Preliminary Exam - Spring 1991

Problem 1   List, to within isomorphism, all the finite groups whose orders do not exceed $5$. Explain why your list is complete and why no two groups on the list are isomorphic.

Problem 2   Let $f$ be a continuous complex valued function on $[0,1]$, and define the function $g$ by

$$% latex2html id marker 667 g(z) = \int_0^1f(t)e^{tz}\,dt \hspace{.5in} (z\in \mbox{$\mathbb{C}\,^{}$}).$$

Prove that $g$ is analytic in the entire complex plane.

Problem 3   For $n$ a positive integer, let $d(n)$ denote the number of positive integers that divide $n$. Prove that $d(n)$ is odd if and only if n is a perfect square.

Problem 4   Let $p$ be a prime number and $R$ a ring with identity containing $p^2$ elements. Prove that $R$ is commutative.

Problem 5   Let $A=(a_{ij})_{i,j=1}^r$ be a square matrix with integer entries.

1. Prove that if an integer $n$ is an eigenvalue of $A$, then $n$ is a divisor of $\det A$, the determinant of $A$.
2. Suppose that $n$ is an integer and that each row of $A$ has sum $n$:
   
   $$\sum_{j=1}^ra_{ij} = n, \qquad 1\leq i \leq r.$$
   
   
   Prove that $n$ is a divisor of $\det A$.

Problem 6   Let the vector field $F$ in $\mathbb{R}^{3}$ have the form

$$F(r) = g(\Vert r\Vert)r \hspace{.5in} (r \neq (0,0,0)),$$

where $g$ is a real valued smooth function on $(0,\infty)$ and $\Vert\cdot\Vert$ denotes the Euclidean norm. ($F$ is undefined at $(0,0,0)$.) Prove that

$$\int_C F\cdot ds = 0$$

for any smooth closed path $C$ in $\mathbb{R}^{3}$ that does not pass through the origin.

Problem 7   Let the function $f$ be analytic in the unit disc, with
$\vert f(z)\vert\leq 1$ and $f(0)=0$. Assume that there is a number $r$ in $(0,1)$ such that $f(r)=f(-r)=0$. Prove that

$$\vert f(z)\vert \leq \vert z\vert\left\vert\frac{z^2-r^2}{1-r^2z^2}\right\vert \cdot$$

Problem 8   Let ${T}$ be a real, symmetric, $n \times n$, tridiagonal matrix:

$$T = \left( \begin{array}{ccccccc} a_1 & b_1 & 0 & 0 & \cdots ... ... \\ 0 & 0 & 0 & 0 & \cdots & b_{n-1} & a_n \end{array} \right)$$

(All entries not on the main diagonal or the diagonals just above and below the main one are zero.) Assume $b_j\neq 0$ for all $j$. _matrix,>tridiagonal

Prove:

1. $\mathrm{rank}\, T\geq n-1$.
2. $T$ has $n$ distinct eigenvalues.

Problem 9   Let $f$ be a continuous function from the ball
$B_n = \{x \in \mbox{$\mathbb{R}^{n}$} \;\vert\; \Vert x\Vert < 1\}$ into itself. (Here, $\Vert\cdot\Vert$ denotes the Euclidean norm.) Assume $\Vert f(x)\Vert<\Vert x\Vert$ for all nonzero $x \in B_n$. Let $x_0$ be a nonzero point of $B_n$, and define the sequence $(x_k)$ by setting $x_k = f(x_{k-1})$. Prove that $\lim x_k = 0$.

Problem 10   Prove that $\mathbb{Q}\,^{}$, the additive group of rational numbers, cannot be written as the direct sum of two nontrivial subgroups.

Note: See also Problems and .

Problem 11   For which real numbers x does the infinite series

$$\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n^x}$$

converge?

Problem 12   Let $A$ be the set of positive integers that do not contain the digit $9$ in their decimal expansions. Prove that

$$\sum_{a\in A}\frac{1}{a} < \infty;$$

that is, $A$ defines a convergent subseries of the harmonic series.

Problem 13   Prove that

$$\lim_{R\to\infty}\int_{-R}^R\frac{\sin x}{x-3i}\,dx$$

exists and find its value.

Problem 14   Let $x(t)$ be a nontrivial solution to the system

$$\frac{dx}{dt} = Ax,$$

where

$$A = \left( \begin{array}{ccc} 1 & 6 & 1 \\ -4 & 4 & 11 \\ -3 & -9 & 8 \end{array} \right).$$

Prove that $\Vert x(t)\Vert$ is an increasing function of t. (Here, $\Vert\cdot\Vert$ denotes the Euclidean norm.)

Problem 15   Let $G$ be a finite nontrivial group with the property that for any two elements $a$ and $b$ in $G$ different from the identity, there is an element $c$ in $G$ such that $b= c^{-1}ac$. Prove that $G$ has order $2$.

Problem 16   Let $A$ be a linear transformation on an n-dimensional vector space over $\mathbb{C}\,^{}$ with characteristic polynomial $(x-1)^n$. Prove that $A$ is similar to $A^{-1}$.

Problem 17   Let the function $f$ be analytic in the punctured disc
$0<\vert z\vert<r_0$, with Laurent series

$$f(z) = \sum_{-\infty}^{\infty}c_nz^n.$$

Assume there is a positive number $M$ such that

$$r^4\int_0^{2\pi}\vert f(re^{i\theta})\vert^2\,d\theta < M, \hspace{.5in} 0 < r < r_0.$$

Prove that $c_n = 0$ for $n< -2$.

Problem 18   Let the real valued function $f$ be defined in an open interval about the point $a$ on the real line and be differentiable at $a$. Prove that if $(x_n)$ is an increasing sequence and $(y_n)$ is a decreasing sequence in the domain of $f$, and both sequences converge to $a$, then

$$\lim_{n\to\infty}\frac{f(y_n)-f(x_n)}{y_n-x_n} = f'(a).$$