Preliminary Exam - Fall 1996

Problem 1   Let $M$ be the set of real valued continuous functions $f$ on $[0,1]$ such that $f'$ is continuous on $[0,1]$, with the norm

$$\Vert f\Vert=\sup_{0\leq x\leq 1} \vert f(x)\vert+\sup_{0\leq x\leq 1} \vert f'(x)\vert \ .$$

Which subsets of $M$ are compact?

Problem 2   A real-valued function $f$ on a closed bounded interval $[a,b]$ is said to be upper semicontinuous provided that for every $\varepsilon > 0$ and $p\in [a,b]$, there is a $\delta=\delta (\varepsilon,p)> 0$ such that if $x\in [a,b]$ and $\vert x-p\vert < \delta$ then $f(x) < f(p)+\varepsilon$. Prove that an upper semicontinuous function is bounded above on $[a,b]$.

Problem 3   Evaluate the integral

$$I=\int^\infty_0 \frac{\sqrt{x}}{1+x^2} \ dx\,.$$

Problem 4   Does there exist a function $f$, analytic in the punctured plane $\mbox{$\mathbb{C}\,^{}$} \setminus \{0\}$, such that

$$\vert f(z)\vert \geq \frac{1}{\sqrt{\vert z\vert}}$$

for all nonzero $z$?

Problem 5   Prove that any linear transformation $T:\mbox{$\mathbb{R}^{3}$} \to \mbox{$\mathbb{R}^{3}$}$ has

1. a one-dimensional invariant subspace

2. a two-dimensional invariant subspace.

Problem 6   Let $A$ and $B$ be real 2$\times$2 matrices such that

$$A^2=B^2=I \ , \qquad AB+BA=0 \ .$$

Show that there exists a real 2$\times$2 matrix $T$ such that

$$TAT^{-1}=\left( \begin{array}{rr} 1 & 0\\ 0 & -1 \end{array}... ...}=\left( \begin{array}{rr} 0 & 1\\ 1 & 0\end{array}\right) \ .$$

Problem 7   Suppose $p$ is a prime. Show that every element of $GL_2(\mbox{\bf {F}}_p)$ has order dividing either $p^2-1$ or $p(p-1)$.

Problem 8   Show the denominator of $\left(\!\!\begin{array}{c} 1/2 \\ n\end{array}\!\! \right)$ is a power of $2$ for all integers $n$.

Problem 9   For positive integers $a$, $b$ and $c$ show that

$$\gcd\left\{a,\mathrm{lcm}\{b,c\}\right\}=\mathrm{lcm}\left\{\gcd \{a,b\}, \gcd \{a,c\} \right\}.$$

Problem 10   If $f$ is a $C^2$ function on an open interval, prove that

$$\lim_{h\to 0} \ \frac{f(x+h)-2f(x)+f(x\!-\! h)}{h^2} = f''(x) \ .$$

Problem 11   Let $f$ be continuous and nonnegative on $[0,1]$ and suppose that

$$f(t)^2\leq 1+2\int^t_0 f(s)ds \ .$$

Prove that $f(t)\leq 1+t$ for $0\leq t\leq 1$.

Problem 12   Find the number of roots, counted with their multiplicities, of

$$z^7-4z^3-11=0$$

which lie between the circles $\vert z\vert=1$ and $\vert z\vert=2$.

Problem 13   Define $F:\mbox{$\mathbb{C}\,^{3}$} \to \mbox{$\mathbb{C}\,^{3}$}$ by

$$F(u,v,w)=(u+v+w, \ uv+vw+uw, \ uvw) \ .$$

Show that $F$ is onto but not one-to-one.

Problem 14   Let $f$ be holomorphic on and inside the unit circle $C$. Let $L$ be the length of the image of $C$ under $f$. Show that

$$L\geq 2\pi \vert f'(0)\vert \ .$$

Problem 15   Is there a real 2$\times$2 matrix $A$ such that

$$A^{20}=\left( \begin{array}{rc} -1 & 0\\ 0 & -1-\varepsilon \end{array}\right) \ ?$$

Exhibit such an $A$ or prove there is none.

Problem 16   Let

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

Show that every real matrix $B$ such that $AB=BA$ has the form

$$B=aI+bA+cA^2$$

for some real numbers $a$, $b$, and $c$.

Problem 17   Let $\mbox{$\mathbb{Z}^{}$}[x]$ be the ring of polynomials in the indeterminate $x$ with coefficients in the ring $\mathbb{Z}^{}$ of integers. Let $\mathfrak{I}\subset \mbox{$\mathbb{Z}^{}$}[x]$ be the ideal generated by $13$ and $x-4$. Find an integer $m$ such that $0\leq m\leq 12$ and

$$(x^{26}+x+1)^{73} -m\in \mathfrak{I} \ .$$

Problem 18   Prove that any finite group is isomorphic to

1. a subgroup of the group of permutations of $n$ objects

2. a subgroup of the group of permutations of $n$ objects which consists only of even permutations.