Preliminary Exam - Spring 1990

Problem 1   Let $y : \mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$ be a $C^{\infty}$ function that satisfies the differential equation

$$y'' + y' - y = 0$$

for $x \in [0,L]$, where $L$ is a positive real number. Suppose that $y(0)=y(L)=0$. Prove that $y \equiv 0$ on $[0,L]$.

Problem 2   Let $A$ be a complex $n \times n$ matrix that has finite order; that is, $A^k=I$ for some positive integer $k$. Prove that $A$ is diagonalizable. _matrix,>finite order

Problem 3   Let $c_0,c_1,\ldots,c_{n-1}$ be complex numbers. Prove that all the zeros of the polynomial

$$z^n + c_{n-1}z^{n-1} + \cdots + c_1z + c_0$$

lie in the open disc with center $0$ and radius

$$\sqrt{1 + \vert c_{n-1}\vert^2 + \cdots + \vert c_1\vert^2 + \vert c_0\vert^2}\, .$$

Problem 4   Let $R$ be a commutative ring with $1$, and $R^*$ be its group of units. Suppose that the additive group of $R$ is generated by $\{u^2 \;\vert\; u \in R^*\}$. Prove that $R$ has, at most, one ideal $\mathfrak{I}$ for which $R/\mathfrak{I}$ has cardinality $3$.

Problem 5   Suppose $x_1,x_2,x_3,\ldots$ is a sequence of nonnegative real numbers satisfying

$$x_{n+1} \leq x_n + \frac{1}{n^2}$$

for all $n\geq 1$. Prove that $\lim_{n\to\infty}x_n$ exists.

Problem 6   Give an example of a continuous function $v : \mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{3}$}$ with the property that $v(t_1)$, $v(t_2)$, and $v(t_3)$ form a basis for $\mathbb{R}^{3}$ whenever $t_1$, $t_2$, and $t_3$ are distinct points of $\mathbb{R}^{}$.

Problem 7   Let $a$ be a positive real number. Evaluate the improper integral

$$\int_0^{\infty}\frac{\sin x}{x(x^2+a^2)}\,dx\, .$$

Problem 8   Let $\mathbb{C}\,^{*}$ be the multiplicative group of nonzero complex numbers. Suppose that $H$ is a subgroup of finite index of $\mathbb{C}\,^{*}$. Prove that $H=\mbox{$\mathbb{C}\,^{*}$}$.

Problem 9   Let the real valued function $f$ on $[0,1]$ have the following two properties:

• If $[a,b] \subset [0,1]$, then $f \left( [a,b] \right)$ contains the interval with endpoints $f(a)$ and $f(b)$ (i.e., f has the Intermediate Value Property).

• For each $c \in \mbox{$\mathbb{R}^{}$}$, the set $f^{-1}(c)$ is closed.

Prove that $f$ is continuous.

Problem 10   Show that there are at least two nonisomorphic nonabelian groups of order $24$, of order $30$ and order $40$.

Problem 11   Let the function $f$ be analytic and bounded in the complex half-plane $\Re z>0$. Prove that for any positive real number $c$, the function $f$ is uniformly continuous in the half-plane $\Re z>c$.

Problem 12   Let $n$ be a positive integer, and let $A= (a_{ij})_{i,j=1}^n$ be the n$\times$n matrix with $a_{ii} = 2,\, a_{i\,i\pm 1} = -1$, and $a_{ij} = 0$ otherwise; that is,

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

Prove that every eigenvalue of $A$ is a positive real number.

Problem 13   Let $f$ be an infinitely differentiable function from $\mathbb{R}^{}$ to $\mathbb{R}^{}$. Suppose that, for some positive integer $n$,

$$f(1) = f(0) = f'(0) = f''(0) = \cdots = f^{(n)}(0) = 0.$$

Prove that $f^{(n+1)}(x)=0$ for some $x$ in $(0,1)$.

Problem 14   Let $A$ and $B$ be subspaces of a finite-dimensional vector space $V$ such that $A+B=V$. Write $n =\dim V$, $a=\dim A$, and $b=\dim B$. Let $S$ be the set of those endomorphisms $f$ of $V$ for which $f(A)\subset~A$ and $f(B)\subset B$. Prove that $S$ is a subspace of the set of all endomorphisms of $V$, and express the dimension of $S$ in terms of $n$, $a$, and $b$.

Problem 15   Find a one-to-one conformal map of the semidisc

$$% latex2html id marker 1030 \{z \in \mbox{$\mathbb{C}\,^{}$}\,\vert\,\Im z>0,\, \vert z-1/2\vert < 1/2\}$$

onto the upper half-plane.

Problem 16   Determine the greatest common divisor of the elements of the set $\{n^{13}-n \;\vert\; n \in \mbox{$\mathbb{Z}^{}$}\}$.

Problem 17   Let $f$ be a differentiable function on $[0,1]$ and let

$$\sup_{0<x<1}\,\vert f'(x)\vert = M < \infty.$$

Let $n$ be a positive integer. Prove that

$$\left\vert \sum_{j=0}^{n-1}\frac{f(j/n)}{n} - \int_0^1f(x)\,dx \right\vert \leq \frac{M}{2n} \cdot$$

Problem 18   Let $z_1,z_2,\ldots,z_n$ be complex numbers. Prove that there exists a subset $J \subset \{1,2,\ldots,n$} such that

$$\left\vert \sum_{j\in J}z_j \right\vert \geq \frac{1}{4\sqrt{2}}\sum_{j=1}^n\vert z_j\vert.$$