A Bound on the Real Stability Radius of Continuous-Time by Bobylev N. A., Bulatov V.

By Bobylev N. A., Bulatov V.

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Extra info for A Bound on the Real Stability Radius of Continuous-Time Linear Infinite-Dimensional Systems

Sample text

Allerdings gilt nat¨ urlich f¨ ur bijektive Abbildungen, dass f −1 ({y}) = {f −1 (y)}. Beispiel Die Umkehrfunktion von f4 : [0, ∞[ → [0, ∞[ , x → x2 ist die Quadratwurzel: f4−1 : [0, ∞[ → [0, ∞[ , x → √ x. Die Komposition von Abbildungen Deﬁnition Seien A, B, X und Y Mengen, und f : A → X und g : Y → B Abbildungen mit f (A) ⊆ Y . Wir deﬁnieren die Komposition (Hintereinanderausf¨ uhrung, Verkettung) von f und g als die Abbildung g◦f: A→B 36 Kapitel 3. Abbildungen (lies: g Kringel f“ oder g nach f“) mit ” ” (g ◦ f )(x) = g(f (x)) f¨ ur alle x ∈ A.

Assoziativit¨ at der Addition: F¨ ur alle a, b, c ∈ K gilt (a + b) + c = a + (b + c). 42 Kapitel 4. K¨ orper und komplexe Zahlen 2. Kommutativit¨ at der Addition: F¨ ur alle a, b ∈ K gilt a + b = b + a. 3. Existenz des neutralen Elements der Addition: Es gibt ein 0 ∈ K, sodass f¨ ur alle a ∈ K gilt: a + 0 = a. 4. Existenz inverser Elemente der Addition: F¨ ur alle a ∈ K gibt es ein −a ∈ K, sodass a + (−a) = 0. 5. Assoziativit¨ at der Multiplikation: F¨ ur alle a, b, c ∈ K gilt (a · b) · c = a · (b · c).

11) f ist genau dann surjektiv, wenn f¨ ur kein x ∈ B gilt: f −1 ({x}) = ∅. II. Welche der folgenden Abbildungen sind bijektiv? (1) f : R → R, f (x) = x3 (2) f : R → R, f (x) = x4 (3) f : [0,1] → [0,1], f (x) = x4 (4) f : N → Z, f (n) = n (5) f : Z → N, f (n) = n2 (6) f : {0,1} → {0,1}, f (n) = 1 − n (7) f : N → N \ {0}, f (n) = n + 1 (8) f : Z → Z, f (n) = −n 4 K¨ orper und komplexe Zahlen Einblick Mit den reellen Zahlen haben Sie bereits intensive Erfahrungen in der Schule gemacht und bereits im Buch hatten wir Beispiele f¨ ur Zahlen gesehen, die gerade keine rationalen Zahlen sind, sodass es n¨otig war, diese zu den reellen Zahlen zu erweitern.