Preliminary Exam - Summer 1981

Problem 1   Let

$$y(h) = 1 - 2\sin^2(2\pi h), \hspace{.2in} f(y)=\frac{2}{1+\sqrt{1-y^2}} \cdot$$

Justify the statement

$$f\left(y(h)\right) = 2 - 4\sqrt{2}\pi\left\vert h\right\vert + O(h^2)$$

where

$$\limsup_{h\to 0}\frac{O(h^2)}{h^2} < \infty.$$

Problem 2   Let $G$ be a finite group, and let $\varphi$ be an automorphism of $G$ which leaves fixed only the identity element of $G$.

1. Show that every element of $G$ may be written in the form $g^{-1}\varphi(g)$.
2. If $\varphi$ has order $2$ (i.e., $\varphi\cdot\varphi=\mathrm{id}$) show that $\varphi$ is given by the formula $g\mapsto g^{-1}$ and that $G$ is an abelian group whose order is odd.

Problem 3   Prove or disprove: The set $\mathbb{Q}\,^{}$ of rational numbers is the intersection of a countable family of open subsets of $\mathbb{R}^{}$.

Problem 4   Let $f:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$ be continuous, with

$$\int_{-\infty}^{\infty}\vert f(x)\vert\,dx < \infty.$$

Show that there is a sequence $(x_n)$ such that $x_n \to \infty$, $x_nf(x_n) \to 0$, and $x_nf(-x_n) \to 0$ as $n \to \infty$.

Problem 5   Let $S$ denote the vector space of real $n \times n$ skew-symmetric matrices. For a nonsingular matrix $A$, compute the determinant of the linear map $T_A:S\to S,\, T_A(X)=AXA^{-1}$.

Problem 6   Let $S \mathbb{O}(3)$ denote the group of orthogonal transformations of $\mathbb{R}^{3}$ of determinant $1$. Let $Q \subset S \mathbb{O}(3)$ be the subset of symmetric transformations $\neq I$. Let $P^2$ denote the space of lines through the origin in $\mathbb{R}^{3}$.

1. Show that $P^2$ and $S \mathbb{O}(3)$ are compact metric spaces (in their usual topologies).

2. Show that $P^2$ and $Q$ are homeomorphic.

Problem 7   Compute

$$\frac{1}{2\pi i}\int_C\frac{dz}{\sin\frac{1}{z}},$$

where $C$ is the circle $\vert z\vert = 1/5$, positively oriented.

Problem 8   Show that $x^{10}+x^9+x^8+\cdots+x+1$ is irreducible over $\mathbb{Q}\,^{}$. How about $x^{11}+x^{10}+\cdots+x+1?$

Problem 9   Let $f:\mbox{$\mathbb{R}^{}$}\to \mbox{$\mathbb{R}^{}$}$ be the function of period $2\pi$ such that $f(x)=x^3$ for $-\pi \leq x < \pi$.

1. Prove that the Fourier series for $f$ has the form $\sum_1^{\infty}b_n\sin nx$ and write an integral formula for $b_n$ (do not evaluate it).

2. Prove that the Fourier series converges for all $x$.

3. Prove
   
   $$\sum_{n=1}^{\infty}b_n^2 = \frac{2\pi^6}{7}\cdot$$
   
   

Problem 10   Let $S$ be a vector space of complex sequences $\{a_n\}_{n=1}^{\infty}$. _Fibonacci numbers Define the map $T : S \to S$ by $T\{a_1,a_2,a_3,\ldots\} = \{a_2,a_3,\ldots\}$.

1. Describe the eigenvectors of $T$.

2. Consider the difference equation $x_{n+2}=x_{n+1}+x_n$. Show that the solutions of this equation define a two-dimensional subspace $E \subset S$ and that $T(E) \subset E$. Find an explicit basis for $E$.

3. The Fibonacci numbers are defined recursively by $f_1=1$, $f_2=1$, and $f_{n+2}=f_{n+1}+f_n$ for $n \geq 1$. Find and explicit formula for $f_n$.

Note: See also Problem .

Problem 11   Show that the equation

$$x\left(1+ \log \left(\frac{1}{\varepsilon\sqrt{x}}\right)\right) =1, \quad x>0, \quad \varepsilon>0\; ,$$

has, for each sufficiently small $\varepsilon>0$, exactly two solutions. Let $x(\varepsilon)$ be the smaller one. Show that

1.

$x(\varepsilon) \to 0$ as $\varepsilon \to 0+$;

yet for any $s>0$,

2.

$\varepsilon^{-s}x(\varepsilon) \to \infty$ as $\varepsilon \to 0+$.

Problem 12   Show that no commutative ring with identity has additive group isomorphic to $\mbox{$\mathbb{Q}\,^{}$}/\mbox{$\mathbb{Z}^{}$}$.

Problem 13   Let $G$ be an additive group, and $u,v: G \to G$ homomorphisms. Show that the map $f: G \to G$, $f(x) = x-v \left( u(x) \right)$ is surjective if the map $h:G\to G$, $h(x)=x-u\left(v(x)\right)$ is surjective.

Problem 14   Let $I \subset \mbox{$\mathbb{R}^{}$}$ be the open interval from $0$ to $1$.
Let $f:I \to \mbox{$\mathbb{C}\,^{ }$}$ be $C^1$ (i.e., the real and imaginary parts are continuously differentiable). Suppose that $f(t) \to 0,\; f'(t) \to C \neq 0$ as $t \to 0+$. Show that the function $g(t)=\vert f(t)\vert$ is $C^1$ for sufficiently small $t>0$ and that $\lim_{t\to 0+}g'(t)$ exists, and evaluate the limit.

Problem 15   Let $V$ be a finite-dimensional vector space over the rationals $\mathbb{Q}\,^{}$ and let $M$ be an automorphism of $V$ such that $M$ fixes no nonzero vector in $V$. Suppose that $M^p$ is the identity map on $V$, where $p$ is a prime number. Show that the dimension of $V$ is divisible by $p-1$.

Problem 16   Let $\{f_n\}$ be a sequence of continuous functions $[0,1] \to \mbox{$\mathbb{R}^{}$}$ such that

$$\int_0^1 \left( f_n(x)-f_m(x)\right)^2\,dx \to 0 \;\; as\;\; n,m \to \infty.$$

Let $K:[0,1] \times [0,1] \to \mbox{$\mathbb{R}^{}$}$ be continuous. Define $g_n:[0,1] \to \mbox{$\mathbb{R}^{}$}$ by

$$g_n(x) = \int_0^1K(x,y)f_n(y)\,dy.$$

Prove that the sequence $\{g_n\}$ converges uniformly.

Problem 17   Suppose that $f(z)$ and $g(z)$ are entire functions such that $\vert f(z)\vert\leq \vert g(z)\vert$ for all $z$. Show that $f(z)=cg(z)$ for some constant $c \in \mbox{$\mathbb{C}\,^{}$}$.

Problem 18   Let $A$ and $B$ be square matrices of rational numbers such that $CAC^{-1}=B$ for some real matrix $C$. Prove that such a $C$ can be chosen to have rational entries.

Problem 19   Prove that the number of roots of the equation $z^{2n}+\alpha^2z^{2n-1}+\beta^2=0$ ($n$ a natural number, $\alpha$ and $\beta$ real, nonzero) that have positive real part is

1. $n$ if $n$ is even, and

2. $n-1$ if $n$ is odd.

Problem 20   Let $y=y(x)$ be a solution of the differential equation $y''=-\vert y\vert$ with $-\infty<x<\infty$, $y(0)=1$, and $y'(0)=0$.

1. Show that $y$ is an even function.

2. Show that $y$ has exactly one zero on the positive real axis.