Preliminary Exam - Fall 1983

Problem 1   Evaluate

$$\int_0^{\infty}(\mathrm{sech}\,x)^2\cos \lambda x\,dx$$

where $\lambda$ is a real constant and

$$\mathrm{sech}\,x = \frac{2}{e^x+e^{-x}} \cdot$$

Problem 2   Let $M_{n\times n}(\mbox{\bf {F}})$ denote the ring of n$\times$n matrices over a field $\mbox{\bf {F}}$. For $n\geq 1$ does there exist a ring homomorphism from $M_{(n+1)\times (n+1)}(\mbox{\bf {F}})$ onto $M_{n\times n}(\mbox{\bf {F}})$?

Problem 3   Let $f:\mbox{$\mathbb{R}^{n}$}\setminus \{0\} \to \mbox{$\mathbb{R}^{}$}$ be a function which is continuously differentiable and whose partial derivatives are uniformly bounded:

$$\left\vert \frac{\partial f}{\partial x_i}(x_1,\ldots,x_n) \right\vert \leq M$$

for all $(x_1,\ldots,x_n)\neq (0,\ldots,0)$. Show that if $n\geq 2$, then $f$ can be extended to a continuous function defined on all of $\mathbb{R}^{n}$. Show that this is false if $n=1$ by giving a counterexample.

Problem 4   Prove or disprove (by giving a counterexample), the following assertion: Every infinite sequence $x_1, x_2, \ldots$ of real numbers has either a nondecreasing subsequence or a nonincreasing subsequence.

Problem 5   Let $A$ be the $n \times n$ matrix which has zeros on the main diagonal and ones everywhere else. Find the eigenvalues and eigenspaces of $A$ and compute $\det(A)$.

Problem 6   Consider the polynomial

$$p(z) = z^5+z^3+5z^2+2.$$

How many zeros (counting multiplicities) does $p$ have in the annular region
$1 < \vert z\vert < 2$?

Problem 7   Let $G$ be a finite group and $G_1 = G \times G$. Suppose that $G_1$ has exactly four normal subgroups. Show that $G$ is simple and nonabelian.

Problem 8   Let $A$ be a linear transformation on $\mathbb{R}^{3}$ whose matrix (relative to the usual basis for $\mathbb{R}^{3}$) is both symmetric and orthogonal. Prove that $A$ is either plus or minus the identity, or a rotation by $180^{\circ}$ about some axis in $\mathbb{R}^{3}$, or a reflection about some two-dimensional subspace of $\mathbb{R}^{3}$.

Problem 9   For which real values of $p$ does the differential equation

$$y''+2py'+y = 3$$

admit solutions $y=f(x)$ with infinitely many critical points?

Problem 10   Let $f:[0,\infty) \to \mbox{$\mathbb{R}^{}$}$ be a uniformly continuous function with the property that

$$\lim_{b\to\infty}\int_0^bf(x)\,dx$$

exists (as a finite limit). Show that

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

Problem 11   Prove or supply a counterexample: If $f$ and $g$ are $C^1$ real valued functions on $(0,1)$, if

$$\lim_{x \to 0}f(x) = \lim_{x \to 0}g(x) = 0,$$

if $g$ and $g'$ never vanish, and if

$$\lim_{x \to 0}\frac{f(x)}{g(x)} = c,$$

then

$$\lim_{x \to 0}\frac{f'(x)}{g'(x)} = c.$$

Problem 12   Let $r_1,r_2,\ldots,r_n$ be distinct complex numbers. Show that a rational function of the form

$$f(z) = \frac{b_0+b_1z+\cdots+b_{n-2}z^{n-2}+b_{n-1}z^{n-1}} {(z-r_1)(z-r_2)\cdots(z-r_n)}$$

can be written as a sum

$$f(z) = \frac{A_1}{z-r_1}+\frac{A_2}{z-r_2}+\cdots+\frac{A_n}{z-r_n}$$

for suitable constants $A_1,\ldots,A_n$.

Problem 13  

1. Let $u(t)$ be a real valued differentiable function of a real variable $t$ which satisfies an inequality of the form
   
   $$u'(t) \leq au(t), \quad t\geq 0, \quad u(0)\leq b,$$
   
   
   where $a$ and $b$ are positive constants. Starting from first principles, derive an upper bound for $u(t)$ for $t>0$.

2. Let $x(t)=(x_1(t),x_2(t),\ldots,x_n(t))$ be a differentiable function from $\mathbb{R}^{}$ to $\mathbb{R}^{n}$ which satisfies a differential equation of the form
   
   $$x'(t) = f(x(t)),$$
   
   
   where $f:\mbox{$\mathbb{R}^{n}$}\to\mbox{$\mathbb{R}^{n}$}$ is a continuous function. Assuming that $f$ satisfies the condition
   
   $$% latex2html id marker 949 \langle f(y), y \rangle \leq \Vert y\Vert^2, \quad y \in \mbox{$\mathbb{R}^{n}$}$$
   
   
   (where $\langle \cdot, \cdot \rangle$ and $\Vert\cdot\Vert$ denote the Euclidean inner product and norm), derive an inequality showing that the norm $\Vert x(t)\Vert$ grows, at most, exponentially.

Problem 14   Let $V$ be a finite-dimensional complex vector space and let $A$ and $B$ be linear operators on $V$ such that $AB = BA$. Prove that if $A$ and $B$ can each be diagonalized, then there is a basis for $V$ which simultaneously diagonalizes $A$ and $B$.

Problem 15  

1. Let $f$ be a complex function which is analytic on an open set containing the disc $\vert z\vert\leq 1$, and which is real valued on the unit circle. Prove that $f$ is constant.

2. Find a nonconstant function which is analytic at every point of the complex plane except for a single point on the unit circle $\vert z\vert=1$, and which is real valued at every other point of the unit circle.

Problem 16   Let $F(t) = \left(f_{ij}(t)\right)$ be an n$\times$n matrix of continuously differentiable functions $f_{ij}:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$, and let

$$u(t) = \mathrm{tr}\left(F(t)^3\right).$$

Show that u is differentiable and

$$u'(t) = 3\,\mathrm{tr}\left(F(t)^2F'(t)\right).$$

Problem 17   Prove that every finite integral domain is a field.

Problem 18   Let $f,g:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ be smooth functions with $f(0)=0$ and $f'(0)\neq 0$. Consider the equation $f(x)=tg(x)$, $t\in\mbox{$\mathbb{R}^{}$}$.

1. Show that in a suitably small interval $\vert t\vert<\delta$, there is a unique continuous function $x(t)$ which solves the equation and satisfies $x(0)=0$.

2. Derive the first order Taylor expansion of $x(t)$ about $t=0$.

Problem 19   Prove that if $p$ is a prime number, then the polynomial

$$f(x) = x^{p-1}+x^{p-2}+\cdots+1$$

is irreducible in $\mathbb{Q}[x]$.

Problem 20   Let m and n be positive integers, with $m<n$. Let $M_{m\times n}$ be the space of linear transformations of $\mathbb{R}^{m}$ into $\mathbb{R}^{n}$ (considered as $n\times m$ matrices) and let $L$ be the set of transformations in $M_{m\times n}$ which have rank m.

1. Show that $L$ is an open subset of $M_{m\times n}$.

2. Show that there is a continuous function $T : L \to M_{m\times n}$ such that
   $T(A)A = I_m$ for all $A$, where $I_m$ is the identity on $\mathbb{R}^{m}$.