Preliminary Exam - Spring 2000

Problem 1   Are the 4$\times$4 matrices

$$A=\left( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0... ...& 1 & -1\\ -1 & 0 & 0 & 0\\ -1 & 0 & 1 & 0\end{array}\right)$$

similar? Explain your reasoning.

Problem 2   Let $(f_n)^{\infty}_{n=1}$ be a uniformly bounded equicontinuous sequence of real-valued functions on the compact metric space $(X,d)$. Define the functions $g_n:X\to\mbox{$\mathbb{R}^{}$}$, for $n\in \mathbb{N}$ by

$$g_n(x)=\max\{f_1(x),\dots ,f_n(x)\} \,.$$

Prove that the sequence $(g_n)^{\infty}_{n=1}$ converges uniformly.

Problem 3   Prove that the group $G=\mathbb{Q}/\mathbb{Z}$ has no proper subgroup of finite index.

Problem 4   Let $f$ and $g$ be entire functions such that ${\displaystyle{\lim_{z\to\infty}}} f(g(z))=\infty$. Prove that $f$ and $g$ are polynomials.

Problem 5   Let $a$ and $x_0$ be positive numbers, and define the sequence $(x_n)^{\infty}_{n=1}$ recursively by

$$x_n=\frac{1}{2}\left( x_{n-1}+\frac{a}{x_{n-1}}\right)\,.$$

Prove that this sequence converges, and find its limit.

Problem 6   Let $A$ be an $n\times n$ matrix over $\mathbb{C}$ whose minimal polynomial $\mu$ has degree $k$.

1. Prove that, if the point $\lambda$ of $\mathbb{C}$ is not an eigenvalue of $A$, then there is a polynomial $p_{\lambda}$ of degree $k-1$ such that $p_{\lambda}(A)=(A-\lambda I)^{-1}$.

2. Let $\lambda_1,\dots ,\lambda_{k}$ be distinct points of $\mathbb{C}$ that are not eigenvalues of $A$. Prove that there are complex numbers $c_1,\dots ,c_{k}$ such that
   
   $$\sum^{k}_{j=1} c_j(A-\lambda_j I)^{-1}=I \, .$$
   
   

Problem 7   Let $f$ be a positive function of class $C^2$ on $(0,\infty)$ such that $f'\leq 0$ and $f''$ is bounded. Prove that $${\lim_{t\to\infty}}f'(t)=0$$.

Problem 8   Find the cardinality of the set of all subrings of $\mathbb{Q}$, the field of rational numbers.

Problem 9   Evaluate

$$I=\int_{\vert z\vert=1}\frac{\cos^3 z}{z^3} \, dz\,,$$

where the direction of integration is counterclockwise.

Problem 10   Let $S$ be an uncountable subset of $\mathbb{R}$. Prove that there exists a real number $t$ such that both sets $S\cap (-\infty,t)$ and $S\cap (t,\infty)$ are uncountable.

Problem 11   Let $A_n$ be the $n\times n$ matrix whose entries $a_{jk}$ are given by

$$a_{jk} = \left\{ \begin{array}{ccc} 1 &\mathrm{if}& \vert j-k\vert=1 \\ 0 & &\mathrm{otherwise}\,. \end{array} \right.$$

Prove that the eigenvalues of $A$ are symmetric with respect to the origin.

Problem 12   Suppose that $H_1$ and $H_2$ are distinct subgroups of a group $G$ such that $[G:H_1]=[G:H_2]=3$. What are the possible values of $[G:H_1\cap H_2]$? Explain.

Problem 13   Let $f$ be a nonconstant entire function whose values on the real axis are real and nonnegative. Prove that all real zeros of $f$ have even order.

Problem 14   Let $I_1,\dots ,I_n$ be disjoint closed nonempty subintervals of $\mathbb{R}$.

1. Prove that if $p$ is a real polynomial of degree less than $n$ such that
   
   $$\int_{I_j} p(x)dx=0 \ , \qquad \mathrm{for}\,\,j=1, \dots ,n \ , \qquad(*)$$
   
   
   then $p=0$.

2. Prove that there is a nonzero real polynomial $p$ of degree $n$ that satisfies $(*)$.

Problem 15   Let $F$, with components $F_1,\dots ,F_n$, be a differentiable map of $\mbox{$\mathbb{R}^{n}$}$ into $\mbox{$\mathbb{R}^{n}$}$ such that $F(0)=0$. Assume that

$$\sum^n_{j,k=1} \left\vert \frac{\partial F_j(0)}{\partial x_k}\right\vert^2 =c<1 \ .$$

Prove that there is a ball $B$ in $\mbox{$\mathbb{R}^{n}$}$ with center $0$ such that $F(B)\subset B$.

Problem 16   Let $A$ be a complex $n\times n$ matrix such that the sequence $(A^n)^{\infty}_{n=1}$ converges to a matrix $B$. Prove that $B$ is similar to a diagonal matrix with zeros and ones along the main diagonal.

Problem 17   Evaluate the integrals

$$I(t)=\int^{\infty}_{-\infty} \frac{e^{itx}}{(x+i)^2} \ dx \ , \qquad -\infty < t <\infty \ .$$

Problem 18   Let $G$ be a finite group and $p$ a prime number. Suppose $a$ and $b$ are elements of $G$ of order $p$ such that $b$ is not in the subgroup generated by $a$. Prove that $G$ contains at least $p^2-1$ elements of order $p$.