Preliminary Exam - Spring 1983

Problem 1   Let $f: \mbox{$\mathbb{R}^{}$}_{+} \to \mbox{$\mathbb{R}^{}$}_{+}$ be a monotone decreasing function, defined on the positive real numbers with

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

Show that

$$\lim_{x\to\infty}xf(x) = 0.$$

Problem 2   Let $A=(a_{ij})$ be an n$\times$n real matrix satisfying the conditions:

$$a_{ii} > 0 \quad (1 \leq i \leq n),$$

$$a_{ij} \leq 0 \quad (i \neq j,\: 1 \leq i,j \leq n),$$

$$\sum_{i=1}^na_{ij} > 0 \quad (1 \leq j \leq n).$$

Show that $\det(A) > 0$.

Problem 3   A fractional linear transformation maps the annulus $r<\vert z\vert<1$ (where $r>0$) onto the domain bounded by the two circles $\vert z-\frac{1}{4}\vert=\frac{1}{4}$ and $\vert z\vert=1$. Find $r$.

Problem 4   In the triangular network in $\mathbb{R}^{2}$ depicted below, the points $P_0$, $P_1$, $P_2$, and $P_3$ are respectively $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Describe the structure of the group of all Euclidean transformations of $\mathbb{R}^{2}$ which leave this network invariant.

file=../Fig/Pr/Sp83-4,width=2.7in

Problem 5   Find all solutions $y: \mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$ to

$$\frac{dy}{dx}=\left( y(y-2) \right)^{1/2}, \hspace{.2in} y(0) = 0.$$

Problem 6   Suppose that $f$ is a continuous function on $\mathbb{R}^{}$ which is periodic with period $1$, i.e., $f(x+1) = f(x)$. Show:

1. The function $f$ is bounded above and below and achieves its maximum and minimum.

2. The function $f$ is uniformly continuous on $\mathbb{R}^{}$.

3. There exists a real number $x_0$ such that
   
   $$f(x_0+\pi ) = f(x_0).$$
   
   

Problem 7   Let $H$ be the group of integers $\bmod\, p$, under addition, where $p$ is a prime number. Suppose that $n$ is an integer satisfying $1\leq n \leq p$, and let $G$ be the group $H\times H \times\cdots\times H$ ($n$ factors). Show that $G$ has no automorphism of order $p^2$.

Problem 8   Suppose that $n>1$ is an integer. Prove that the sum

$$1 + \frac{1}{2}+ \cdots +\frac{1}{n}$$

is not an integer.

Problem 9   Suppose that $F: \mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{n}$}$ is continuous and satisfies

$$\Vert F(x)-F(y)\Vert \geq \lambda \Vert x-y\Vert$$

for all $x, y \in \mbox{$\mathbb{R}^{n}$}$ and some $\lambda >0$. Prove that $F$ is one-to-one, onto, and has a continuous inverse.

Note: See also Problem .

Problem 10   Evaluate

$$\int_0^{\infty}\left(\frac{\sin x}{x}\right)^2\,dx$$

and justify your calculations.

Problem 11   Let $M$ be an invertible real $n\times n$ matrix. Show that there is a decomposition $M = UT$ in which $U$ is an $n\times n$ real orthogonal matrix and $T$ is upper-triangular with positive diagonal entries. Is this decomposition unique?

Problem 12   Determine all the complex analytic functions $f$ defined on the unit disc $\mathbb{D}$ which satisfy

$$f''\left(\frac{1}{n}\right) + f\left(\frac{1}{n}\right) = 0$$

for $n = 2, 3, 4, \ldots$.

Problem 13   Let $\lambda_1,\lambda_2,\ldots,\lambda_n,\ldots$ be real numbers. Show that the infinite series

$$\sum_{n=1}^{\infty}\frac{e^{i\lambda_nx}}{n^2}$$

converges uniformly over $\mathbb{R}^{}$ to a continuous limit function $f:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{C}\,^{}$}$. Show, further, that the limit

$$\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(x)\,dx$$

exists.

Problem 14   Let $G$ be an abelian group which is generated by, at most, $n$ elements. Show that each subgroup of $G$ is again generated by, at most, $n$ elements.

Problem 15   Let $f$ and $g$ be complex polynomials with the degree of $g$ at least two more than the degree of $f$. Show that there is a positive number $r$ such that

$$\int_C\frac{f(z)}{g(z)}\,dz = 0$$

for each simple closed curve $C$ which does not intersect $\{z \;\vert\; \vert z\vert \leq r\}$.

Problem 16   Let $V$ be a real vector space of dimension $n$, and let $S : V\times V \to \mbox{$\mathbb{R}^{}$}$ be a nondegenerate bilinear form. Suppose that $W$ is a linear subspace of $V$ such that the restriction of $S$ to $W\times W$ is identically $0$. Show that we have $\dim W \leq n/2$.

Problem 17   Evaluate

$$\int_{-\infty}^{\infty}\frac{\sin x}{x(x-\pi)}\,dx\, .$$

Problem 18   Let $\mbox{\bf {F}}$ be the field with seven elements. How many 3$\times$3 matrices with coefficients in $\mbox{\bf {F}}$ have determinant $2$? How many have determinant $3$?

Problem 19   Show that the initial value problem

$$x' = 1 + 5\cos x , \hspace{.2in} x(0) = 7$$

has a solution defined on all of $\mathbb{R}^{}$.

Problem 20   Show that the interval $[0,1]$ cannot be written as a countably infinite disjoint union of closed subintervals of $[0,1]$.