Preliminary Exam - Spring 1979

Problem 1   Let $f:\mbox{$\mathbb{R}^{n}$}\setminus \{0\} \to \mbox{$\mathbb{R}^{}$}$ be differentiable. Suppose

$$\lim_{x \to 0} \frac{\partial f}{\partial x_j}(x)$$

exists for each $j = 1, \ldots, n$.

1. Can $f$ be extended to a continuous map from $\mathbb{R}^{n}$ to $\mathbb{R}^{}$?
2. Assuming continuity at the origin, is $f$ differentiable from $\mathbb{R}^{n}$ to $\mathbb{R}^{}$?

Problem 2   Let $E$ denote a finite-dimensional complex vector space with a Hermitian inner product $\langle x, y \rangle$.

1. Prove that $E$ has an orthonormal basis.

2. Let $f: E \to \mbox{$\mathbb{C}\,^{}$}$ be such that $f(x,y)$ is linear in $x$ and conjugate linear in $y$. Show there is a linear map $A : E \to E$ such that $f(x,y)=\langle Ax, y \rangle$.

Problem 3   Let $S_7$ be the group of permutations of a set of seven elements. Find all $n$ such that some element of $S_7$ has order $n$.

Problem 4   Prove that every compact metric space has a countable dense subset.

Problem 5   Find all solutions to the differential equation

$$y' = \sqrt{y}, \hspace{.3in} y(0) = 0\, .$$

Problem 6   Prove that if $1 < \lambda < \infty$, the function

$$f_{\lambda}(z) = z + \lambda - e^z$$

has only one zero in the half-plane $\Re z<0$, and that this zero is real.

Problem 7   Evaluate

$$\int_{0}^{\infty}\frac{x^2+1}{x^4+1}\,dx\,.$$

Problem 8   Let $M$ be a real nonsingular 3$\times$3 matrix. Prove there are real matrices $S$ and $U$ such that $M = SU = US$, all the eigenvalues of $U$ equal $1$, and $S$ is diagonalizable over $\mathbb{C}\,^{}$.

Problem 9   Let $M$ be an $n\times n$ complex matrix. Let $G_M$ be the set of complex numbers $\lambda$ such that the matrix $\lambda M$ is similar to $M$.

1. What is $G_M$ if
   
   $$M = \left( \begin{array}{ccc} 0 & 0 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\:?$$
   
   

2. Assume $M$ is not nilpotent. Prove $G_M$ is finite.

Problem 10   Let $f(x)$ be a polynomial over $\mbox{$\mathbb{Z}^{}$}_p$, the field of integers $\bmod p$. Let $g(x)=x^p-x$. Show that the greatest common divisor of $f(x)$ and $g(x)$ is the product of the distinct linear factors of $f(x)$.

Problem 11   Let $S$ be a collection of abelian groups, each of order $720$, no two of which are isomorphic. What is the maximum cardinality $S$ can have?

Problem 12   Let $G$ be a group with three normal subgroups $N_1$, $N_2$, and $N_3$. Suppose $N_i \cap N_j$ = {e} and $N_i N_j = G$ for all $i,j$ with $i\neq j$. Show that $G$ is abelian and $N_i$ is isomorphic to $N_j$ for all $i,j$.

Problem 13   Consider the system of differential equations:

$$\begin{eqnarray*} \frac{dx}{dt} & = & y + tz \\ \frac{dy}{dt} & = & z + t^2x \\ \frac{dz}{dt} & = & x + e^ty. \end{eqnarray*}$$

Prove there exists a solution defined for all $t \in [0,1]$, such that

$$\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 &... ...ight)= \left( \begin{array}{c} 0 \\ 0 \\ 0\end{array} \right)$$

and also

$$\int_{0}^{1}\left( x(t)^2 + y(t)^2 + z(t)^2 \right) dt = 1.$$

Problem 14   Let $M_{n \times n}$ denote the vector space of $n\times n$ real matrices for $n\geq 2$. Let $\det:M_{n\times n}\to\mbox{$\mathbb{R}^{}$}$ be the determinant map.

1. Show that $\det$ is $C^{\infty}$.

2. Show that the derivative of $\det$ at $A \in M_{n\times n}$ is zero if and only if $A$ has rank $\leq n - 2$.

Problem 15   Which of the following matrices are similar as matrices over $\mathbb{R}^{}$?

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

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

Problem 16   For which $z \in \mbox{$\mathbb{C}\,^{}$}$ does

$$\sum_{n=0}^{\infty}\left( \frac{z^n}{n!} + \frac{n^2}{z^n} \right)$$

converge?

Problem 17   Let $P$ and $Q$ be complex polynomials with the degree of $Q$ at least two more than the degree of $P$. Prove there is an $r>0$ such that if $C$ is a closed curve outside $\vert z\vert=r$, then

$$\int_C\frac{P(z)}{Q(z)}\,dz = 0.$$

Problem 18   Show that for any continuous function $f:[0,1] \to \mbox{$\mathbb{R}^{}$}$ and $\varepsilon > 0$, there is a function of the form

$$g(x) = \sum_{k=0}^{n}C_kx^{4k}$$

for some $n \in \mbox{$\mathbb{Z}^{}$}$, where $C_0,\ldots, C_n \in \mbox{$\mathbb{Q}\,^{}$}$ and $\vert g(x) - f(x)\vert<\varepsilon$ for all x in $[0,1]$.

Problem 19   Let $P$ be a 9$\times$9 real matrix such that $x^{t}Py = - y^{t}Px$ for all column vectors $x,y$ in $\mathbb{R}^{9}$. Prove that $P$ is singular.

Problem 20   Give an example of a function $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ having all three of the following properties:

• $f(x)=0$ for $x<0$ and $x>2$,
• $f'(1)=1$,
• f has derivatives of all orders.