Rutgers - Graduate Program in Mathematics Written Qualifying Examination

Fall 1995

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 X = ℝ and let τ = {G ⊂ X|X \ G is countable} ∪ {∅}. Show i) τ is a topology on X; ii) in the topological space (X, τ) the collection U_{0} of neighborhoods of 0 does not have

a countable base.

2. Use contour integration to compute

∫_{0}^{∞} \frac{dx}{x^{4} + 1}.

3. Suppose that

A = [ \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ] .

Find a 3 × 3 orthogonal matrix U such that U^{−1}AU is diagonal.

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

4. a) How many Sylow 5-subgroups does the alternating group A_{5} have?

b) How many Sylow 5-subgroups does the alternating group A_{6} have?

c) How many Sylow 5-subgroups does a simple group of order 360 have? (\underline{Hint}: use part b).)

5. Let {f_{n}} and {g_{n}} be sequences of measurable functions on ℝ, and let p > 1. Assume i) each f_{n} is in L_{p} and f_{n} ⟶ f in L_{p}; ii) g_{n} ⟶ g a.e.; iii) |g_{n}| ≤ M < ∞ for each n.

Show that g_{n}f_{n} ⟶ gf in L_{p}.

6. Consider the topology on the set of integers ℤ for which {n + 7^{k}ℤ|k ∈ ℤ} is a basis for the neighborhoods of n ∈ ℤ.

i) Show that this is a Hausdorff topology;

ii) Show there is a sequence {n_{j}} in ℤ such that (n_{j})^{2} ⟶ 2 in this topology.

7. Consider the spiral S in the plane which in polar coordinates has the equation r = θ for θ ≥ 0. Let G = ℂ \ S. Show there is a holomorphic function L : G ⟶ ℂ such that e^{L(z)} = z^{2} for all z ∈ G.

8. Let ℤ_{2} be the field of two elements. Consider the matrix

A = [ \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ]

over ℤ_{2}.

a) Show that the characteristic polynomial of A factors as a product of linear polynomials over ℤ_{2}.

b) Find a non-singular matrix P over ℤ_{2} so that P^{−1}AP is in Jordan form.

9. Let v(x, t) be a C^{∞} vector field and ρ(x, t) a C^{∞} real-valued function defined for x = (x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} and t ∈ ℝ. Assume that for any ball B ⊂ ℝ^{3}

\frac{d}{dt}∫∫∫_{B} ρdx = −∫∫_{∂B} ρv · ndA

holds for all t where n is the exterior unit normal on ∂B and dA is the element of surface area. Show that

\frac{dρ}{dt} + ▽ · (ρv) = 0

for all (x, t).

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

1. Let f : ℝ ⟶ ℝ be a differentiable function. Assume that f(x) ⟶ L as x ⟶ ∞.

a) Show that if lim_{x→∞} f′(x) exists then lim_{x→∞} f′(x) = 0.

b) Is a) true if “lim_{x→∞} f′(x) exists” is replaced by “f′(x) is bounded”?

2. For a ring R let M_{n}(R) denote the ring of n × n matrices with entries in r. Prove that if R is a ring with multiplicative identity 1, then every two-sided ideal of M_{n}(R) is of the form M_{n}(I), where I is an ideal of R.

3. Let m be the Lebesgue measure on ℝ. Let A ⊂ ℝ be a measurable set such that there exists a number b with 0 < b < 1 and

m(A ∩ I) < b m(I)

for all open intervals I. Show that m(A) = 0.

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

4. Let {f_{n}} be a sequence of Lebesgue measurable functions on the interval (0, 1) such that

i) sup ∫_{0}^{1} |f_{n}|dx < ∞
ii) f_{n} ⟶ 0 in measure.

Show that

∫_{0}^{1} \sqrt{|f_{n}|} dx ⟶ 0.

5. Let (X, τ) be a compact metric space. Prove that if {U_{1}, U_{2}, …, U_{k}} is an open cover of X, then there exists a closed cover {C_{1}, C_{2}, …, C_{k}} with C_{i} ⊂ U_{i} for each i.

6. Let Q = {z ∈ ℂ|Re z > 0, Im z > 0} and let E = {z ∈ ℂ||z| < 1}. Give explicitly a holomorphic function f : Q ⟶ E which is one-to-one and onto.

7. Let 𝕁 denote the set of Gaussian integers; that is,

𝕁 = {α + βi|α, β ∈ ℤ}.

Let u and v be nonzero Gaussian integers, and let γ̄ denote the conjugate of the complex number γ.

a) Show that there exist Gaussian integers x, y such that uv̄ = xvv̄ + y and yȳ < (vv̄)^{2}.

b) Show that 𝕁 is a Euclidean domain whose degree function is complex absolute value squared.

c) Find the greatest common divisor (in 𝕁) of the two Gaussian integers 7 − 6i and 7 − 11i.

8. For what values of λ will the solution x(t) to

x′(t) = x^{3}(t) + λx^{2}(t)
x(0) = 1

exist for all t ≥ 0 and be uniformly bounded. Justify your answer fully.

9. Find two 4 × 4 matrices A and B such that:

i) A and B have the same characteristic polynomial; ii) A and B have the same minimal polynomial; iii) A and B are not similar.