Rutgers - Graduate Program in Mathematics Written Qualifying Examination

Fall 1996

This exam will be given over two days, in two three hour sessions. Each session will consist of 3 required questions and a choice of 3 out of 6 remaining questions. The basic idea is to ensure that all students at least attempt a range of questions, but one area of weakness should not be overly magnified.

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First Day – Part I: Answer each of the following three questions.

1. Let U be an open connected subset of ℝ^{n}. Show that U is pathwise connected, i.e. that any two points of U may be connected by a continuous path lying entirely in U.

2. Find a conformal map taking the disk {z | |z| < 1/2} to the half-plane {z | Re(z) > 1}. Justify the correctness of your solution.

3. Let M be an n×n matrix over C. For 1 ≤ i ≤ n, let a_{i} = ∑_{j=1}^{n} |M_{ij}|. Prove that if λ is

an eigenvalue of M, then |λ| ≤ max_{1≤i≤n} a_{i}.

First Day – Part II: Answer \underline{three} out of the following six questions.

4. Let f : [0, 1] → [0, 1] be a continuous, nondecreasing function such that f(0) = 0 and f(1) = 1. For such an f, f′(x) exists for almost every x in [0, 1]. Use Fatou’s lemma to show that

∫_{0}^{1} f′(x) dx ≤ 1.

Give an example to show that ∫_{0}^{1} f′(x) dx < 1 is a possibility.

5. The orthogonal group O_{2}(ℝ) is the set of all 2 × 2 matrices A over ℝ satisfying

AA^{t} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ). Let G be a finite subgroup of O_{2}(ℝ). Show that G is either a cyclic

group or a dihedral group.

6. Show that there are at least 3 non-abelian groups of order 16. Make sure to show the groups are different.

7. Let a_{1}, a_{2}, … be a sequence of positive real numbers satisfying a_{i} + a_{j} ≥ a_{i+j} for all i, j ≥ 1. Prove that lim sup_{i→∞} \frac{a_{i}}{i} ≤ \frac{a_{j}}{j} for all j ≥ 1. Show that \{a_{i}/i : i ≥ 1\} converges to a

finite limit.

8. Let M denote the surface in ℝ^{3} defined by the equation x^{2} + y^{3} + z^{5} = 0, and let S^{2} be the sphere of radius 1 about the origin. Show that M ∩ S is a curve.

9. Let f be a function which is analytic everywhere on the complex plane except at the points 0, i, −i, and ∞, where it has poles. Show that

f(z) = \frac{P (z)}{(z(z^{2} + 1))^{m}}

for some polynomial P and integer m.

Second Day – Part I: Answer each of the following three questions.

1. In this problem, measurable means Lebesgue measurable. Let f be a real-valued function such that f^{−1}([a, ∞)) is measurable for every real number a. Show that f^{−1}(B) is measurable for every Borel subset B of ℝ.

2. Let U ⊂ ℝ^{n} be open and suppose x_{0} ∈ U. Let f be a function which is continuous on U and continuously differentiable on U − {x_{0}}. If

ℓ_{i} = lim_{x→x_{0}} \frac{∂f}{∂x_{i}} (x)

exists and is finite for 1 ≤ i ≤ n, show that f is continuously differentiable on U.

3. Let f(x) = x^{5} − 5x^{2} + 1. Prove that f has exactly 3 real roots and is irreducible over the rational numbers ℚ.

Second Day – Part II: Answer \underline{three} of the following six questions.

4. Let f be analytic in a neighborhood of the origin in the complex plane. Suppose that there is a sequence {a_{n}} of distinct real numbers converging to 0 and that f(a_{n}) is real for each n. Show that f(z) = \overline{f(z̄)} in a neighborhood of 0.

5. Prove that for every prime p, the set of non-zero elements of ℤ/pℤ form a cyclic group of order p − 1 under multiplication.

6. Let A be a 2×2 matrix over C. Prove that the number of matrices X satisfying X^{2} = A is 0, 2, 4, or ∞.

7. Let X_{1}, X_{2}, … be a sequence of finite sets, each endowed with the discrete topology, and consider the infinite product ∏_{1}^{∞} X_{i} as a topological space with the product topology. Suppose that for every n > 1, f_{n} is a function from X_{n} to X_{n−1}. Let X be the subset of ∏_{1}^{∞} X_{i} consisting of all sequences (x_{1}, x_{2}, …) with f_{n}(x_{n}) = x_{n−1} for every n > 1. Show that X is compact in the subspace topology.

8. Let A be a 3×3 symmetric matrix over ℝ with det A < 0 and a_{11} + a_{22} + a_{33} > 0. Show that there is a matrix P such that

P AP^{T} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) .

9. Let f : ℝ → ℝ be a Lebesgue measurable function such that ∫_{−∞}^{∞} |f(x)| dx < ∞.

Assume that ∫_{a}^{b} f(x) dx = 0 for all a < b. Prove that f(x) = 0 for almost every x.