Preliminary Exam - Spring 1999

Problem 1   Let $d(X, Y)$ the distance between $X$ and $Y$ as defined in Problem . Give a proof or counterexample for each of the following statements, for disjoint sets $X$ and $Y$.

1. If $X$ and $Y$ are closed in $M$ then $d(X, Y) > 0$.
2. If $X$ and $Y$ are compact then $d(X, Y) > 0$.
3. If $X$ is closed and $Y$ compact then $d(X, Y) > 0$.

Problem 2   Suppose that a sequence of functions $f_n : \mbox{$\mathbb{R}^{}$} \rightarrow \mbox{$\mathbb{R}^{}$}$ converges uniformly on $\mbox{$\mathbb{R}^{}$}$ to a function $f : \mbox{$\mathbb{R}^{}$} \rightarrow \mbox{$\mathbb{R}^{}$}$, and that $${c_n = \lim_{x \rightarrow \infty} f_n(x)}$$ exists for each positive integer $n$. Prove that $${\lim_{n \rightarrow \infty} c_n}$$ and $${\lim_{x \rightarrow \infty} f(x)}$$ both exist and are equal.

Problem 3   Suppose that $f$ is a twice differentiable real-valued function on $\mbox{$\mathbb{R}^{}$}$ such that $f(0) = 0$, $f'(0) > 0$, and $f''(x) \geq f(x)$ for all $x \geq 0$. Prove that $f(x) > 0$ for all $x > 0$.

Problem 4   Evaluate $${\int_0^{\infty}\frac{dx}{x^c(x + 1)}}$$ for each real number $c \in (0, 1)$.

Problem 5  

1. Prove that if $f$ is holomorphic on the unit disc $\mathbb{D}$ and $f(z) \neq 0$ for all $z \in \mathbb{D}$, then there is a holomorphic function $g$ on $\mathbb{D}$ such that $f(z) = e^{g(z)}$ for all $z \in \mathbb{D}$.

2. Does the conclusion of Part 1 remain true if $\mathbb{D}$ is replaced by an arbitrary connected open set in $\mbox{$\mathbb{C}\,^{}$}$?

Problem 6   Let $p$, $q$, $r$ and $s$ be polynomials of degree at most 3. Which, if any, of the following two conditions is sufficient for the conclusion that the polynomials are linearly dependent?

1. At 1 each of the polynomials has the value 0.
2. At 0 each of the polynomials has the value 1.

Problem 7   Suppose that the minimal polynomial of a linear operator $T$ on a seven-dimensional vector space is $x^2$. What are the possible values of the dimension of the kernel of T?

Problem 8   Let $M$ be a $3 \times 3$ matrix with entries in the polynomial ring $\mbox{$\mathbb{R}^{}$}[t]$ such that $${ M^3 = \left( \begin{array}{ccc} t & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t \\ \end{array} \right)}$$. Let $N$ be the matrix with real entries obtained by substituting $t = 0$ in $M$. Prove that $N$ is similar to $${\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} \right)}$$.

Problem 9   Let $G$ be a finite group, with identity $e$. Suppose that for every $a, b \in G$ distinct from $e$, there is an automorphism $\sigma$ of $G$ such that
$\sigma(a) = b$. Prove that $G$ is abelian.

Problem 10   Suppose that $f$ is a twice differentiable real function such that $f''(x) > 0$ for all $x \in [a, b]$. Find all numbers $c \in [a, b]$ at which the area between the graph $y = f(x)$, the tangent to the graph at $(c, f(c))$, and the lines $x = a$, $x = b$, attains its minimum value.

Problem 11   Prove that if $n$ is a positive integer and $\alpha$, $\varepsilon$ are real numbers with $\varepsilon > 0$, then there is a real function $f$ with derivatives of all orders such that

1. $\vert f^{(k)}(x)\vert \leq \varepsilon$ for $k = 0, 1, \dots n - 1$ and all $x \in \mbox{$\mathbb{R}^{}$}$,

2. $f^{(k)}(0) = 0$ for $k = 0, 1, \dots n - 1$,

3. $f^{(n)}(0) = \alpha$.

Problem 12   Suppose that $f$ is holomorphic in some neighborhood of $a$ in the complex plane. Prove that either $f$ is constant on some neighborhood of $a$, or there exist an integer $n > 0$ and real numbers $\delta, \varepsilon > 0$ such that for each complex number $b$ satisfying $0 < \vert b - f(a)\vert < \varepsilon$, the equation $f(z) = b$ has exactly $n$ roots in $\{z \in \mbox{$\mathbb{C}\,^{}$} \vert \;\vert z - a\vert < \delta\}$.

Problem 13   Let $b_1$, $b_2$, ...be a sequence of real numbers such that $b_k \geq b_{k + 1}$ for all $k$ and $${\lim_{k \rightarrow \infty} b_k = 0}$$. Prove that the power series $${\sum_{k = 1}^{\infty} b_k z^k}$$ converges for all complex numbers $z$ such that $\vert z\vert \leq 1$ and $z \neq 1$.

Problem 14   Let $A = \left(a_{i j}\right)$ be a $n \times n$ complex matrix such that
$a_{i j} \neq 0$ if $i = j + 1$ but $a_{i j} = 0$ if $i \geq j + 2$. Prove that $A$ cannot have more than one Jordan block for any eigenvalue.

Problem 15   Let $M$ be a square complex matrix, and let
$S = \{XMX^{-1} \vert \; X \mbox{is non-singular}\}$ be the set of all matrices similar to $M$. Show that $M$ is a nonzero multiple of the identity matrix if and only if no matrix in $S$ has a zero anywhere on its diagonal.

Problem 16   Let $\Vert x\Vert$ denote the Euclidean length of a vector $x$. Show that for any real $m \times n$ matrix $M$ there is a unique non-negative scalar $\sigma$, and (possibly non-unique) unit vectors $u \in \mbox{$\mathbb{R}^{n}$}$ and $v \in \mbox{$\mathbb{R}^{m}$}$ such that

1. $\Vert Mx\Vert \leq \sigma\Vert x\Vert$ for all $x \in \mbox{$\mathbb{R}^{n}$}$,
2. $Mu = \sigma v$,
3. $M^Tv = \sigma u$ (where $M^T$ is the transpose of $M$).

Problem 17   Let $f(x) \in \mbox{$\mathbb{Q}\,^{}$}[x]$ be an irreducible polynomial of degree $n \geq 3$. Let $L$ be the splitting field of $f$, and let $\alpha \in L$ be a zero of $f$. Given that $[L : \mbox{$\mathbb{Q}\,^{}$}] = n!$, prove that $\mbox{$\mathbb{Q}\,^{}$}(\alpha^4) = \mbox{$\mathbb{Q}\,^{}$}(\alpha)$.

Problem 18   Let $G$ be a finite simple group of order $n$. Determine the number of normal subgroups in the direct product $G \times G$.