Preliminary Exam - Fall 1995

Problem 1   Let $G$ be a group generated by $n$ elements. Find an upper bound $N(n,k)$ for the number of subgroups $H$ of $G$ with the index $[G:H]=k$.

Problem 2   Let $A$ be a finite subset of the unit disc in the plane, and let $N(A,r)$ be the set of points at distance $\leq r$ from $A$, where $0 < r < 1$. Show that the length of the boundary $N(A,r)$ is, at most, $C/r$ for some constant $C$ independent of $A$.

Problem 3   Find the radius of convergence $R$ of the Taylor series about $z=1$ of the function $f(z)=1/(1+z^2+z^4+z^6+z^8+z^{10})$. Express your answer in terms of real numbers and square roots only.

Problem 4   Suppose $A$ and $B$ are real $n\times n$ matrices and $C$ is a complex $n\times n$ matrix such that

$$CAC^{-1}=B \ .$$

Find a real $n\times n$ matrix $D$ such that $DAD^{-1}=B$.

Problem 5   Prove that $\mbox{$\mathbb{Q}\,^{}$}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain and that its field of fractions is isomorphic to the field of rational functions $\mbox{$\mathbb{Q}\,^{}$}(t)$.

Problem 6   Determine all real numbers $L > 1$ so that the boundary value problem

$$x^2y''(x)+y(x)=0, \qquad 1\leq x\leq L$$

$$y(1)=y(L)=0$$

has a nonzero solution.

Problem 7   Let $g(z)=\sum^\infty_{n=0} g_nz^n$ and $h(z)=\sum^\infty_{n=0}h_nz^n$ be entire functions. Find a formula for the coefficients $f_n$ in the Taylor expansion about $z=0$ of

$$f(z) = \frac{1}{2\pi i} \int_{\vert w\vert=1} g(z/w)h(w) \ \frac{dw}{w} \; \cdot$$

Problem 8   Show that an $n\times n$ matrix of complex numbers $A$ satisfying

$$\vert a_{ii}\vert > \sum_{j\neq i} \vert a_{ij}\vert$$

for $1\leq i\leq n$ must be invertible.

Problem 9   Let $x_1$ be a real number, $0 < x_1 < 1$, and define a sequence by $x_{n+1}=x_n-x_n^{n+1}$. Show that $\liminf_{n\to\infty} x_n > 0$.

Problem 10   Let $\mbox{\bf {F}}$ be a field and $\mbox{\bf {F}}^*$ be the multiplicative group of nonzero elements. Let $G$ be a subgroup of $\mbox{\bf {F}}^*$ of finite order $n$. Show that $G$ is cyclic.

Problem 11   Let $f(z)=u(z)+iv(z)$ be holomorphic in $\vert z\vert < 1$, $u$ and $v$ real. Show that

$$\int^{2\pi}_0 u(re^{i\theta})^2d\theta = \int^{2\pi}_0 v(re^{i\theta})^2d\theta$$

for $0 < r < 1$ if $u(0)^2=v(0)^2$.

Problem 12   Let $f$ and $f'$ be continuous on $[0,\infty)$ and $f(x)=0$ for $x\geq 10^{10}$. Show that

$$\int^\infty_0 f(x)^2dx \leq 2 \sqrt{\int^\infty_0 x^2f(x)^2dx} \ \sqrt{\int^\infty_0 f'(x)^2dx} \ \ .$$

Problem 13   Show that

$$(1+z+z^2+\cdots +z^9)\! (1+z^{10}+z^{20}+\cdots +z^{90})\! (1+z^{100}+z^{200}+\cdots +z^{900})\cdots\!=\!\frac{1}{1\!-\!z}$$

for $\vert z\vert < 1$.

Problem 14   Let $f(x)\in \mbox{$\mathbb{Q}\,^{}$}[x]$ be a polynomial with rational coefficients. Show that there is a $g(x)\in \mbox{$\mathbb{Q}\,^{}$}[x]$, $g\neq 0$, such that $f(x)g(x)=a_2x^2+a_3x^3+a_5x^5+\cdots +a_px^p$ is a polynomial in which only prime exponents appear.

Problem 15   Let $f:\mbox{$\mathbb{R}^{}$}\to \mbox{$\mathbb{R}^{}$}$ be a $C^\infty$ function. Assume that $f(x)$ has a local minimum at $x=0$. Prove there is a disc centered on the $y$ axis which lies above the graph of $f$ and touches the graph at $(0,f(0))$.

Problem 16   Let $A$ and $B$ be nonsimilar $n\times n$ complex matrices with the same minimal and the same characteristic polynomial. Show that $n\geq 4$ and the minimal polynomial is not equal to the characteristic polynomial.

Problem 17   Let $f_1,f_2,\dots ,f_n$ be continuous real valued functions on $[a,b]$. Show that the set $\{ f_1,\dots ,f_n\}$ is linearly dependent on $[a,b]$ if and only if

$$\det\left( \int^b_a f_i(x)f_j(x)dx\right)=0 \;.$$

Problem 18   Let $f:\mbox{$\mathbb{R}^{}$}\to \mbox{$\mathbb{R}^{}$}$ be a nonzero $C^\infty$ function such that $f(x)f(y)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$ and that $f(x)\to 0$ as $\vert x\vert\to\infty$.

1. Prove that $f$ is an even function and that $f(0)$ is $1$.

2. Prove that $f$ satisfies the differential equation $f'(x)=f''(0)xf(x)$, and find the most general function satisfying the given conditions.