Preliminary Exam - Spring 1988

Problem 1   Suppose that $f(x)$, $-\infty < x < \infty$, is a continuous real valued function, that $f'(x)$ exists for $x \neq 0$, and that $\lim_{x \to 0}f'(x)$ exists. Prove that $f'(0)$ exists.

Problem 2   Determine the last digit of

$$23^{23^{23^{23}}}$$

in the decimal system.

Problem 3   If a finite homogeneous system of linear equations with rational coefficients has a nontrivial complex solution, need it have a nontrivial rational solution? Give a proof or a counterexample.

Problem 4   True or false: A function $f(z)$ analytic on $\vert z-a\vert<r$ and continuous on $\vert z-a\vert\leq r$ extends, for some $\delta > 0$, to a function analytic on $\vert z-a\vert<r+\delta$? Give a proof or a counterexample.

Problem 5   Let $D$ be a group of order $2n$, where $n$ is odd, with a subgroup $H$ of order $n$ satisfying $xhx^{-1} =h^{-1}$ for all h in $H$ and all x in $D\setminus H$. Prove that $H$ is commutative and that every element of $D\setminus H$ is of order $2$.

Problem 6   Prove or disprove: There is a real $n\times n$ matrix ${A}$ such that

$$A^2 + 2A + 5I = 0$$

if and only if $n$ is even.

Problem 7   Let $R$ be a commutative ring with unit element and $a\in R$. Let $n$ and $m$ be positive integers, and write $d=\gcd\{n,m\}$. Prove that the ideal of $R$ generated by $a^n-1$ and $a^m-1$ is the same as the ideal generated by $a^d-1$.

Problem 8   For $a>1$ and $n= 0, 1, 2, \ldots$, evaluate the integrals

$$C_n(a) = \int_{-\pi}^{\pi}\frac{\cos n\theta}{a - \cos\theta}... ...t_{-\pi}^{\pi}\frac{\sin n\theta}{a - \cos\theta}\,d\theta \,.$$

Problem 9   Prove that the integrals

$$\int_{0}^{\infty}\cos x^2 \,dx \hspace{.2in} and \hspace{.2in} \int_{0}^{\infty}\sin x^2 \,dx$$

converge.

Problem 10  

1. Let $G$ be an open connected subset of the complex plane, f an analytic function in $G$, not identically $0$, and n a positive integer. Assume that f has an analytic $n^{th}$ root in $G$; that is, there is an analytic function g in $G$ such that $g^n = f$. Prove that f has exactly n analytic $n^{th}$ roots in $G$.

2. Give an example of a continuous real valued function on $[0,1]$ that has more than two continuous square roots on $[0,1]$.

Problem 11   Let $S_9$ denote the group of permutations of $\{1,2,\ldots,9\}$ and let $A_9$ be the subgroup consisting of all even permutations. Denote by $1\in S_9$ the identity permutation. Determine the minimum of all positive integers $m$ such that every $\sigma\in S_9$ satisfies $\sigma^m=1$. Determine also the minimum of all positive integers $m$ such that every $\sigma\in A_9$ satisfies $\sigma^m=1$.

Problem 12   For each real value of the parameter $t$, determine the number of real roots, counting multiplicities, of the cubic polynomial
$p_t(x) = (1+t^2)x^3 - 3t^3x + t^4$.

Problem 13   Find all groups $G$ such that every automorphism of $G$ is trivial.

Problem 14   Let the function f be analytic in the open unit disc of the complex plane and real valued on the radii $[0,1)$ and $[0,e^{i\pi \sqrt{2}}\,)$. Prove that f is constant.

Problem 15   Compute ${A}^{10}$ for the matrix

$$A = \left( \begin{array}{rrr} 3 & 1 & 1 \\ 2 & 4 & 2 \\ -1 & -1 & 1\end{array} \right).$$

Problem 16   Let $X$ be a set and $V$ a real vector space of real valued functions on $X$ of dimension $n$, $0 < n < \infty$. Prove that there are $n$ points $x_1, x_2, \ldots, x_n$ in $X$ such that the map $f \to (f(x_1), \ldots, f(x_n))$ of $V$ to $\mathbb{R}^{n}$ is an isomorphism (i.e., one-to-one and onto). (The operations of addition and scalar multiplication in $V$ are assumed to be the natural ones.)

Problem 17  

1. Let $f$ be an analytic function that maps the open unit disc, $\mathbb{D}$, into itself and vanishes at the origin. Prove that
   $\vert f(z)+f(-z)\vert\leq 2\vert z\vert^2$ in $\mathbb{D}$.

2. Prove that the inequality in Part 1 is strict, except at the origin, unless $f$ has the form $f(z) = \lambda z^2$ with $\lambda$ a constant of absolute value one.

Problem 18   For which positive integers $n$ is there a 2$\times$2 matrix

$$A= \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$$

with integer entries and order $n$; that is, $A^n=I$ but $A^k\neq I$ for $0 < k < n$? _matrix,>order n

Note: See also Problem .

Problem 19   Show that you can represent the set of nonnegative integers, $\mbox{$\mathbb{Z}^{}$}_+$, as the union of two disjoint subsets $N_1$ and $N_2$ $(N_1 \cap N_2 = \emptyset,\; N_1 \cup N_2 = \mbox{$\mathbb{Z}^{}$}_+)$ such that neither $N_1$ nor $N_2$ contains an infinite arithmetic progression.

Problem 20   Does there exist a continuous real valued function $f(x)$, $0\leq x \leq 1$, such that

$$\int_0^1xf(x)\,dx = 1 \quad and \quad \int_0^1x^nf(x)\,dx = 0$$

for $n = 0, 2, 3, 4, \ldots$? Give an example or a proof that no such $f$ exists.