Rutgers University - Graduate Program in Mathematics Written Qualifying Examination

Spring 1998

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 A be an n×n-matrix over an algebraically closed field k. Let P (x) be a polynomial over k. Show that λ ∈ k is an eigenvalue of P (A) if and only if λ = P (γ) where γ is an eigenvalue of A.

2. Let D be an open connected domain in the complex plane ℂ and f(z) an analytic function defined on D. Assume that there is a smooth curve Γ in ℂ such that f(D) ⊆ Γ. Prove that f is constant.

3. Let f be a real valued function on an interval I ⊆ ℝ such that f′(x) exists for all x ∈ I. Show, that f′ has the intermediate value property, i.e., for all a, b ∈ I and c ∈ ℝ with f′(a) < c < f′(b) there is ξ in the interval bounded by a and b such that f′(ξ) = c.

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

4. Show that there is no simple group of order 224.

5. Let E be a Lebesgue measurable subset of ℝ with 0 < m(E) < ∞. Let χ_{E}(x) denote the characteristic function of E.

a) Prove that

φ(x) = ∫_{ℝ} χ_{E}(y)χ_{E}(x + y)dy.

is a continuous function on ℝ. [Hint: Observe that φ(x) is an L^{2}-scalar product.]

b) Let F = {x − y | x, y ∈ E}. Prove that there exists δ > 0 such that (−δ, δ) ⊆ F.

6. Compute

∫_{−∞}^{∞} \frac{cos x}{x^{2} + a^{2}} dx

for a > 0. Hint: cos x = Re(e^{ix}).

7. Let X be a metric space. A point x ∈ X is called isolated if the set {x} is open. Show that the following two properties are equivalent:

a) Every subset of X is open or closed.

b) X contains at most one non-isolated point.

8. Let f be an analytic mapping of the open unit disk D into itself. Suppose there are two points z_{1} ≠ z_{2} ∈ D with f(z_{1}) = z_{1} and f(z_{2}) = z_{2}. Show that f is the identity map.

9. Let A and B be real symmetric n × n-matrices. Assume that A is positive definite. Show that there is an invertible real n × n-matrix S such that both SAS^{t} and SBS^{t} are diagonal. (Here S^{t} is the transpose of S.)

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

1. Let f(x) be a continuous function on ℝ with compact support and α > 1. Show that

g_{N}(x) = ∑_{n=0}^{N} \frac{f(x − n)}{(n + 1)^{α}}

converges in L^{2} as N → ∞.

2. For N a positive integer, determine

∫_{|z|=π(N+ \frac{1}{2} )} e^{iz}cot(z) dz.

3. Let A be the ring of C^{∞}-functions on ℝ.

a) Show that

P := \{f ∈ A | \frac{d^{k}f}{dx^{k}} (0) = 0 for all k ≥ 0\}

is a prime ideal of A.

b) Show that

Q := {f ∈ A | there is a neighborhood U of 0 with f|_{U} ≡ 0}

is an ideal but not a prime ideal of A.

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

4. Let P be the group of all permutations of ℤ. For φ ∈ P define its support as Supp(φ) := {x ∈ ℤ | φ(x) ≠ x}. Let A_{∞} be the set of all φ ∈ P with S := Supp(φ) finite and such that φ|_{S} is an even permutation of S.

a) Show that A_{∞} is a subgroup of P.

b) Show that A_{∞} is a simple group.

5. A sequence of L^{2}-functions {f_{n}} on ℝ is said to converge weakly to f ∈ L^{2} if for every g ∈ L^{2}

∫ f_{n}g ⟶ ∫ fg for n → ∞.

Give an example (with proof) of a sequence of functions which converges weakly to 0 but does not converge to 0 in the L^{2}-norm.

6. Let S_{n} be the symmetric group on n letters.

a) Show: an automorphism of S_{n} is inner if and only if it maps transpositions to transpositions.

b) Show (assuming part a): every automorphism of S_{7} is inner.

7. Let f and g be functions which are holomorphic in a neighborhood of the closed unit disk |z| ≤ 1. Assume |g(z)| < |f(z)| for all z with |z| = 1. Prove that f and f + g have the same number of zeroes (counted with multiplicities) in the closed unit disk. [Hint: consider f + tg for 0 ≤ t ≤ 1.]

8. Let f(z) be an analytic function on ℂ such that there exists a constant C > 0 and an integer k ≥ 0 with

|f(z)| ≤ C(1 + |z|)^{k}

for all z ∈ ℂ. Prove that f is a polynomial in z.

9. Let f be a non-negative measurable function on the interval I = [0, 1] such that ∫_{I} fg < ∞ for every non-negative measurable function g on I with ∫_{I} g < ∞. Show that there is a constant M > 0 with f < M almost everywhere.