Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Problem 1. Let C > 0. Find all nonnegative differentiable f on [0, +∞) which vanish in 0 and satisfy Cf (t) |ln f (t)| , f (t) > 0, d ∀t ≥ 0 . f (t) ≤ 0, dt f (t) = 0, Problem 2. Let C > 0. Find all nonnegative differentiable f on [0, +∞) which vanish in 0 and satisfy the following inequality for all t ≥ 0 and ∈ (0, 1), d C f (t) ≤ f 1− (t) . dt Problem 3. Let A ⊂ R. We call a point x ∈ A isolated if there is ρ > 0 with Bρ (x) ∩ A = {x} (Bρ (x) = y ∈ R |y − x| < ρ = (x − ρ, x + ρ)). If A is closed and contains no isoated points, then A is called perfect. Prove that every perfect subset A of R is uncountable. Problem 4. (a) Prove the following identity for all n ∈ N \ {0} and x ∈ R: ! 2 n X k x(1 − x) n −x xk (1 − x)n−k = . n n k k=0 (b) Let f : [0, 1] → R be continuous and define for every n ∈ N \{0} the Polynomial ! n X k n Sn (x) = f xk (1 − x)n−k . n k k=0 Use (a) to prove that Sn converges uniformly to f on [0, 1]. Problem 5. Find all continuously differentiable functions u : [0, +∞) × R2 → R, (t, x, y) 7→ u(t, x, y), such that ∂u ∂u ∂u +2 +3 = 0, ∂t ∂x ∂y − x2 +y 2 u(0, x, y) = e , auf [0, +∞) × R2 , on R2 . Problem 6. Consider the map X : R2 → R2 , (x, y) 7→ ! X1 (x, y) = X2 (x, y) 1 3 2 ! −x − 4xy , −16y 3 − 4x2 y 2 and assume that γ : [0, +∞) → R2 , t 7→ γ(t) = γ1 (t), γ2 (t) is a continuously differentiable curve that, at every t ≥ 0, satisfies the equations γ10 (t) = X1 γ1 (t), γ2 (t) γ 0 (t) = X γ (t), γ (t) . 2 1 2 2 Prove that γ [0, +∞) is a bounded subset of R2 . Problem 7. Let n, k ∈ N \ {0} be given and assume that f : Rn → R is a function, which is k times differentiable with continuous derivatives of order k. How many different partial derivatives of order k is it necessary to compute, so to know all the partial derivatives of order k? Problem 8. Let U ⊂ Rn (n ≥ 1) be open and bounded and let f : U → R be a continuous function, which is two times differentiable in U and satisfies the following equation n X ∂2 f (x) < 0 ∀x ∈ U . 4f (x) = 2 ∂x i i=1 Prove that then f takes its minimum on ∂U . Problem 9. Prove that there is no function u ∈ C 1 [0, +∞) × R which solves ∂ u(t, x) + 1 ∂ u(t, x) 2 = 0, (t > 0, x ∈ R), ∂t 2 ∂x u(0, x) = e−x2 , (x ∈ R). Problem 10. Let c > 0 and f, g ∈ C 2 ([−1, 1]) with f (±1) = g(±1) = 0. Prove that there is at most one solution u ∈ C 2 [0, +∞) × [−1, 1] of the system 2 ∂ ∂2 u(t, x) = c2 ∂x (t > 0, x ∈ (−1, 1)), 2 u(t, x), ∂t2 u(t, −1) = u(t, 1) = 0, (t ≥ 0), (x ∈ [−1, 1]), u(0, x) = f (x), ∂ u(0, t) = g(x), ∂t (x ∈ [−1, 1]). Problem 11. Consider the map f : [0, +∞) → R, ˆ x 7→ 0 Is f continuous in 0? Is f differentiable in 0? Problem 12. +∞ cos xt dt. 1 + t2 3 (1) Let k ∈ N \ {0} and assume that U ⊂ Rk is open and closed. Prove then that either U = ∅ or U = Rk ; (2) Let k ∈ N \ {0} and u : Rk → Rk be continuously differentiable, with kdukop ≤ L < 1. Prove then that the following map is a diffeomorphism: Φ : Rk → Rk , x 7→ x − u(x) . Problem 13. Let F (x, y) = −x4 y 6 + y 4 + 2x3 y 3 − 2y 2 − x2 + 1. Prove the existence of an interval I ⊂ R with 0 ∈ I and of four different continuously differentiable functions φi : I → R (i = 1, . . . , 4) with F x, φi (x) = 0, ∀x ∈ I, i ∈ {1, . . . , 4}. Problem 14. Prove the following inequality for every n × n–Matrix A: v n uX Y u n t |det A| ≤ |Aij |2 . j=1 i=1