vault backup: 2026-04-16 11:48:52

.gitignore

@@ -1 +1,3 @@
 .trash/
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.obsidian/workspace.json

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00 Inbox/29605546 - 29593852 - Strings.md

@@ -0,0 +1,71 @@
+---
+created: 2026-04-08 08:52
+course: "[[29593850 - Automationtheory]]"
+topic: strings
+related: "[[29593929 - Alphabets]]"
+type: lecture
+status: 🔴
+tags:
+  - university
+---
+## 📌 Summary
+
+> [!abstract]
+> Overview of lecture 1 on `Wednesday, 2026/Apr/08`
+
+---
+
+## 📝 Content
+
+A word (or _string_) is a finite sequence $w = a_1 a_2 ... a_n$ if characters from $Sigma$.
+
+> [!CONVENTION]
+> We will use small letters to describe strings that are part of a language.
+
+> [!EXAMPLE]
+> $"aa", "ab", "bba"$ and $"baab"$ are strings over $Sigma = {a, b}.
+
+#### Length of a string
+The _length_ $abs(x)$ of a string $x = a_1 ... a_n$ is its number $abs(x) = n$ of characters.
+#### Empty String
+The empty string is denoted by $epsilon$, this is the neutral element.
+-> $abs(epsilon) = 0$
+
+## String Operations
+### Concatenation
+String can be concatenated, where one string is appended to another. 
+For strings $x = a_1 ... a_n$ and $y  = b_1 ... b_m$ over alphabets $Sigma_x$ and $Sigma_y$, their _concatenation_ over the alphabet $Sigma = Sigma_x union Sigma_y$ is the string
+$$x circle.small y = x y = a_1 a_2 ... a_n b_1 b_2 ... b_m$$
+> This string is of the length $abs(x y) = n + m$
+
+> [!EXAMPLE]
+> $x = "apple"$
+> $y = "pie"$ 
+> $x circle.small y = "applepie"$
+
+Order of operations / Brackets do _not matter_. (Concatenation is associative but **not** commutative $x y eq.not y x$)
+> $(x circle.small y) circle.small z = x circle.small (y circle.small z)$
+
+Any string concatenated with the empty string $epsilon$ will result in itself.
+> $x circle.small epsilon = x = epsilon circle.small x$ 
+
+### Exponentiation
+
+The $n^"th"$ power $x^n$ of a string $x$ is the $(n-1)$-fold concatenation of $x$ with itself.
+> $x^0 := epsilon$
+> $x^n := x^(n-1) circle.small x$ for $n in NN$
+
+> [!Example]
+> $x^4 = x x x x$
+> $(a b)^3 = a b a b a b$
+
+### Reversing / Mirroring
+For a string $x = a_1 a_2 ... a_(n-1) a_n$ of length $n$, it's _mirrored string_ is given by 
+$$ x^("Rev") = a_n a_(n-1)...a_2 a_1$$
+## Substrings
+A string $x$ is a _substring_ of a string $y$ if $y = u x v$, where $u$ and $v$ can be arbitrary strings.
+- If $u = epsilon$ then $x$ is a _prefix_ of $y$.
+- If $v = epsilon$ then $x$ is a suffix of $y$.
+
+For strings $x$ and $y$ the quantity $abs(y)_x$ is the number of times that $x$ is a substring of $y$.
+

00 Inbox/29605547 - 29605397 - OOP.md

@@ -0,0 +1,18 @@
+---
+created: 2026-04-16 11:47
+course: "[[29605547 - 29605397 - OOP]]"
+tags:
+  - university
+topic:
+related:
+type: lecture
+status: 🔴
+---
+## 📌 Summary
+
+> [!abstract]
+> 
+
+---
+
+## 📝 Content

00 Inbox/29605548 - 29605397 - OOP.md

@@ -0,0 +1,18 @@
+---
+created: 2026-04-16 11:48
+course: "[[29605548 - 29605397 - OOP]]"
+tags:
+  - university
+topic:
+related:
+type: lecture
+status: 🔴
+---
+## 📌 Summary
+
+> [!abstract]
+> 
+
+---
+
+## 📝 Content

10 Courses/02 - SoSe 2026/OOP/29605321 - Classes.md

@@ -1,6 +1,6 @@
 ---
 created: 2026-04-16 08:01
-course: "[[29605397 - OOP]]"
+course: "[[29605548 - 29605397 - OOP]]"
 topic: classes
 related:
 type: lecture

10 Courses/02 - SoSe 2026/OOP/29605397 - OOP.md

@@ -1,6 +0,0 @@
----
-created: 2026-04-16 09:17
-course: "[[29605397 - OOP]]"
-tags:
-  - university
----