Preliminary Exam - Fall 1984

Problem 1   Let $G$ be a group and $H$ a subgroup of index $n < \infty$. Prove or disprove the following statements:

1. If $a \in G$, then $a^n \in H$.

2. If $a \in G$, then for some $k$, $0 < k \leq n$, we have $a^k \in H$.

Problem 2   Let $A$ and $B$ be n$\times$n real matrices, and k a positive integer. Find

1. 
   
   $$\lim_{t \to 0}\frac{1}{t} \left( (A + tB)^k - A^k \right) .$$
   
   

2. 
   
   $$\left. \frac{d}{dt}\mathrm{tr}\,(A + tB)^k \right\vert _{t=0} .$$
   
   

Problem 3   Prove or supply a counterexample: If $f$ is a nondecreasing real valued function on $[0,1]$, then there is a sequence of continuous functions on $[0,1]$, $\{f_n\}$, such that for each $x \in [0,1]$,

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

Problem 4   Evaluate

$$\int_{0}^{\infty}\frac{x - \sin x}{x^3}\,dx\,.$$

Problem 5   Consider the differential equation

$$\frac{dy}{dx} = 3xy + \frac{y}{1 + y^2}\cdot$$

Prove

1. For each $n=1, 2, \ldots$, there is a unique solution $y = f_n(x)$ defined for $0 \leq x \leq 1$ such that $f_n(0) = 1/n$.

2. $\lim_{n \to \infty}f_n(1) = 0.$

Problem 6   Let $\gcd$ abbreviate greatest common divisor and $\mathrm{lcm}$ abbreviate least common multiple. For three nonzero integers a, b, c, show that

$$\gcd\left\{a,\mathrm{lcm}\{b,c\}\right\}=\mathrm{lcm}\left\{\gcd \{a,b\}, \gcd \{a,c\} \right\}.$$

Problem 7   Let $\mathbb{R}^{}$ $[x_1,\ldots,x_n]$ be the polynomial ring over the real field $\mathbb{R}^{}$ in the $n$ variables $x_1,\ldots,x_n$. Let the matrix $A$ be the $n \times n$ matrix whose $i^{th}$ row is $(1, x_i, x_i^2, \ldots ,x_i^{n-1})$, $i = 1, \ldots , n$. Show that

$${\det A} = \prod_{i > j}(x_i - x_j) .$$

Problem 8   Let $a$, $b$, $c$, and $d$ be real numbers, not all zero. Find the eigenvalues of the following 4$\times$4 matrix and describe the eigenspace decomposition of $\mathbb{R}^{4}$:

$$\left( \begin{array}{cccc} aa & ab & ac & ad \\ ba & bb & b... ...\ ca & cb & cc & cd \\ da & db & dc & dd\end{array} \right).$$

Problem 9   Let $f$ and $g$ be analytic functions in the open unit disc, and let $C_r$ denote the circle with center $0$ and radius $r$, oriented counterclockwise.

1. Prove that the integral
   
   $$\frac{1}{2\pi i}\int_{C_r}\frac{1}{w}f(w)g\left(\frac{z}{w}\right)\,dw$$
   
   
   is independent of $r$ as long as $\vert z\vert< r < 1$ and that it defines an analytic function $h(z)$, $\vert z\vert< 1$.

2. Prove or supply a counterexample: If $f \not\equiv 0$ and $g \not\equiv 0$, then $h \not\equiv 0$.

Problem 10   Prove or supply a counterexample:

1. 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
   
   $$\lim_{x \to 0}\frac{f(x)}{g(x)} = c.$$
   
   

2. Do the same question for complex valued $f$ and $g$.

Problem 11   Show that all groups of order $\leq 5$ are commutative. Give an example of a noncommutative group of order $6$.

Problem 12   Let $\theta$ and $\varphi$ be fixed, $0 \leq \theta \leq 2\pi$, $0 \leq \varphi \leq 2\pi$ and let $R$ be the linear transformation from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ whose matrix in the standard basis $\vec{\imath}$, $\vec{\jmath}$, and $\vec{k}$ is

$$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos\theta & \sin\theta \\ 0 & -\sin\theta & \cos\theta \end{array} \right)\,.$$

Let $S$ be the linear transformation of $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ whose matrix with respect to the basis

$$\left\{\frac{1}{\sqrt{2}}(\vec{\imath} + \vec{k}), \vec{\jmath}, \frac{1}{\sqrt{2}}(\vec{\imath} - \vec{k})\right\}$$

is

$$\left( \begin{array}{ccc} \cos\varphi & \sin\varphi & 0 \\ -... ...\varphi & \cos\varphi & 0 \\ 0 & 0 & 1 \end{array} \right)\,.$$

Prove that $T = R\circ S$ leaves a line invariant.

Problem 13   Show that if $f$ is a homeomorphism of $[0,1]$ onto itself, then there is a sequence $\{p_n\}$, $n = 1, 2, 3, \ldots\,$ of polynomials such that $p_n \to f$ uniformly on $[0,1]$ and each $p_n$ is a homeomorphism of $[0,1]$ onto itself.

Problem 14   Evaluate

1. 
   
   $$\int_{-\infty}^{\infty}\frac{dx}{(1+x+x^2)^2}$$
   
   

2. 
   
   $$\int_{0}^{2\pi}\frac{d\theta}{a + \sin \theta} \quad where \quad a>1.$$
   
   

Problem 15   Consider the differential equation

$$\frac{dx}{dt} = y, \hspace{.1in} \frac{dy}{dt} = -ay - x^3 - x^5, \;\; where\;\; a > 0.$$

1. Show that
   
   $$F(x,y) = \frac{y^2}{2} + \frac{x^4}{4} + \frac{x^6}{6}$$
   
   
   decreases along solutions.

2. Show that for any $\varepsilon > 0$, there is a $\delta > 0$ such that whenever
   $\Vert \left( x(0), y(0) \right)\Vert < \delta$, there is a unique solution $\left( x(t), y(t) \right)$ of the given equations with the initial condition $\left( x(0), y(0) \right)$ which is defined for all $t \geq 0$ and satisfies $\Vert \left( x(t), y(t) \right) \Vert < \varepsilon$.

Problem 16   Let $a$ be an element in a field $\mbox{\bf {F}}$ and let $p$ be a prime. Assume a is not a $p^{th}$ power. Show that the polynomial $x^p-a$ is irreducible in $\mbox{\bf {F}}[x]$.

Problem 17   Let $M$ be the $n\times n$ matrix over a field $\mbox{\bf {F}}$ all of whose entries are equal to $1$. Find the Jordan Canonical Form of $M$ and discuss the extent to which the Jordan form depends on the characteristic of the field $\mbox{\bf {F}}$.

Problem 18   Let $P_n$ be the vector space of all real polynomials with degrees at most n. Let $D:P_n \to P_n$ be given by differentiation: $D(p) = p'$. Let $\pi$ be a real polynomial. What is the minimal polynomial of the transformation $\pi(D)$?

Problem 19   Prove or supply a counterexample: If $f$ is a continuous complex valued function defined on a connected open subset of the complex plane and if $f^2$ is analytic, then $f$ is analytic.

Problem 20   Let $f$ be a $C^2$ function on the real line. Assume $f$ is bounded with bounded second derivative. Let

$$% latex2html id marker 1199 A=\sup_{x \in \mbox{$\mathbb{R}... ...{.1in} B=\sup_{x \in \mbox{$\mathbb{R}^{}$}}\vert f''(x)\vert.$$

Prove that

$$% latex2html id marker 1200 \sup_{x \in \mbox{$\mathbb{R}^{}$}}\vert f'(x)\vert \leq 2\sqrt{AB}.$$