Rutgers - Graduate Program in Mathematics Written Qualifying Examination

Spring 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. Compute

\frac{1}{2πi} ∫_{|z|=100.5} \frac{πz}{sin πz} dz.

2. Let the function y = f(x) satisfy

\{ \begin{matrix}y′′ = λ(x)y for 0 ≤ x ≤ 1 \\ y(0) = y(1) = 0, \end{matrix}

where λ(x) > 0 for 0 ≤ x ≤ 1. Show that f(x) = 0 for all x ∈ [0, 1].

3. Let p be a prime integer. Prove that a group of order p^{2} is Abelian.

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

4. Suppose that {A_{n} | n = 1, 2, … } is a sequence of real d × d matrices such that lim(A_{n}) exists and equals A. Determine whether each of the following two statements is true or false. If the statement is true, prove it. If it is false, give a counterexample.

a) ∃n_{0}, ∀n ≥ n_{0}, rank(A_{n}) ≥ rank(A);

b) ∃n_{0}, ∀n ≥ n_{0}, rank(A_{n}) ≤ rank(A).

5. Let f_{n} : [0, 1] ⟶ [0, ∞) be a Lebesgue integrable function for n = 1, 2, …. Assume that

∫_{0}^{1} f_{n}(x)dx = 1 and ∫_{1/n}^{1} f_{n}(x)dx < 1/n

for all n. Let g : [0, 1] ⟶ [0, ∞] be defined by

g(x) = sup{f_{n}(x) | n = 1, 2, 3, … }.

Prove that

∫_{0}^{1} g(x)dx = ∞.

6. Write E = {z ∈ ℂ : |z| = 1}, ∂E = {z ∈ ℂ : |z| = 1}, and Ē = ∂E ∪ E. Let f : Ē → Ē be continuous and holomorphic on E. Assume that f(∂E) is contained in ∂E. Assume also that f′(0) ≠ 0 and that f(z) = 0 ⇔ z = 0. Show that f : E → E is one-to-one and onto.

7. Let A be the 3-dimensional algebra over ℂ with basis 1, x, x^{2} and x^{3} = 1. Let B be the 3-dimensional commutative algebra over ℂ with basis 1, x, y and x^{2} = xy = y^{2} = 0. Show that A has only finitely many ideals, but that B has infinitely many.

8. Let X be a metric space with no isolated points. Assume that the set X is countably infinite. Show that X cannot be compact.

9. Let R be a commutative ring with unity 1, and let f(x) = 1 + a_{1}x + ⋯ + a_{n}x^{n}, g(x) = b_{0} + b_{1}x + ⋯ + b_{m}x^{m} be polynomials with coefficients a_{i}, b_{j} ∈ R. If n > 0 and f(x)g(x) = 1, show that a_{n}^{r+1}b_{m−r} = 0 for all r and that the elements a_{1}, a_{2}, …, a_{n} are nilpotent in R.

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

1. Let f be a measurable function on ℝ with

∫_{−∞}^{∞}(1 + |x|)|f(x)|dx < ∞.

Show that

F (ξ) = ∫_{−∞}^{∞} e^{ixξ}f(x)dx

is differentiable at each ξ ∈ ℝ.

2. The least upper bound axiom for the real numbers ℝ states that every non-empty S ⊂ ℝ which is bounded above has a least upper bound. Derive from this axiom, without citing any theorems, that the interval [0, 1] is a compact subset of ℝ.

3. Let e_{1}, e_{2}, …, e_{n+2} be idempotent n×n matrices over a field. (A matrix e is idempotent if e^{2} = e.) Show that at least two of these matrices are conjugate.

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

4. Let M be a 6 × 6 real symmetric matrix whose minimal polynomial has degree 2. Let Z(M) denote the linear subspace of ℝ^{6×6} consisting of all symmetric matrices that commute with M. Determine the set of integers d that are possible values for the dimension of Z(M).

5. Show that there is a branch of \sqrt{z^{2} − 1} which is holomorphic on ℂ \ {x ∈ ℝ : −1 ≤ x ≤ 1}.

6. Let H be a subgroup of the group G = GL_{3}(𝔽_{p}) of invertible 3 × 3 matrices over the field 𝔽_{p} of order p. If H has order p^{2}, show that there is some g ∈ G such that every matrix in gHg^{−1} is upper triangular.

7. A plane graph is a subset E of the plane ℝ^{2} which can be written as a finite union of subspaces e_{1}, e_{2}, …, e_{q} (called edges), each of which is homeomorphic to the unit interval [0, 1], such that distinct edges e_{i} and e_{j} intersect in at most one point, which is an endpoint of each edge. The set V = {v_{1}, v_{2}, …, v_{p}} of endpoints of edges is called the set of vertices of the graph. Show that the complement ℝ^{2} \ E has c = q + 2 − p connected components.

8. Let F : ℝ^{3} → ℝ^{3} be a vector field of the form F(r) = g(|r|^{2})r for r = (x, y, z) ∈ ℝ^{3}, where g : ℝ^{3} → ℝ is a C^{1} function. Let Σ be a smooth surface in ℝ^{3} whose boundary is a simple closed curve γ. Show that the line integral

∫_{γ} F · dr = 0.

9. For n = 1, 2, … let f_{n} : ℝ → ℝ be continuous, and assume that f_{n}(x) → f(x) as n → ∞ for each x ∈ ℝ. Let a ∈ ℝ. Show that {x ∈ ℝ|f(x) > a} is a countable union of closed sets.