Preliminary Exam - Summer 1982

Problem 1   Determine the Jordan Canonical Form of the matrix

$$A = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{array} \right).$$

Problem 2   Compute the integral

$$\int_0^{\infty}\frac{x^{50}}{x^{100}+1}\,dx\, .$$

Problem 3  

Let $K$ be a nonempty compact set in a metric space with distance function $d$. Suppose that $\varphi\colon K\to K$ satisfies

$$d(\varphi(x),\varphi(y))<d(x,y)$$

for all $x\ne y$ in $K$. Show there exists precisely one point $x\in K$ such that
$x=\varphi(x)$.

Problem 4   Let $G$ be a group with generators $a$ and $b$ satisfying

$$a^{-1}b^2a = b^3, \hspace{.3in} b^{-1}a^2b = a^3.$$

Is $G$ trivial?

Problem 5   Let $0<a_0\leq a_1\leq \cdots \leq a_n$. Prove that the equation

$$a_0z^n + a_1z^{n-1}+\cdots +a_n = 0$$

has no roots in the disc $\vert z\vert < 1$.

Problem 6   Suppose $f$ is a differentiable real valued function such that $f'(x)>f(x)$ for all $x \in \mbox{$\mathbb{R}^{}$}$ and $f(0)=0$. Prove that $f(x)>0$ for all positive $x$.

Problem 7   Let $V$ be the vector space of all real 3$\times$3 matrices and let ${A}$ be the diagonal matrix

$$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array} \right).$$

Calculate the determinant of the linear transformation ${T}$ on $V$ defined by
$T(X) = \frac{1}{2}(AX + XA)$.

Problem 8   Let n be a positive integer.

1. Show that the binomial coefficient
   
   $$c_n = \left(\!\!\begin{array}{c} 2n \\ n\end{array}\!\! \right)$$
   
   
   is even.

2. Prove that $c_n$ is divisible by $4$ if and only if n is not a power of $2$.

Problem 9   Determine the complex numbers $z$ for which the power series

$$\sum_{n=1}^{\infty}\frac{z^n}{n^{\log n}}$$

and its term by term derivatives of all orders converge absolutely.

Problem 10   For complex numbers $\alpha_1,\alpha_2,\ldots,\alpha_k$, prove

$$\limsup_n\left\vert \sum_{j=1}^k\alpha_j^n\right\vert^{1/n} = \;\;\sup_j\vert\alpha_j\vert.$$

Note: See also Problem .

Problem 11   Let $s(y)$ and $t(y)$ be real differentiable functions of $y$, $-\infty < y < \infty$, such that the complex function

$$f(x+iy) = e^x\left(s(y)+it(y)\right)$$

is complex analytic with $s(0)=1$ and $t(0)=0$. Determine $s(y)$ and $t(y)$.

Problem 12   Determine (with proofs) which of the following polynomials are irreducible over the field $\mathbb{Q}\,^{}$ of rationals.

1. $x^2+3$

2. $x^2-169$

3. $x^3+x^2+x+1$

4. $x^3+2x^2+3x+4$.

Problem 13   Let $f:[0,\pi] \to \mbox{$\mathbb{R}^{}$}$ be continuous and such that

$$\int_0^{\pi}f(x)\sin nx \,dx = 0$$

for all integers $n \geq 1$. Is $f$ identically $0$?

Problem 14   Let $A$ be a real $n \times n$ matrix such that $\langle Ax, x \rangle \geq 0$ for every real $n$-vector $x$. Show that $Au=0$ if and only if $A^tu=0$.

Problem 15   Let $f(z)$ be analytic on the open unit disc $\mathbb{D}= \{z \;\vert\; \vert z\vert < 1\}$. Prove that there is a sequence $(z_n)$ in $\mathbb{D}$ such that $\vert z_n\vert \to 1$ and $(f(z_n))$ is bounded.

Problem 16   A square matrix $A$ is nilpotent if $A^k=0$ for some positive integer $k$.

1. If $A$ and $B$ are nilpotent, is $A + B$ nilpotent? Proof or counterexample.
2. Prove: If $A$ is nilpotent, then $I-A$ is invertible.

Problem 17   Let $f:\mbox{$\mathbb{R}^{3}$} \to \mbox{$\mathbb{R}^{2}$}$ and assume that $0$ is a regular value of f (i.e., the differential of $f$ has rank $2$ at each point of $f^{-1}(0)$). Prove that $\mbox{$\mathbb{R}^{3}$}\setminus f^{-1}(0)$ is arcwise connected.

Problem 18   Let $E$ be the set of all continuous real valued functions satisfying

$$\vert u(x) - u(y)\vert \leq \vert x - y\vert, \quad 0 \leq x,y \leq 1, \quad u(0) = 0.$$

Let $\varphi : E \to \mbox{$\mathbb{R}^{}$}$ be defined by

$$\varphi(u) = \int_{0}^{1} \left( u(x)^2 - u(x) \right) \,dx\, .$$

Show that $\varphi$ achieves its maximum value at some element of $E$.

Problem 19   Let $V$ be a finite-dimensional vector space over the rationals $\mathbb{Q}\,^{}$ and let ${M}$ be an automorphism of $V$ such that ${M}$ fixes no nonzero vector in $V$. Suppose that ${M}^p$ is the identity map on V, where $p$ is a prime number. Show that the dimension of $V$ is divisible by $p-1$.

Problem 20   Let $M_{2\times 2}$ be the four-dimensional vector space of all 2$\times$2 real matrices and define $f:M_{2\times 2} \to M_{2\times 2}$ by $f(X)=X^2$.

1. Show that $f$ has a local inverse near the point
   
   $$X = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right).$$
   
   

2. Show that $f$ does not have a local inverse near the point
   
   $$X = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right).$$