Rutgers University - Graduate Program in Mathematics Written Qualifying Examination

Spring 1997

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. Evaluate

∫_{−∞}^{∞} \frac{dx}{(1 + x^{2})^{n+1}} (n = 0, 1, 2, ⋯)

using complex analysis.

2. Assume that f(x) is differentiable and f′(x) is strictly increasing for x ≥ 0. If f(0) = 0, prove that f(x)/x is strictly increasing for x > 0.

3. Let A be an n × n matrix with rational entries satisfying the equation A^{2} − 2I = 0. Prove that n is even.

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

4. Let Ω ⊂ C be a bounded domain with smooth boundary. Let f be analytic in Ω and continuous up to the boundary. Assume that |f(z)| is a constant for all z on the boundary of Ω. Prove that either f is identically equal to some constant in Ω, or f(z) = 0 has a solution in Ω.

5. Let G be a group with finitely many subgroups. Prove that G is finite.

6. Evaluate the sum of the infinite series

1^{2}/0! + 2^{2}/1! + 3^{2}/2! + ⋯ + (n + 1)^{2}/n! + ⋯

7. Let A, B, S be n × n complex matrices satisfying AS = SB and S ≠ 0. Prove that A and B have a common eigenvalue.

8. Let f be any Lebesgue measurable function on R^{n}. Let |E| denote the Lebesgue measure of a set E that is measurable. Prove that for 0 < p < ∞ we have

p ∫_{0}^{∞} λ^{p−1}|{x : |f(x)| > λ}| dλ = ∫_{R^{n}} |f|^{p} dx.

9. Let f : R^{n} → R^{n} be a continuously differentiable function. Assume furthermore that the Jacobian of f is not zero at any point. Assume that f is proper (the inverse image of a compact set is compact). Prove that f is onto. Give an example in R^{2} to show that one cannot drop the assumption of being proper.

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

1. We are given a 3×3 matrix A with real entries. Furthermore, we are given that A is not triangulable over the real numbers (not similar to an upper or lower triangular matrix). Prove that A is diagonalizable over the complex numbers.

2. Given that:

I = ∫_{0}^{∞} \frac{sin x}{x} dx,

(a)Show that I is finite as an appropriate improper integral, by simple calculus.

(b) Compute I using complex analysis.

3. Let u = u(x, y) be a smooth function defined on R^{2} which is 2π-periodic in each variable, i.e.,

u(x + 2nπ, y + 2mπ) = u(x, y) for all x, y ∈ R, n, m ∈ Z.

Prove:

∫_{0}^{2π} ∫_{0}^{2π}(u_{xx}u_{yy} − u_{xy}^{2}) dxdy = 0.

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

4. Let f(x) = c_{1}e^{r_{1}x} +c_{2}e^{r_{2}x} +⋯+c_{n}e^{r_{n}x}, where c_{i} ≠ 0. Assume that the r_{i} are distinct. Prove that f cannot have n real zeros.

5. Let f be a holomorphic function in the unit disk D = {z : |z| < 1}, which is continuous on the closed disk. Suppose that f vanishes on the arc, given by the points{e^{iθ} : 0 ≤ θ ≤ π}. Show f(z) = 0 for all z ∈ D.

6. Let A and B be disjoint compact subsets of a Hausdorff topological space. Prove the existence of disjoint open sets U and V satisfying A ⊂ U and B ⊂ V.

7. Let G be a group of order 90. Prove that G is solvable.

8. Let f(x) = 2^{−n} if x = k2^{−n} where k is odd and n is a natural number, and f(x) = 0 otherwise. Prove that f^{2} has bounded variation on the interval [0, 1].

9. Give an example of a measurable function f : R × R → R such that the two double integrals

∫_{R} ∫_{R} f(x, y) dxdy and ∫_{R} ∫_{R} f(x, y) dydx

are defined, but are not equal. Fubini’s Theorem tells us that the condition ∫ ∫ |f(x, y)| dxdy < ∞ cannot be satisfied by such a function.