(x^2-1)f(x)$ for all $% latex2html id marker 885 $x\in\mbox{$\mathbb{R}^{}$}$$ , and that $f(0)=1,$ $f'(0)=0$ . Show that $f(x)\to 0$ as $x\to\infty$ .

Problem 12 Evaluate the integral

$\begin{displaymath}\int_{-\infty}^{\infty}\frac{\sin^2x}{x^2}dx\,.\end{displaymath}$

Problem 13 Suppose that $% latex2html id marker 930 $f:\mbox{$\mathbb{C}\,^{}$}\to\mbox{$\mathbb{C}\,^{}$}$$ is injective and everywhere holomorphic. Prove that there exist $% latex2html id marker 931 $a,\,b\in\mbox{$\mathbb{C}\,^{}$}$$ with $a\neq 0$ such that $f(z)=az+b$ for all $% latex2html id marker 934 $z\in\mbox{$\mathbb{C}\,^{}$}.$$

Problem 14 Show that

$\begin{displaymath} % latex2html id marker 969\det (\exp (M))=e^{\mathrm{tr}(M)} \end{displaymath}$

for any complex $n\times n$ matrix $M$ , where $% latex2html id marker 965 $\exp (M)$$ is defined as in Problem

Problem 15 Suppose that $P$ and $Q$ are $n\times n$ matrices such that
$P^2=P$ , $Q^2=Q$ , and $1-(P+Q)$ is invertible. Show that $P$ and $Q$ have the same rank.

Problem 16 Suppose that $A$ is a commutative algebra with identity over $% latex2html id marker 1010 $\mbox{$\mathbb{C}\,^{}$}$$ (i.e., $A$ is a commutative ring containing $% latex2html id marker 1012 $\mbox{$\mathbb{C}\,^{}$}$$ as a subring with identity). Suppose further that $a^2\neq 0$ for all nonzero elements $a\in A$ . Show that if the dimension of $A$ as a vector space over $% latex2html id marker 1016 $\mbox{$\mathbb{C}\,^{}$}$$ is finite and at least two, then the equations $a^2=a$ is satisfied by at least three distinct elements $a\in A$ .

Problem 17 Let $% latex2html id marker 1054 $GL_2(\mbox{$\mathbb{Z}^{}$}_m)$$ denote the multiplicative group of invertible $2\times 2$ matrices over the ring of integers modulo $m$ . Find the order of $% latex2html id marker 1057 $GL_2(\mbox{$\mathbb{Z}^{}$}_{p^n})$$ for each prime $p$ and positive integer $n.$

Problem 18 Let $H$ be the quotient of an abelian group $G$ by a subgroup $K$ . Prove or disprove each of the following statements:

If $H$ is finite cyclic then $G$ is isomorphic to the direct product of $H$ and $K.$
If $H$ is a direct product of infinite cyclic groups then $G$ is isomorphic to the direct product of $H$ and $K.$