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10 Courses/02 - SoSe 2026/Automatentheorie und formale Sprachen/29595454 - Komplexe Zahlen und Eigenwertaufgaben.md

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+---
+created: 2026-04-09 11:34
+course: "[[29595454 - Mathematik II]]"
+topic: komplexe Zahlen, Eigenwert
+related:
+type: lecture
+status: 🔴
+tags:
+  - university
+---
+## 📌 Summary
+
+> [!abstract]
+> 
+
+---
+
+## 📝 Content
+
+## Imaginäre Einheit
+Die imaginäre Einheit $i$ ist definiert über $i^2 = -1$.
+
+## Menge der komplexen Zahlen
+Die Elemente der menge $CC := {x + i y : x, y in RR}$ nennt man _komplexe Zahlen_.
+
+## Komplexe Zahlen
+Zu einer komplexen Zahl $z = x + i y in CC$ mit $x, y in RR$ heißt
+- $"Re" z := x$ _Realteil_ von $z$
+- $"Im" z := y$ _Imaginärteil_ von $z$
+- $overline(z) := x - i y$ die _konjugiert komplexe Zahl_ zu $z$
+- $abs(z) := sqrt(x^2 + y^2)$ der _Betrag_ von $z$
+- $phi in [0, 2pi]$ _Argument_ oder _Phase_ von $z$
+
+### Operationen
+
+#### Addition
+Die Addition wird (komponentenweise) wie für Vektoren definiert: 
+$a = vec(x, y), b = vec(x_2, y_2)$
+
+#### Multiplikation

10 Courses/02 - SoSe 2026/Automatentheorie und formale Sprachen/29615600 - Chomsky Hierarchy.md

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+---
+created: 2026-04-28 14:25
+course:
+topic:
+related:
+type: lecture
+status: 🔴
+tags:
+  - university
+---
+## 📌 Summary
+
+> [!abstract]
+> 
+
+---
+
+## 📝 Content
+
+A containment hierarchy of classes of formal grammars. Grammars are classified into four types with different limitations.
+
+### Type 0
+no restrictions
+
+### Type 1
+Each rule $sans(w)_1 -> sans(w_2)$ satisfies $sans(abs(w_1) <= abs(w_2))$ with the exception that $sans(S) -> epsilon$ is allowed if $sans(S)$ does not occur on any right hand side of rules.
+$sans(abs(w_1) <= abs(w_2))$ implies that no shortening rules are allowed.
+
+### Type 2 (context free)
+Same restriction as [[#Type 1|type 1]] and additionally for each rule $sans(w_1 -> w_2)$ the string $sans(w_1)$ contains only a single variable (i.e. $sans(w_1 in V)$)
+
+### Type 3 (regular)
+Same restriction as [[#Type 2|type 2]] and additionally $sans(w_2 in Sigma union Sigma V)$), i.e. the right hand side is either a single terminal symbol or a terminal symbol followed by a variable.
+
+> [!REMARK]
+> The order of the variable and the terminal symbol is chosen arbitrarily.
+> Important is just that within a regular grammar only a single order occurs. If for all rules $sans(w_2 in Sigma union V Sigma)$ it would also be a regular grammar.
+
+## Definition
+A language $sans(L subset Sigma^*)$ is said to be of type 0 (1,2,3) if there exist a type 0 (1,2,3) grammar $sans(G)$ for which $sans(L(G) = L)$.
+
+> [!REMARK]
+> In order to show that a language is op type 0 (1,2,3) one thus just has to find a corresponding grammar.
+> In order to show that a grammar is not of type 0 (1,2,3) one has to prove that no type 0 (1,2,3) grammar can exist generating the corresponding language.
+