Preliminary Exam - Spring 1993

Problem 1   Let $(a_{n})$ and $(\varepsilon_{n})$ be sequences of positive numbers. Assume that $\lim_{n \to \infty}\varepsilon_{n}=0$ and that there is a number $k$ in $(0,1)$ such that $a_{n+1} \leq k a_{n} + \varepsilon_{n}$ for every $n$. Prove that $\lim_{n \to \infty}a_{n}=0$.

Problem 2   Let $A =(a_{ij})$ be an $n \times n$ matrix such that $\sum_{j=1}^{n} \vert a_{ij}\vert < 1$ for each $i$. Prove that $I-A$ is invertible.

Problem 3   Evaluate

$$\int_{- \infty}^{\infty} \frac{x^{3} \sin x}{(1+x^{2})^{2}}\,dx\, .$$

Problem 4   Suppose that the group $G$ is generated by elements $x$ and $y$ that satisfy $x^{5}y^{3}=x^{8}y^{5}=1$. Does it follow that $G$ is the trivial group?

Problem 5   Let $k$ be a positive integer. For which values of the real number $c$ does the differential equation

$$\frac{d^{2}x}{dt^{2}}-2 c \frac{dx}{dt}+x=0$$

have a solution satisfying $x(0)=x(2 \pi k)=0$?

Problem 6   Find a list of real matrices, as long as possible, such that

• the characteristic polynomial of each matrix is $(x-1)^{5}(x+1)$,
• the minimal polynomial of each matrix is $(x-1)^{2}(x+1)$,
• no two matrices in the list are similar to each other.

Problem 7   Let $f_{n}:[0, 4] \to \mathbb{R}\: \: (n=1, 2, \ldots)$ be continuous functions that are twice continuously differentiable on $(0, 4)$ and satisfy

• $f_{n}(1)=f'_{n}(1)=0\,$;
• $\vert f'_{n}(x)\vert \leq C$ for all $x$ in $(0, 4)$, where $C$ is a constant, independent of $n$.

Prove that the sequence $\{f_{n}\}$ has a uniformly convergent subsequence.

Problem 8   Classify up to isomorphism all groups of order $45$.

Problem 9   Let $a$ be a complex number and $\varepsilon$ a positive number. Prove that the function $f(z)= \sin z + \frac{1}{z-a}$ has infinitely many zeros in the strip $\vert \Im z\vert < \varepsilon$.

Problem 10   Let $f$ be a real valued $C^{1}$ function on $[0, \infty )$ such that the improper integral $\int_{1}^{\infty}\vert f'(x)\vert dx$ converges. Prove that the infinite series
$\sum_{n=1}^{\infty} f(n)$ converges if and only if the integral $\int_{1}^{\infty}f(x)dx$ converges.

Problem 11   Let $P$ be the vector space of polynomials over $\mathbb{R}$. Let the linear transformation $E : P \to P$ be defined by $Ef=f+f'$, where $f'$ is the derivative of $f$. Prove that $E$ is invertible.

Problem 12   Prove that for any fixed complex number $\zeta$,

$$\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{2 \zeta \cos \theta}d \th... ...m_{n=0}^{\infty} \left( \frac{\zeta ^{n}}{n!} \right)^{2} \cdot$$

Problem 13   Prove that no commutative ring with identity has additive group isomorphic to $\mathbb{Q}/\mathbb{Z}$.

Problem 14   Prove that every solution $x(t)\; (t \geq 0)$ of the differential equation

$$\frac{dx}{dt}=x^{2}-x^{6}$$

with $x(0) > 0$ satisfies $${\lim_{t \to \infty} x(t)=1}$$.

Problem 15   Let $\Lambda$ be the set of 2$\times$2 matrices of the form

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

where a and b are elements of a given field $\mbox{\bf {F}}$. Prove that $\Lambda$, with the usual matrix operations, is a commutative ring with identity. For which of the following fields $\mbox{\bf {F}}$ is $\Lambda$ a field? $\mbox{\bf {F}}= \mathbb{Q}, \; \mathbb{C}, \; \mbox{$\mathbb{Z}^{}$}_5, \; \mbox{$\mathbb{Z}^{}$}_7$.

Problem 16   Prove that $${\frac{x^{2}+y^{2}}{4} \leq e^{x+y-2}}$$ for $x \geq 0\,,\; y \geq 0\,$.

Problem 17   Prove that if $G$ is a group containing no subgroup of index $2$, then any subgroup of index $3$ in $G$ is a normal subgroup.

Problem 18   Let $f$ be an analytic function in the unit disc, $\vert z\vert < 1$.

1. Prove that there is a sequence $(z_{n})$ in the unit disc with $\lim_{n \to \infty} \vert z_{n}\vert=1$ and $\lim_{n \to \infty} f(z_{n})$ exists (finitely).
2. Assume $f$ nonconstant. Prove that there are two sequences $(z_{n})$ and $(w_{n})$ in the disc such that $\lim_{n \to \infty} \vert z_{n}\vert=\lim_{n \to \infty} \vert w_{n}\vert=1$, and such that both limits $\lim_{n \to \infty} f(z_{n})$ and $\lim_{n \to \infty} f(w_{n})$ exist (finitely) and are not equal.