Day 1 — Required

1–1. Let A be an n × n (real) symmetric matrix.

(a) Prove that all eigenvalues of A are real.

(b) Prove that two eigenvectors x and y of A with distinct eigenvalues λ_{x} and λ_{y} are orthogonal.

(c) If x is an eigenvector of A, find an (n − 1)-dimensional A-invariant subspace V of ℝ^{n} such that x ∉ V. Recall that a subspace V of ℝ^{n} is A-invariant iff for all v ∈ V, Av ∈ V.

1–2. Find all possible values of

∮_{Γ} \frac{e^{2πz}}{(z − 1) (z − i)^{2}} dz

where Γ ranges over the class of simple, closed, smooth curves with Γ ⊂ ℂ\ {1, i}.

1–3. Let f ∈ L^{1} (−∞, ∞). Show that the functions f (x) (sin x)^{n}, n = 1, 2, ⋯, are measurable and integrable on (−∞, ∞) and

lim_{n→∞} ∫_{−∞}^{∞} f (x) (sin x)^{n} dx = 0.

Day 1 — Select 3

1–4. Let 𝔽_{2} be the field with 2 elements, and let G = GL (3, 𝔽_{2}) be the general linear group of 3 × 3 invertible matrices over 𝔽_{2}. Show that G has an element of order 7, and find all possible minimal polynomials for elements of order 7 in G.

1–5. Let {f_{n} (z)} be a sequence of holomorphic functions defined in the open unit disk D = {z : |z| < 1}. Assume ∥f_{n}∥_{L^{2}(D)} ≤ 1. Show that {f_{n}} has a subsequence which converges uniformly on every compact subset of D.

1–6. Let {E_{k}}_{k=1}^{∞} be a collection of Lebesgue measurable subsets of ℝ^{n} with ∑_{k=1}^{∞} |E_{k}| < ∞. Show that the set consisting of all points which belong to infinitely many of the E_{k} has measure 0.

1–7. Let R = ℤ ⊕ 2xℤ [x] be the subring of the ring of polynomials ℤ [x] given by

R = {f (x) = a_{0} + 2a_{1}x + 2a_{2}x^{2} + ⋯ + 2a_{n}x^{n} : a_{i} ∈ ℤ} .

That is, the constant term of a polynomial in R is any integer, but the coefficients of positive powers of x are even integers. Show that R is not Noetherian.

1–8. Let z = x + iy and f (z) = x (x^{2} − 3y^{2}) + iy (3x^{2} + y^{2}). State whether each of the following is true or false and give a reason.

(a) f′ (0) exists.

(b) f satisfies the Cauchy-Riemann equations at z = 0.

(c) f is analytic (holomorphic) in a neighborhood of z = 0.

1–9. Let R > 0 and let f be an infinitely differentiable real-valued function on ℝ^{n} with support in a ball B_{R} of radius R centered at the origin. Recall that the support of a function f(x) is the closure of the set of all x in the domain of f for which f(x) ≠ 0.

(a) Show that |f (x)| ≤ c ∫ _{B_{R}} \frac{|∇f (y)|}{|x − y|^{n−1}} dy for some constant c which depends on n but

not on f, x or B_{R}. Hint: start by applying the fundamental theorem of calculus along rays issuing from x.

(b) Using the above, show that if n < p < ∞, then there is a constant c (n, p) such that

|f (x)| ≤ c (n, p) R^{1−n/p} (∫ _{B_{R}} |∇f (y)|^{p} dy)^{1/p}.

Day 2 — Required

2–1. Consider the system of equations

t^{2} + x^{3} + y^{3} + z^{3} = 0
t + x^{2} + y^{2} + z^{2} = 2
t + x + y + z = 0

and the point P with coordinates (t, x, y, z) = (0, −1, 1, 0).

(a) Show that the above equations determine a differentiable curve which can be expressed

as

x = x (t) , y = y (t) , z = z (t)

in some neighborhood of the point P. Here “determine a differentiable curve” means that there is a differentiable function F with F (t) = (x(t), y(t), z(t)) and F (0) = (−1, 1, 0).

(b) Determine a nonzero tangent vector to the curve in (2–1.a) at the point P.

2–2. (a) Show that a compact metric space is complete.

(b) Two metrics d_{1} and d_{2} on a space X are said to be “equivalent” if they generate identical topologies on X. If d_{1} and d_{2} are equivalent on X, does completeness of (X, d_{1}) imply completeness of (X, d_{2})? Prove it or give a counterexample.

2–3. For n ≥ 2 let A_{n} denote the alternating group of even permutations of the set {1, 2, …, n}. Let K be the subgroup of A_{n} fixing a given element, say the element 1. If H is any subgroup of A_{n} with [A_{n} : H] = n, show that there is an isomorphism φ of A_{n} with itself such that φ (H) = K.

Day 2 — Select 3

2–4. Let {f_{n}} and f be Lebesgue measurable functions on ℝ, and let m denote Lebesgue measure on ℝ. Determine whether or not each of the following statements is true or false, giving a proof if it is true, a counterexample if false.

(a) If lim_{n→∞} f_{n}(x) = f(x) for a.e. x, then f_{n} → f in measure as n → ∞.

(b) If lim_{n→∞} ∫_{ℝ} |f_{n} − f| dm = 0, then f_{n} → f in measure as n → ∞.

(c) Let f_{n} be integrable for each n, and let g(x) = lim inf_{n→∞} f_{n}(x). Then

∫ g(x) dm ≤ lim inf_{n→∞} ∫ f_{n} dm.

2–5. Find a conformal mapping of the sector

\frac{π}{6} < arg z < \frac{π}{3}

of the complex plane onto the open disk |z| < 1. Explain your answer.

2–6. Let u⃗ (x, y, z) denote the vector field

u⃗ (x, y, z) = (\frac{x}{(x^{2} + y^{2} + z^{2})^{3/2}} , \frac{y}{(x^{2} + y^{2} + z^{2})^{3/2}} , \frac{z}{(x^{2} + y^{2} + z^{2})^{3/2}} )

on ℝ^{3} \ {(0, 0, 0)}, and let D denote the annular region

1 < x^{2} + y^{2} + (z − 2)^{2} < 9.

(a) Compute the divergence of u⃗.

(b) Evaluate ∫ ∫_{∂D} u⃗ · n⃗ ds, where ds denotes the element of surface area of ∂D and n⃗ denotes the unit outer normal to ∂D.

2–7. Suppose that f_{n}(x) is a continuous real-valued function on [a, b] for n = 1, 2, …, and that for each x ∈ [a, b], {f_{n}(x)} is a bounded sequence. Prove that there is a subinterval of [a, b] where {f_{n}(x)} is uniformly bounded.

2–8. Let R be the subring of ℂ generated by the integers ℤ and \sqrt{−5} = i\sqrt{5}, that is,

R = ℤ [\sqrt{−5}] .

(a) Show that the element x = 2 + \sqrt{−5} is not a product of two nonunits in R, that is, if 2 + \sqrt{−5} = rs for some r and s in R, then either r or s has an inverse in R.

(b) Show that the ring R/xR is not an integral domain, where x = 2 + \sqrt{−5}.

(c) Show that, for m and n nonzero integers, m + 0\sqrt{−5} and n + 0\sqrt{−5} have a greatest common divisor in R.

2–9. The exponential of a complex matrix A is the power series

exp (A) = ∑_{i=0}^{∞} \frac{(A)^{i}}{i!} .

You may use the properties that if A and B are commuting matrices, then exp (A + B) = exp (A) · exp (B), but this in general fails if AB ≠ BA.

(a) Show that any matrix A is similar to a matrix B which is a sum B = D + N of two commuting matrices D and N with D diagonal and N nilpotent, that is, N has some power equal to 0.

(b) It is a fact that the exponential exp (At) has columns which form a basis for the solution space of the system of differential equations dX⃗/ dt = AX⃗. Use this to find a basis for the solution space of the system

[ \begin{matrix}dx_{1}/ dt \\ dx_{2}/ dt \\ dx_{3}/ dt \end{matrix} ] = [ \begin{matrix}1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & −1 & 1 \end{matrix} ] [ \begin{matrix}x_{1} \\ x_{2} \\ x_{3} \end{matrix} ]