Rutgers University - Graduate Program in Mathematics

Written Qualifying Examination

Fall 1999

Day 1

This exam will be given in two three-hour sessions, one today and one tomorrow. At each session the exam will have two parts. Answer all three of the questions on Part I and three of the six questions on Part II. If you work on more than three questions on Part II, indicate clearly which three should be graded.

First Day—Part I: Answer each of the following three questions.

1. Let p(x) and q(x) be relatively prime polynomials with real coefficients, of strictly positive degrees d_{1} and d_{2}, respectively. Let r(x) be a polynomial of degree less than

d_{1} + d_{2}. Show that the rational function \frac{r(x)}{p(x)q(x)} has a partial fraction decomposition

of the form

\frac{a(x)}{p(x)} + \frac{b(x)}{q(x)} ,

where a(x) and b(x) are polynomials of degrees less than d_{1} and d_{2}, respectively.

2. A function f : C → C is said to satisfy a (global) Lipschitz condition if there exists a constant K for which |f(z) − f(w)| ≤ K|z − w| for all z, w ∈ C. Show that an entire (i.e., holomorphic) function that satisfies a global Lipschitz condition must have the form f(z) = az + b for some complex constants a and b.

3. Let U ⊂ IR be a (possibly infinite) open interval and f : U → IR a differentiable function whose derivative is nondecreasing: ξ_{1} < ξ_{2} ⇒ f′(ξ_{1}) ≤ f′(ξ_{2}). Prove carefully that for every [a, b] ⊂ U,

(a) max{ f(x) | a ≤ x ≤ b } = max{ f(a), f(b) };

(b) If ℓ(x) is the linear function such that ℓ(a) = f(a) and ℓ(b) = f(b), then f(x) ≤ ℓ(x) holds for all a ≤ x ≤ b.

First Day—Part II: Answer \underline{three} of the following six questions. If you work on more than three questions, indicate clearly which three should be graded.

4. Suppose that the finite group G acts on the finite set Ω. Let n = |G| and m = |Ω|. Assume that G has exactly r orbits on Ω, and that their cardinalities are m_{1}, …, m_{r}. An element α ∈ Ω is chosen randomly, and independently of this an element g ∈ G is chosen randomly, with all choices equally likely in each case. Determine, with a proof, the probability that gα = α.

5. Suppose that g is a bounded measurable function defined on IR, and that for z ∈ C with Re z > 0, f(z) is defined by

f(z) = ∫_{−∞}^{∞} \frac{g(y)}{1 + zy^{2}} dy.

Prove that f is holomorphic in the right half plane { z ∈ C | Re z > 0 }.

6. Define f : IR^{2} → IR by f(x, y) = \frac{x^{2}y}{x^{2} + y^{2}} for (x, y) ≠ (0, 0), f(0, 0) = 0.

(a) Prove that f is continuous on IR^{2} and that (D_{1}f)(x, y) ≡ \frac{∂f}{∂x}(x, y) and (D_{2}f)(x, y) ≡ \frac{∂f}{∂y} (x, y) exist for all (x, y) ∈ IR^{2} and are bounded on IR^{2}.

(b) Prove that for every unit vector u = (u_{1}, u_{2}) ∈ IR^{2}, the directional derivative

(D_{u}f)(0, 0) ≡ lim_{t→0} \frac{f(tu) − f(0, 0)}{t} exists.

(c) Prove that f is not differentiable at (0, 0).

7. Define the exponential e^{A} of a square matrix A with complex entries by the standard power series:

e^{A} = I + A + \frac{1}{2!}A^{2} + \frac{1}{3!}A^{3} + · · · .

Show:

(a) For any such matrix A, this infinite series converges.

(b) If A and B are similar matrices, then e^{A} and e^{B} are similar matrices.

(c) det(e^{A}) = e^{tr(A)}. (Here tr denotes the trace.)

8. Suppose that f(z) is holomorphic in the unit disc D = { z ∈ C | |z| < 1 }, that f(0) = 0, and that Re[f(z)] < 1 for z ∈ D. Show that

|f(z)| ≤ \frac{2|z|}{1 − |z|}

for all z ∈ D.

9. Evaluate the following limits rigorously, justifying your answers with the aid of appropriate theorems. (a)

lim_{n→∞} ∫_{0}^{∞} (1 + \frac{x}{n})^{−n} cos (\frac{x}{n}) dx.

(Hint: Prove that (1 + \frac{x}{n})^{−n} ≤ (1 + x + \frac{x^{2}}{4} )^{−1} for x ≥ 0 and n ≥ 2.)

(b)

lim_{n→∞} ∫_{0}^{n} (1 − \frac{x}{n})^{n} e^{x/2} dx.

(Hint: Prove that (1 − \frac{x}{n})^{n} ≤ e^{−x} for n ≥ 1 and 0 ≤ x ≤ n.)

Fall 1999

Day 2

This is the second of two three-hour exam sessions; you should have taken the first session yesterday. The exam has two parts. Answer all three of the questions on Part I and three of the six questions on Part II. If you work on more than three questions on Part II, indicate clearly which three should be graded.

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

1. This problem concerns Lebesgue integration of functions from IR to IR. For such functions, a trivial form of Hölder’s inequality says that if f ∈ L^{∞}(IR) and g ∈ L^{1}(IR) then

|∫_{IR} fg| ≤ ∥f∥_{∞}∥g∥_{1}.  (∗)

(a) Prove that for each g ∈ L^{1}(IR) there is an f ∈ L^{∞}(IR) with ∥f∥_{∞} = 1 such that equality holds in (∗).

(b) Prove that for each f ∈ L^{∞}(IR) and each ϵ > 0 there is a g ∈ L^{1}(IR) with ∥g∥_{1} = 1 such that

|∫_{IR} fg| ≥ ∥f∥_{∞} − ϵ.

2. Let P (z) be a polynomial in z of degree two or higher. Let C_{R} be the positively oriented semicircle of radius R, with center at the origin, in the upper half plane. Show that

lim_{R→∞} ∫_{C_{R}} \frac{e^{ikz}}{P (z)} dz = \{ \begin{matrix}0, & if k ≥ 0, \\ 2πi ∑_{n=1}^{m} Res_{ζ_{n}} \frac{e^{ikz}}{P (z)}, & if k < 0, \end{matrix}

where ζ_{1}, …, ζ_{m} are the distinct zeros of P (z) and Res_{ζ}f(z) is the residue of f(z) at ζ.

3. Let G be a group and let H be the subgroup of G generated by the set { x^{2} | x ∈ G } of all squares in G.

(a) Show that H is a normal subgroup of G.

(b) Show that G/H is abelian.

(c) Let a, b, c, and d be elements of G. Using (b), deduce that

abcdcccbadacdcad

is an element of H.

Second Day—Part II: Answer \underline{three} of the following six questions. If you work on more than three questions, indicate clearly which three should be graded.

4. A function f : IR → IR is called 1-periodic if f(x + 1) = f(x) for all x ∈ IR. Let V be the vector space of all 1-periodic C^{∞} functions f : IR → IR, let h : IR → IR be a fixed 1-periodic C^{∞} function, and consider the linear differential operator T : V → V defined by

(T f)(x) = f′′(x) + h(x)f(x).

Show that any finite-dimensional T -invariant subspace of V is the span of certain onedimensional T -invariant subspaces. (Hint: use the inner product (f, g) = ∫_{0}^{1} f(x)g(x) dx.) Recall that a subspace W of V is T -invariant if and only if T (W) ⊆ W.

5. Suppose that a_{k} = ∫_{0}^{1} \frac{cos kπx}{\sqrt{x}} dx for k = 0, 1, 2 . . .. Show that ∑_{k=0}^{∞} a_{k}^{2} = ∞.

6. Let P (z) be a polynomial, with P (0) ≠ 0, whose complex zeros, enumerated according

to multiplicity, are \{a_{k}\}_{k=1}^{n}. Find expressions for the sums ∑_{k=1}^{n} \frac{1}{a_{k}} and ∑_{k=1}^{n} \frac{1}{a_{k}^{2}} in terms of

P (0), P′(0), and P′′(0). (Hint: Begin by finding an expression for ∑_{k=1}^{n} \frac{1}{z − a_{k}}.)

7. A metric space is called separable if it contains a countable dense set. Let A be a subset of a separable metric space (X, d). Prove that A (with the metric inherited from X) is separable.

8. This problem concerns polynomials and rational functions in two variables, t and u, over the complex field C. Suppose that

f(t, u) = ∑_{i=0}^{n} α_{i}(u)t^{i}  and  g(t, u) = ∑_{j=0}^{m} β_{j}(u)t^{j} ,

where all α_{i} and β_{j} are rational functions of u (α_{i}, β_{j} ∈ C(u)) and α_{n}(u) = β_{m}(u) = 1. Show that

fg ∈ C[t, u] ⟺ f ∈ C[t, u] and g ∈ C[t, u].

9. Let θ : [0, 1] → [0, 1] be a strictly increasing, continuously differentiable function with θ(0) = 0 and θ(1) = 1. In this problem we write V [F; a, b] for the total variation of the function F over the interval [a, b].

(a) Let f : [0, 1] → [0, 1] be a function of bounded variation and let g = θ^{∘} f ^{∘} θ^{−1}. Show, directly from the definition of total variation, that there exists a constant K depending only on θ such that

V [g; 0, 1] ≤ K V [f; 0, 1].

(b) Determine, with a proof, the constant K in (a) which is optimal in the sense that for any any K′ < K there exists an f with V [g; 0, 1] > K′ V [f; 0, 1].