Preliminary Exam - Fall 1987

Problem 1   Prove that $(\cos \theta )^p \leq \cos (p\,\theta )$ for $0 \leq \theta \leq \pi /2$ and $0 < p < 1$.

Problem 2   Suppose that $\{f_n\}$ is a sequence of nondecreasing functions which map the unit interval into itself. Suppose that

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

pointwise and that $f$ is a continuous function. Prove that $f_n(x) \to f(x)$ uniformly as $n \to \infty$, $0 \leq x \leq 1$. Note that the functions $f_n$ are not necessarily continuous.

Problem 3   Show that the following limit exists and is finite:

$$\lim_{t \to 0^+}\left( \int_{0}^{1} \frac{dx}{\left(x^4 + t^4\right)^{1/4}} + \log t \right) .$$

Problem 4   Let u and v be two real valued $C^1$ functions on $\mathbb{R}^{2}$ such that the gradient $\nabla u$ is never $0$, and such that, at each point, $\nabla v$ and $\nabla u$ are linearly dependent vectors. Given $p_0 = (x_0,y_0) \in \mbox{$\mathbb{R}^{2}$}$, show that there is a $C^1$ function $F$ of one variable such that $v(x,y) = F\left( u(x,y)\right)$ in some neighborhood of $p_0$.

Problem 5   Calculate $A^{100}$ and $A^{-7}$, where

$$A = \left( \begin{array}{cc} 3/2 & 1/2 \\ -1/2 & 1/2 \end{array} \right).$$

Problem 6   Let $G$ and $H$ be finite groups of relatively prime order. Show that $\mathrm{Aut}(G \times H)$, the group of automorphisms of $G \times H$, is isomorphic to the direct product of $\mathrm{Aut}(G)$ and $\mathrm{Aut}(H)$.

Problem 7   Let $A$ and $B$ be real n$\times$n symmetric matrices with $B$ positive definite. Consider the function defined for $x\neq 0$ by

$$G(x)=\frac{\langle Ax, x \rangle }{\langle Bx, x \rangle }\cdot$$

1. Show that $G$ attains its maximum value.
2. Show that any maximum point $U$ for $G$ is an eigenvector for a certain matrix related to $A$ and $B$ and show which matrix.

Problem 8   Let $R$ be the set of 2$\times$2 matrices of the form

$$\left( \begin{array}{cc} a & -b \\ b & a\end{array} \right)$$

where a, b are elements of a given field $\mbox{\bf {F}}$. Show that with the usual matrix operations, $R$ is a commutative ring with identity. For which of the following fields $\mbox{\bf {F}}$ is $R$ a field: $F = \mbox{$\mathbb{Q}\,^{}$},\, \mbox{$\mathbb{C}\,^{}$},\, \mbox{$\mathbb{Z}^{}$}_5,\, \mbox{$\mathbb{Z}^{}$}_7$ ?

Problem 9   Evaluate the integral

$$I = \int_{0}^{2\pi}\frac{\cos^2 3\theta}{5-4\cos 2\theta} \, d\theta \, .$$

Problem 10   If $f(z)$ is analytic in the open disc $\vert z\vert<1$, and $\vert f(z)\vert\leq 1/(1-\vert z\vert)$, show that

$$\vert a_n\vert = \vert f^{(n)}(0)/n!\vert \leq (n+1)(1+1/n)^n < e(n+1).$$

Problem 11   Let $V$ be a finite-dimensional vector space and $T : V \to V$ a diagonalizable linear transformation. Let $W \subset V$ be a linear subspace which is mapped into itself by $T$. Show that the restriction of $T$ to $W$ is diagonalizable.

Problem 12   Given two real $n\times n$ matrices $A$ and $B$, suppose that there is a nonsingular complex matrix $C$ such that $CAC^{-1} = B$. Show that there exists a real nonsingular $n\times n$ matrix $C$ with this property.

Problem 13   Let $A$ be the group of rational numbers under addition, and let $M$ be the group of positive rational numbers under multiplication. Determine all homomorphisms $\varphi : A \to M$.

Problem 14   Show that $\mbox{$M_{n \times n}$}(\mbox{\bf {F}})$, the ring of all $n\times n$ matrices over the field $\mbox{\bf {F}}$, has no proper two sided ideals.

Problem 15   Let $f(z)$ be analytic for $z \neq 0$, and suppose that
$f(1/z) = f(z)$. Suppose also that $f(z)$ is real for all $z$ on the unit circle $\vert z\vert=1$. Prove that $f(z)$ is real for all real $z \neq 0$.

Problem 16   How many zeros (counting multiplicities) does the polynomial

$$2z^5 - 6z^3 + z + 1$$

have in the annular region $1 \leq \vert z\vert \leq 2$?

Problem 17   Let $u$ be a positive harmonic function on $\mathbb{R}^{2}$; that is,

$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0.$$

Show that $u$ is constant. _function,>harmonic

Problem 18   Find a curve $C$ in $\mathbb{R}^{2}$, passing through the point $(3,2)$, with the following property: Let $L(x_0,y_0)$ be the segment of the tangent line to $C$ at $(x_0,y_0)$ which lies in the first quadrant. Then each point $(x_0,y_0)$ of $C$ is the midpoint of $L(x_0,y_0)$.

file=../Fig/Pr/Fa87-18,width=2.7in

Problem 19   Define a sequence of positive numbers as follows. Let
$x_0 > 0$ be any positive number, and let $x_{n+1} = (1 + x_n)^{-1}$. Prove that this sequence converges, and find its limit.

Problem 20   Let $S$ be the set of all real $C^1$ functions $f$ on $[0,1]$ such that $f(0) = 0$ and

$$\int_{0}^{1}f'(x)^2\,dx \leq 1.$$

Define

$$J(f) = \int_{0}^{1}f(x)\,dx\, .$$

Show that the function $J$ is bounded on $S$, and compute its supremum. Is there a function $f_0 \in S$ at which $J$ attains its maximum value? If so, what is $f_0$?