├── TUM
├── .obsidian
│ ├── app.json
│ ├── hotkeys.json
│ ├── appearance.json
│ └── core-plugins.json
├── 1. Semester
│ ├── 1. Semester.md
│ └── ERA
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│ │ ├── Digitale Schaltungen.md
│ │ ├── Speicherhierarchie.md
│ │ ├── Multicycle Prozessoren.md
│ │ ├── Einfuehrung in die Rechnerarchitekur.md
│ │ ├── Endliche Automaten.md
│ │ ├── Befehlsverarbeitung.md
│ │ ├── Systembefehle.md
│ │ ├── Das Von-Neumann Konzept.md
│ │ ├── Binaere Entscheidungsdiagramme.md
│ │ ├── Speicherbausteine.md
│ │ ├── Speicherverwaltung.md
│ │ ├── Hauptspeicher.md
│ │ ├── Instruction Set Architecture.md
│ │ ├── Multithreading.md
│ │ ├── Moderne Prozessoren.md
│ │ └── Speicheraufbau.md
├── 5. Semester
│ ├── 5. Semester.md
│ ├── RTS
│ │ ├── Real-Time Systems.md
│ │ ├── Images
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│ │ ├── Real-Time Hardware.md
│ │ └── Time and Clocks.md
│ └── ML
│ │ ├── Images
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│ │ ├── Maschinelles Lernen.md
│ │ ├── Classification Performance.md
│ │ ├── k-Nearest Neighbors.md
│ │ └── Hyperparameters.md
├── 3. Semester
│ ├── Applied Reg
│ │ ├── Applied Regression.md
│ │ └── Images
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│ ├── GBS
│ │ ├── GBS.apkg
│ │ ├── Images
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│ │ ├── Grundlagen Betriebssysteme und Systemsoftware.md
│ │ ├── Betriebssysteme.md
│ │ ├── Scheduling.md
│ │ └── Modellierung paralleler Systeme.md
│ ├── GDB
│ │ ├── GDB.apkg
│ │ ├── Images
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│ │ ├── Grundlagen Datenbanken.md
│ │ ├── Konzeptuelle Modellierung.md
│ │ └── Datenintegritaet.md
│ ├── ITSec
│ │ ├── ITSec.apkg
│ │ ├── Images
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│ │ ├── IT Sicherheit.md
│ │ └── Netzwerksicherheit.md
│ ├── NumProg
│ │ ├── NumProg.apkg
│ │ ├── Images
│ │ │ ├── Pasted image 20231102170305.png
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│ │ └── Numerisches Programmieren.md
│ └── 3. Semester.md
├── 6. Semester
│ ├── Markovketten
│ │ ├── Markovketten.md
│ │ └── Linear Algebra in Markov Chains.md
│ └── IDSST
│ │ ├── Images
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│ │ ├── Introduction to Data Science and Statistical Thinking.md
│ │ ├── Probability.md
│ │ └── Numerical Data.md
├── TUM.md
├── 2. Semester
│ ├── FPV
│ │ ├── Pasted image 20230417211426.png
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│ │ ├── Funktionale Programmierung und Verifikation.md
│ │ ├── Assertions and Verification.md
│ │ ├── Termination.md
│ │ └── Correctness.md
│ ├── GAD
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│ │ ├── Grundlagen Algorithmen und Datenstrukturen.md
│ │ ├── Pattern Matching.md
│ │ └── Datenstrukturen fuer Sequenzen.md
│ ├── 2. Semester.md
│ └── LinAlg
│ │ ├── Orthogonal- und Orthonormalsystem.md
│ │ ├── Lineare Algebra.md
│ │ ├── Saetze.md
│ │ ├── Skalarprodukt.md
│ │ ├── Gaussches Eliminationsverfahren.md
│ │ ├── Matrixnormen.md
│ │ ├── Basen von Vektorraeumen.md
│ │ ├── Konstruktion der komplexen Zahlen.md
│ │ ├── Erzeugendensysteme.md
│ │ ├── Definitheit von Matrizen.md
│ │ ├── Singulaerwertszerlegung.md
│ │ ├── Komplexe Zahlen und Polarkoordinaten.md
│ │ ├── Kryptologie.md
│ │ ├── Lineare Ausgleichsrechnung.md
│ │ ├── Komplexe Zahlen und Kartesische Koordinaten.md
│ │ └── Determinante.md
└── 4. Semester
│ ├── DWT
│ ├── Images
│ │ └── Pasted image 20240708102651.png
│ └── Diskrete Wahrscheinlichkeitstheorie.md
│ ├── GRnVS
│ ├── Images
│ │ ├── Pasted image 20240416151100.png
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│ └── Grundlagen Rechnernetze und Verteilte Systeme.md
│ ├── ModSim
│ ├── Images
│ │ ├── Pasted image 20240511114119.png
│ │ ├── Pasted image 20240516155357.png
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│ ├── Modellbildung und Simulation.md
│ ├── Nash Equilibrium and Banzhaf Power Index.md
│ ├── Directional Fields.md
│ └── Job Scheduling.md
│ ├── Theo
│ ├── Images
│ │ ├── Pasted image 20240430153859.png
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│ └── Einfuehrung in die Theoretische Informatik.md
│ └── 4. Semester.md
├── Books
├── Network Architecture.md
├── Introduction to Networking.md
├── Store-and-Forward Networks.md
├── Packets.md
└── Link Layer.md
├── .idea
├── .gitignore
└── vcs.xml
├── Pasted image 20250629145140.png
├── Pasted image 20250629151847.png
├── .obsidian
├── community-plugins.json
├── plugins
│ ├── obsidian-completr
│ │ ├── blacklisted_suggestions.txt
│ │ └── manifest.json
│ ├── obsidian-quickshare
│ │ ├── styles.css
│ │ └── manifest.json
│ └── obsidian-relative-line-numbers
│ │ └── manifest.json
├── themes
│ ├── Pisum
│ │ └── manifest.json
│ ├── Comfort color dark
│ │ └── manifest.json
│ ├── Pine Forest Berry
│ │ └── manifest.json
│ ├── Obsidian Nord
│ │ └── manifest.json
│ ├── PLN
│ │ └── manifest.json
│ └── Catppuccin
│ │ └── manifest.json
├── appearance.json
├── core-plugins-migration.json
├── core-plugins.json
└── hotkeys.json
├── .gitmodules
├── .gitignore
└── README.md
/TUM/.obsidian/app.json:
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1 | {}
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/TUM/.obsidian/hotkeys.json:
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1 | {}
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/Books/Network Architecture.md:
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1 | [[Link Layer]]
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/TUM/.obsidian/appearance.json:
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1 | {
2 | "accentColor": ""
3 | }
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/.idea/.gitignore:
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1 | # Default ignored files
2 | /shelf/
3 | /workspace.xml
4 |
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/TUM/1. Semester/1. Semester.md:
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1 | ### [[Einfuehrung in die Rechnerarchitekur]]
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/TUM/5. Semester/5. Semester.md:
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1 | ### [[Maschinelles Lernen]]
2 | ### [[Real-Time Systems]]
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/TUM/5. Semester/RTS/Real-Time Systems.md:
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1 | ### [[Time and Clocks]]
2 | ### [[Real-Time Hardware]]
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/TUM/3. Semester/Applied Reg/Applied Regression.md:
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1 | ### [[Statistical Basics]]
2 | ### [[Simple Regression]]
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/TUM/6. Semester/Markovketten/Markovketten.md:
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1 | ### [[Definitions]]
2 | ### [[Linear Algebra in Markov Chains]]
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/Books/Introduction to Networking.md:
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1 | [[Store-and-Forward Networks]]
2 | [[Packets]]
3 | [[Network Architecture]]
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/TUM/3. Semester/GBS/GBS.apkg:
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/.obsidian/community-plugins.json:
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1 | [
2 | "obsidian-relative-line-numbers",
3 | "hot-reload",
4 | "obsidian-completr"
5 | ]
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/TUM/3. Semester/NumProg/NumProg.apkg:
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/TUM/TUM.md:
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1 | ### [[1. Semester]]
2 | ### [[2. Semester]]
3 | ### [[3. Semester]]
4 | ### [[4. Semester]]
5 | ### [[5. Semester]]
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/.obsidian/plugins/obsidian-completr/blacklisted_suggestions.txt:
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1 | gebracht
2 | gebildet
3 | TOO
4 | RAID
5 | \Uppi
6 | validate
7 | validation
8 | Validation
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/.idea/vcs.xml:
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1 |
2 |
3 |
4 |
5 |
6 |
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/TUM/6. Semester/IDSST/Introduction to Data Science and Statistical Thinking.md:
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1 | ### [[Data Basics]]
2 | ### [[Numerical Data]]
3 | ### [[Probability]]
4 | ### [[Predictive Modeling]]
5 | ### [[Inference]]
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/TUM/2. Semester/2. Semester.md:
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1 | ### [[Funktionale Programmierung und Verifikation]]
2 | ### [[Lineare Algebra]]
3 | ### [[Grundlagen Algorithmen und Datenstrukturen]]
4 | ### [[Einfuehrung in die Softwaretechnik]]
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/TUM/3. Semester/3. Semester.md:
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1 | ### [[Applied Regression]]
2 | ### [[IT Sicherheit]]
3 | ### [[Numerisches Programmieren]]
4 | ### [[Grundlagen Datenbanken]]
5 | ### [[Grundlagen Betriebssysteme und Systemsoftware]]
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/.obsidian/themes/Obsidian Nord/manifest.json:
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1 | {
2 | "name": "Obsidian Nord",
3 | "version": "0.2.0",
4 | "minAppVersion": "0.16.0",
5 | "author": "insanum",
6 | "authorUrl": "https://insanum.com"
7 | }
8 |
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/.obsidian/themes/PLN/manifest.json:
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1 | {
2 | "name": "PLN",
3 | "version": "1.9.0",
4 | "minAppVersion": "1.4.16",
5 | "author": "PipeItToDevNull",
6 | "authorUrl": "https://blog.dev0.sh"
7 | }
8 |
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/TUM/4. Semester/DWT/Diskrete Wahrscheinlichkeitstheorie.md:
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1 | ### [[Diskrete Wahrscheinlichkeitsraeume]]
2 | ### [[Kontinuierliche Wahrscheinlichkeitsraeume]]
3 | ### [[Induktive Statistik]]
4 | ### [[Stochastische Prozesse]]
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/TUM/4. Semester/4. Semester.md:
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1 | ### [[Grundlagen Rechnernetze und Verteilte Systeme]]
2 | ### [[Einfuehrung in die Theoretische Informatik]]
3 | ### [[Modellbildung und Simulation]]
4 | ### [[Diskrete Wahrscheinlichkeitstheorie]]
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/.obsidian/themes/Catppuccin/manifest.json:
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1 | {
2 | "name": "Catppuccin",
3 | "version": "0.4.18",
4 | "minAppVersion": "1.0.0",
5 | "author": "Marshall Beckrich",
6 | "authorUrl": "https://github.com/catppuccin/obsidian"
7 | }
8 |
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/TUM/4. Semester/Theo/Einfuehrung in die Theoretische Informatik.md:
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1 | ### [[Grundlagen formaler Sprache]]
2 | ### [[Regulaere Sprachen]]
3 | ### [[Kontextfreie Sprachen]]
4 | ### [[Berechenbarkeit und Entscheidbarkeit]]
5 | ### [[Komplexitaetstheorie]]
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/TUM/4. Semester/GRnVS/Grundlagen Rechnernetze und Verteilte Systeme.md:
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1 | ### [[Physikalische Schicht]]
2 | ### [[Sicherungsschicht]]
3 | ### [[Vermittlungsschicht]]
4 | ### [[Transportschicht]]
5 | ### [[Sitzungs-, Darstellungs- und Anwendungsschicht]]
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/.gitignore:
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1 | # Obsidian Generated
2 | .obsidian/app.json
3 | .obsidian/workspace.json
4 | .obsidian/graph.json
5 | .obsidian/workspace-mobile.json
6 |
7 | # Plugin Generated
8 | .obsidian/plugins/obsidian-completr/scanned_words.txt
9 |
10 | TUM/2. Semester/EIST
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/TUM/2. Semester/FPV/Funktionale Programmierung und Verifikation.md:
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1 | ### [[Assertions and Verification]]
2 | ### [[Weakest Pre-Condition]]
3 | ### [[Correctness]]
4 | ### [[Termination]]
5 | ### [[Verification in Ocaml]]
6 | ### [Flashcards](https://ankiweb.net/shared/info/754030607)
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/TUM/3. Semester/NumProg/Numerisches Programmieren.md:
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1 | ### [[Fliesskommazahlen und Rundung]]
2 | ### [[Interpolation]]
3 | ### [[Integration]]
4 | ### [[Loesen linearer Gleichungssysteme]]
5 | ### [[Differentialgleichungen]]
6 | ### [[Iterative Verfahren]]
7 | ### [[Eigenwertprobleme]]
8 | ### Download Flashcards
9 |
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/TUM/5. Semester/ML/Maschinelles Lernen.md:
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1 | ### [[k-Nearest Neighbors]]
2 | ### [[Decision Trees]]
3 | ### [[Probabalistic Inference]]
4 | ### [[Linear Regression]]
5 | ### [[Linear Classification]]
6 | ### [[Optimization]]
7 | ### [[Deep Learning 1]]
8 | ### [[Deep Learning 2]]
9 | ### [[SVMs and Kernels]]
10 | ### [[Dimensionality Reduction]]
11 | ### [[Clustering]]
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/.obsidian/appearance.json:
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1 | {
2 | "accentColor": "#5e81ac",
3 | "cssTheme": "Obsidian Nord",
4 | "theme": "obsidian",
5 | "translucency": false,
6 | "nativeMenus": false,
7 | "showViewHeader": true,
8 | "baseFontSize": 14,
9 | "enabledCssSnippets": [
10 | "extension"
11 | ],
12 | "interfaceFontFamily": "Inter",
13 | "textFontFamily": "Inter"
14 | }
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/TUM/2. Semester/GAD/Grundlagen Algorithmen und Datenstrukturen.md:
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1 | ### [[Effizienzmetrik]]
2 | ### [[Laufzeitanalyse]]
3 | ### [[Datenstrukturen fuer Sequenzen]]
4 | ### [[Hashing]]
5 | ### [[Sortieren]]
6 | ### [[Priority Queues]]
7 | ### [[Suchstrukturen]]
8 | ### [[Graphen]]
9 | ### [[Pattern Matching]]
10 | ### [Flashcards](https://ankiweb.net/shared/info/1983647928)
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1 | {
2 | "id": "obsidian-relative-line-numbers",
3 | "name": "Relative Line Numbers",
4 | "version": "2.0.1",
5 | "minAppVersion": "0.13.14",
6 | "description": "Enables relative line numbers in editor mode",
7 | "author": "Nadav Spiegelman",
8 | "authorUrl": "https://nadav.is",
9 | "isDesktopOnly": true
10 | }
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/TUM/.obsidian/core-plugins.json:
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1 | [
2 | "file-explorer",
3 | "global-search",
4 | "switcher",
5 | "graph",
6 | "backlink",
7 | "outgoing-link",
8 | "tag-pane",
9 | "page-preview",
10 | "daily-notes",
11 | "templates",
12 | "note-composer",
13 | "command-palette",
14 | "editor-status",
15 | "starred",
16 | "outline",
17 | "word-count",
18 | "file-recovery"
19 | ]
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/TUM/3. Semester/ITSec/IT Sicherheit.md:
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1 | ### [[Schwachtellen und Angriffe]]
2 | ### [[Kryptographie]]
3 | ### [[Kryptographische Hashfunktionen]]
4 | ### [[Schluesselmanagement]]
5 | ### [[Authentisierung]]
6 | ### [[Public Key Infrastructure]]
7 | ### [[Netzwerksicherheit]]
8 | ### [[Rechtemanagement]]
9 | ### [[Systemsicherheit]]
10 | ### Download Flashcards
--------------------------------------------------------------------------------
/TUM/3. Semester/GDB/Grundlagen Datenbanken.md:
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1 | ### [[Konzeptuelle Modellierung]]
2 | ### [[Relationales Datenmodell]]
3 | ### [[SQL]]
4 | ### [[Datenintegritaet]]
5 | ### [[Relationale Entwurfstheorie]]
6 | ### [[Physische Datenorganisation]]
7 | ### [[Anfragebearbeitung]]
8 | ### [[Transaktionen]]
9 | ### [[Mehrbenutzersynchronisation]]
10 | ### Download Flashcards
--------------------------------------------------------------------------------
/TUM/4. Semester/ModSim/Modellbildung und Simulation.md:
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1 | ### [[Calculus]]
2 | ### [[Nash Equilibrium and Banzhaf Power Index]]
3 | ### [[Zero-Sum Games and Elections]]
4 | ### [[Job Scheduling]]
5 | ### [[Population Dynamics]]
6 | ### [[Directional Fields]]
7 | ### [[Ordinary Differential Equations]]
8 | ### [[Fuzzy Control]]
9 | ### [[Heat Conduction]]
10 | ### [[Markov Chains]]
11 | ### [[Traffic Modelling]]
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/.obsidian/plugins/obsidian-completr/manifest.json:
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1 | {
2 | "id": "obsidian-completr",
3 | "name": "Completr",
4 | "version": "3.2.0",
5 | "minAppVersion": "1.0.0",
6 | "description": "This plugin provides advanced auto-completion functionality for LaTeX, Frontmatter and standard writing.",
7 | "author": "tth05",
8 | "authorUrl": "https://github.com/tth05",
9 | "isDesktopOnly": true
10 | }
11 |
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1 | {
2 | "id": "obsidian-quickshare",
3 | "name": "QuickShare",
4 | "version": "1.0.1",
5 | "minAppVersion": "0.13.25",
6 | "description": "Securely share your Obsidian notes with one click. Notes are end-to-end encrypted. No API keys or configuration required.",
7 | "author": "Maxime Cannoodt (@mcndt)",
8 | "authorUrl": "https://mcndt.dev",
9 | "isDesktopOnly": false
10 | }
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/TUM/3. Semester/GBS/Grundlagen Betriebssysteme und Systemsoftware.md:
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1 | ### [[Betriebssysteme]]
2 | ### [[Prozessverwaltung]]
3 | ### [[Scheduling]]
4 | ### [[Parallele Systeme und Synchronisation]]
5 | ### [[Modellierung paralleler Systeme]]
6 | ### [[Interprozesskommunikation]]
7 | ### [[Speichermanagement]]
8 | ### [[Dateisysteme]]
9 | ### [[Ein- und Ausgabe]]
10 | ### [[Virtualisierung]]
11 | ### Download Flashcards
--------------------------------------------------------------------------------
/TUM/1. Semester/ERA/Digitale Schaltungen.md:
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1 | ## Repraesentation von Funktionen
2 | - Boolesche Operationen, wie AND, OR oder NOT koennen mithilfe von Schaltungen durch Gatter repraesentiert werden
3 | - Ausgehend von diesen grundlegenden Operationen koennen [[Arithmetische und Logische Operationen]], sowie [[Speicherbausteine]] implementiert werden
4 | - Um Schaltungen zu modellieren, verwendet man [[VHDL]]
5 | #### Beispiel
6 | ![[Pasted image 20221122205448.png]]
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/TUM/5. Semester/RTS/Real-Time Hardware.md:
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1 | ## Microcontroller
2 | - A microcontroller consists of a few different parts, that are all connected to each other on a single board
3 | #### Microprocessor
4 | - A microprocessor consists of a control unit, an ALU and a few registers
5 | - This microprocessor can be connected to IO, timers and memory via busses to form a microcomputer
6 | ###### Example
7 | ![[Pasted image 20241107161304.png]]
8 | #### Example
9 | ![[Pasted image 20241107161317.png]]
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/Books/Store-and-Forward Networks.md:
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1 | - Originally, wiring was used to connect computers to one another
2 | - However, in order to send data across large distances, expensive, long-distance connections had to be used
3 | - Instead of relying on these connections, intermediate computers would store and forward data across shorter distances, one hop and one message at a time
4 | - This resulted in cheaper connections, however, large sets of data could easily block entire networks
5 | - This problem was solved with the introduction of [[Packets]]
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/TUM/1. Semester/ERA/Speicherhierarchie.md:
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1 | ## Allgemeines
2 | - Da die Arbeit alleine mit dem [[Hauptspeicher]] zu langsam ist, wird das Speicherwerk in der Regel hierarchisch unterteilt
3 | ## Aufteilung
4 | - Je naeher ein Speicherkonstrukt am Prozessor ist, desto geringer ist seine Kapazitaet und desto schneller ist seine Zugriffszeit
5 | #### Speicherkonstrukte
6 | - Register
7 | - [[Cache|Caches]]
8 | - Es gibt unterschiedliche Level von Caches
9 | - Je niedriger das Level, desto geringer die Kapazitaet und die Zugriffszeit
10 | - [[Hauptspeicher]]
11 | - Hintergrundspeicher
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/TUM/1. Semester/ERA/Multicycle Prozessoren.md:
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1 | ## Allgemeines
2 | - Statt ganze Instruktionen in einem Takt abzuarbeiten, wird die [[Befehlsverarbeitung]] in mehrere Schritte mit kuerzerem Takt unterteilt
3 | - Eine Instruktion benoetigt somit mehrere Takte zum durchlaufen
4 | ## Realisierung
5 | - Multicycle Prozessoren werden mithilfe von [[Endliche Automaten|endlichen Automaten]] realisiert
6 | - Je nach auszufuehrendem Befehl, wechselt der Prozessor in unterschiedliche Zustaende
7 | #### Zustaende
8 | ![[Pasted image 20221216141452.png]]
9 | #### Beispiel
10 | ![[Pasted image 20221213172856.png]]
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/TUM/2. Semester/LinAlg/Orthogonal- und Orthonormalsystem.md:
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1 | ## Allgemeines
2 | - Es sei $(V, \langle \cdot, \cdot \rangle)$ ein beliebiger euklidischer Vektorraum
3 | #### Orthogonalbasis
4 | - $B \subseteq V$ ist ein Orthogonalsystem, falls gilt:
5 | $$\forall \space b_i, b_j \in B, i \neq j: b_i \perp b_j$$
6 | - Bildet $B$ eine Basis, so bezeichnet man diese als Orthogonalbasis
7 | #### Orthonormalbasis
8 | - $B \subseteq V$ ist ein Orthonormalsystem, falls gilt:
9 | $$\forall \space b_i, b_j \in B, i \neq j: b_i \perp b_j$$
10 | $$\forall \space b \in B: ||b|| = 1$$
11 | - Bildet $B$ eine Basis, so bezeichnet man diese als Orthonormalbasis
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/TUM/6. Semester/Markovketten/Linear Algebra in Markov Chains.md:
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1 | ## Transition Probabilities
2 | - Given an initial distribution $\mu$ and the transition matrix $\Pi$ of a Markov chain $(X_n)_{n=0}^{\infty}$, the distribution of $X_n$ can be computed as:
3 | $$\mu^T \Pi^{n}$$
4 | - Additionally, the following holds for the same Markov chain:
5 | $$P(X_{n + m} = j \mid X_n = i) = \Pi^m(i, j)$$
6 | ## Properties
7 | - Since $\Pi$ is a stochastic matrix, all of it's Eigenvalues are at most $|1|$
8 | - In fact any $\Pi$ of a Markov Chain has an Eigenvector $\hat{\mu}$, whose Eigenvalue is $1$
9 | - Because of this, the distribution of $\lim_{n \to \infty} X_n = \hat{\mu}$
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/README.md:
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1 | ## Notes
2 | - This is a collection of my notes for all sorts of things
3 | #### File Structure
4 | - The most interesting folder for you will likely be TUM
5 | - Inside it, you will find folders organized into semesters and courses
6 | - Inside each course folder there is a file with the same name as the folder
7 | - This file functions as a directory for the course with links to relevant pages
8 | #### Tools
9 | - While GitHub offers markdown support and pages can thus be viewed directly in the browser, the use of [Obsidian](https://obsidian.md/download) is highly recommended
10 | - Inside Obsidian, helpful images are displayed and links behave like intended
11 |
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/TUM/2. Semester/LinAlg/Lineare Algebra.md:
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1 | ### [[Komplexe Zahlen und Kartesische Koordinaten]]
2 | ### [[Komplexe Zahlen und Polarkoordinaten]]
3 | ### [[Lineare Gleichungssysteme]]
4 | ### [[Rechnen mit Matrizen]]
5 | ### [[Restklassengruppen]]
6 | ### [[Kryptologie]]
7 | ### [[Erzeugendensysteme]]
8 | ### [[Basen von Vektorraeumen]]
9 | ### [[Lineare Codes]]
10 | ### [[Determinante]]
11 | ### [[Orthogonalithaet]]
12 | ### [[Lineare Ausgleichsrechnung]]
13 | ### [[Lineare Abbildungen]]
14 | ### [[Eigenwerte und Eigenvektoren]]
15 | ### [[Singulaerwertszerlegung]]
16 | ### [[Definitheit von Matrizen]]
17 | ### [[Matrixnormen]]
18 | ### [Flashcards](https://ankiweb.net/shared/info/921866146)
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/Books/Packets.md:
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1 | - In order to prevent larger messages from blocking the entire network, a message would be split into packets
2 | - Those packets could then be sent across the network seperately and more quickly
3 | - In order to reassemble the packets at the correct destination, in the correct order, each packet has to contain information about its source and destination adress, as well as its offset
4 | - Routers were developed, which are computers, specialized in sending packets along their shortest route
5 | - Now, computers which were part of a Local-Area-Network, could send data across the Wide-Area-Network by connecting to a router
6 | - The term "Internet" thus captures the idea of connecting multiple local networks together
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/TUM/5. Semester/ML/Classification Performance.md:
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1 | ## Binary Classification Performance
2 | - In order to determine the performance of a binary classification algorithm, a confusion table is used
3 | #### Confusion Table
4 | - A confusion table for a binary classification algorithm has the following form:
5 | ![[Pasted image 20241018104453.png]]
6 | #### Attributes
7 | - Based on the amounts of true positives and negatives and false ones, a number of attributes can be defined:
8 | - Accuracy: $\frac{TP + TN}{TP + TN + FP + FN}$
9 | - Precision: $\frac{TP}{TP + FP}$
10 | - Sensitivity: $\frac{TP}{TP + FN}$
11 | - Specificity: $\frac{TN}{FP + TN}$
12 | - Depending on the type of problem at hand, certain attributes are more relevant than others
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/TUM/2. Semester/LinAlg/Saetze.md:
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1 | ## Satz von Lagrange
2 | - Ist $(\mathbb{G}, \cdot)$ eine endliche [[Algebraische Strukturen|Gruppe]], und $(\mathbb{U}, \cdot)$ eine Untergruppe dieser Gruppe, so gilt:
3 | $$|\mathbb{U}| \space | \space |\mathbb{G}|$$
4 | ## Satz von Euler
5 | - Ist $(\mathbb{G}, \cdot)$ eine endliche Gruppe, so gilt:
6 | $$a^{|\mathbb{G}|} = e$$
7 | ## Kleiner Satz von Fermat
8 | - Falls $p \in \mathbb{N}$ prim, so gilt, falls $p \nmid a$
9 | $$\forall \space a \in \mathbb{Z}: a^{p-1} = 1 \mod p$$
10 | - Und falls $p \mid a$:
11 | $$\forall \space a \in \mathbb{Z}: a^p = a \mod p$$
12 | ## Determinanten vom Transponierten
13 | - Es sei $A \in \mathbb{K}^{n \times n}$ beliebig
14 | - Fuer $det(A)$ gilt:
15 | $$det(A) = det(A^T)$$
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/TUM/2. Semester/LinAlg/Skalarprodukt.md:
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1 | ## Allgemeines
2 | - Es wird eine Abbildung $\langle \cdot, \cdot \rangle$ wird definiert durch:
3 | $$\langle \cdot, \cdot \rangle: \begin{cases}
4 | V \times V \rightarrow \mathbb{R} \\
5 | (v, w) \mapsto \langle v, w \rangle \\
6 | \end{cases}$$
7 | - Diese Abbildung bezeichnet man als Skalarprodukt, falls sie bilinear, symmetrisch und positiv definit ist:
8 | $$\forall \space v, v', w, \in V: \forall \space \lambda \in \mathbb{R}: \langle \lambda v + v', w \rangle = \lambda \langle v, w \rangle + \langle v', w \rangle$$
9 | $$\forall \space v, w \in V: \langle v, w \rangle = \langle w, v \rangle$$
10 | $$\forall \space v \in V: \langle v, v \rangle \geq 0$$
11 | $$\forall v \in V: \langle v, v \rangle = 0 \equiv v = 0$$
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/TUM/1. Semester/ERA/Einfuehrung in die Rechnerarchitekur.md:
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1 | ### [[Das Von-Neumann Konzept]]
2 | ### [[Hauptspeicher]]
3 | ### [[Befehlsverarbeitung]]
4 | ### [[Instruction Set Architecture]]
5 | ### [[Assembler]]
6 | ### [[Compilierung und Assemblierung]]
7 | ### [[Speicherhierarchie]]
8 | ### [[Speicherverwaltung]]
9 | ### [[Systembefehle]]
10 | ### [[Digitale Schaltungen]]
11 | ### [[Arithmetische und Logische Operationen]]
12 | ### [[Speicherbausteine]]
13 | ### [[RISC-V]]
14 | ### [[Aufbau eines Prozessors]]
15 | ### [[Endliche Automaten]]
16 | ### [[Multicycle Prozessoren]]
17 | ### [[Pipelining]]
18 | ### [[Logiksynthese und - minimierung]]
19 | ### [[Binaere Entscheidungsdiagramme]]
20 | ### [[Speicheraufbau]]
21 | ### [[Cache]]
22 | ### [[Moderne Prozessoren]]
23 | ### [[Multithreading]]
24 | ### [[VHDL]]
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/TUM/2. Semester/LinAlg/Gaussches Eliminationsverfahren.md:
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1 | ## Eliminationsverfahren
2 | - Ein [[Lineare Gleichungssysteme|LGS]] kann geloest werden, indem es in seine Zeilenstufenform ueberfuehrt wird
3 | - Die Umformung erfolgt mithilfe elementarer Zeilenumformungen
4 | - Nach Erreichen der Zeilenstufenform koennen die Loesungen des LGS durch Rueckwaertseinsetzen bestimmt werden
5 | #### Zeilenumformungen
6 | - Vertauschen zweier Zeilen
7 | - Multiplikation einer Zeile mit einem $\lambda \neq 0$
8 | - Addition des $\lambda$-fachen einer Zeile zu einer anderen Zeile
9 | #### Beispiel
10 | $$\begin{pmatrix}
11 | 1 & 1 & 2 &| & 2 \\
12 | 2 & 2 & -1 &| & 1 \\
13 | 3 & 4 & 2 & | & 2
14 | \end{pmatrix}
15 | \leadsto
16 | \begin{pmatrix}
17 | 1 & 1 & 2 & | & 2 \\
18 | 0 & 1 & -4 & | & -4 \\
19 | 0 & 0 & -5 & | & -3
20 | \end{pmatrix}
21 | $$
22 |
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/TUM/1. Semester/ERA/Endliche Automaten.md:
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1 | ## Allgemeines
2 | - Ein Automat ist ein Konstrukt, welches ausgehend von einem Ausgangszustand und einer Eingabe in einen Nachfolgezustand wechselt und eine Ausgabe produziert
3 | ## Darstellung
4 | #### Mathematisch
5 | ![[Pasted image 20221212122849.png]]
6 | #### Graphisch
7 | ![[Pasted image 20221212125721.png]]
8 | ## Realisierung
9 | - Um einen Automaten als [[Digitale Schaltungen|digitale Schaltung]] zu realisieren, muessen die erforderlichen Signale kodiert werden
10 | - Ein Automat erhaelt am Eingang ein Zustands- und ein Eingabesignal
11 | - Ein neuer Zustand, sowie ein Ausgabesignal koennen ausgehend von Logikgattern ermittelt werden
12 | - Dieser neue Zustand wird mithilfe von Flipflops gespeichert und im naechsten Takt als Eingang verwendet
13 | #### Beispiel
14 | ![[Pasted image 20221212132121.png]]
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/TUM/1. Semester/ERA/Befehlsverarbeitung.md:
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1 | ## Phasen der Befehlsverarbeitung
2 | - Die Ausführung von Befehlen lässt sich in die Interpretations- und Ausführungsphase einteilen
3 | ### Interpretationsphase
4 | - Die Adresse des auszuführenden Befehls wird im Befehlszähler gespeichert
5 | - Über MAR und MDR wird der Befehl in das Instruktionsregister geladen und anschließend dekodiert und ausgeführt
6 | - Nach dem Ausführen wird der Befehlszähler inkrementiert
7 | ### Ausführungsphase
8 | - Die, im Befehl enthaltenen, Daten werden über MAR und MDR aus dem [[Hauptspeicher]] geholt und in Register geladen
9 | - Über die ALU werden die Operanden verrechnet und das Ergebnis im Akkumulator abgespeichert
10 | - Mithilfe von weiteren Befehlen kann das Ergebnis in den [[Hauptspeicher]] zurückgeschrieben werden
11 | ### Beispiel
12 | ![[Pasted image 20221024164610.png]]
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/TUM/2. Semester/LinAlg/Matrixnormen.md:
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1 | ## Allgemeines
2 | - Fuer eine Matrix $A \in \mathbb{R}^{n \times n}$ wird die die Norm mithilfe der Abbildung $||\cdot||$ definiert:
3 | $$||\cdot|| = \begin{cases}
4 | V \rightarrow \mathbb{R} \\
5 | v \mapsto ||v||
6 | \end{cases}$$
7 | #### Einsnorm
8 | - Fuer Spaltenvektoren $v$ wird die Norm $||v||_1$ definiert ueber:
9 | $$||v||_1 = |v_1| + ... + |v_n|$$
10 | - Die Einsnorm einer Matrix ist die maximale Spaltensumme
11 | #### Zweinorm
12 | - Fuer Spaltenvektoren $v$ wird die Norm $||v||_2$ definiert ueber:
13 | $$||v||_2 = \sqrt{{v_1}^2 + ... + {v_n}^2}$$
14 | - Die Zweinorm einer Matrix wird mithilfe der Spektralnorm bestimmt
15 | #### Unendlichkeitsnorm
16 | - Fuer Spaltenvektoren $v$ wird die Norm $||v||_{\infty}$ definiert ueber:
17 | $$||v||_{\infty} = max\{|v_1|, ..., |v_n|\}$$
18 | - Die Unendlichkeitsnorm einer Matrix ist die maximale Zeilensumme
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/TUM/2. Semester/FPV/Assertions and Verification.md:
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1 | ## Assertions
2 | - In order to verify the correctness of a given program, assertions can be used
3 | - Assertions throw an exception if the program state does not imply the assertion
4 | - In order to verify a program, every program points needs to be annotated with an assertion
5 | - The assertion before an operation is called a pre-condition
6 | - The assertion after an operation is called a post-condition
7 | #### Example
8 | ![[Pasted image 20230417211426.png]]
9 | ## Verification
10 | - Each program points needs to be annotated with an assertion with the start being annotated with true
11 | - For each statement (or conditional ) $s$ the given assertion $A$ needs to imply the [[Weakest Pre-Condition|weakest pre-condition]], derived from the post-condition $B$ (or $B_0$ and $B_1$) of $s$
12 | - If this is fulfilled, the assertions is considered locally consistent
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/TUM/1. Semester/ERA/Systembefehle.md:
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1 | ## Aufgaben des Betriebssystems
2 | - Ueber das Betriebssystem werden Datein, Prozesse und Speicher verwaltet ([[Speicherverwaltung]])
3 | - Um alle Rechnerkomponenten verwalten zu koennen, verfuegt das Betriebssystem ueber einen priviligierten Modus
4 | #### Moduswechsel
5 | - Um vom Benutzer- zum Systemmodus zu wechseln, kann ein System Call ausgefuehrt werden
6 | - Bei einem System Call wird an eine vordefinierte Adresse im Betriebssystem gesprungen, eine Routine ausgefuehrt und anschliessend in den Benutzermodus zurueckgekehrt
7 | #### Interrupts
8 | - Der Prozessor kann ueber Interrupts auf externe, asynchrone Ereignisse reagieren
9 | - Die Ruecksprungadresse und Registerinhalte werden gesichert, bevor der Interrupt-Handler des Betriebssystems ausgefuehrt wird
10 | #### Exceptions
11 | - Beim Auftreten einer Exception wird das Programm mithilfe des Exception-Handlers des Betriebssystems korrigiert oder abgebrochen
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/TUM/2. Semester/LinAlg/Basen von Vektorraeumen.md:
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1 | ## Allgemeines
2 | - Es sei $V$ ein $\mathbb{K}$-[[Vektorraeume|Vektorraum]], so bezeichnet man $B \subseteq V$ als Basis von $V$, falls gilt:
3 | - $B$ ist [[Erzeugendensysteme|linear unabhaengig]]
4 | - $B$ erzeugt $V$
5 | - Jeder $\mathbb{K}$-Vektorraum besitzt eine Basis
6 | #### Basisverkuerzung und -erweiterung
7 | - Ist $B$ eine Basis von $V$, so gilt:
8 | - $B$ ist ein minimales Erzeugendensystem von $V$
9 | - $B$ ist eine maximale, linear unabhaengige Teilmenge von $V$
10 | - Jede linear unabhaengige Teilmenge von $V$ kann zu einer Basis von $V$ ergaenzt werden
11 | - Jedes Erzeugendensystem von $V$ kann zu einer Basis von $V$ verkuerzt werden
12 | #### Maechtigkeit von Basen
13 | - Alle Basen von $V$ haben dieselbe Maechtigkeit
14 | - Falls $n = dim(V)$, so gilt:
15 | - Jede linear unabhaengige Menge und jedes Erzeugendensystem mit $n$ Elementen ist eine Basis von $V$
16 | - Jede Menge mit mehr als $n$ Elementen ist linear abhaengig
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/TUM/5. Semester/RTS/Time and Clocks.md:
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1 | ## Time Measurements in Computers
2 | - Computers measure time based on a multiple of a periodically created microtick
3 | - The granularity of a clock describes the size of the intervals between microticks
4 | ![[Pasted image 20241031162537.png]]
5 | #### Measurement Errors
6 | - Assuming there is a perfect clock $z$, and all other clocks are compared against it
7 | - Given a clock $k$ with a granularity of $g^k$ which is equal to $10$ microticks of $z$, or $n^k$
8 | - The drift of this clock is described by:
9 | $$\text{drift}_i^k = \frac{z (\text{microtick}^k_{i + 1}) - z(\text{microtick}^k_i)}{n^k}$$
10 | - The drift is thereforce the relation of the microticks performed by $k$ to the microticks it is supposed to perform
11 | - From this, the drift rate $\rho$ can be computed:
12 | $$\rho_i^k = | \text{drift}_i^k - 1|$$
13 | - A perfect clock therefore has a drift rate of $0$
14 | ###### Example
15 | ![[Pasted image 20241031163456.png]]
16 | ![[Pasted image 20241031163504.png]]
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/TUM/2. Semester/FPV/Termination.md:
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1 | ## General
2 | - It is not guranteed, that an [[Assertions and Verification|assertion]] is always reached, since a program might not terminate
3 | - The termination of a given program must therefore be proven
4 | #### Approach
5 | ###### Indicator Variable
6 | - For any loop, an indicator variable $r$ needs to be determined
7 | - $r$ ist often calculated using an expression $e_r$, composed of local variables
8 | ###### Control Flow
9 | - The necessary operations for creating and updating $r$ are inserted into the program
10 | - The loop invariant is formed around $r$
11 | ###### Proof
12 | - Locally consistent assertions are calculated inside the loop via [[Weakest Pre-Condition|weakest pre-conditions]], starting with the assertion $r > e_r$
13 | - If $r$ fulfills all necessary criteria, a given loop will terminate:
14 | - The assertion at the start of the loop implies $r \geq 0$
15 | - The assertion at the end of the loop implies $r > e_r$
16 | ###### Example
17 | ![[Pasted image 20230627195811.png]]
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/Books/Link Layer.md:
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1 | ## Tasks of the Link Layer
2 | - The link layer is responsible for connecting a device to a local area network and moving data across a single hop
3 | - Link layer technologies, such as cellphone towers and wifi networks, are often used my multiple devices at the same location
4 | - Some technologies, such as fiber optic cables are not shared and only used to send data across routers
5 | - The link layer thus has to deal with encoding, transmitting and coordinating the data traffic
6 | ## Packet Transmission
7 | - In order to enable each computer to send packets across the internet, they implement the Carrier Sense Multiple Access with Collision Detection (CSMA/CD)
8 | - Each computer listens, to see if the network is busy
9 | - If it is not, a computer starts sending data, while simultaneously listening to see, if it can receive it's own data
10 | - If it cannot, the network is busy and the computer waits before trying again
11 | - Once a computer has finished sending a packet, it pauses, so that other devices get a chance to send theirs
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/TUM/4. Semester/ModSim/Nash Equilibrium and Banzhaf Power Index.md:
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1 | ## Nash Equilibrium
2 | - A Nash equilibrium is reached, if every player's strategy is optimal given the strategies of all other players
3 | - As such, a player would have no interest in changing strategies, since it would only result in a worse result for themselves
4 | ## Banzhaf Power Index
5 | - The Banzhaf power index is used to describe the influence of a party in a political system
6 | - In this system, all parties have a weight $ and can either vote yes, or no
7 | - A vote is passed, if the weights of the parties voting yes exceeds a specific quorum $q$
8 | #### Computing the Index
9 | - The Banzhaf power index of a party is the number of scenarios, in which it would sway the vote
10 | - More specifically given a set of parties $\{S_1, ..., S_n\}$ with weights $w_1, ..., w_n$, the Banzhaf power index of $S_i$ is defined as
11 | $$|T|, \; \; T \subseteq S \setminus \{S_i\}: \sum_{k \in T} w_k < q \leq \sum_{k \in T \cup \{S_i\}} w_k$$
12 | ###### Relative Index
13 | - The relative Banzhaf power index of a party is it's absolute index, divided by the sum of power indices of all parties
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/TUM/1. Semester/ERA/Das Von-Neumann Konzept.md:
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1 | ## Sieben Prinzipien eines Rechners
2 | 1. Die Struktur eines Rechners ist problemunabhängig
3 | 2. Ein Rechner besteht aus vier Werken
4 | - Speicherwerk (speichert Programme und Daten)
5 | - Leitwerk (interpretiert Programme)
6 | - Rechenwerk (führt arithmetische Operationen aus)
7 | - Ein-/Ausgabewerk (kommuniziert mit der Umgebung)
8 | 4. Der [[Hauptspeicher]] ist in adressierte, gleich große Zellen unterteilt
9 | 5. Programme und Daten stehen im selben [[Hauptspeicher]] und können durch den Rechner verändert werden
10 | 6. Der Rechner verwendet das Binärsystem
11 | 7. Ein Programm besteht aus einer Folge von Befehlen, welche sequenziell verarbeitet werden ([[Befehlsverarbeitung]])
12 | 8. Von der Befehlsfolge kann über bedingte und unbedingte Sprünge abgewichen werden
13 |
14 | ## Abweichungen vom Konzept
15 | - Das Leit- und Rechenwerk werden zum Prozessor zusammengefasst
16 | - Das Speicherwerk ist hierarchisch unterteilt ([[Speicherhierarchie]])
17 | - Auf einem Prozessor befinden sich mehrere Rechenwerke
18 | - Programme und Daten werden in größtenteils getrennten Speichern verwaltet
19 | ## Beispiel
20 | ![[Pasted image 20230205164416.png]]
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/TUM/2. Semester/FPV/Correctness.md:
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1 | ## Correctness of a Program
2 | #### Program State
3 | - The state of any given program $\sigma$ can be described by it's variables and their values
4 | - A program state $\sigma$ satisfies a statement $A$, if the statement is true for the current values of all variables
5 | - The correctness of a statement can thus be determined by substituting in all current values
6 | ###### Example
7 | ![[Pasted image 20230424132008.png]]
8 | #### Execution Trace
9 | - The execution trace $\pi$ of a program is a series of program states $\sigma_0 ... \sigma_m$ with their designated program points $u_0 ... u_m$
10 | - An execution trace $\pi$ can therefore be described by $$(u_0, \sigma_0)s_1(u_1, \sigma_1)...s_m(u_m, \sigma_m)$$where $s_i$ represents statements, such as basic operations and assignments in the program
11 | #### Theorem
12 | - Using execution traces, the correctness of a program can be verified as follows:
13 | - Given a program with the execution trace $\pi$ that ranges from the program point $u$ to $v$
14 | - The program point $u$ is annotated with the Assertion $A$ and $v$ with $B$
15 | - If the initial state of $\pi$ satisfies $A$, then it's final state will satisfy $B$
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/TUM/3. Semester/GDB/Konzeptuelle Modellierung.md:
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1 | ## Entity-Relationship Modell
2 | - Fuer den Konzeptuellen Entwurf einer Datenbank werden Entity-Relationship Modelle verwendet
3 | #### Entities
4 | - Verschiedene Entities stehen in Beziehungen zueinander
5 | - Entities werden zudem mithilfe von Attributen beschrieben
6 | - Das Schluesselattribut ist ein Attribut, welches den Entity eindeutig definiert
7 | ###### Schwacher Entitytyp
8 | - Alle Entities eines schwachen Typs nehmen an einer Beziehung Teil
9 | - Der Schluessel eines schwachen Entities ergibt sich aus seinem partiellen Schluessel und dem des Entites zu dem die Beziehung besteht
10 | ###### Beispiel
11 | ![[Pasted image 20231018114626.png]]
12 | ![[Pasted image 20231018124624.png]]
13 | #### Beziehungen
14 | ###### n-stellige Beziehungen
15 | - Man unterscheidet 1:1, 1:n, sowie n:m Beziehungen
16 | - Hierbei wird jeweils ausgedrueckt wieviele Entities eines Typs in einer Beziehung mit Entities eines anderen Typs stehen
17 | ###### Min-Max Beziehungen
18 | - Bei einer Beziehung wird ausgedrueckt, an wievielen Beziehungen eine Entity mindestens und hoechstens teilnehmen kann
19 | ###### Beispiel
20 | ![[Pasted image 20231018123352.png]]
21 | ![[Pasted image 20231018123412.png]]
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/TUM/1. Semester/ERA/Binaere Entscheidungsdiagramme.md:
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1 | ## Aufbau
2 | - Das Entscheidungsdiagramm ist ein Baum, wobei jeder Knoten einer Funktion entspricht, welche entweder zu 1, oder zu 0 evaluiert
3 | - Eine Kante entspricht dem Wert, den die Funktion annimmt und verbindet sie mit einer weitern Funktion
4 | - Blattknoten, oder Terminale, entsprechen dem Wert, zu dem die Funktion mit der gegebenen Belegung evaluiert
5 | #### Beispiel
6 | ![[Pasted image 20230114152702.png]]
7 | ## Reduktion
8 | #### Isomorphismus
9 | - Sind zwei Knoten isomorph, so koennen sie unter einem zusammengefasst werden
10 | ###### Beispiel
11 | ![[Pasted image 20230114152724.png]]
12 | #### Elimination
13 | - Deuten beide ausgehende Kanten eines Knoten auf denselbsen Knoten, so kann dieser entfernt werden
14 | ###### Beispiel
15 | ![[Pasted image 20230114152747.png]]
16 | #### Shannon Zerlegung
17 | - Eine boolsche Funktion wird in eine Funktion bestehend aus einem Literal und einer aus den Restlichen zerlegt
18 | - Im Entscheidungsdiagramm evaluiert das einzelne Literal zu 1, beziehungsweise zu 0
19 | - Die darauffolgende Funktion ist die aus den restlichen Literalen
20 | - Dieser Vorgang kann rekursiv wiederholt werden, bis das Entscheidungsdiagramm konstruiert wurde
21 | ###### Beispiel
22 | ![[Pasted image 20230114152843.png]]
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/TUM/2. Semester/LinAlg/Konstruktion der komplexen Zahlen.md:
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1 | ## Vorgehen
2 | - Zur Konstruktion von $\mathbb{C}$ wird die Menge $\mathbb{R}^2=\{(a, b) | a, b \in \mathbb{R}\}$ betrachtet
3 | - Die Elemente von $\mathbb{R}^2$ sind somit Punkte in einer zweidimensionalen Ebene
4 | - Diese Ebene besitzt eine reelle und eine imaginaere Achse
5 | - Befinden sich Elemente auf der reellen Achse, so entsprechen Addition und Multiplikation dieser Elemente in $\mathbb{R}^2$ den Operationen in $\mathbb{R}$
6 | #### Definition von Operationen
7 | - Fuer $\mathbb{R}^2$ werden Addition und Multiplikation definiert
8 | - $\mathbb{R^2}$ bildet somit einen [[Algebraische Strukturen|Koerper]]
9 | ###### Addition
10 | - Die Addition von $(a, b)$ und $(c, d)$ wird ueber $(a, b) + (c, d) = (a+c, b+d)$ definiert
11 | - Das neutrale Element bezueglich der Addition ist $(0, 0)$
12 | - Jedes Element besitzt ein Inverses
13 | ###### Multiplikation
14 | - Die Multiplikation von $(a, b)$ und $(c, d)$ wird ueber $(a,b) \cdot(c, d)=(ac - bd, ad + bc)$ definiert
15 | - Das neutrale Element bezueglich der Multiplikation ist $(1, 0)$
16 | - Ausser dem Nullvektor hat jedes Element ein Inverses
17 | #### Notation
18 | - Ein Element $(a, 0)$ wird durch $a$ beschrieben
19 | - Das Element $(0, 1)$ wird durch $i$ beschrieben
20 | - Ein beliebiges Element $(a, b)$ kann somit als $a + bi$ dargestellt werden
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/TUM/1. Semester/ERA/Speicherbausteine.md:
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1 | ## Allgemeines
2 | - Speicherbausteine basieren auf dem Konzept der Rueckkopplung
3 | - Indem der Ausgang einer Schaltung als Eingang wiederverwendet wird, wird Speicherverhalten simuliert
4 | #### Beispiel
5 | ![[Pasted image 20221129161417.png]]
6 | ## RS-Latch
7 | - Indem zwei NOR Gatter parallel rueckgekoppelt werden, kann mithilfe eines SET und eines RESET Signals ein Bit gespeichert werden
8 | #### Beispiel
9 | ![[Pasted image 20221129162651.png]]
10 | ![[Pasted image 20221129162707.png]]
11 | ## D-Latch
12 | - Anstelle eines RS-Latches koennen sowohl SET, als auch RESET von einem Eingangssignal abgeleitet werden
13 | - Um das Latch anzusteuern, wird ein ENABLE Signal mit dem Eingangssignal verundet
14 | #### Taktgesteuerte D-Flipflops
15 | - Im Gegensatz zu Latches, koennen Flipflops nur zu steigender Taktflanke angesteuert werden
16 | - Ueber OR Gatter kann das Signal des Takts verzoegert werden, sodass ein Speicherbaustein nur zum Zeitpunkt der aufsteigenden Flanke angesteuert werden kann
17 | #### Beispiel
18 | ![[Pasted image 20221129163006.png]]
19 | ![[Pasted image 20221129165727.png]]
20 | ## Mehrstellige Speicher
21 | - Mehrstellige Binaerzahlen koennen mithilfe einer Kombination aus Adressbits, sowie D-Latches und Multiplexern ueberschrieben und ausgelesen werden
22 | #### Beispiel
23 | ![[Pasted image 20221129164144.png]]
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/TUM/2. Semester/GAD/Pattern Matching.md:
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1 | ## Problem
2 | - In einem Text der Laenge $n$ wird nach einem Teilwort der Laenge $m \leq n$ gesucht
3 | - Die Indizes dieses Teilworts muessen zurueckgegeben werden
4 | ## Knuth-Morris-Pratt Algorithmus
5 | - Um den Suchbereich zu optimieren, wird der Rand eines Wortes verwendet
6 | #### Rand
7 | - Ein Wort $r$ ist der Rand eines Wortes $w$, falls $r$ sowohl Praefix, als auch Suffix von $w$ ist
8 | - $r$ ist der eigentliche Rand, falls er der groesste moegliche Rand, nach $w$ selbst ist
9 | #### Shifts
10 | - Tritt ein Mismatch auf, so wird das Suchwort so weit verschoben, bis der gematchte Praefix im Suffix liegt, in dem das Mismatch aufgetreten ist
11 | ###### Beispiel
12 | ![[Pasted image 20230722205453.png]]
13 | ###### Sichere Shifts
14 | - Es muss garantiert werden, dass bei einem Shift kein Vorkommen des Teilworts uebersprungen wird
15 | - Hierfuer wird ein Array $borders$ erstellt, welches am Index $i$ die Laenge des eigentlichen Rands des Praefix der Laenge $i$ des Teilworts enthaelt
16 | - Indem bei einem Mismatch an der Stelle $j$ um $j - borders[j]$ geshiftet wird, wird sichergestellt, dass erneut am fruehesten moeglichen Vorkommen des Teilworts gesucht wird
17 | ![[Pasted image 20230722215215.png]]
18 | #### Laufzeit
19 | - Der Algorithmus benoetigt maximal $2n + m$ Vergleiche um ein Teilwort der Laenge $m$ in einem Text der Laenge $n$ zu finden
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/TUM/4. Semester/ModSim/Directional Fields.md:
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1 | ## Visualizing Differential Equations
2 | - In order to visualize a differential equation, a directional field, consisting of arrows in a field, can be used
3 | - Each arrow describes how a sepecific point will change over time
4 | #### Construction
5 | - In order to construct a vector field, a few steps are required
6 | ###### Nullclines
7 | - Nullclines are linear lines, at which a differential equation equals zero
8 | - Those nullclines can be thought of as the axes of directional fields
9 | ![[Pasted image 20240722172000.png]]
10 | ###### Steady States
11 | - Steady states are the intersections of nullclines
12 | - Directional fields, in a certain sense, exist only to identify steady states as either attractive, repulsive, or saddle points
13 | ![[Pasted image 20240722172251.png]]
14 | ###### Considering Signs
15 | - The direction of arrows depends on the sign of a differential equation in a sector
16 | - These sectors are defined by the nullclines
17 | - For each sector, the signs of all differential equations are combined
18 | ![[Pasted image 20240722173211.png]]
19 | ![[Pasted image 20240722173226.png]]
20 | ###### Construct Arrows
21 | - Once the direction of arrows has been defined for every sector, they can be drawn
22 | - It is important to consider the fact, that arrows might cross nullclines and don't have to be in line with them
23 | ![[Pasted image 20240722173539.png]]
24 | #### Example
25 | ![[Pasted image 20240722163726.png]]
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/TUM/3. Semester/GDB/Datenintegritaet.md:
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1 | ## Referentielle Integritaet
2 | - Fremdschluessel einer Relation muessen auf existierende Tupel verweisen, oder null sein
3 | - Die Intigritaet muss erhalten bleiben, falls sich das referenzierte Tupel aendert oder geloescht wird
4 | - Hierfuer gibt es unterschiedliche Herangehensweisen
5 | #### Cascade
6 | - Mithilfe des **cascade** Schluesselworts werden die Aenderungen kaskadiert
7 | - Wird das referenzierte Tupel beispielsweise geloescht, so wird auch das referenzierende Tupel geloescht
8 | #### Set Null
9 | - Mithilfe des **set null** Schluesselworts wird der Fremdschluessel bei Aenderungen auf null gesetzt
10 | ## Statische Integritaetsbedingungen
11 | - Mithilfe von statischen Integritaetsbedingungen wird festgelegt, welche Werte ein Attribut annehmen kann
12 | - Diese Integritaetsbedingungen werden mithilfe von **check** definiert
13 | ###### Beispiel
14 | ```sql
15 | create table Users(
16 | name varchar(30),
17 | ...
18 | age integer check(age > 0)
19 | );
20 | ```
21 | - Es wird sichergestellt, dass das Alter eines Nutzers positiv ist
22 | ```sql
23 | create table Likes(
24 | id integer,
25 | tweet_id integer,
26 | ...
27 | creation_date date check(not exists(
28 | select * from Tweet t
29 | where t.id = tweet_id
30 | and creation_date < t.creation_date
31 | )
32 | )
33 | foreign key (tweet_id) references Tweet(id)
34 | )
35 | ```
36 | - Es wird sichergestellt, dass ein Like nicht vor seinem entsprechenden Tweet erstellt werden kann
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/TUM/1. Semester/ERA/Speicherverwaltung.md:
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1 | ## Speicherorganisation
2 | - Jeder Prozess hat einen eigenen Platz im [[Hauptspeicher]], samt Stack, Heap, Data- und Codesegment
3 | - Laufen mehrere Prozesse parallel, so muss das Betriebssystem, den Speicher eines Prozesses vor allen anderen schuetzen
4 | #### Virtueller Adressraum
5 | - Mithilfe eines virtuellen Adressraums kann der Speicher eines Prozesses geschuetzt werden
6 | - Ein Prozess kann seinen virtuellen Adressraum beliebig nutzen, der Adressraum anderer Prozesse bleibt allerdings verborgen
7 | - Fuer tatsaechliche Speicherzugriffe werden allerdings die physikalischen Adressen benoetigt
8 | #### Memory Management Unit
9 | - Um virtuelle Adressen in Physikalische zu uebersetzen, wird die Memory Management Unit verwendet
10 | - Der Uebersetzungsprozess erfolgt mithilfe von Page Tables
11 | - Ein [[Cache]] kann wahlweise vor, oder nach dem Zugriff auf die MMU verwendet werden
12 | ###### Pages
13 | - Ein [[Hauptspeicher]] wird in Pages fester Groesse eingeteilt
14 | - In Page Tables werden virtuelle Adressen und ihre physikalischen Gegenstuecke gespeichert
15 | ###### Page Directory
16 | - Ueber ein Page Directory wird auf die Page Tables eines Prozesses verwiesen
17 | - Ein Spezialregister verweist wiederum auf das Page Directory
18 | - Bei einem Prozesswechsel wird somit nur dieses Spezialregister veraendert
19 | ###### Swapping
20 | - Um bei der Ueberschreitung der Speicherkapazitaet einen Absturz zu vermeiden, koennen Pages auf den externen Speicher ausgelagert werden
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/TUM/2. Semester/LinAlg/Erzeugendensysteme.md:
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1 | ## Linearkombinationen
2 | - Es sei $V$ ein $\mathbb{K}$-Vektorraum mit $v_1, ..., v_n \in V$ und $\lambda_1, ..., \lambda_n \in \mathbb{K}$
3 | - Man bezeichnet einen Vektor $v \in V$ als Linearkombination von $v_1, ..., v_n$, falls er dargestellt werden kann durch:
4 | $$v = \lambda_1 v_1 + ... + \lambda_n v_n$$
5 | - Um zu bestimmen ob ein gegebener Vektor $v$ als Linearkombination von $v_1, ..., v_n$ dargestellt werden kann, kann ein [[Lineare Gleichungssysteme|homogenes lineares Gleichungssystem]] verwendet werden
6 | #### Das Erzeugnis
7 | - Fuer $X \subseteq V$ wird definiert:
8 | $$\langle X \rangle = lin(X) = span(X) = \Big \{ \sum_{i = 1}^{n} \lambda_i v_i \mid n \in \mathbb{N}, \lambda_1, ..., \lambda_n \in \mathbb{K}, v_1, ..., v_n \in \mathbb{X} \Big \}$$
9 | - Gilt $\langle X \rangle = U$, so bezeichnet man $X$ als Erzeugendensystem von $U$
10 | #### Lineare Unabhaengigkeit
11 | - Die Vektoren $v_1, ..., v_n$ sind linear unabhaengig, falls gilt:
12 | $$\forall \space T \subset \{v_1, ..., v_n\}: \langle T \rangle \subset \langle v_1, ..., v_n \rangle$$
13 | - Umgekehrt sind die Vektoren linear abhaengig, falls gilt:
14 | $$\exists \space T \subset \{v_1, ..., v_n\}: \langle T \rangle = \langle v_1, ..., v_n \rangle$$
15 | ###### Kriterium
16 | - Das lineare Gleichungssystem $\lambda_1 v_1 + ... + \lambda_n v_n = 0$ wird aufgestellt
17 | - Hat das LGS genau eine Loesung, so sind die Vektoren linear unabhaengig
18 | - Hat es mehr als eine Loesung, so sind die Vektoren linear abhaengig
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/TUM/2. Semester/LinAlg/Definitheit von Matrizen.md:
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1 | ## Allgemeines
2 | - Es sei $A \in \mathbb{R}^{n \times n}$ symmetrisch
3 | - $A$ ist definit, falls sie unterschiedliche Eigenschaften erfuellt
4 | - Dies kann anhand der [[Eigenwerte und Eigenvektoren|Eigenwerte]] von $A$ ermittelt werden
5 | #### Positiv Definit
6 | - $A$ ist positiv definit, falls gilt:
7 | $$\forall \space v \in \mathbb{R}^n \setminus \{0\}: v^TAv > 0$$
8 | - Dies ist erfuellt, falls fuer die Eigenwerte $\lambda_i$ von $A$ gilt:
9 | $$\lambda_i > 0$$
10 | #### Negativ Definit
11 | - $A$ ist negativ definit, falls gilt:
12 | $$\forall \space v \in \mathbb{R}^n \setminus \{0\}: v^TAv < 0$$
13 | - Dies ist erfuellt, falls fuer die Eigenwerte $\lambda_i$ von $A$ gilt:
14 | $$\lambda_i < 0$$
15 | #### Semi Positiv Definit
16 | - $A$ ist semi positiv definit, falls gilt:
17 | $$\forall \space v \in \mathbb{R}^n \setminus \{0\}: v^TAv \geq 0$$
18 | - Dies ist erfuellt, falls fuer die Eigenwerte $\lambda_i$ von $A$ gilt:
19 | $$\lambda_i \geq 0$$
20 | #### Semi Negativ Definit
21 | - $A$ ist semi negativ definit, falls gilt:
22 | $$\forall \space v \in \mathbb{R}^n \setminus \{0\}: v^TAv \leq 0$$
23 | - Dies ist erfuellt, falls fuer die Eigenwerte $\lambda_i$ von $A$ gilt:
24 | $$\lambda_i \leq 0$$
25 | #### Indefinit
26 | - $A$ ist indefinit, falls gilt:
27 | $$\exists \space v, w \in \mathbb{R}^n: v^TAv > 0, \space w^TAw < 0$$
28 | - Dies ist erfuellt, falls fuer die Eigenwerte $\lambda_1, ..., \lambda_n$ von $A$ gilt:
29 | $$\exists \space \lambda_i, \lambda_j \in \{\lambda_1, ..., \lambda_n\}: \lambda_i > 0, \space \lambda_j < 0$$
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/TUM/3. Semester/ITSec/Netzwerksicherheit.md:
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1 | ## Transportsicherheit
2 | - Der Transport von Datenpaketen in einem Netzwerk muss sicher gestaltet werden
3 | - Diese Sicherheitsziele werden auf verschiedene Weisen realisiert
4 | #### Transport Layer Security
5 | - Mithilfe von TLS kann ein sicherer Kommunikationskanal zwischen zwei Entitaeten erzeugt werden
6 | ###### Handshake
7 | - Der Client sendet eine Zufallszahl, eine Menge von Ciphersuites, die entsprechenden [[Schluesselmanagement|Diffie-Hellman]] Parameter und eine Menge von Signaturverfahren an den Server
8 | - Der Server sendet eine Zufallszahl, eine Ciphersuite, einen Diffie-Hellman Parameter, das Zertifikat und $Sig(H(\nabla))$, sowie $HMAC(\nabla)$, verschluesselt mit $k_{SC}$
9 | - Nun besitzen sowohl der Client, als auch der Server das shared secret und der Client versendet abschliessend $HMAC(\nabla)$, verschluesselt mit $k_{CS}$
10 | ![[Pasted image 20240109151035.png]]
11 | #### Firewall Architekturen
12 | - Um das Versenden und Empfangen ungewollter Nachrichten zu verhindern, werden Firewalls verwendet
13 | - Firewalls koennen unterschiedliche Formen annehmen
14 | ###### Paketfilter
15 | - Anhand der Headerdaten eines Pakets wird entschieden, ob es weitergeleitet wird, oder nicht
16 | ###### Deep Package Inspection
17 | - Sowohl die Headerdaten, als auch die Payload eines Pakets werden analysiert
18 | - Somit kann Malware blockiert werden
19 | ###### Application Layer Gateway
20 | - Die Kommunikation wird auf Proxys in der Firewall umgeleitet
21 | - Die Firewall hat Einblick in die Applikation und nutzt diesen Hintergrund, um Pakete besser zu filtern
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/TUM/2. Semester/LinAlg/Singulaerwertszerlegung.md:
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1 | ## Allgemeines
2 | - Zu einer Matrix $A \in \mathbb{R}^{m \times n}$ werden die orthogonalen Matrizen $U \in \mathbb{R}^{m \times m}$ und $V \in \mathbb{R}^{n \times n}$, sowie die Diagonalmatrix $\Sigma \in \mathbb{R}^{m \times n}$ bestimmt, sodass gilt:
3 | $$A = U\Sigma V^T$$
4 | - Falls $m \leq n$, so hat $\Sigma$ das Format:
5 | $$\Sigma = \begin{pmatrix}
6 | \sigma_1 & ... & 0 & 0 & ... & 0 \\
7 | & \ddots & & \vdots & & \vdots \\
8 | 0 & ... & \sigma_r & 0 & ... & 0
9 | \end{pmatrix}$$
10 | - Falls $n \leq m$, so hat $\Sigma$ das Format:
11 | $$\Sigma = \begin{pmatrix}
12 | \sigma_1 & & 0 \\
13 | & \ddots & \\
14 | 0 & & \sigma_r \\
15 | 0 & ... & 0 \\
16 | \vdots & & \vdots \\
17 | 0 & ... & 0
18 | \end{pmatrix}$$
19 | - Hierbei ist $r = min\{m, n\}$
20 | #### Bestimmen von $\Sigma$
21 | - Die Eigenwerte $\lambda_1, ..., \lambda_r, ..., \lambda_n$ von $A^TA$ werden bestimmt
22 | - Von den Eigenwerten werden $\lambda_1, ..., \lambda_r$ betrachtet und absteigend sortiert
23 | - Fuer die Singulaerwerte $\sigma_i$ von $\Sigma$ gilt:
24 | $$\sigma_i = \sqrt{\lambda_i}$$
25 | #### Bestimmen von V
26 | - Da $A^TA$ symmetrisch ist, bilden die Eigenvektoren $v_1, ..., v_n$ eine [[Orthogonal- und Orthonormalsystem|ONB]]
27 | - Fuer $V$ gilt hierbei:
28 | $$V = (v_1, ..., v_n)$$
29 | #### Bestimmen von U
30 | - Die Vektoren $u_1, ..., u_r$ werden bestimmt ueber:
31 | $$u_i = \frac{1}{\sigma_i}Av_i$$
32 | - Falls $r < m$, werden $u_1, ..., u_r$ mithilfe des [[Orthogonalithaet|Gram-Schmidtverfahrens]] zu einer ONB ergaenzt
33 | - Fuer $U$ gilt hierbei:
34 | $$U = (u_1, ..., u_m)$$
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/TUM/1. Semester/ERA/Hauptspeicher.md:
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1 | ## Aufbau
2 | - Ein Speicher ist eine Menge von Zellen fester Wortlänge
3 | - Sämtliche Zellen haben eine Adresse und einen Wert
4 | - Ein Speicher wird in einen Stack, Heap, Data und Code Segment eingeteilt
5 | - Ruecksprungadressen und Registerinhalte koennen im Stack gespeichert werden
6 | - Speicher im Heap kann dynamisch angefragt und freigegeben werden
7 | - Das Data Segment enthaelt festgelegte, globale Daten
8 | - Das Code Segment enthaelt das eigentliche Programm
9 | #### Beispiel
10 | ![[Pasted image 20221025170103.png]]
11 | ## Datenzugriff
12 | - Zugriffe auf Daten werden über das Memory Access Register und das Memory Data Register verwaltet
13 | - Über das MAR wird die gewünschte Zelle adressiert
14 | - Über das MDR können Daten gelesen oder überschrieben werden
15 | - Obwohl ein Speicher in Worten organisiert ist, sind Lese- und Schreibzugriffe auf nicht ausgerichtete Daten möglich
16 | #### Beispiel
17 | ![[Pasted image 20221024152907.png]]
18 |
19 | ## Ausrichtung von Daten
20 | - Ein Datum der Größe n Byte ist korrekt ausgerichtet, falls für seine Adresse A gilt: A mod n = 0
21 | - Da ein Speicher in Worten organisiert ist, werden bei korrekter Ausrichtung die Anzahl der Speicherzugriffe minimiert
22 | - Inkorrekte Ausrichtung bietet effizientere Speichernutzung, führt allerdings neben mehr Speicherzugriffen auch zu einem höheren Hardwareaufwand
23 | #### Endianness
24 | - Endianness beschreibt die Reihenfolge von Bytes in einem Datum
25 | - Bei Little Endian stehen signifikantere Bytes an höheren Adressen
26 | - Bei Big Endian stehen signifikantere Bytes an niedrigeren Adressen
27 |
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/TUM/1. Semester/ERA/Instruction Set Architecture.md:
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1 | ## Allgemeines
2 | - Die Instruction Set Architecture bietet die Schnittstelle zwischen dem Programmierer und dem Rechner
3 | - Die Kommunikation erfolgt mithilfe von [[Assembler]]
4 | - Die ISA umfasst Informationen ueber Ausfuehrungsmodelle, den Prozessor und seine Register, sowie den Befehlssatz
5 | ## Ausfuehrungsmodelle
6 | - Das Ausfuehrungsmodell beschreibt die Anzahl der Operanden, die in einem Befehl kodiert sind
7 | - **Stack Modell** (Alle Operanden sind durch den Stack impliziert)
8 | - **Akkumulator Modell** (Ein Operand, sowie das Ergebnis werden in den Akkumulator geladen)
9 | - **Register Modell** (Beide Operanden werden uebergeben, wobei einer vom Ergebnis ueberschrieben wird)
10 | - **Register-Register Modell** (Alle Operanden, sowie das Ergebnisregister werden separat spezifiziert)
11 | ## Befehlssatz
12 | - Der Befehlssatz ist die Menge an Befehlen, mit der der Programmierer mit einem Rechner interagieren kann
13 | - Man unterscheidet drei Klassen von Befehlen
14 | - Arithmetische und logische Operationen
15 | - Befehle fuer den Datentransfer, welche Daten aus und in den [[Hauptspeicher]] laden
16 | - Sprungbefehle, welche den Befehlszaehler ausgehend von Statusregistern aendern koennen
17 | #### Complex Instruction Set Architecture (CISC)
18 | - Grosser, komplexer Befehlssatz
19 | - CISC Rechner werden durch Mikroprogramme realisiert
20 | - Viele Befehle bleiben allerdings ungenutzt und sind meist langsamer
21 | #### Reduced Instruction Set Architecture (RISC)
22 | - Kleiner, elementarer Befehlssatz
23 | - RISC Rechner werden durch feste Verdrahtung realisiert
24 | - Die Befehle sind einfach und schnell ([[RISC-V]])
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/TUM/2. Semester/LinAlg/Komplexe Zahlen und Polarkoordinaten.md:
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1 | ## Grundlagen der Polardarstellung
2 | - Die komplexen Zahlen bilden die [[Konstruktion der komplexen Zahlen|Gausssche Zahlenebene]] und koennen alternativ zur kartesischen Darstellung $(a, b)$ mithilfe von Polarkoordinaten dargestellt werden
3 | - Eine Zahl z wird durch ihre Laenge $r = |z|$ und ihr Argument, den Winkel $\varphi \in (-\pi, \pi]$ zur positiven reellen Achse, beschrieben
4 | #### Umrechnung von Koordinaten
5 | - Aus $(a, b)$ laesst sich $(r, \varphi)$ folgendermassen ermitteln:
6 | $$r = \sqrt{a^2 + b^2}$$
7 | $$\varphi =
8 | \begin{cases}
9 | arccos(\frac{a}{r}) &\text{, falls $b \geq 0$} \\
10 | -arccos(\frac{a}{r}) &\text{, falls $b < 0$}
11 | \end{cases}
12 | $$
13 | - Umgekehrt gilt:
14 | $$a = cos(\varphi) \cdot r$$
15 | $$b = sin(\varphi) \cdot r$$
16 | $$z = cos(\varphi)r + isin(\varphi)r = r(cos(\varphi) + isin(\varphi))$$
17 | - Letztere Form bezeichnet man als Polardarstellung
18 | #### Arithmetik in Polardarstellung
19 | - Es seien $z_1 = r_1(cos(\varphi_1) + isin(\varphi_1))$ und $z_2 = r_2(cos(\varphi_2) + isin(\varphi_2))$ zwei komplexe Zahlen
20 | ###### Multiplikation
21 | $$z_1 \cdot z_2 = r_1r_2(cos(\varphi_1 + \varphi_2) + isin(\varphi_1 + \varphi_2))$$
22 | - Um das Produkt zu bilden, werden somit die Laengen der Zahlen multipliziert und ihre Argumente addiert
23 | ###### Potenz
24 | $$z^n = r^n(cos(n\varphi) + isin(n\varphi))$$
25 | ###### Wurzel
26 | - Fuer jede komplexe Zahl $z$ und jedes $n \in \mathbb{N}$ sind $z_k$ mit $k \in \{0, 1, ..., n-1\}$ genau die $n$-ten Wurzeln von $z$
27 | - Fuer in $z_k$ gilt:
28 | $$z_k = \sqrt[n]{r}(cos(\frac{\varphi + 2k\pi}{n}) + isin(\frac{\varphi + 2k\pi}{n}))$$
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/TUM/2. Semester/LinAlg/Kryptologie.md:
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1 | ## Pohlig-Hellman Verfahren
2 | - Es sei $S$ der Sender einer Nachricht $N \in \mathbb{Z}^{\star}_p$ und $R$ ihr Empfaenger, wobei $p$ eine beliebige, aber grosse Primzahl ist
3 | #### Verschluesselung
4 | - Es wird ein $e$ gewaehlt, sodass $ggT(e, p-1) = 1$ ist
5 | - Somit ist $e \in \mathbb{Z}^{\star}_{p-1}$ und besitzt ein Inverses
6 | - $N$ wird ueber $e$ zu $N^e$ verschluesselt
7 | #### Entschluesselung
8 | - Da $e$ ein Inverses besitzt, kann ueber den euklidischen Algorithmus ein $d$ bestimmt werden, sodass gilt:
9 | $$ed = 1 \mod p-1$$
10 | - Mithilfe von $d$ kann $N^e$ entschluesselt werden:
11 | $$(N^e)^d = N^{ed} = N^{1 + r(p-1)} = N * N^{r(p-1)} = N * (N^{p-1})^r = N * 1^r = N$$
12 | ## RSA Verfahren
13 | - Es sei $S$ der Sender einer Nachricht $N \in \mathbb{Z}^{\star}_n$ und R ihr Empfaenger
14 | - Zudem stellt $R$ einen public Key $(n, e)$ zur Verfuegung
15 | #### Public Key
16 | - $n$ setzt sich folgendermassen aus dem beiden Primzahlen $p$ und $q$ zusammen:
17 | $$n = pq$$
18 | - Fuer die [[Restklassengruppen|prime Restklassengruppe]] $\mathbb{Z}^{\star}_n$ gilt somit:
19 | $$|\mathbb{Z}^{\star}_n| = \varphi(n) = (p-1)(q-1)$$
20 | - $e$ wird so bestimmt, dass gilt:
21 | $$gTT(e, (p-1)(q-1)) = 1$$
22 | - Somit ist $e \in \mathbb{Z}^{\star}_{(p-1)(q-1)}$ und bestizt ein Inverses
23 | - $N$ wird ueber $e$ zu $N^e$ verschluesselt
24 | #### Private Key
25 | - Da $e$ ein Inverses besitzt, kann ueber den euklidischen Algorithmus ein $d$ bestimmt werden, sodass gilt:
26 | $$ed = 1 \mod (p-1)(q-1)$$
27 | - Dieses $d$ wird geheim gehalten und kann $N^e$ entschluesseln:
28 | $$(N^e)^d = N^{ed} = N^{1 + r(p-1)(q-1)} = N^1 * (N^{(p-1)(q-1)})^r = N * 1^r = N$$
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/TUM/3. Semester/GBS/Betriebssysteme.md:
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1 | ## Aufgaben
2 | - Das Betriebssystem bietet Schnittstellen zur Hardware, wie beispielsweise das Lesen und Schreiben von Daten
3 | - Adressraeume und CPU werden ebenfalls von dem Betriebssystem verwaltet
4 | ## Prozesse
5 | - Ein Prozess ist ein Programm, welches ausgefuehrt wird
6 | #### Anforderungen
7 | - Prozesse benoetigen Ressourcen, wie CPU oder Ein- und Ausgabe, um laufen zu koennen
8 | - Jeder Prozess besitzt einen eigenen, abgeschotteten Adressraum, welcher virtualisiert wird
9 | - Ist ein Prozess multi-threaded, so benoetigt er zusaetzliche Register und Befehlszaehler
10 | ## Arbeitsmodi
11 | - Das Betriebssystem bietet unterschiedliche Modi, um Sicherheit zu garantieren
12 | - Programme laufen hauptsaechlich im Benutzermodus und koennen mithilfe von System Calls kontrolliert in den Systemmodus wechseln
13 | - Im Systemmodus kann direkt auf Hardwareressourcen zugegriffen werden
14 | #### Beispiel
15 | ![[Pasted image 20231023150812.png]]
16 | ## Architekturen
17 | - Ein Betriebssystem kann auf unterschiedliche Weisen realisiert werden
18 | #### Monolithisches System
19 | - Das Betriebssystem umfasst eine grosse Menge an Funktionen welche alle im Systemmodus ausgefuehrt werden
20 | - Das System wird hierdurch jedoch unuebersichtlich und schwer zu warten
21 | ###### Beispiel
22 | ![[Pasted image 20231023150752.png]]
23 | #### Mikrokernel System
24 | - Das Betriebssystem wird in mehrere, kleine Module aufgeteilt
25 | - Der Mikrokern laeuft als einziges Modul im Systemmodus und verwaltet Scheduling
26 | - Alle anderen Systemdienste laufen als Serverprozess im Benutzermodus
27 | - Benoetigt ein Programm einen Dienst, so erfolgt die Kommunikation ueber den Mikrokern
28 | ###### Beispiel
29 | ![[Pasted image 20231023152522.png]]
30 |
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/TUM/1. Semester/ERA/Multithreading.md:
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1 | ## Allgemeines
2 | - Ein Hardware-Thread ist eine Ausfuehrungseinheit, welche aus einem Befehlszaehler und Registern besteht
3 | - Andere Ressourcen, wie Speicher und Rechenwerke werden von Threads geteilt
4 | - Ein Kern ist die Kombination aus Threads und den geteilten Ressourcen
5 | - Kerne teilen sich wiederum die Cachehierarchie und arbeiten groesstenteils unabhaengig
6 | - Multithreading Architekturen unterscheiden sich darin, wie sich Threads, Ressourcen teilen
7 | ## Architekturen
8 | - Einfaches Multithreading
9 | - Threads wechseln sich zu einem festen Takt ab
10 | - Simultaneous Multithreading
11 | - Threads laufen echt parallel, falls sie unterschiedliche Teile der Ressourcen benoetigen
12 | #### Beispiel
13 | ![[Pasted image 20230125155911.png]]
14 | ## Cacheorganisation
15 | - Manche [[Cache|Caches]] werden von mehreren Kernen geteilt
16 | - Bei Schreibzugriffen geht die Kohaerenz der Daten verloren
17 | - Die Cachekohaerenz kann mithilfe von Protokollen gewahrt werden
18 | #### MESI Protokoll
19 | - Bei dem MESI Protokoll wird die Kohaerenz von Caches mithilfe eines Automaten gewahrt
20 | ###### Zustaende
21 | - Modified
22 | - Eine Cachezeile wurde lokal veraendert
23 | - Exclusive
24 | - Wert liegt in genau einer Cachezeile vor
25 | - Shared
26 | - Wert liegt in mindestens zwei Cachezeilen vor
27 | - Invalid
28 | - Wert ist nicht gueltig
29 | ###### Beispiel
30 | ![[Pasted image 20230130122858.png]]
31 | #### False Sharing
32 | - Schreiben mehrere Threads auf nicht ausgerichtete Daten, so kommt es zu zusaetzlichen Misses im Cache, da die Daten staendig ueberschrieben und neu geladen werden
33 | - False Sharing kann durch die Ausrichtung von Daten vermieden werden
34 | ###### Beispiel
35 | ![[Pasted image 20230130125256.png]]
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/TUM/6. Semester/IDSST/Probability.md:
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1 | ## Hypothesis Testing
2 | - In order to make draw conclusions from data, hypotheses need to be made
3 | #### Procedure
4 | - In general, we make a null hypothesis $H_0$, which states that there is nothing to be gained from the data and any observed correlation is coincidence
5 | - Under the assumption that $H_0$ is true we then compute some summary statistic $T(X)$ of our data and check how likely it is that it can even occur
6 | - If there's enough evidence that the observed $T(X)$ is not a coincidence, we can reject $H_0$, otherwise, we assume $H_0$ holds
7 | ## Normal Distribution
8 | - The [[Kontinuierliche Wahrscheinlichkeitsraeume|Normal Distribution]] is very commonly used for statistical analysis
9 | #### Quantiles
10 | - An observation $x$ is considered the $q$-quantile of a distribution function $F$, if the following holds:
11 | $$F(x) = Pr[X \leq x] = q$$
12 | - Which $q$-quantile an observation is can be obtained using the $\texttt{pnorm}$ function
13 | - Which observation a given $q$-quantile corresponds to can be obtained using the $\texttt{qnorm}$ function
14 | #### The 68-95-99.7 Rule
15 | - If a given dataset follows a normal distribution, about $68$% of the data falls within one standard deviation , $95$% within two standard deviations and $99.7$% within three standard deviations from the mean
16 | #### Normal Probability Plots
17 | - A normal probability plot has the number of standard deviations from the mean on the x-axis and the corresponding sample values on the y-axis
18 | - Samples from the dataset are then drawn on this plot
19 | - If the position of the samples on the plot represents a linear relationship, the data follows a normal distribution
20 | ###### Examples
21 | ![[Pasted image 20250601152707.png]]
22 | ![[Pasted image 20250601152811.png]]
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/TUM/5. Semester/ML/k-Nearest Neighbors.md:
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1 | ## Basic Concepts
2 | - Given a training dataset of categorized vectors, the $k$ nearest neighbors approach yields the category for new vectors
3 | - To determine the category of a new vector, the categories of its $k$ nearest neighbors are counted and the majority is chosen
4 | - The [[Classification Performance|performance of the classification]] can be improved by weighing a vector's neighbors based on their distance from it
5 | #### Distance Measuring
6 | - A number of distance measures can be used, in order to determine the distance between two neighbors
7 | #### Hyperparameter $k$
8 | - The number $k$ of the neighbors the model considers is a hyperparameter, that needs to be [[Hyperparameters|tuned]]
9 | ###### Example
10 | ![[Pasted image 20241015122632.png]]
11 | ## Classification Pitfalls
12 | - Certain classification errors can appear, if a few of the common pitfalls aren't avoided
13 | #### Standardization
14 | - Many models are sensitive to differences in scale
15 | - To work around this sensitivity, the training data is standardized, so that the mean is always $0$ and variance always $1$
16 | - Given a set of training vectors $x_i$, the training set can be standardized as follows:
17 | $$x_{i, \text{std}} = \frac{x_i - \mu_i}{\sigma_i}$$
18 | ###### Handling Input and Output
19 | - Once the model has been trained, and is being used, inputs will not adhere to the standardization automatically
20 | - As such, they need to be standardized before being fed to the model and it's output needs to be destandardized accordingly
21 | #### Dimensionality
22 | - The more dimensions the input space has, the less of the space is covered by the same amount of vectors
23 | - For a precise classification, the number of vectors thus needs to grow exponentially with the number of features
24 | ###### Example
25 | ![[Pasted image 20241018160305.png]]
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/TUM/1. Semester/ERA/Moderne Prozessoren.md:
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1 | ## Out-of-Order Execution
2 | - Bei der sequentiellen Befehlsverarbeitung in einem pipelined Prozessor kann es zu Hazards kommen
3 | - Manche Hazards koennen ueber das umsortieren von Befehlen behoben werden
4 | - Out-of-Order Prozessoren sind deutlich komplexer, bieten allerdings auch eine effizientere Nutzung von Parallelitaet
5 | ## Parallelitaet
6 | #### Superskalaritaet
7 | - Mehrere unabhaengige Maschinenbefehle koennen parallel ausgefuehrt werden
8 | - Hierfuer verwendet der Prozessor mehrere Funktionseinheiten und Dekodierer
9 | - Zudem werden Pipelinestufen nach der Art des Befehls unterschieden
10 | ###### Beispiel
11 | ![[Pasted image 20230125122843.png]]
12 | #### VLIW Prozessoren
13 | - Ein Instruktionswort enthaelt mehrere Befehle
14 | - Diese Befehle sind unabhaengig voneinander und koennen parallel bearbeitet werden
15 | ###### Beispiel
16 | ![[Pasted image 20230125123922.png]]
17 | #### SIMD
18 | - Bei Single Instruction Multiple Data wird eine Operation parallel auf mehrere Datenstroeme angewandt
19 | - Dies eignet sich besonders fuer Vektoroperationen
20 | ###### Beispiel
21 | ![[Pasted image 20230125125524.png]]
22 | #### [[Multithreading]]
23 | ## Speedup
24 | - Der Speedup eines parallelen Programms ergibt sich aus der Relation der sequentiellen und der parallelen Laufzeit
25 | - Ein paralleles Programm besteht stets aus sequentiellen und parallelen Abschnitten
26 | #### Amdahl's law
27 | - Beschreibt den moeglichen Speedup bei konstanter Problemgroesse
28 | ###### Berechnung
29 | ![[Pasted image 20230127141037.png]]
30 | ![[Pasted image 20230127141139.png]]
31 | ![[Pasted image 20230127141151.png]]
32 | ![[Pasted image 20230127141203.png]]
33 | #### Gustafson's law
34 | - Beschreibt den moeglichen Speedup bei proportionaler Problemgroesse
35 | ###### Berechnung
36 | ![[Pasted image 20230127143143.png]]
37 | ![[Pasted image 20230127143154.png]]
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/TUM/5. Semester/ML/Hyperparameters.md:
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1 | ## Tuning Hyperparameters
2 | - The goal of tuning hyperparameters is to develop the model, that performs best on unseen data
3 | #### Approach
4 | - To tune hyperparameters, a dataset is split into a training set, a validation set and a test set
5 | - Using the training set and validation set, the optimal hyperparameters can be determined
6 | - These hyperparameters can then be tested against the test set
7 | ###### Example
8 | ![[Pasted image 20241015122517.png]]
9 | #### Overfitting
10 | - Overfitting occurs, when the model is trained too closely on the training data
11 | - In training, the model therefore shows very good results, but since it is generalized poorly, it underperforms on unseen data
12 | ###### Batteling Overfitting
13 | - The model is optimized using the validation data in a training loop
14 | - At the end of training, the model is tested against the test data set
15 | - This test data set must only be used once, to avoid meta optimization on the test data
16 | ###### Example
17 | ![[Pasted image 20241021144659.png]]
18 | #### K-fold Cross-Validation
19 | - In order to optimize training, when the underlying dataset is small, the dataset is split into $K$ sections, where $K - 1$ sections are used for training and one for validation
20 | - A different section is used for training on every training round
21 | - Over mutliple rounds, each section will have been used once for validation and the optimal hyperparameters can be obtained by averaging the best parameters for each round
22 | ###### Leave-one-out cross-validation
23 | - In leave-one-out cross-validation, there are as many sections, as there vectors
24 | - This means that the validation data consists of just a single data vector
25 | - This approach yields the optimal model but is very expensive so it should only be used on small datasets
26 | ###### Example
27 | ![[Pasted image 20241021150209.png]]
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/TUM/2. Semester/LinAlg/Lineare Ausgleichsrechnung.md:
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1 | ## Orthogonalraum
2 | - Es sei $V$ ein [[Orthogonalithaet|euklidischer Vektorraum]] mit einem Untervektorraum $U$, wobei gilt $dim(V) = n$, $dim(U) = r$
3 | - Der Orthogonalraum von $U$ wird definiert durch:
4 | $$U^{\perp} = \{v \in V \mid \forall \space u \in U: v \perp u \space \} \leq V$$
5 | - Fuer den Orthogonalraum $U^{\perp}$ gilt:
6 | $$U \cap U^{\perp} = \{0\}$$
7 | $$\forall \space v \in V, \exists_1 \space u \in U, \exists_1 \space u' \in U^{\perp}: v = u + u'$$
8 | $$dim(U^{\perp}) = n - r$$
9 | ## Orthogonale Projektion
10 | - Es sei $V = \mathbb{R}^n$ ein und $U \leq V$
11 | - Die orthogonale Projektion $P_U$ wird beschrieben durch:
12 | $$P_U = \begin{cases}
13 | V \rightarrow U \\
14 | v = u + u^{\perp} \mapsto u \\
15 | \end{cases}$$
16 | - Hierbei ist $u$ das Element aus $U$, fuer das die Distanz $||v - u||$ minimal ist
17 | - Es sei $U = \langle b_1, ..., b_r \rangle$ mit $b_i \in \mathbb{R}^n$ und $A = (b_1, ..., b_r) \in \mathbb{R}^{n \times r}$
18 | - $u$ kann somit mithilfe von $Ax$ mit $x \in \mathbb{R}^r$ beschrieben werden
19 | - Ueber ein [[Lineare Gleichungssysteme|lineares Gleichungssystem]] ist $x$ so zu bestimmen, dass $||v - Ax||$ minimal ist
20 | - Fuer $x$ gilt hierbei die Normalengleichung:
21 | $$A^TAx = A^Tv$$
22 | #### Methode der kleinsten Quadrate
23 | - Es sei eine Punktwolke $(x_i, y_i)$ gegeben
24 | - Alle Punkte $(x_i, y_i)$ sollen mithilfe der Funktion $f(x) = \beta_1 f_1(x)+ ... + \beta_r f_r(x)$ moeglichst genau approximiert werden
25 | - Es werden die Matrix $A$, sowie die Vektoren $x$ und $v$ festgelegt, sodass gilt:
26 | $$A = \begin{pmatrix}
27 | f_1(x_1) & ... & f_r(x_1) \\
28 | \vdots & & \vdots \\
29 | f_1(x_n) & ... & f_r(x_n)
30 | \end{pmatrix}$$
31 | $$x = (\beta_1, ..., \beta_r)^T$$
32 | $$v = (y_1, ..., y_n)^T$$
33 | - Mithilfe der Normalengleichung kann nun $x$ so bestimmt werden, dass $||v - Ax||$ minimal ist
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/TUM/4. Semester/ModSim/Job Scheduling.md:
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1 | ## Scheduling
2 | - A set of tasks $A_1, ..., A_n$ is run on a set of machines $M_1, ..., M_m$, where each task takes a certain amount of time
3 | - The task execution must be ordered such that the resulting schedule is optimal
4 | - Precedences define, which task needs to be executed before which, creating an ordering of tasks
5 | #### Graph View
6 | - Using tasks as vertices and precedences as edges, a graph view of a schedule can be constructed
7 | - A virtual initial vertex $A_S$ and a virtual final vertex $A_E$ serve as the start and end points of the graph
8 | - If tasks are assigned to different machines, they can be iterated through concurrently
9 | ###### Schedule Construction
10 | - One option to construct a schedule, is to start every task, as soon as all preceding tasks have been completed
11 | - Alternatively, the completion time of $A_E$ can be used to determine the latest possible time, a task needs to finish at, in order to still meet the completion time of $A_E$
12 | - Those approaches are the forward and backward step and they both result in an optimal schedule
13 | ###### Critical Paths
14 | - A path in a schedule is critical, if there can be no downtime between tasks on it
15 | ###### Example
16 | ![[Pasted image 20240516155357.png]]
17 | #### Job Shop
18 | - In a job shop model, tasks are divided into subtasks, which need to run on a specific machine and are all dependent on one another
19 | ###### Disjunctive Edges
20 | - If the subtasks $A_{x, y}$ and $B_{i, j}$ of two different tasks happen to be executed on the same machine, a clear ordering cannot be established immediately
21 | - Instead, disjunctive edges, which face from $A_{x, y}$ to $B_{i, j}$ and the other way around, are created
22 | - In a disjunctive edge assignment, each of the edges is tested out, removing the corresponding one
23 | - Using this assignment, an optimal schedule can be determined
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/TUM/2. Semester/LinAlg/Komplexe Zahlen und Kartesische Koordinaten.md:
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1 | ## Grundlagen von komplexen Zahlen
2 | - Die [[Konstruktion der komplexen Zahlen]] erfolgt mithilfe der Menge $\mathbb{R}^2$
3 | - Es sei $z$ eine beliebige Zahl aus $\mathbb{C} = \{a + bi | a, b \in \mathbb{R}\}$
4 | - Man bezeichnet $Re(z) = a$ als Realteil und $Im(z) = b$ als Imaginaerteil von $z$
5 | - Eine beliebige Zahl $w$ ist zu $z$ aequivalent, falls Real- und Imaginaeteil der beiden Zahlen identisch sind
6 | #### Konjugiert komplexe Zahlen
7 | - Man bezeichnet $\overline{z}=a -bi$ als die zu $z$ konjugiert komplexe Zahl
8 | - $\overline{z}$ entsteht somit durch die Spiegelung von $z$ an der reellen Achse
9 | - Fuer $\overline{z}$ gilt:
10 | $$z\overline{z} = (a + bi) \cdot (a - bi) = a^2 + b^2 \in \mathbb{R}$$
11 | $$\overline{z+w} = \overline{z} + \overline{w}$$
12 | $$\overline{z \cdot w} = \overline{z} \cdot \overline{w}$$
13 | #### Betrag einer komplexen Zahl
14 | - Der Abstand $|z|$ einer komplexen Zahl $z$ zum Ursprung bezeichnet man als Betrag von $z$
15 | - Der Abstand von $z$ zu einer komplexen Zahl $w$ wird durch $|z - w|$ beschrieben
16 | - Fuer $|z|$ gilt:
17 | $$|z| = \sqrt{a^2 + b^2} = \sqrt{z\overline{z}}$$
18 | #### Multiplikative Inverse einer komplexen Zahl
19 | - Sei $q = \frac{z}{w}$ eine komplexe Zahl
20 | - Fuer das multiplikative Inverse von $q$ gilt:
21 | $$q^{-1}=q^{-1} \cdot \frac{\overline{w}}{\overline{w}}$$
22 | #### Dreiecksungleichung
23 | - Fuer zwei komplexe Zahlen $z$ und $w$ gilt:
24 | $$|zw| = |z| \cdot |w|$$
25 | $$|z + w| \leq |z| + |w|$$
26 | ## Fundamentalsatz der Algebra
27 | - Jedes Polynom $f = a_nx^n + ... + a_1x + a_0$ mit komplexen Koeffizienten $a_n ... a_0$ kann folgendermassen in Linearfaktoren zerlegt werden:
28 | $$f = a_n(x - z_n)^{v_n} + ... + (x - z_1)^{v_1}$$
29 | - Hierbei sind $z_n ... z_1$ Nullstellen von $p$ mit den Vielfachheiten $v_n ... v_1$
30 | - Die Zerlegung kann durch wiederholtes Bestimmen der Nullstellen und Polynomdivision ermittelt werden
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/TUM/3. Semester/GBS/Scheduling.md:
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1 | ## Multiprogramming
2 | - Da unterschiedliche Prozesse unterschiedliche Ressourcen benoetigen, wird ihre Ausfuehrungsreihenfolge misachtet, um die CPU Auslastung zu erhoehen
3 | - Somit koennen CPU intensive Prozesse ausgefuehrt werden, waehrend ein anderer Prozess mit der I/O interagiert
4 | ## Strategien
5 | - Je nach Anforderungen sind unterschiedliche Scheduling Strategien geeignet
6 | #### Batch Systeme
7 | - Prozesse werden anhand der Reihenfolge ihrer Erzeugung oder nach ihrer verbleibenden Laufzeit ausgefuehrt
8 | - In First-Come-First-Served und Shortest-Job-First werden Prozesse bis zu ihrer Terminierung ausgefuehrt
9 | - In Shortest-Remaining-Time-Next wird ein Prozess unterbrochen, falls einer mit einer kuerzeren Laufzeit erstellt wurde
10 | #### Interaktive Systeme
11 | - Damit jeder Prozess auf moegliche Eingaben reagieren kann, werden sie abwechselnd ausgefuehrt
12 | ###### Implementierungen
13 | - Bei Round-Robin wechseln sich Prozesse zu einem festen Zeitintervall ab
14 | - Durch statisches Priority-Scheduling wird vorab fuer jeden Prozess eine Proritaet berechnet, wodurch Prozesse jedoch verhungern koennen
15 | - In dynamischem Priority-Scheduling werden die Prioritaeten periodisch neu ermittelt
16 | #### Echtzeit Systeme
17 | - Manche Systeme muessen strikte Zeitanforderungen erfuellen
18 | - Prozesse besitzen somit Deadlines, die erfuellt werden muessen
19 | ###### Implementierungen
20 | - Bei einem Rate-Monotonic-System Scheduling muessen Prozesse periodisch ausgefuhert werden
21 | - Prozesse mit der kuerzesten Periode werden priorisiert
22 | ## Multicore Scheduling
23 | - Sind mehrere CPUs verfuegbar, so muss die Rechenlast gleichmaessig auf sie verteilt werden
24 | #### Rechenlastverteilung
25 | - Es werden eine globale Warteschlange, sowie separate Warteschlangen fuer jede CPU angelegt
26 | - Ist eine CPU idle, so wird uber work stealing die Rechenlast gleichmaessig verteilt
27 | #### Beispiel
28 | ![[Pasted image 20231116215352.png]]
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/TUM/1. Semester/ERA/Speicheraufbau.md:
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1 | ## Allgemeines
2 | - Ein Random Access Memory kann auf verschiedene Arten implementiert werden
3 | - Hierfuer werden Matrizen aus Static RAM Zellen und Dynamic RAM Zellen verwendet
4 | ## SRAM
5 | - Eine SRAM Zelle wird mithilfe von zwei Transistoren und zwei NOT Gates realisiert
6 | - SRAM wird fuer Register und [[Cache|Caches]] verwendet
7 | #### Schreiben
8 | - Das Select Signal wird auf 1 gesetzt und die gewuenschten Bits an den Eingaengen der Zelle angelegt
9 | - Die Eingaenge werden ueber die Transistoren in einen Zyklys aus NOT Gates geleitet und somit gespeichert
10 | #### Lesen
11 | - Beide Eingaenge der Zelle, sowie das Select Signal werden auf 1 gesetzt
12 | - Durch den Spannungsabfall auf einer Seite der Schaltung, koennen die Bits ausgelesen werden
13 | #### Vor- und Nachteile
14 | - SRAM Speicher sind schnell und koennen Daten speichern, solange Strom anliegt
15 | - Sie sind allerdings gross und energieintensiv
16 | #### Beispiel
17 | ![[Pasted image 20230117113227.png]]
18 | ![[Pasted image 20230117114108.png]]
19 | ## DRAM
20 | - Eine DRAM Zelle wird mithilfe eines Transistors und eines Kondensators realisiert
21 | - DRAM wird fuer den [[Hauptspeicher]] verwendet
22 | #### Schreiben
23 | - Das Select Signal wird gesetzt und das gewuenschte Bit an den Eingang angelegt
24 | - Das Bit wird ueber einen Transistor in einen Kondensator geleitet und dort gespeichert
25 | #### Lesen
26 | - Es werden immer ganze Zeilen eines DRAM Speichers ausgelesen
27 | - Das Select Signal wird gesetzt und eine "0.5" an den Eingang angelegt, wodurch sich der Kondensator laedt, beziehungsweise entlaedt und das Bit ausgelesen wird
28 | - Der Inhalt einer Zelle wird somit durch das Auslesen zerstoert und der Kondensator entlaedt sich mit der Zeit
29 | - Der Speicher muss somit regelmaessig und bei jedem Auslesen neu beschrieben werden
30 | #### Vor- und Nachteile
31 | - DRAM Speicher sind klein und einfach
32 | - Speicherzugriffe sind recht zeitintensiv
33 | #### Beispiel
34 | ![[Pasted image 20230117120833.png]]
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/TUM/2. Semester/LinAlg/Determinante.md:
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1 | ## Symmetrische Gruppe
2 | - Es sei die Menge an Abbildungen $S_n$ definiert durch:
3 | $$S_n = \{ \sigma : \{1, ..., n\} \rightarrow \{1, ..., n\} \mid \sigma \text{ ist Bijektion} \}$$
4 | - $\sigma$ sind hierbei Permutationen
5 | - Fuer die Kardinalitaet von $S_n$ gilt:
6 | $$|S_n| = n!$$
7 | - $(S_n, \circ)$ bildet hierbei die symmetrische Gruppe
8 | #### Fehlstaende
9 | - $(j, i)$ ist ein Fehlstand der Permutation $\sigma$, falls gilt:
10 | $$i < j, \space \sigma(i) > \sigma(j)$$
11 | - $f$ beschreibt die Anzahl der Fehlstaende von $\sigma$
12 | #### Signum
13 | - Das Signum einer Permutation $\sigma$ mit $f$ Fehlstaenden wird beschrieben durch:
14 | $$sng(\sigma) = (-1)^f$$
15 | ## Determinante einer Matrix
16 | - Fuer jede quadratische Matrix $A \in \mathbb{K}^{n \times n}$ wird die Determinante von $A$ definiert durch:
17 | $$|A| = det(A) = \sum_{\sigma \in S_n} sgn(\sigma) \prod_{i = 1}^{n} a_{i\sigma(i)}$$
18 | - Eine Matrix $A$ ist genau dann invertierbar, falls $det(A) \neq 0$ ist
19 | #### Umformungen
20 | - Fuer zwei beliebige Matrizen $A, B \in \mathbb{K}^{n \times n}$ gilt:
21 | $$det(AB) = det(A) \cdot det(B)$$
22 | - Fuer das Transponierte und die $k$-te Potenz einer Matrix $A \in \mathbb{K}^{n \times n}$ gilt:
23 | $$det(A) = det(A^T)$$
24 | $$det(A^k) = det(A)^k$$
25 |
26 | - Durch das Addieren des $\lambda$-fachen einer Zeile zu einer anderen Zeilen aendert sich die Determinante nicht
27 | ###### Laplace Entwicklung
28 | - Fuer die Entwicklung nach der $i$-ten Zeile gilt:
29 | $$det(A) = \sum_{j=1}^n(-1)^{i + j} \cdot a_{ij} \cdot det(A_{ij})$$
30 | - Fuer die Entwicklung nach der $j$-ten Spalte gilt:
31 | $$det(A) = \sum_{i=1}^n(-1)^{i + j} \cdot a_{ij} \cdot det(A_{ij})$$
32 | #### Dreiecksblockgestalt
33 | - Hat $A$ die Dreiecksblockgestalt, so kann sie dargestellt werden durch:
34 | $$A = \begin{pmatrix}
35 | B & 0 \\
36 | C & D \\
37 | \end{pmatrix}$$
38 | $$A = \begin{pmatrix}
39 | B & C \\
40 | 0 & D \\
41 | \end{pmatrix}$$
42 | - Fuer die Determinante von $A$ gilt:
43 | $$det(A) = det(B) \cdot det(D)$$
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/TUM/6. Semester/IDSST/Numerical Data.md:
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1 | ## Measures
2 | - To be able to analyze numerical data it must be measured
3 | - In general, a measure of center and a measure of distribution are needed in order to be able to grasp the behavior of a distribution
4 | #### Measures of Center
5 | - Popular measures of center are the median and the mean
6 | - The median describes the point at the center of the dataset which splits it in half, or the average of the two center points, if there is an even number of datpoints
7 | - The mean is the average value from all of the datapoints
8 | #### Measures of Variability
9 | - Popular measures of variability are the variance and standard deviation
10 | - The empirical variance is described like so:
11 | $$s^2_{x, n} = \frac{\sum_{i = 1}^n (x_i - \overline{x}_n)^2}{n - 1}$$
12 | - The empirical standard deviation $s_{x, n}$ is then simply the square root of $s^2_{x, n}$
13 | ###### Percentiles
14 | - The $k$th percentile is the value in the dataset, below which $k\%$ and above which $100\% - k\%$ of the data resides
15 | - The median is therefore the $50$th percentile
16 | - The interquartile range is the difference between the $75$th and $25$th percentile and can also be used as a measure of variability in the dataset, as it describes the range taken up by the "center" of the dataset
17 | #### Measures of Correlation
18 | - In order to compute how much two variables correlate with each other, the empirical correlation coefficient can be computed:
19 | $$r_{(x, y), n} = \frac{\sum_{i = 1}^n (x_i - \overline{x}_n)(y_i - \overline{y}_n)}{(n - 1)\cdot s_{x, n} \cdot s_{y, n}}$$
20 | #### Robust Statistics
21 | - In general, if the dataset is skewed towards a certain direction, using the median and interquartile range makes the analysis less affected by outliers
22 | ## Visualization
23 | - In general, when comparing two numerical attributes, something like a scatterplot is most useful
24 | - When comparing two categorical attributes, a table or heatmap is best suited
25 | - For a combination of categorical and numerical attributes, boxplots can be used
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/TUM/3. Semester/GBS/Modellierung paralleler Systeme.md:
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1 | ## Petri Netze
2 | - Parallele Systeme koennen mit einem Petri Netz modelliert werden
3 | - Der Kontrollfluss eines parallelen Systems wird durch Tokens repraesentiert, die das Netz durchlaufen
4 | #### Aufbau
5 | - Ein Petri Netz setzt sich aus Stellen, Transitionen und Flussrelationen zusammen
6 | - Stellen und Transitionen dienen als Konten und Flussrelationen als Kanten in einem Graph
7 | ###### Stellen
8 | - Werden als Kreise modelliert und repraesentieren passive Elemente
9 | - Stellen besitzen eine Kapazitaet, welche beschreibt, wieviele Tokens aufgenommen werden koennen
10 | ###### Transitionen
11 | - Werden als Rechtecke modelliert und repraesentieren aktive Elemente
12 | ###### Flussrelationen
13 | - Werden als Kanten modelliert und verbinden Stellen mit Transitionen
14 | - Flussrelationen besitzen Gewichte, welche angeben, wieviele Transitionen ueber sie transportiert werden
15 | #### Schalten
16 | - Der Zustand eines Petri Netzes aendert sich durch das Schalten der Transitionen
17 | - Eine Transition schaltet, falls die Anzahl der Tokens in den eingehenden Stellen mindestens so hoch ist wie das Gewicht der entsprechenden Flussrelation
18 | - Zusaetzlich darf durch das Schalten der Transition die Kapazitaet der ausgehenden Stellen nicht ueberschritten werden
19 | ###### Beispiel
20 | ![[Pasted image 20231128164152.png]]
21 | #### Erreichbarkeit
22 | - Mithilfe eines Erreichbarkeitsgraphen werden alle moeglichen Zustaende notiert, die ein Petri Netz annehmen kann
23 | - Ob ein gegebener Zustand erreichbar ist, kann mithilfe eines Erreichbarkeitsgraphen ueberprueft werden
24 | ###### Beispiel
25 | ![[Pasted image 20231128164922.png]]
26 | #### Nebenlaeufigkeit
27 | - Transitionen koennen unabhaengig voneinander schalten
28 | - Hierdurch kann es zu Konflikten kommen
29 | ###### Beispiel
30 | ![[Pasted image 20231128164552.png]]
31 | #### Lebendigkeit
32 | - Ein Petri Netz ist lebendig, falls jeder Zustand von jedem anderen Zustand erreicht werden kann
33 | - Eine Verklemmung, ist ein Zustand, in dem keine Transition schalten kann
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/TUM/2. Semester/GAD/Datenstrukturen fuer Sequenzen.md:
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1 | ## Anforderungen
2 | - Eine Datenstruktur fuer eine Sequenz muss Operationen zum Einfuegen, Lesen, Aendern und Loeschen fuer all ihre Elemente bieten
3 | - Verschiedene Implementierungen bieten unterschiedliche Vor- und Nachteile
4 | ## Arrays
5 | - Ein Array ist ein festgelegtes, kontinuierliches Stueck Speicher, auf welches beliebig zugegriffen werden kann
6 | #### Vorteile
7 | - Alle Operationen werden sehr effizient ausgefuehrt, da auf alle Elemente gleich schnell zugegriffen werden kann
8 | #### Nachteile
9 | - Die Groesse eines Arrays muss bei der Initialisierung festgelegt werden
10 | #### Dynamisches Array
11 | - Ist das urspruengliche Array voll, so wird ein neues Array mit groesserer Laenge angelegt
12 | - Die Elemente des urspruenglichen Arrays muessen in das Neue kopiert werden, was in $O(n)$ erfolgt
13 | - Mithilfe einer amortisierten [[Laufzeitanalyse|Aufwandsanalyse]] kann jedoch ermittelt werden, dass die tatsaechliche Laufzeit fuer das Einfuegen $O(1)$ betraegt
14 | #### Beispiel
15 | ![[Pasted image 20230513192742.png]]
16 | ## Doppelt verkettete Liste
17 | - Die Daten werden in Elementen abgespeichert, welche jeweils ihren Vorgaenger und Nachfolger referenzieren
18 | #### Vorteile
19 | - Da sich eine Liste dynamisch veraendern laesst, werden keine teueren Kopier-Operationen benoetigt
20 | - Bestimmte Operationen, wie das Splicing, sind nur auf Listen moeglich
21 | #### Nachteile
22 | - Such-Operationen laufen in $O(n)$
23 | #### Splicing
24 | - Eine doppelt verkettete Liste bietet die splice Methode, welche eine Sequenz von Elementen $a, ..., b$ ausschneidet und hinter einem Element $t$ einfuegt
25 | ###### Beispiel
26 | ![[Pasted image 20230510133347.png]]
27 | #### Beispiel
28 | ![[Pasted image 20230510132900.png]]
29 | ## Sortierte Sequenzen
30 | - Fuer Listen und Arrays kann ein sortiertes Einfuegen und Loeschen bereitgestellt werden
31 | - Die [[Effizienzmetrik|Laufzeit]] fuer diese Operationen betraegt zwar $O(n)$, jedoch kann hierdurch binary search implementiert werden, was das Finden von Elementen in $O(log(n))$ ermoeglicht
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