Preliminary Exam - Fall 1991

Problem 1   Prove that every finite group of order at least $3$ has a nontrivial automorphism.

Problem 2   Let $f$ be a continuous function from $\mathbb{R}^{}$ to $\mathbb{R}^{}$ such that $\vert f(x)-f(y)\vert\geq \vert x-y\vert$ for all $x$ and $y$. Prove that the range of $f$ is all of $\mathbb{R}^{}$.

Note: See also Problem .

Problem 3   Evaluate the integral

$$I = \frac{1}{2\pi i}\int_C \frac{z^{n-1}}{3z^n-1}\,dz,$$

where n is a positive integer, and $C$ is the circle $\vert z\vert=1$, with counterclockwise orientation.

Problem 4  

1. Prove that any real n$\times$n matrix $M$ can be written as $M=A+S+cI$, where $A$ is antisymmetric, $S$ is symmetric, $c$ is a scalar, $I$ is the identity matrix, and $\mathrm{tr} \, S=0$.

2. Prove that with the above notation,

   
   

   $$\mathrm{tr}(M^2) =\mathrm{tr}(A^2) +\mathrm{tr}(S^2) + \frac{1}{n}(\mathrm{tr}\,M)^2.$$
   
   

Problem 5   Let $f$ be an infinitely differentiable function from $\mathbb{R}^{}$ to $\mathbb{R}^{}$. Suppose that, for some positive integer $n$,

$$f(1) = f(0) = f'(0) = f''(0) = \cdots = f^{(n)}(0) = 0.$$

Prove that $f^{(n+1)}(x)=0$ for some $x$ in $(0,1)$.

Problem 6   Let the function f be analytic in the disc $\vert z\vert<1$ of the complex plane. Assume that there is a positive constant $M$ such that

$$\int_0^{2\pi}\vert f'(re^{i\theta})\vert\,d\theta \leq M, \hspace{.3in} ( 0\leq r < 1).$$

Prove that

$$\int_{[0,1)}\vert f(x)\vert\,dx < \infty.$$

Problem 7   Consider the vector differential equation

$$\frac{dx(t)}{dt} = A(t)x(t)$$

where $A$ is a smooth $n\times n$ function on $\mathbb{R}^{}$. Assume $A$ has the property that $\langle {\rm A}(t)y, y \rangle \leq c\Vert y\Vert^2$ for all $y$ in $\mathbb{R}^{n}$ and all $t$, where $c$ is a fixed real number. Prove that any solution $x(t)$ of the equation satisfies $\Vert x(t)\Vert\leq e^{ct}\Vert x(0)\Vert$ for all $t>0$.

Problem 8   Let $a_1,a_2,a_3,\ldots$ be positive numbers.

1. Prove that $\sum a_n < \infty$ implies $\sum \sqrt{a_na_{n+1}} < \infty$.

2. Prove that the converse of the above statement is false.

Problem 9   Let $G$ be a group of order $2p$, where $p$ is an odd prime. Assume that $G$ has a normal subgroup of order $2$. Prove that $G$ is cyclic.

Problem 10   Let $\mbox{\bf {F}}$ be a finite field of order $p$. Compute the order of $SL_3(\mbox{\bf {F}})$, the group of 3$\times$3 matrices over $\mbox{\bf {F}}$ of determinant $1$.

Problem 11   Let $X$ and $Y$ be metric spaces and $f$ a continuous map of $X$ into $Y$. Let $K_1,K_2,\ldots$ be nonempty compact subsets of $X$ such that $K_{n+1} \subset K_n$ for all $n$, and let $K=\bigcap K_n$. Prove that $f(K)= \bigcap f(K_n)$.

Problem 12   Let $p$ be a nonconstant complex polynomial whose zeros are all in the half-plane $\Im z>0$.

1. Prove that $\Im(p'/p)>0$ on the real axis.
2. Find a relation between $\deg p$ and
   
   $$\int_{-\infty}^{\infty} \Im \frac{p'(x)}{p(x)}\,dx\, .$$
   
   

Problem 13   Let $A = (a_{ij})_{i,j=1}^n$ be a real $n\times n$ matrix with nonnegative entries such that

$$\sum_{j=1}^n a_{ij} = 1 \hspace{.5in} (1\leq i \leq n).$$

Prove that no eigenvalue of $A$ has absolute value greater than $1$.

Problem 14   Let ${\cal B}$ denote the unit ball of $\mathbb{R}^{3}$, ${\cal B}=\{r\in\mbox{$\mathbb{R}^{3}$} \;\vert\; \Vert r\Vert\leq 1\}$. Let $J=(J_1,J_2,J_3)$ be a smooth vector field on $\mathbb{R}^{3}$ that vanishes outside of ${\cal B}$ and satisfies $\nabla \cdot \vec{J} = 0$.

1. For $f$ a smooth, scalar-valued function defined on a neighborhood of ${\cal B}$, prove that
   
   $$\int_{\cal B} \left(\nabla f\right)\cdot \vec{J}\,dx dy dz = 0.$$
   
   

2. Prove that
   
   $$\int_{\cal B} J_1\,dxdydz = 0.$$
   
   

Problem 15   Let $\mathfrak{I}$ be the ideal in the ring $\mbox{$\mathbb{Z}^{}$}[x]$ generated by $x-7$ and $15$. Prove that the quotient ring $\mbox{$\mathbb{Z}^{}$}[x]/\mathfrak{I}$ is isomorphic to $\mbox{$\mathbb{Z}^{}$}_{15}$.

Problem 16   Let $M_{n \times n}$ be the space of real n$\times$n matrices. Regard it as a metric space with the distance function

$$d(A, B) = \sum_{i,j=1}^n\left\vert a_{ij}-b_{ij}\right\vert \hspace{.3in} \left( A=( a_{ij}), B=(b_{ij}) \right) .$$

Prove that the set of nilpotent matrices in $M_{n \times n}$ is a closed set.

Problem 17   Let $f$ be a $C^1$ function from the interval $(-1,1)$ into $\mathbb{R}^{2}$ such that $f(0)=0$ and $f'(0)\neq 0$. Prove that there is a number $\varepsilon$ in $(0,1)$ such that $\Vert f(t)\Vert$ is an increasing function of $t$ on $(0,\varepsilon)$.

Problem 18   Let the function $f$ be analytic in the entire complex plane and satisfy the inequality $\vert f(z)\vert\leq\vert\Re z\vert^{-1/2}$ off the imaginary axis. Prove that $f$ is constant.