Preliminary Exam - Summer 1980

Problem 1   Exhibit a real 3$\times$3 matrix having minimal polynomial $(t^2+1)(t-10)$, which, as a linear transformation of $\mathbb{R}^{3}$, leaves invariant the line $L$ through $(0,0,0)$ and $(1,1,1)$ and the plane through $(0,0,0)$ perpendicular to $L$.

Problem 2   Which of the following matrix equations have a real matrix solution X? (It is not necessary to exhibit solutions.)

1. 
   
   $$X^3 = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 2 & 3 & 0 \end{array} \right),$$
   
   

2. 
   
   $$2X^5 + X=\left( \begin{array}{ccc} 3 & 5 & 0 \\ 5 & 1 & 9 \\ 0 & 9 & 0 \end{array} \right),$$
   
   

3. 
   
   $$X^6 + 2X^4 + 10X=\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right),$$
   
   

4. 
   
   $$X^4 = \left( \begin{array}{ccc} 3 & 4 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -3 \end{array} \right).$$
   
   

Problem 3   Let $T:V \to V$ be an invertible linear transformation of a vector space $V$. Denote by $G$ the group of all maps $f_{k,a}: V \to V$ where $k \in \mbox{$\mathbb{Z}^{}$}$, $a \in V$, and for $x \in V$,

$$f_{k,a}(x) = T^kx + a \hspace{.2in} (x \in V).$$

Prove that the commutator subgroup $G'$ of $G$ is isomorphic to the additive group of the vector space $(T-I)V$, the image of $T-I$. ($G'$ is generated by all $ghg^{-1}h^{-1}$, $g$ and $h$ in $G$.)

Problem 4   Let $G$ be a finite group and $H\subset G$ a subgroup.

1. Show that the number of subgroups of $G$ of the form $xHx^{-1}$ for some $x\in G$ is $\leq$ the index of $H$ in $G$.
2. Prove that some element of $G$ is not in any subgroup of the form $xHx^{-1}$, $x\in G$.

Problem 5   Consider the differential equations

$$\frac{dx}{dt} = -x + y, \hspace{.2in} \frac{dy}{dt} = \log(20+x) - y.$$

Let $x(t)$ and $y(t)$ be a solution defined for all $t\geq 0$ with $x(0)>0$ and $y(0)>0$. Prove that $x(t)$ and $y(t)$ are bounded.

Problem 6   Let $C$ denote the positively oriented circle $\vert z\vert = 2$,
$z\in \mbox{$\mathbb{C}\,^{}$}$. Evaluate the integral

$$\int_C\sqrt{z^2-1}\,dz$$

where the branch of the square root is chosen so that $\sqrt{2^2-1}>0.$

Problem 7   Exhibit a conformal map from the set $\{z \in \mbox{$\mathbb{C}\,^{}$} \;\vert\; \vert z\vert<1, \Re z > 0 \}$ onto $\mathbb{D}=\{z\in\mbox{$\mathbb{C}\,^{}$}\;\vert\;\vert z\vert<1\}$.

Problem 8   Give an example of a subset of $\mathbb{R}^{}$ having uncountably many connected components. Can such a subset be open? Closed?

Problem 9   For each $(a,b,c) \in \mbox{$\mathbb{R}^{3}$}$, consider the series

$$\sum_{n=3}^{\infty}\frac{a^n}{n^b(\log n)^c} \cdot$$

Determine the values of $(a,b,c)$ for which the series

1. converges absolutely;

2. converges but not absolutely;

3. diverges.

Problem 10   Let $f:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$ be a function whose partial derivatives of order $\leq 2$ are everywhere defined and continuous.

1. Let $a \in \mbox{$\mathbb{R}^{n}$}$ be a critical point of f (i.e., $\frac{\partial f}{\partial x_j}(a)=0, \; i = 1,\ldots,n$). Prove that a is a local minimum provided the Hessian matrix
   
   $$\left( \frac{\partial^2f}{\partial x_i \partial x_j} \right)$$
   
   
   is positive definite at $x=a$.

2. Assume the Hessian matrix is positive definite at all x. Prove that f has, at most, one critical point.

Problem 11   Prove that every finite group is isomorphic to

1. A group of permutations;

2. A group of even permutations.

Problem 12   Let $S \subset \mbox{$\mathbb{R}^{3}$}$ denote the ellipsoidal surface defined by

$$2x^2+(y-1)^2+(z-10)^2=1.$$

Let $T \subset \mbox{$\mathbb{R}^{3}$}$ be the surface defined by

$$z = \frac{1}{x^2+y^2+1} \cdot$$

Prove that there exist points $p \in S,\; q \in T$, such that the line $\overline{pq}$ is perpendicular to $S$ at $p$ and to $T$ at $q$.

Problem 13   Let $\mathfrak{J}$ be the ideal in the ring $\mbox{$\mathbb{Z}^{}$}[x]$ (of polynomials with integer coefficients) generated by $x-5$ and $14$. Find $n \in \mbox{$\mathbb{Z}^{}$}$ such that $0\leq n\leq 13$ and $(x^3+2x+1)^{50}-n \in \mathfrak{J}$.

Problem 14   Let $A$ and $B$ be real 2$\times$2 matrices with $A^2=B^2 =I,\; AB + BA=0$. Prove there exists a real nonsingular matrix $T$ with

$$TAT^{-1} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{arr... ... \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right).$$

Problem 15   Let $E$ be a finite-dimensional vector space over a field $\mbox{\bf {F}}$. Suppose $B:E \times$E $\to \mbox{\bf {F}}$ is a bilinear map (not necessarily symmetric). Define subspaces

$$E_1=\{x \in E \;\vert\; B(x,y) = 0 \; for\;\; all\; y \in E\},$$

$$E_2=\{y \in E \;\vert\; B(x,y) = 0 \; for\;\; all\; x \in E\}$$

Prove that $\dim E_1=\dim E_2$.

Problem 16   Let $(a_n)$ be a sequence of nonzero real numbers. Prove that the sequence of functions $f_n : \mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$

$$f_n(x) = \frac{1}{a_n}\sin(a_nx) + \cos(x+a_n)$$

has a subsequence converging to a continuous function.

Problem 17   Let $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ be monotonically increasing (perhaps discontinuous). Suppose $0<f(0)$ and $f(100)<100$. Prove $f(x)=x$ for some $x$.

Problem 18   How many zeros does the complex polynomial

$$3z^9 + 8z^6 + z^5 + 2z^3 + 1$$

have in the annulus $1 < \vert z\vert < 2$?

Problem 19   Let $f$ be a meromorphic function on $\mathbb{C}\,^{}$ which is analytic in a neighborhood of $0$. Let its Maclaurin series be

$$\sum_{k=0}^{\infty}a_kz^k$$

with all $a_k \geq 0$. Suppose there is a pole of modulus $r>0$ and no pole has modulus $< r$. Prove there is a pole at $z=r$.

Problem 20   Prove that the initial value problem

$$\frac{dx}{dt} = 3x+85\cos x, \hspace{.2in} x(0) = 77$$

has a solution $x(t)$ defined for all $t \in \mbox{$\mathbb{R}^{}$}$.