1. Let z ∈ R, and consider the function fz : R→ R x 7→ x+ z We have fz is Borel-measurable since for all c ∈ R, {x ∈ R|fz(x) ≤ c} = {x ∈ R|x+ z ≤ c} = (−∞, c− z) ∈ B. We also observe that if E ⊆ R, then f−1 z (E) = {x ∈ R|x + z ∈ E} = {−z + (x + z)|x + z ∈ E} = −z + E. Thus we have: if E is a Borel set, then f−1 −y (E) = y + E is a Borel set. Conversely, if y + E is a Borel set, then −y + (y + ...

THEOREM 1. Let f(z) be a polynomial of degree N with complex coefficients, and let / (r) (z) denote the rth derivative of f(z). If f{z) has two or more distinct roots, N Yl / (r) (z) has at least N+l distinct roots.r=0 plicity of z s as a root of/ (r) (z), for r = 0,. be the sum of the mth powers of the roots of/ (r) (z), and let A m (r) be the elementary symmetric polynomial of degree m in the...

For simplicity, we follow the rules: n denotes a natural number, a, b, r denote real numbers, x, y, z denote points of En T , and e denotes a point of En. The following propositions are true: (1) x − y − z = x − z − y. (2) If x + y = x + z, then y = z. (3) If n is non empty, then x 6= x + 1.REALn. (4) For every set x such that x = (1 − r) · y + r · z holds x = y iff r = 0 or y = z and x = z iff...

We prove that to every rational function R(z) satisfying R(−z)R(z) = 1, there exists a symplectic Runge-Kutta method with R(z) as stability function. Moreover, we give a surprising relation between the poles of R(z) and the weights of the quadrature formula associated with a symplectic Runge-Kutta method.