Preliminary Exam - Spring 1984

Problem 1   Evaluate

$$\int_0^{\infty}\frac{\log x}{a^2+x^2}\,dx$$

for $a>0$.

Problem 2   For a $p$-group of order $p^4$, assume the center of $G$ has order $p^2$. Determine the number of conjugacy classes of $G$.

Problem 3   Let $f : [0,1] \to \mbox{$\mathbb{R}^{}$}$ be continuous, with $f(0)=f(1)=0$. Assume that $f''$ exists on $0<x<1$, with $f''+2f'+f\geq 0$. Show that $f(x)\leq 0$ for all $0\leq x \leq 1$.

Problem 4   Which number is larger, $\pi^3$ or $3^{\pi}$?

Problem 5   Let $A$ and $B$ be complex $n\times n$ matrices such that $AB=BA^2$, and assume $A$ has no eigenvalues of absolute value $1$. Prove that $A$ and $B$ have a common (nonzero) eigenvector.

Problem 6   Let $a$ be a positive real number. Define a sequence $(x_n)$ by

$$x_0 = 0, \quad x_{n+1} = a + x_{n}^2, \quad n \geq 0\, .$$

Find a necessary and sufficient condition on $a$ in order that a finite limit $${\lim_{n \to \infty} x_n}$$ should exist.

Problem 7   Find the number of roots of

$$z^7 - 4z^3 - 11 = 0$$

which lie between the two circles $\vert z\vert=1$ and $\vert z\vert=2$.

Problem 8   Show that the system of differential equations

$$\frac{d}{dt} \left( \begin{array}{c} x \\ y \\ z \end{array} ... ... \right)\left( \begin{array}{c} x \\ y \\ z \end{array} \right)$$

has a solution which tends to $\infty$ as $t \to -\infty$ and tends to the origin as $t \to +\infty$.

Problem 9   Let $A$ be a real $m\times n$ matrix with rational entries and let b be an $m$-tuple of rational numbers. Assume that the system of equations Ax = b has a solution x in complex $n$-space $\mathbb{C}\,^{n}$. Show that the equation has a solution vector with rational components, or give a counterexample.

Problem 10   Let $R$ be a principal ideal domain and let $\mathfrak{I}$ and $\mathfrak{J}$ be nonzero ideals in $R$. Show that $\mathfrak{IJ} = \mathfrak{I} \cap \mathfrak{J}$ if and only if $\mathfrak{I} + \mathfrak{J} = R$.

Problem 11   Prove the following statement or supply a counterexample: If $A$ and $B$ are real $n\times n$ matrices which are similar over $\mathbb{C}\,^{}$, then $A$ and $B$ are similar over $\mathbb{R}^{}$.

Problem 12   Consider the equation

$$\frac{dy}{dx} = y - \sin y.$$

Show that there is an $\varepsilon > 0$ such that if $\vert y_0\vert< \varepsilon$, then the solution $y = f(x)$ with $f(0) = y_0$ satisfies

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

Problem 13   Let $I$ be an open interval in $\mathbb{R}^{}$ containing zero. Assume that $f'$ exists on a neighborhood of zero and $f''(0)$ exists. Show that

$$f(x) = f(0) + f'(0)\sin x + \frac{1}{2}f''(0)\sin^2 x + o(x^2)$$

($o(x^2)$ denotes a quantity such that $o(x^2)/x^2 \to 0$ as $x \to 0$).

Problem 14   Let $\mbox{\bf {F}}$ be a field and let $X$ be a finite set. Let $R(X,\mbox{\bf {F}})$ be the ring of all functions from $X$ to $\mbox{\bf {F}}$, endowed with the pointwise operations. What are the maximal ideals of $R(X,\mbox{\bf {F}})$?

Problem 15   Let $F$ be a continuous complex valued function on the interval $[0,1]$. Let

$$f(z) = \int_0^1 \frac{F(t)}{t-z}\,dt,$$

for $z$ a complex number not in $[0,1]$.

1. Prove that $f$ is an analytic function.

2. Express the coefficients of the Laurent series of $f$ about $\infty$ in terms of $F$. Use your result to show that $F$ is uniquely determined by $f$.

Problem 16   Prove, or supply a counterexample: If $A$ is an invertible $n \times n$ complex matrix and some power of $A$ is diagonal, then $A$ can be diagonalized.

Problem 17   Prove that the Taylor coefficients at the origin of the function

$$f(z) = \frac{z}{e^z-1}$$

are rational numbers.

Problem 18   Prove or supply a counterexample: If the function $f$ from $\mathbb{R}^{}$ to $\mathbb{R}^{}$ has both a left limit and a right limit at each point of $\mathbb{R}^{}$, then the set of discontinuities of $f$ is, at most, countable.

Problem 19   Let $f(x)=x\log\left( 1+x^{-1} \right)$, $0<x<\infty$.

1. Show that $f$ is strictly monotonically increasing.

2. Compute $\lim f(x)$ as $x \to 0$ and $x \to \infty$.

Problem 20   Determine all finitely generated abelian groups $G$ which have only finitely many automorphisms.