Preliminary Exam - Fall 1988

Problem 1   Let $R$ be a finite ring. Prove that there are positive integers m and n with $m>n$ such that $x^m=x^n$ for every $x$ in $R$.

Problem 2   Determine the group ${\rm Aut}(\mbox{$\mathbb{C}\,^{}$})$ of all one-to-one analytic maps of $\mathbb{C}\,^{}$ onto $\mathbb{C}\,^{}$.

Problem 3   Let the real valued functions $f_1,\ldots,f_{n+1}$ on $\mathbb{R}^{}$ satisfy the system of differential equations

$$\begin{eqnarray*} f_{k+1}' + f_k'& = & (k+1)f_{k+1} - kf_k, \hspace{.1in} k = 1,\ldots,n \\ f_{n+1}' & = & -(n+1)f_{n+1}. \end{eqnarray*}$$

Prove that for each $k$,

$$\lim_{t\to\infty}f_k(t) = 0.$$

Problem 4   Find the Jordan Canonical Form of the matrix

$$\left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 ... ... & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1\end{array} \right).$$

Problem 5   Let $f$ be a continuous, strictly increasing function from $[0,\infty)$ onto $[0,\infty)$ and let $g=f^{-1}$. Prove that

$$\int_0^af(x)\,dx + \int_0^bg(y)\,dy \geq ab$$

for all positive numbers $a$ and $b$, and determine the condition for equality.

Problem 6   Let $f$ be a function from $[0,1]$ into itself whose graph

$$G_f = \left\{\left(x,f(x)\right) \;\vert\; x \in [0,1] \right\}$$

is a closed subset of the unit square. Prove that $f$ is continuous.

Note: See also Problem .

Problem 7   Find all abelian groups of order $8$, up to isomorphism. Then identify which type occurs in each of

1. $\left(\mbox{$\mathbb{Z}^{}$}_{15}\right)^*$,

2. $\left(\mbox{$\mathbb{Z}^{}$}_{17}\right)^*/(\pm 1)$,

3. the roots of $z^8-1$ in $\mathbb{C}\,^{}$,

4. $\mbox{\bf {F}}_8^+$,

5. $\left(\mbox{$\mathbb{Z}^{}$}_{16}\right)^*$.

$\mbox{\bf {F}}_8$ is the field of eight elements, and $\mbox{\bf {F}}_8^+$ is its underlying additive group; $R^*$ is the group of invertible elements in the ring $R$, under multiplication.

Problem 8   Do the functions $f(z)=e^z+z$ and $g(z)=ze^z+1$ have the same number of zeros in the strip $-\frac{\pi}{2} < \Im z < \frac{\pi}{2}$ ?

Problem 9   Let $A$ and $B$ be real symmetric n$\times$n matrices. Assume that the eigenvalues of $A$ all lie in the interval $[a_1,a_2]$ and those of $B$ all lie in the interval $[b_1,b_2]$. Prove that the eigenvalues of $A + B$ all lie in the interval $[a_1+b_1, a_2+b_2]$.

Problem 10   Find (up to isomorphism) all groups of order $2p$, where $p$ is a prime ($p\geq 2$).

Problem 11   Let $f$ be an analytic function on a disc $D$ whose center is the point $z_0$. Assume that $\vert f'(z)-f'(z_0)\vert<\vert f'(z_0)\vert$ on D. Prove that $f$ is one-to-one on $D$.

Problem 12   Let n be a positive integer and let $f$ be a polynomial in $\mbox{$\mathbb{R}^{}$}[x]$ of degree $n$. Prove that there are real numbers $a_0,a_1,\ldots,a_n$, not all equal to zero, such that the polynomial

$$\sum_{i=0}^{n}a_ix^{2^i}$$

is divisible by f.

Problem 13   Let $A$ be a complex $n\times n$ matrix, and let $C(A)$ be the commutant of $A$; that is, the set of complex $n\times n$ matrices $B$ such that
$AB = BA$. (It is obviously a subspace of $M_{n\times n}$, the vector space of all complex $n\times n$ matrices.) Prove that $\dim C(A) \geq n$. _matrix,>commutant of a

Problem 14   Let the group $G$ be generated by two elements, $a$ and $b$, both of order $2$. Prove that $G$ has a subgroup of index $2$.

Problem 15   Prove that a real valued $C^3$ function $f$ on $\mathbb{R}^{2}$ whose Laplacian,

$$\frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2}\,,$$

is everywhere positive cannot have a local maximum.

Problem 16   Let $n$ be a positive integer. Prove that the polynomial

$$f(x)=\sum_{i=0}^n\frac{x^i}{i!} = 1+x+\frac{x^2}{2}+\cdots+\frac{x^n}{n!}$$

in $\mbox{$\mathbb{R}^{}$}[x]$ has $n$ distinct complex zeros, $z_1,z_2,\ldots,z_n$, and that they satisfy

$$\sum_{i=1}^nz_i^{-j} = 0 \quad for \quad 2 \leq j \leq n.$$

Problem 17   Prove that

$$\int_0^{\infty}\frac{x}{e^x-e^{-x}}\,dx = \frac{\pi^2}{8} \cdot$$

Problem 18   Let $g$ be a continuous real valued function on $[0,1]$. Prove that there exists a continuous real valued function $f$ on $[0,1]$ satisfying the equation

$$f(x) - \int_0^xf(x-t)e^{-t^2}\,dt = g(x).$$