Preliminary Exam - Fall 1997

Problem 1   Define a sequence of real numbers $(x_n)$ by

$$x_0 = 1, \quad\quad x_{n + 1} = \frac{1}{2 + x_n}\quad \mathrm{for}\quad n \geq 0.$$

Show that $(x_n)$ converges, and evaluate its limit.

Problem 2   Let $f$ be a real valued function that is differentiable on an open interval containing $[a, b]$. Prove that if $f'(a) < 0$ and $f'(b) > 0$ then there is a point $c \in (a, b)$ such that $f'(c) = 0$.

Problem 3   Let $f$ be an entire function such that, for all $z$, $\vert f(z)\vert = \vert\sin z \vert$. Prove that there is a constant $C$ of modulus $1$ such that $f(z) = C \sin z$.

Problem 4   Evaluate the integral

$$\int_{-\infty}^{\infty} \frac{dx}{1 + x^{2n}}$$

where $n > 0$ is an integer.

Problem 5   Let $\mathbb{D}= \{z \;\vert\; \vert z\vert < 1\}$, the open unit disc in the complex plane. Suppose that $f : \mathbb{D}\rightarrow \mathbb{D}$ is analytic, and that there exist two distinct points $a, b \in \mathbb{D}$ with $f(a) = a$, $f(b) = b$. Prove that $f(z) = z$ for all $z \in \mathbb{D}$.

Problem 6   Let $\alpha_1, \alpha_2, \dots ,\alpha_n$ be distinct real numbers. Show that the $n$ exponential functions $e^{\alpha_1t}, e^{\alpha_2t}, \dots ,e^{\alpha_nt}$ are linearly independent over the real numbers.

Problem 7   Define the index of a real symmetric matrix $A$ _matrix,>index to be the number of strictly positive eigenvalues of $A$ minus the number of strictly negative eigenvalues. Suppose $A$, and $B$ are real symmetric $n \times n$ matrices such that $x^tAx \leq x^tBx$ for all $n \times 1$ matrices $x$. Prove the the index of $A$ is less than or equal to the index of $B$.

Problem 8   Suppose $H_i$ is a normal subgroup of a group $G$ for
$1 \leq i \leq k$, such that $H_i \cap H_j = \{1\}$ for $i \neq j$. Prove that $G$ contains a subgroup isomorphic to $H_1 \times H_2 \times \cdots \times H_k$ if $k = 2$, but not necessarily if $k \geq 3$.

Problem 9   Prove that if $p$ is prime then every group of order $p^2$ is abelian.

Problem 10   Prove that for all $x > 0$, $\sin x > x - x^3/6$.

Problem 11   Let $f : \mbox{$\mathbb{R}^{}$} \rightarrow \mbox{$\mathbb{R}^{}$}$ be twice differentiable, and suppose that for all $x \in \mbox{$\mathbb{R}^{}$}$, $\vert f(x)\vert \leq 1$ and $\vert f''(x)\vert \leq 1$. Prove that $\vert f'(x)\vert \leq 2$ for all $x \in \mbox{$\mathbb{R}^{}$}$.

Problem 12   A map $f : \mbox{$\mathbb{R}^{m}$} \rightarrow \mbox{$\mathbb{R}^{n}$}$ _map>compact _map>closed is proper if it is continuous and $f^{-1}(B)$ is compact for each compact subset $B$ of $\mbox{$\mathbb{R}^{n}$}$; $f$ is closed if it is continuous and $f(A)$ is closed for each closed subset $A$ of $\mbox{$\mathbb{R}^{m}$}$.

1. Prove that every proper map $f : \mbox{$\mathbb{R}^{m}$} \rightarrow \mbox{$\mathbb{R}^{n}$}$ is closed.
2. Prove that every one-to-one closed map $f : \mbox{$\mathbb{R}^{m}$} \rightarrow \mbox{$\mathbb{R}^{n}$}$ is proper.

Problem 13   Conformally map the region inside the disc $\{z \in \mbox{$\mathbb{C}\,^{}$} \;\vert\; \vert z - 1\vert \leq 1\}$ and outside the disc $\{z \in \mbox{$\mathbb{C}\,^{}$} \;\vert\; \vert z - \frac{1}{2}\vert \leq \frac{1}{2}\}$ onto the upper half-plane.

Problem 14   Evaluate the integral

$$\int_{-\infty}^{\infty}\frac{\cos kx }{1 + x + x^2}\,dx$$

where $k \geq 0$.

Problem 15   Let $M_{n\times n}(\mbox{\bf {K}})$ be the vector space of $n \times n$ matrices over a field $\mbox{\bf {K}}$. Find the dimension of the subspace of $M_{n\times n}(\mbox{\bf {K}})$ spanned by $\{XY - YX\,\vert\, X, Y \in M_{n\times n}(\mbox{\bf {K}})\}$.

Problem 16   Prove that if $A$ is a 2$\times$2 matrix over the integers such that $A^n = I$ for some strictly positive integer $n$, then $A^{12} = I$.

Problem 17   A group $G$ is generated by two elements $a, b$, each of order $2$. Prove that $G$ has a cyclic subgroup of index $2$.

Problem 18   A finite abelian group $G$ has the property that for each positive integer $n$ the set $\{x \in G \,\vert\, x^n = 1\}$ has at most $n$ elements. Prove that $G$ is cyclic, and deduce that every finite field has cyclic multiplicative group.