MATH 395 PROBLEMS 10

IGOR KRIZ

Regular problems:

1. Solve

y′ = ( \begin{matrix}λ & 1 & 0 \\ 0 & λ & 1 \\ 0 & 0 & λ \end{matrix} ) y

for a general λ ∈ ℝ.

2. Which of the following sets are countable:

(a) The set of all finite sequences of rational numbers (a_{1}, ...a_{n}) for all possible positive integers n

(b) The set of all real numbers not containing the digits 4, 5, 6 in their decimal expansion. Can you describe this set?

(c) The set of all numbers which are roots of non-zero polynomials with integer coefficients.

3. Using Fubini’s theorem, calculate

∫_{[0,π]×[0,π]} | cos(x + y)|dxdy

4. Recall that Z is the set of all continuous functions f : ℝ → ℝ with compact support. Recall also that Z^{inc} resp. Z^{dec} is the set of all functions f : ℝ → ℝ ∪ {−∞, ∞} which are limits of an increasing (resp. decreasing) sequence of functions from Z. Now let

g(x) = \{ \begin{matrix}\frac{1}{\sqrt[3]{x}} & if x ∈ (0, 1] \\ 0 & if x ∉ (0, 1]. \end{matrix}

(a) Prove that g(x) ∈ Z^{inc}. Is g(x) ∈ Z^{dec}?

(b) Choosing an increasing sequence f_{n} of functions from Z converging to g, calculate

lim_{n→∞} ∫_{−∞}^{∞} f_{n}(x)dx.

Challenge problem:

5. Reducing (L) to (L̃). Let

y′ = Ay.

Reduce this to the problem (L̃) of the form

y^{(n)} + a_{1}y^{(n−1)} + … a_{n}y = 0

in the following steps:

(a) It suffices (at least theoretically) to consider the case when A is a Jordan block

( \begin{matrix}λ & 1 & … & 0 \\ 0 & λ & … & 0 \\ … & … & … & … \\ 0 & 0 & … & λ \end{matrix} ) .

Why?

(b) Write down the matrix B of the problem (L̃) with characteristic polynomial (x − λ)^{n}, rewritten in the form of (L) with y_{1} = y, y_{2} = y′, ..., y_{n} = y^{(n−1)}. It suffices to exhibit P such that

P^{−1}AP = B.

Why?

(c) Assume you can find vectors u_{1}, …, u_{n} ∈ ℝ^{n} such that

Bu_{1} = λu_{1}, Bu_{i} = λu_{i} + u_{i−1}, i > 1.

Then you can find P. How?

(d) Find the vectors u_{1}, … u_{n}. [This is the tough part. Try it for n = 1, 2, 3 and try to guess the general pattern.]