Rutgers University - Graduate Program in Mathematics Written Qualifying Examination

Fall 1998 Day 1

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. Suppose μ^{∗} is an outer measure on X and A_{i}, i = 1, 2, …, is a countable sequence of disjoint μ^{∗}-measurable sets. Show that for any E ⊆ X,

μ^{∗}(E ∩ (∪_{i}A_{i})) = ∑_{i} μ^{∗}(E ∩ A_{i}).

2. Explain carefully how to evaluate

∫_{0}^{∞} \frac{cos x}{(A + x^{2})^{2}} dx

for A > 0 using the Residue Theorem. What happens to the value of this integral as A → ∞? What happens to the value of this integral as A → 0^{+}?

3. (a) Suppose that G is a simple group and that G has a subgroup H such that [G : H] = n > 1. Show that G is isomorphic to a subgroup of the symmetric group S_{n} on n letters.

(b) Prove that there is no simple group of order 300.

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

4. Let A, B be n × n real matrices. Suppose that A has n distinct real eigenvalues and that AB = BA. Show that B can be expressed as a polynomial in A.

5. Explicitly describe a function meromorphic in all of the complex plane ℂ whose set of poles is exactly all the Gaussian integers: {m + ni | m & n are integers}. Prove that your function has the desired properties.

6. The Cantor set C is the subset of the unit interval consisting of those x ∈ [0, 1] such that x has a base-3 expansion ∑ a_{j}3^{−j} with a_{j} ≠ 1 for all j. Alternatively C = ⋂_{j=1}^{∞} C_{j}, where C_{0} = [0, 1] and C_{j+1} is obtained from C_{j} by deleting the open middle-third of each subinterval of C_{j}. Prove that

(a) C is compact, totally disconnected, and nowhere dense.

(b) C has measure zero.

(c) C is uncountable.

7. (a) Show that the principal ideal ( 3 ) is not a prime ideal in ℤ[\sqrt{−5} ].

(b) Find a prime ideal P such that ( 3 ) ⊂ P ⊂ ℤ[\sqrt{−5} ].

8. Let m and μ be, respectively, the Lebesgue measure and the counting measure on the Borel σ-algebra on [0, 1].

(a) Show that m << μ but dm ≠ fdμ for any f.

(b) State the Lebesgue-Radon-Nikodym Theorem for signed measures, and explain why part (a) is not a counter-example!

9. (a) Define T_{t}f(z) = f(z + t). Suppose that f is an entire function and that T_{t}f → f \underline{uniformly} for all z ∈ ℂ as t → 0. Prove that f is a linear function.

(b) Give an example of a continuous function g defined on ℝ which is \underline{not} linear and which has the property that T_{t}g → g uniformly for all x ∈ ℝ as t → 0.

Rutgers University - Graduate Program in Mathematics Written Qualifying Examination

Fall 1998 Day 2

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.

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

1. Let μ be a measure on the σ-algebra of Lebesgue measurable subsets of ℝ. Assume that μ is translation invariant and that μ( [0, 1] ) < ∞. Prove that μ is a multiple of the Lebesgue measure. By giving a counterexample, show that the condition μ( [0, 1] ) < ∞ is essential.

2. Suppose that A and B are 6 × 6 nilpotent matrices over some field F. Assume that A and B have the same rank and the same minimal polynomial. Show that A and B are similar. Would the same be true for 7 × 7 nilpotent matrices? [ Hint: consider the Jordan normal forms of A and B.]

3. (a) Suppose that f is holomorphic in {z ∈ ℂ | |z| < 5} and that sup_{|z|<5}{|f(z)|} < 8. Find

A > 0 so that sup_{|z|<3}{|f′(z)|} ≤ A for all such f. Prove that your bound is valid.

(b) Find an example of a real-valued differentiable function g defined in the interval (−5, 5)

so that sup_{−5<x<5} {|g(x)|} < 8 and sup_{−3<x<3} {|g′(x)|} ≥ 1,000.

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

4. Let m be the Lebesgue measure on the real line ℝ. Suppose f_{i} → f, g_{i} → g a.e. with f_{i}, g_{i}, f, g ∈ L^{1}(ℝ). Also suppose that |f_{i}| ≤ |g_{i}|, and that ∫_{ℝ} g_{i} dm → ∫_{ℝ} g dm. Prove that ∫_{ℝ} f_{i} dm → ∫_{ℝ} f dm.

5. Prove that if F is a finite field, then every element of F can be expressed as a sum of two squares.

6. Let F_{b} be the entire function defined by F_{b}(z) = z^{4} + 100z^{2} + e^{−b(z+2)}. Prove that if b is a sufficiently large positive real number, then F_{b} has two roots (counting multiplicity) inside the unit circle.

7. Find and classify all finite singularities of the complex function

f(z) = \frac{cos z − z − \frac{π}{2}}{sin z + 1} .

8. Suppose that G_{1}, G_{2} are finite groups such that G_{1} ≠ 1 and G_{2} ≠ 1. Prove that the following statements are equivalent.

(a) Whenever H is a subgroup of the direct product G_{1} × G_{2}, then there exist subgroups H_{1}, H_{2} of G_{1}, G_{2} respectively such that H = H_{1} × H_{2}.

(b) The orders |G_{1}| and |G_{2}| are relatively prime.

9. Let m be the Lebesgue measure on the real line ℝ. Let f : ℝ → ℝ and g : ℝ → ℝ be positive functions such that f ∈ L^{p}(ℝ), g ∈ L^{q}(ℝ), where 1/p + 1/q = 1 and 1 < p < ∞, 1 < q < ∞. Suppose that ∫_{ℝ} fg dm = ||f||_{p}||g||_{q}. Prove that there is a constant C such that f(x)^{p} = Cg(x)^{q} for almost all x.