RUTGERS UNIVERSITY

GRADUATE PROGRAM IN MATHEMATICS

Written Qualifying Examination

Spring 2000, Day 1

This examination will be given in two three-hour sessions, one today and one tomorrow. At each session the examination will have two parts. Answer all three of the questions in Part I (numbered 1–3) and three of the six questions in Part II (numbered 4–9). If you work on more than three questions in Part II, indicate clearly which three should be graded. No additional credit will be given for more than three partial solutions. If no three questions are indicated, then the first three questions attempted in the order in which they appear in the examination book(s), and only those, will be the ones graded.

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

1. Let p and q be prime numbers, such that p < q but p does not divide q − 1. Prove that any group of order pq is commutative.

2. Let A_{1} ⊆ A_{2} ⊆ A_{3} ⊆ ⋯ be subsets of [0, 1]. We do not assume that these sets are measurable. Prove that

lim_{n→∞} m^{∗}(A_{n}) = m^{∗} ( ⋃_{n=1}^{∞} A_{n})

where m^{∗} denotes Lebesgue outer measure.

3. Prove that the formula f(z) = ∑_{n=1}^{∞} [1 − cos (\frac{z}{n})] defines an entire function (i.e., a function defined and holomorphic on the entire complex plane ℂ.

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

4. Suppose that T is a square complex matrix whose minimal polynomial is μ_{T} (t) = t^{3} − 1 and whose characteristic polynomial is χ_{T} (t) = (t^{3} − 1)^{2}. Suppose that S is a complex matrix with the same number of rows and columns as T, such that the only (complex) eigenvalue of S is 0 and such that ST = T S. Prove that S^{2} = 0.

5. Let G be a group which is also a topological space, and assume that for each g ∈ G the translation mapping λ_{g}(h) = gh, is continuous. Prove that any subgroup of G which is open is necessarily closed, but give an example to show that the converse is false in general.

6. Let ℝ and ℝ^{+} denote the real numbers and the nonnegative real numbers respectively, as usual. Let γ and k be real constants. Solve the initial-value problem for a function x : ℝ^{+} → ℝ satisfying

\begin{matrix}\frac{d^{2}x(t)}{dt^{2}}+ & 2γ \frac{dx(t)}{dt} + k^{2}x(t) = 0 for t > 0, and \\ x(0) = & 1 \\ x′(0) = & 0 \end{matrix}

by using the Laplace-transform technique, indicating the way in which you employ this technique.

7. (a) Consider the space L^{2}[0, 1] = {f : ∫_{0}^{1} |f(x)|^{2} dx < ∞}, and consider the functions

f_{r}(x) = e^{irx} (where r ∈ ℝ^{+}) as elements of this space. Prove that f_{r} converges weakly to zero in L^{2}[0, 1] as r → ∞.

(b) Now consider the space L^{2}([0, 1], x dx) = {f : ∫_{0}^{1} |f(x)|^{2}x dx < ∞}; prove that f_{r}

converges weakly to zero as r → ∞ in this space also.

8. Let 𝔻 = \{z ∈ ℂ : |z| < 1\}. Suppose that f : \overline{𝔻} → \overline{𝔻} is a continuous function that is holomorphic and nonconstant on 𝔻 and such that |f(z)| < 1 for |z| = 1. Prove that there is one and only one fixed point of the function f in 𝔻, i.e., one and only one point z_{0} for which f(z_{0}) = z_{0}.

9. Let {f_{n} : [0, 1] → [0, ∞)}_{n=1}^{∞} be a sequence of continuous functions and let f : [0, 1] → [0, ∞) be a function for which (a) and (b) below hold:

(a) If m ≠ n and x ∈ [0, 1] then f_{m}(x) ≠ f_{n}(x);

(b) {f_{n}(x)}_{n=1}^{∞} converges to f(x) for (Lebesgue-)almost all x ∈ [0, 1].

Prove that lim_{n→∞} ∫_{0}^{1} f_{n}(x)dx = ∫_{0}^{1} f(x)dx.

RUTGERS UNIVERSITY

GRADUATE PROGRAM IN MATHEMATICS

Written Qualifying Examination

Spring 2000, Day 2

This examination will be given in two three-hour sessions, today’s being the second part. At each session the examination will have two parts. Answer all three of the questions in Part I (numbered 1–3) and three of the six questions in Part II (numbered 4–9). If you work on more than three questions in Part II, indicate clearly which three should be graded. No additional credit will be given for more than three partial solutions. If no three questions are indicated, then the first three questions attempted in the order in which they appear in the examination book(s), and only those, will be the ones graded.

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

1. Given an example of a sequence {f_{n} : [0, 1] → ℝ}_{n=1}^{∞} of Lebesgue-integrable functions, such that f_{n} → 0 in L^{1} norm but lim_{n→∞} f_{n}(x) does not exist for any x ∈ [0, 1].

2. Let V be a finite-dimensional vector space over a field 𝔽, and let T : V → V be a linear transformation. Make V into a module for the polynomial ring 𝔽[t] by defining tv = T v for all v ∈ V. Assume that

V ≅ \frac{𝔽[t]}{(f_{1})} ⊕ \frac{𝔽[t]}{(f_{2})} ⊕ ⋯ ⊕ \frac{𝔽[t]}{(f_{m})} as 𝔽[t] − modules

where f_{1},f_{2},…,f_{m} ∈ 𝔽[t]. No divisibility assumption is made about f_{1},f_{2},…,f_{m}. Express the following in terms of f_{1},f_{2},…,f_{m}, and prove that your answers are correct:

(a) dim V;

(b) The minimal polynomial μ_{T} (t) of T;

(c) The characteristic polynomial χ_{T} (t) of T.

3. Let f(x) = ∑_{n=0}^{∞}a^{n} cos(b^{n}πx), where 0 < a < 1 and b is an odd natural number. Prove

that f(x) is continuous for all x ∈ ℝ.

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

4. Prove that the limit γ = lim_{n→∞} [∑_{k=1}^{n} \frac{1}{k} − log n] exists and is positive. For full credit,

also prove that ∑_{k=1}^{n} \frac{1}{k} = γ + log n + O (\frac{1}{n}), where O (\frac{1}{n}) denotes a “remainder term”

whose quotient by 1/n remains bounded as n → ∞.

5. Find the (real) value of each of the following integrals (in (b), for all values of the real parameter k):

(5a) ∫_{0}^{∞} \frac{dx}{1 + x^{3}} ;

(5a) ∫_{0}^{∞} \frac{sin(kx)}{x}dx .

6. Let A be an invertible complex matrix. Prove that for any natural number k there exists a matrix X such that X^{k} = A.

7. (a) Exhibit a power series ∑_{n=0}^{∞} a_{n}z^{n} that has all of the following three properties: (i)

the series converges for all complex |z| < 1; (ii) ∑_{n=0}^{∞} |a_{n}| diverges; (iii) ∑_{n=0}^{∞} |a_{n}|^{2} converges.

(b) Prove that for any sequence of coefficients such that ∑_{n=0}^{∞} |a_{n}|^{2} converges (to a finite

value), the series f(z) = ∑_{n=0}^{∞} a_{n}z^{n} must converge for all (complex) |z| < 1 and the function

it defines must satisfy lim_{|z|→1^{−}} \sqrt{1 − |z|} · f(z) = 0.

8. (a) If K ⊂ ℝ^{n} is a closed, bounded, nonempty set, define what it means for K to be connected.

(b) Suppose that {K_{i}}_{i=1}^{∞} is a sequence of closed, bounded, nonempty, connected sets in ℝ^{n} that is decreasing, i.e., satisfies K_{1} ⊇ K_{2} ⊇ ·. Prove that K = ⋂_{i=1}^{∞} K_{i} is also nonempty and connected. (You may use well-known fundamental properties of ℝ and ℝ^{n}, but you must indicate explicitly what properties you use.)

9. Say that a group G is revolutionary if and only if there exist a natural number n and a chain of subgroups

{e} = G_{0} ◁ G_{1} ◁ G_{2} ◁ ⋯ G_{n} = G ,

each normal in the next, such that the order of G_{i}/G_{i−1} divides either 1848 = 2^{3} · 3 · 7 · 11 or 1917 = 3^{3} · 71 for each i = 1, …, n. (Here e is the identity element of G.)

(a) Prove that if N is a normal subgroup of a group G such that both N and G/N are revolutionary, then G is revolutionary.

(b) Prove that every subgroup of a revolutionary group is revolutionary.