Preliminary Exam - Spring 1989

Problem 1   Let $a_1, a_2, \ldots$ be positive numbers such that

$$\sum_{n=1}^{\infty}a_n<\infty\, .$$

Prove that there are positive numbers $c_1, c_2, \ldots$ such that

$$\lim_{n \to \infty}c_n = \infty \quad and \quad \sum_{n=1}^{\infty}c_{n}a_n<\infty \, .$$

Problem 2   Let $\mbox{\bf {F}}$ be a field, $n$ and $m$ positive integers, and $A$ an $n\times n$ matrix with entries in $\mbox{\bf {F}}$ such that $A^m = 0$. Prove that $A^n = 0$.

Problem 3   Let f be a continuous real valued function on $[0,1]\times[0,1]$. Let the function g on $[0,1]$ be defined by

$$g(x) = \max\left\{f(x,y) \;\vert\; y \in [0,1] \right\}.$$

Prove that g is continuous.

Problem 4   Prove that if $1 < \lambda < \infty$, the function

$$f_{\lambda}(z) = z + \lambda - e^z$$

has only one zero in the half-plane $\Re z<0$, and this zero is on the real axis.

Problem 5   Let $G$ be a group whose order is twice an odd number. For $g$ in $G$, let $\lambda_g$ denote the permutation of $G$ given by $\lambda_g(x) = gx$ for $x\in G$.

1. Let $g$ be in $G$. Prove that the permutation $\lambda_g$ is even if and only if the order of $g$ is odd.

2. Let $N=\{g\in G \;\vert\; \mathrm{order}(g)\; is\, odd\}$. Prove that $N$ is a normal subgroup of $G$ of index $2$.

Problem 6   Let $S$ be the subspace of $M_{n\times n}$ (the vector space of all real $n\times n$ matrices) generated by all matrices of the form $AB - BA$ with $A$ and $B$ in $M_{n\times n}$. Prove that $\dim(S) = n^2-1$.

Problem 7   Let

$$A = \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & ... ...& 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array} \right)$$

Find the general solution of the matrix differential equation $dX/dt=AXB$ for the unknown 4$\times$4 matrix function $X(t)$.

Problem 8   Let $f$ be an analytic function mapping the open unit disc, $D$, into itself. Assume that there are two points $a$ and $b$ in $D$, with $a \neq b$, such that $f(a) = a$ and $f(b) = b$. Prove that $f$ is the identity function ($f(z)=z$).

Problem 9   For $G$ a group and $H$ a subgroup, let $C(G,H)$ denote the collection of left cosets of $H$ in $G$. Prove that if $H$ and $K$ are two subgroups of $G$ of infinite index, then $G$ is not a finite union of cosets from $C(G,H) \cup C(G,K)$.

Problem 10   Let the real $2n\times 2n$ matrix $X$ have the form

$$\left( \begin{array}{ccc} A & B \\ C & D \end{array} \right)$$

where $A$, $B$, $C$, and $D$ are $n\times n$ matrices that commute with one another. Prove that $X$ is invertible if and only if $AD - BC$ is invertible.

Problem 11   Let $R$ be a ring with at least two elements. Suppose that for each nonzero $a$ in $R$ there is a unique $b$ in $R$ (depending on $a$) with $aba = a$. Show that $R$ is a division ring.

Problem 12   Evaluate

$$\int_C(2z-1)e^{z/(z-1)}\,dz$$

where $C$ is the circle $\vert z\vert = 2$ with counterclockwise orientation.

Problem 13   Let $D_n$ be the dihedral group, the group of rigid motions of a regular n-gon $(n\geq 3)$. (It is a noncommutative group of order $2n$.) Determine its center $C=\{c \in D_n \;\vert\; cx = xc \; for\; all\; x \in D_n\}$.

Problem 14   Suppose $f$ is a continuously differentiable map of $\mathbb{R}^{2}$ into $\mathbb{R}^{2}$. Assume that $f$ has only finitely many singular points, and that for each positive number $M$, the set $\{z \in \mbox{$\mathbb{R}^{2}$} \;\vert\; \vert f(z)\vert \leq M\}$ is bounded. Prove that $f$ maps $\mathbb{R}^{2}$ onto $\mathbb{R}^{2}$.

Problem 15   Let ${B} =(b_{ij})_{i,j=1}^{20}$ be a real $20\times 20$ matrix such that

$$b_{ii} = 0 \quad for \quad 1 \leq i \leq 20,$$

$$b_{ij} \in \{1,-1\} \quad for \quad 1 \leq i,j \leq 20, \hspace{.1in} i\neq j.$$

Prove that ${B}$ is nonsingular.

Problem 16   Let $f$ and $g$ be entire functions such that
${\displaystyle{\lim_{z\to\infty}}} f(g(z))=\infty$. Prove that $f$ and $g$ are polynomials.

Problem 17  

1. Let $R$ be a commutative ring with $1$ containing an element $a$ with $a^3=a+1$. Further, let $\mathfrak{I}$ be an ideal of $R$ of index $< 5$ in $R$. Prove that $\mathfrak{I} = R$. _ring>index
2. Show that there exists a commutative ring with $1$ that has an element $a$ with $a^3=a+1$ and that contains an ideal of index $5$.

Note: The term index is used here exactly as in group theory; namely the index of $\mathfrak{I}$ in $R$ means the order of $R/\mathfrak{I}$.

Problem 18   Let $f$ be a real valued function on $\mathbb{R}^{2}$ with the following properties:

1. For each $y_0$ in $\mathbb{R}^{}$, the function $x \mapsto f(x,y_0)$ is continuous.

2. For each $x_0$ in $\mathbb{R}^{}$, the function $y \mapsto f(x_0,y)$ is continuous.

3. $f(K)$ is compact whenever $K$ is a compact subset of $\mathbb{R}^{2}$.

Prove that $f$ is continuous.