Preliminary Exam - Fall 1990

Problem 1   Find all pairs of integers $a$ and $b$ satisfying $0<a<b$ and $a^b=b^a$.

Problem 2   Evaluate the integral

$$I = \frac{1}{2\pi i}\int_{C}\frac{dz}{(z-2)(1+2z)^2(1-3z)^3}$$

where $C$ is the circle $\vert z\vert=1$ with counterclockwise orientation.

Problem 3   Let $R$ be a ring with $1$, and let $\mathfrak{I}$ be the left ideal of $R$ generated by $\{ab-ba \;\vert\; a, b \in R\}$. Prove that $\mathfrak{I}$ is a two-sided ideal.

Problem 4   Suppose $f$ is a continuous real valued function. Show that

$$\int_0^1f(x)x^2\,dx = \frac{1}{3}f(\xi)$$

for some $\xi\in [0,1]$.

Problem 5   Let ${A}$ be a real symmetric $n\times n$ matrix that is positive definite. Let $y \in \mbox{$\mathbb{R}^{n}$}$, $y\neq0$. Prove that the limit

$$\lim_{m\to\infty}\frac{y^{t} {A}^{m+1}y} {y^{t} {A}^{m} y}$$

exists and is an eigenvalue of ${A}$.

Problem 6   Let the function $f$ be analytic in the entire complex plane, and suppose that $f(z)/z \to 0$ as $\vert z\vert \to \infty$. Prove that $f$ is constant.

Problem 7   Let $G$ be a group and $N$ be a normal subgroup of $G$ with $N \neq G$. Suppose that there does not exist a subgroup $H$ of $G$ satisfying $N \subset H \subset G$ and $N \neq H \neq G$. Prove that the index of $N$ in $G$ is finite and equal to a prime number.

Problem 8   Let $f$ be a continuous real valued function satisfying $f(x) \geq 0$, for all x, and

$$\int_0^{\infty}f(x)\,dx < \infty.$$

Prove that

$$\frac{1}{n}\int_0^nxf(x)\,dx \to 0$$

as $n \to \infty$.

Problem 9   Let $\mathbb{R}^{3}$ be 3-space with the usual inner product, and
$(a,b,c) \in \mbox{$\mathbb{R}^{3}$}$ a vector of length $1$. Let $W$ be the plane defined by $ax+by+cz=0$. Find, in the standard basis, the matrix representing the orthogonal projection of $\mathbb{R}^{3}$ onto $W$.

Problem 10   Determine the Jordan Canonical Form of the matrix

$$\left( \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{array} \right).$$

Problem 11   Suppose that $f$ maps the compact interval $I$ into itself and that

$$\vert f(x)-f(y)\vert < \vert x-y\vert$$

for all $x, y \in I$, $x\neq y$. Can one conclude that there is some constant $M<1$ such that, for all $x, y \in I$,

$$\vert f(x)-f(y)\vert \leq M\vert x-y\vert\,?$$

Problem 12   Let $A$ be an additively written abelian group, and $f, g:A \to A$ two group homomorphisms. Define the group homomorphisms $i,j:A \to A$ by

$$i(a)= a-g \left( f(a) \right), \hspace{.2in} j(a)=a-f \left( g(a) \right) \hspace{.2in} (a \in A).$$

Prove that the kernel of i is isomorphic to the kernel of j.

Problem 13   Suppose that $f$ is analytic on the open upper half-plane and satisfies $\vert f(z)\vert\leq 1$ for all $z$, $f(i)=0$. How large can $\vert f(2i)\vert$ be under these conditions?

Problem 14   Prove that $\sqrt{2}+\sqrt[3]{3}$ is irrational.

Problem 15   Let $n$ be a positive integer and let $P_{2n+1}$ be the vector space of real polynomials whose degrees are, at most, $2n+1$. Prove that there exist unique real numbers $c_1,\ldots,c_n$ such that

$$\int_{-1}^1p(x)\,dx = 2p(0)+ \sum_{k=1}^nc_k(p(k)+p(-k)-2p(0))$$

for all $p \in P_{2n+1}$.

Problem 16   Evaluate the limit

$$\lim_{n\to\infty}\cos\frac{\pi}{2^2}\cos\frac{\pi}{2^3} \cdots\cos\frac{\pi}{2^n} \cdot$$

Problem 17   Does the set $G=\{a \in \mbox{$\mathbb{R}^{}$} \;\vert\; a>0, a\neq 1\}$ form a group with the operation $a*b=a^{\log b}$?

Problem 18   Let the function $f$ be analytic in the entire complex plane and satisfy

$$\int_0^{2\pi}\vert f(re^{i\theta})\vert\,d\theta \leq r^{17/3}$$

for all $r>0$. Prove that $f$ is the zero function.