Preliminary Exam - Fall 1980

Problem 1   Define

$$F(x) = \int_{\sin x}^{\cos x}e^{(t^2+xt)}\,dt.$$

Compute $F'(0)$.

Problem 2   Let

$$% latex2html id marker 680 {\rm A} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 1 \\ -1 & 0 & 2 \end{array} \right).$$

Is $A$ similar to

$$% latex2html id marker 681 {\rm B} = \left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right)?$$

Problem 3   Do there exist functions $f(z)$ and $g(z)$ that are analytic at $z=0$ and that satisfy

1. $f\left(1/n\right) = f\left(-1/n\right) = 1/n^2$, $n = 1, 2, \ldots$,

2. $g\left(1/n\right) = g\left(-1/n\right) = 1/n^3$, $n = 1, 2, \ldots$?

Problem 4   Let $G$ be the group of orthogonal transformations of $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ with determinant $1$. Let $v \in \mbox{$\mathbb{R}^{3}$},\; \vert v\vert=1$, and let $H_v = \{T \in G \;\vert\; Tv=v\}$.

1. Show that $H_v$ is a subgroup of $G$.

2. Let $S_v=\{T \in G \;\vert\; T$ is a rotation through $180^{\circ}$ about a line orthogonal to $v\}$. Show that $S_v$ is a coset of $H_v$ in $G$.

Problem 5   Evaluate

$$\int_0^{\infty}\frac{x^{m-1}}{1+x^n}\,dx$$

where $n$ and $m$ are positive integers and $0<m<n$.

Problem 6   Let $M$ be the ring of real 2$\times$2 matrices and $S \subset M$ the subring of matrices of the form

$$\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right).$$

1. Exhibit (without proof) an isomorphism between $S$ and $\mathbb{C}\,^{}$.

2. Prove that
   
   $$A = \left( \begin{array}{cc} 0 & 3 \\ -4 & 1 \end{array} \right)$$
   
   
   lies in a subring isomorphic to $S$.

3. Prove that there is an $X \in M$ such that $X^4 + 13X = A$.

Problem 7   Let $g$ be $2\pi$-periodic, continuous on $[-\pi,\pi]$ and have Fourier series

$$\frac{a_0}{2} + \sum_{n=1}^{\infty}(a_n\cos nx + b_n\sin nx)\,.$$

Let $f$ be $2\pi$-periodic and satisfy the differential equation

$$f''(x) + kf(x) = g(x)$$

where $k\neq n^2, n= 1, 2, 3,\ldots$. Find the Fourier series of $f$ and prove that it converges everywhere.

Problem 8   Let $\mbox{\bf {F}}_2 = \{0,1\}$ be the field with two elements. Let $G$ be the group of invertible 2$\times$2 matrices with entries in $\mbox{\bf {F}}_2$. Show that $G$ is isomorphic to $S_3$, the group of permutations of three objects.

Problem 9   For a real 2$\times$2 matrix

$$X=\left( \begin{array}{cc} x & y \\ z & t \end{array} \right),$$

let $\Vert X\Vert = x^2+y^2+z^2+t^2$, and define a metric by $d(X,Y) = \Vert X-Y\Vert$. Let $\Sigma = \{X \;\vert\; \det(X) = 0\}$. Let

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

Find the minimum distance from $A$ to $\Sigma$ and exhibit an $S \in \Sigma$ that achieves this minimum.

Problem 10   Show that there is an $\varepsilon>0$ such that if $A$ is any real 2$\times$2 matrix satisfying $\vert a_{ij}\vert\leq \varepsilon$ for all entries $a_{ij}$ of $A$, then there is a real 2$\times$2 matrix $X$ such that $X^2+X^t=A$, where $X^{t}$ is the transpose of $X$. Is $X$ unique?

Problem 11   Let $f(z)$ be an analytic function defined for $\vert z\vert\leq 1$ and let

$$u(x,y)= \Re f(z), \hspace{.2in} z=x+iy.$$

Prove that

$$\int_C\frac{\partial u}{\partial y}\,dx - \frac{\partial u}{\partial x}\,dy = 0$$

where $C$ is the unit circle, $x^2+y^2=1$.

Problem 12   Prove that any group of order $6$ is isomorphic to either $\mbox{$\mathbb{Z}^{}$}_6$ or $S_3$ (the group of permutations of three objects).

Problem 13   Let $f(x)=\frac{1}{4}+x-x^2$. For any real number $x$, define a sequence $(x_n)$ by $x_0=x$ and $x_{n+1}=f(x_n)$. If the sequence converges, let $x_{\infty}$ denote the limit.

1. For $x=0$, show that the sequence is bounded and nondecreasing and find $x_{\infty}=\lambda$.

2. Find all $y \in \mbox{$\mathbb{R}^{}$}$ such that $y_{\infty}=\lambda$.

Problem 14   Exhibit a set of 2$\times$2 real matrices with the following property: A matrix $A$ is similar to exactly one matrix in $S$ provided $A$ is a 2$\times$2 invertible matrix of integers with all the roots of its characteristic polynomial on the unit circle.

Problem 15   Consider the differential equation $x''+x'+x^3=0$ and the function $f(x,x')= (x+x')^2+(x')^2+x^4$.

1. Show that $f$ decreases along trajectories of the differential equation.

2. Show that if $x(t)$ is any solution, then $(x(t),x'(t))$ tends to $(0,0)$ as $t \to \infty$.

Problem 16   Suppose that $A$ and $B$ are real matrices such that $A^t=A$,

$$v^t A v \ge 0$$

for all $v \in \mbox{$\mathbb{R}^{n}$}$ and

$$AB + BA = 0.$$

Show that $AB=BA=0$ and give an example where neither $A$ nor $B$ is zero.

Problem 17   Let $P_n$ be a sequence of real polynomials of degree $\leq D$, a fixed integer. Suppose that $P_n(x) \to 0$ pointwise for $0\leq x \leq 1$. Prove that $P_n \to 0$ uniformly on $[0,1]$.

Problem 18   Suppose that $f$ is analytic inside and on the unit circle $\vert z\vert=1$ and satisfies $\vert f(z)\vert<1$ for $\vert z\vert=1$. Show that the equation $f(z)=z^3$ has exactly three solutions (counting multiplicities) inside the unit circle.

Problem 19   Let $X$ be a compact metric space and $f : X \to X$ an isometry. Show that $f(X) = X$.

Problem 20   Let $R$ be a ring with multiplicative identity $1$. Call $x \in R$ a unit if $xy=yx=1$ for some $y \in R$. Let $G(R)$ denote the set of units. _ring>unit

1. Prove $G(R)$ is a multiplicative group.

2. Let $R$ be the ring of complex numbers $a+bi$, where $a$ and $b$ are integers. Prove $G(R)$ is isomorphic to $\mbox{$\mathbb{Z}^{}$}_4$ (the additive group of integers modulo $4$).