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18 Commits
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daf634e355
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3897babce3 |
1
.gitignore
vendored
Normal file
1
.gitignore
vendored
Normal file
@@ -0,0 +1 @@
|
|||||||
|
.trash/
|
||||||
4
.obsidian/community-plugins.json
vendored
4
.obsidian/community-plugins.json
vendored
@@ -8,5 +8,7 @@
|
|||||||
"highlightr-plugin",
|
"highlightr-plugin",
|
||||||
"obsidian-file-color",
|
"obsidian-file-color",
|
||||||
"darlal-switcher-plus",
|
"darlal-switcher-plus",
|
||||||
"dataview"
|
"dataview",
|
||||||
|
"obsidian-vimrc-support",
|
||||||
|
"emoji-shortcodes"
|
||||||
]
|
]
|
||||||
9
.obsidian/plugins/emoji-shortcodes/data.json
vendored
Normal file
9
.obsidian/plugins/emoji-shortcodes/data.json
vendored
Normal file
@@ -0,0 +1,9 @@
|
|||||||
|
{
|
||||||
|
"immediateReplace": true,
|
||||||
|
"suggester": true,
|
||||||
|
"historyPriority": true,
|
||||||
|
"historyLimit": 100,
|
||||||
|
"history": [
|
||||||
|
":gear:"
|
||||||
|
]
|
||||||
|
}
|
||||||
2071
.obsidian/plugins/emoji-shortcodes/main.js
vendored
Normal file
2071
.obsidian/plugins/emoji-shortcodes/main.js
vendored
Normal file
File diff suppressed because it is too large
Load Diff
11
.obsidian/plugins/emoji-shortcodes/manifest.json
vendored
Normal file
11
.obsidian/plugins/emoji-shortcodes/manifest.json
vendored
Normal file
@@ -0,0 +1,11 @@
|
|||||||
|
{
|
||||||
|
"id": "emoji-shortcodes",
|
||||||
|
"name": "Emoji Shortcodes",
|
||||||
|
"version": "2.2.0",
|
||||||
|
"minAppVersion": "1.0.0",
|
||||||
|
"description": "This Plugin enables the use of Markdown Emoji Shortcodes :smile:",
|
||||||
|
"author": "phibr0",
|
||||||
|
"authorUrl": "https://github.com/phibr0",
|
||||||
|
"isDesktopOnly": false,
|
||||||
|
"fundingUrl": "https://ko-fi.com/phibr0"
|
||||||
|
}
|
||||||
31
.obsidian/plugins/emoji-shortcodes/styles.css
vendored
Normal file
31
.obsidian/plugins/emoji-shortcodes/styles.css
vendored
Normal file
@@ -0,0 +1,31 @@
|
|||||||
|
a[href="https://ko-fi.com/phibr0"] > img
|
||||||
|
{
|
||||||
|
height: 3em;
|
||||||
|
}
|
||||||
|
|
||||||
|
a[href="https://ko-fi.com/phibr0"]
|
||||||
|
{
|
||||||
|
transform: translate(0, 5%);
|
||||||
|
}
|
||||||
|
|
||||||
|
.ES-suggester-container {
|
||||||
|
display: flex;
|
||||||
|
place-content: space-between;
|
||||||
|
}
|
||||||
|
|
||||||
|
.ES-shortcode {
|
||||||
|
margin-right: 8px;
|
||||||
|
}
|
||||||
|
|
||||||
|
.ES-suggestion-item {
|
||||||
|
border-top: solid var(--background-secondary) 1px;
|
||||||
|
padding-left: 10px;
|
||||||
|
}
|
||||||
|
|
||||||
|
.ES-sub-setting {
|
||||||
|
padding-left: 2em;
|
||||||
|
}
|
||||||
|
.ES-sub-setting + .ES-sub-setting {
|
||||||
|
padding-left: 0;
|
||||||
|
margin-left: 2em;
|
||||||
|
}
|
||||||
14
.obsidian/plugins/obsidian-vimrc-support/data.json
vendored
Normal file
14
.obsidian/plugins/obsidian-vimrc-support/data.json
vendored
Normal file
@@ -0,0 +1,14 @@
|
|||||||
|
{
|
||||||
|
"vimrcFileName": ".obsidian.vimrc",
|
||||||
|
"displayChord": true,
|
||||||
|
"displayVimMode": true,
|
||||||
|
"fixedNormalModeLayout": false,
|
||||||
|
"capturedKeyboardMap": {},
|
||||||
|
"supportJsCommands": false,
|
||||||
|
"vimStatusPromptMap": {
|
||||||
|
"normal": "🟢",
|
||||||
|
"insert": "🟠",
|
||||||
|
"visual": "🟡",
|
||||||
|
"replace": "🔴"
|
||||||
|
}
|
||||||
|
}
|
||||||
1247
.obsidian/plugins/obsidian-vimrc-support/main.js
vendored
Normal file
1247
.obsidian/plugins/obsidian-vimrc-support/main.js
vendored
Normal file
File diff suppressed because it is too large
Load Diff
10
.obsidian/plugins/obsidian-vimrc-support/manifest.json
vendored
Normal file
10
.obsidian/plugins/obsidian-vimrc-support/manifest.json
vendored
Normal file
@@ -0,0 +1,10 @@
|
|||||||
|
{
|
||||||
|
"id": "obsidian-vimrc-support",
|
||||||
|
"name": "Vimrc Support",
|
||||||
|
"version": "0.10.2",
|
||||||
|
"description": "Auto-load a startup file with Obsidian Vim commands.",
|
||||||
|
"minAppVersion": "0.15.3",
|
||||||
|
"author": "esm",
|
||||||
|
"authorUrl": "",
|
||||||
|
"isDesktopOnly": false
|
||||||
|
}
|
||||||
164
.obsidian/snippets/matugen.css
vendored
164
.obsidian/snippets/matugen.css
vendored
@@ -6,113 +6,113 @@
|
|||||||
.theme-dark, .theme-light {
|
.theme-dark, .theme-light {
|
||||||
|
|
||||||
/* ── Material You RGB helpers ──────────────────────────── */
|
/* ── Material You RGB helpers ──────────────────────────── */
|
||||||
--mat-bg-rgb: 14, 21, 13;
|
--mat-bg-rgb: 19, 18, 29;
|
||||||
--mat-surface-rgb: 14, 21, 13;
|
--mat-surface-rgb: 19, 18, 29;
|
||||||
--mat-on-surface-rgb: 221, 229, 215;
|
--mat-on-surface-rgb: 228, 224, 240;
|
||||||
--mat-primary-rgb: 88, 225, 104;
|
--mat-primary-rgb: 195, 192, 255;
|
||||||
--mat-on-primary-rgb: 0, 57, 13;
|
--mat-on-primary-rgb: 29, 0, 165;
|
||||||
|
|
||||||
/* ── Core Backgrounds ──────────────────────────────────── */
|
/* ── Core Backgrounds ──────────────────────────────────── */
|
||||||
--background-primary: #0e150d;
|
--background-primary: #13121d;
|
||||||
--background-primary-alt: #0e150d;
|
--background-primary-alt: #13121d;
|
||||||
--background-secondary: #161d15;
|
--background-secondary: #1b1a25;
|
||||||
--background-secondary-alt: #1a2219;
|
--background-secondary-alt: #1f1e29;
|
||||||
|
|
||||||
/* ── Titlebar ──────────────────────────────────────────── */
|
/* ── Titlebar ──────────────────────────────────────────── */
|
||||||
--titlebar-background: #0e150d;
|
--titlebar-background: #13121d;
|
||||||
--titlebar-background-focused: #161d15;
|
--titlebar-background-focused: #1b1a25;
|
||||||
--titlebar-text-color: #dde5d7;
|
--titlebar-text-color: #e4e0f0;
|
||||||
|
|
||||||
/* ── Borders & Dividers ────────────────────────────────── */
|
/* ── Borders & Dividers ────────────────────────────────── */
|
||||||
--background-modifier-border: #3d4a3b;
|
--background-modifier-border: #464557;
|
||||||
--background-modifier-border-focus: #879583;
|
--background-modifier-border-focus: #918fa3;
|
||||||
--background-modifier-border-hover: #879583;
|
--background-modifier-border-hover: #918fa3;
|
||||||
|
|
||||||
/* ── Text Colors ───────────────────────────────────────── */
|
/* ── Text Colors ───────────────────────────────────────── */
|
||||||
--text-normal: #dde5d7;
|
--text-normal: #e4e0f0;
|
||||||
--text-muted: #bccbb7;
|
--text-muted: #c7c4da;
|
||||||
--text-faint: #879583;
|
--text-faint: #918fa3;
|
||||||
--text-on-accent: #00390d;
|
--text-on-accent: #1d00a5;
|
||||||
--text-selection: rgba(88, 225, 104, 0.25);
|
--text-selection: rgba(195, 192, 255, 0.25);
|
||||||
|
|
||||||
/* ── Accent & Interactive ──────────────────────────────── */
|
/* ── Accent & Interactive ──────────────────────────────── */
|
||||||
--interactive-accent: #58e168;
|
--interactive-accent: #c3c0ff;
|
||||||
--interactive-accent-hover: #16af3e;
|
--interactive-accent-hover: #3c24f6;
|
||||||
--interactive-accent-rgb: 88, 225, 104;
|
--interactive-accent-rgb: 195, 192, 255;
|
||||||
--text-accent: #58e168;
|
--text-accent: #c3c0ff;
|
||||||
--text-accent-hover: #16af3e;
|
--text-accent-hover: #3c24f6;
|
||||||
|
|
||||||
/* ── Hover & Active Modifiers ──────────────────────────── */
|
/* ── Hover & Active Modifiers ──────────────────────────── */
|
||||||
--background-modifier-hover: rgba(var(--mat-on-surface-rgb), 0.06);
|
--background-modifier-hover: rgba(var(--mat-on-surface-rgb), 0.06);
|
||||||
--background-modifier-active-hover: rgba(var(--mat-primary-rgb), 0.15);
|
--background-modifier-active-hover: rgba(var(--mat-primary-rgb), 0.15);
|
||||||
--background-modifier-success: #fe5f8c;
|
--background-modifier-success: #a32200;
|
||||||
--background-modifier-error: #93000a;
|
--background-modifier-error: #93000a;
|
||||||
--background-modifier-error-hover: #ffb4ab;
|
--background-modifier-error-hover: #ffb4ab;
|
||||||
|
|
||||||
/* ── Obsidian Color Scale (--color-base-XX) ────────────── */
|
/* ── Obsidian Color Scale (--color-base-XX) ────────────── */
|
||||||
--color-base-00: #0e150d;
|
--color-base-00: #13121d;
|
||||||
--color-base-05: #0e150d;
|
--color-base-05: #13121d;
|
||||||
--color-base-10: #091008;
|
--color-base-10: #0d0d17;
|
||||||
--color-base-20: #161d15;
|
--color-base-20: #1b1a25;
|
||||||
--color-base-25: #1a2219;
|
--color-base-25: #1f1e29;
|
||||||
--color-base-30: #242c23;
|
--color-base-30: #292934;
|
||||||
--color-base-35: #2f372d;
|
--color-base-35: #34343f;
|
||||||
--color-base-40: #3d4a3b;
|
--color-base-40: #464557;
|
||||||
--color-base-50: #879583;
|
--color-base-50: #918fa3;
|
||||||
--color-base-60: #bccbb7;
|
--color-base-60: #c7c4da;
|
||||||
--color-base-70: #dde5d7;
|
--color-base-70: #e4e0f0;
|
||||||
--color-base-100: #dde5d7;
|
--color-base-100: #e4e0f0;
|
||||||
|
|
||||||
/* ── Semantic Colors ───────────────────────────────────── */
|
/* ── Semantic Colors ───────────────────────────────────── */
|
||||||
--color-red: #ffb4ab;
|
--color-red: #ffb4ab;
|
||||||
--color-orange: #ffb1c1;
|
--color-orange: #ffb4a3;
|
||||||
--color-yellow: #98d695;
|
--color-yellow: #c3c0ff;
|
||||||
--color-green: #fe5f8c;
|
--color-green: #a32200;
|
||||||
--color-cyan: #1a5422;
|
--color-cyan: #3e3a96;
|
||||||
--color-blue: #58e168;
|
--color-blue: #c3c0ff;
|
||||||
--color-purple: #98d695;
|
--color-purple: #c3c0ff;
|
||||||
--color-pink: #ffb1c1;
|
--color-pink: #ffb4a3;
|
||||||
|
|
||||||
/* ── Headings ──────────────────────────────────────────── */
|
/* ── Headings ──────────────────────────────────────────── */
|
||||||
--h1-color: #58e168;
|
--h1-color: #c3c0ff;
|
||||||
--h2-color: #58e168;
|
--h2-color: #c3c0ff;
|
||||||
--h3-color: #98d695;
|
--h3-color: #c3c0ff;
|
||||||
--h4-color: #ffb1c1;
|
--h4-color: #ffb4a3;
|
||||||
--h5-color: #bccbb7;
|
--h5-color: #c7c4da;
|
||||||
--h6-color: #879583;
|
--h6-color: #918fa3;
|
||||||
|
|
||||||
/* ── Links ─────────────────────────────────────────────── */
|
/* ── Links ─────────────────────────────────────────────── */
|
||||||
--link-color: #58e168;
|
--link-color: #c3c0ff;
|
||||||
--link-color-hover: #000601;
|
--link-color-hover: #faf6ff;
|
||||||
--link-external-color: #ffb1c1;
|
--link-external-color: #ffb4a3;
|
||||||
--link-unresolved-color: #879583;
|
--link-unresolved-color: #918fa3;
|
||||||
|
|
||||||
/* ── Tags ──────────────────────────────────────────────── */
|
/* ── Tags ──────────────────────────────────────────────── */
|
||||||
--tag-color: #000601;
|
--tag-color: #faf6ff;
|
||||||
--tag-background: #16af3e;
|
--tag-background: #3c24f6;
|
||||||
--tag-border-color: #58e168;
|
--tag-border-color: #c3c0ff;
|
||||||
--tag-color-hover: #00390d;
|
--tag-color-hover: #1d00a5;
|
||||||
--tag-background-hover: #58e168;
|
--tag-background-hover: #c3c0ff;
|
||||||
|
|
||||||
/* ── Checkboxes ────────────────────────────────────────── */
|
/* ── Checkboxes ────────────────────────────────────────── */
|
||||||
--checkbox-color: #58e168;
|
--checkbox-color: #c3c0ff;
|
||||||
--checkbox-color-hover: #16af3e;
|
--checkbox-color-hover: #3c24f6;
|
||||||
--checkbox-border-color: #879583;
|
--checkbox-border-color: #918fa3;
|
||||||
--checkbox-marker-color: #00390d;
|
--checkbox-marker-color: #1d00a5;
|
||||||
|
|
||||||
/* ── Code Blocks ───────────────────────────────────────── */
|
/* ── Code Blocks ───────────────────────────────────────── */
|
||||||
--code-background: #161d15;
|
--code-background: #1b1a25;
|
||||||
--code-normal: #dde5d7;
|
--code-normal: #e4e0f0;
|
||||||
--code-comment: #879583;
|
--code-comment: #918fa3;
|
||||||
--code-function: #58e168;
|
--code-function: #c3c0ff;
|
||||||
--code-important: #ffb4ab;
|
--code-important: #ffb4ab;
|
||||||
--code-keyword: #98d695;
|
--code-keyword: #c3c0ff;
|
||||||
--code-operator: #ffb1c1;
|
--code-operator: #ffb4a3;
|
||||||
--code-property: #bccbb7;
|
--code-property: #c7c4da;
|
||||||
--code-punctuation: #3d4a3b;
|
--code-punctuation: #464557;
|
||||||
--code-string: #ffb1c1;
|
--code-string: #ffb4a3;
|
||||||
--code-tag: #ffb4ab;
|
--code-tag: #ffb4ab;
|
||||||
--code-value: #98d695;
|
--code-value: #c3c0ff;
|
||||||
|
|
||||||
/* ── Scrollbar ─────────────────────────────────────────── */
|
/* ── Scrollbar ─────────────────────────────────────────── */
|
||||||
--scrollbar-thumb-bg: rgba(var(--mat-on-surface-rgb), 0.12);
|
--scrollbar-thumb-bg: rgba(var(--mat-on-surface-rgb), 0.12);
|
||||||
@@ -121,16 +121,16 @@
|
|||||||
|
|
||||||
/* ── Inputs ────────────────────────────────────────────── */
|
/* ── Inputs ────────────────────────────────────────────── */
|
||||||
--input-shadow: none;
|
--input-shadow: none;
|
||||||
--input-shadow-hover: 0 0 0 2px #879583;
|
--input-shadow-hover: 0 0 0 2px #918fa3;
|
||||||
|
|
||||||
/* ── Graph View ────────────────────────────────────────── */
|
/* ── Graph View ────────────────────────────────────────── */
|
||||||
--graph-node: #58e168;
|
--graph-node: #c3c0ff;
|
||||||
--graph-node-unresolved: #879583;
|
--graph-node-unresolved: #918fa3;
|
||||||
--graph-node-focused: #000601;
|
--graph-node-focused: #faf6ff;
|
||||||
--graph-node-tag: #98d695;
|
--graph-node-tag: #c3c0ff;
|
||||||
--graph-node-attachment: #ffb1c1;
|
--graph-node-attachment: #ffb4a3;
|
||||||
--graph-line: #3d4a3b;
|
--graph-line: #464557;
|
||||||
--graph-background: #0e150d;
|
--graph-background: #13121d;
|
||||||
|
|
||||||
}
|
}
|
||||||
|
|
||||||
|
|||||||
115
.obsidian/workspace.json
vendored
115
.obsidian/workspace.json
vendored
@@ -4,35 +4,49 @@
|
|||||||
"type": "split",
|
"type": "split",
|
||||||
"children": [
|
"children": [
|
||||||
{
|
{
|
||||||
"id": "af3dc6a9ef643c24",
|
"id": "81f0ebb32620dead",
|
||||||
"type": "tabs",
|
"type": "tabs",
|
||||||
"children": [
|
"children": [
|
||||||
{
|
{
|
||||||
"id": "0e89c1d1d5ec049c",
|
"id": "140d404d9b2faf63",
|
||||||
"type": "leaf",
|
"type": "leaf",
|
||||||
"state": {
|
"state": {
|
||||||
"type": "markdown",
|
"type": "markdown",
|
||||||
"state": {
|
"state": {
|
||||||
"file": "99 Templates/New File Template.md",
|
"file": "10 Courses/02 - SoSe 2026/AT/29593852 - Strings.md",
|
||||||
"mode": "source",
|
"mode": "source",
|
||||||
"source": false
|
"source": false
|
||||||
},
|
},
|
||||||
"icon": "lucide-file",
|
"icon": "lucide-file",
|
||||||
"title": "New File Template"
|
"title": "29593852 - Strings"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
{
|
{
|
||||||
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|
||||||
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||||||
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||||||
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|
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|
||||||
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|
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|
||||||
|
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|
||||||
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|
||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
"title": "29593952 - Regular Languages"
|
||||||
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|
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|
||||||
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|
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|
||||||
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|
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|
||||||
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|
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|
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---
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created: 2026-04-07 16:31
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course:
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topic:
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type: lecture
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status: 🔴
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tags:
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- university
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---
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||||||
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||||||
> [!abstract]
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>
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## 📝 Content
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---
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created: 2026-04-07 16:32
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course:
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tags:
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- university
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---
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||||||
> [!abstract]
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>
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---
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created: 2026-04-07 16:37
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> [!abstract]
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>
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> [!abstract]
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>
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>
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|
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|
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## 📌 Summary
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> [!abstract]
|
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>
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---
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## 📝 Content
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||||||
@@ -1,320 +0,0 @@
|
|||||||
# Arithmetic
|
|
||||||
|
|
||||||
## Asymptotic Equivalence Classes (Big-O)
|
|
||||||
|
|
||||||
The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
|
|
||||||
|
|
||||||
$f asymp g <==> f in O(g) and g in O(f)$
|
|
||||||
|
|
||||||
Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
|
|
||||||
|
|
||||||
### The "Dominant Term" Rule
|
|
||||||
To find which class a function belongs to, simplify it to its core growth rate:
|
|
||||||
1. **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
|
|
||||||
2. **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
|
|
||||||
3. **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
|
|
||||||
|
|
||||||
## Euclidian Algorithm
|
|
||||||
Purpose is to find the **GCD** (Greatest Common Divisor).
|
|
||||||
|
|
||||||
### Core Rule
|
|
||||||
Fill out this formula:
|
|
||||||
$$"Dividend" = ("Quotient" * "Divisor") + "Remainder"$$
|
|
||||||
1. **Divide** the bigger number by the smaller one
|
|
||||||
2. **How many times** does it fit -> $"Quotient"$
|
|
||||||
3. Find out **whats leftover** -> $"Remainder"$
|
|
||||||
4. **Shift to left** and repeat ($"Old Divisor" -> "Dividend"$, $"Remainder" -> "Divisor"$)
|
|
||||||
|
|
||||||
Result is the last $"Remainder"$ that is not $0$.
|
|
||||||
|
|
||||||
### Bezout Coefficients
|
|
||||||
**Goal:** find a $x$ and $y$ so that $"Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")$
|
|
||||||
|
|
||||||
1. **Rewrite Euclid (above) equations** to solve for remainder ($"Remainder" = "Old Remainder" - "Dividend" * "Divisor"$)
|
|
||||||
2. **Substitute remainders** -> $$
|
|
||||||
|
|
||||||
---
|
|
||||||
|
|
||||||
# Counting
|
|
||||||
|
|
||||||
## Inclusion-Exclusion Principle
|
|
||||||
Principle that dictates that when combining / overlapping sets, you have to make sure to not include elements that occur in multiple sets multiple times.
|
|
||||||
|
|
||||||
### Example:
|
|
||||||
How many integers between $1$ and $10^6$ are of the form $x^2$ or $x^5$ for some $x in NN$?
|
|
||||||
#### How many $x^2$?
|
|
||||||
$sqrt(10^6) = 10^3 = 1.000$
|
|
||||||
#### How many $x^5$?
|
|
||||||
By estimation:
|
|
||||||
$$
|
|
||||||
&15^5 = 759,375 && "--- in the range / below" 10^6 \
|
|
||||||
&16^5 = 1,048,576 && "--- outside the range / above" 10^6 \
|
|
||||||
&=> 15 "numbers in the form "x^5"exist" &&
|
|
||||||
$$
|
|
||||||
> [!warning]
|
|
||||||
> Now, we can't add $1,000$ and $15$, since there are numbers that match both, so we need to subtract these duplicates.
|
|
||||||
|
|
||||||
#### How many $x^2$ and $x^5$ / $x^10$?
|
|
||||||
$$
|
|
||||||
& 3^10 = 59,049 \
|
|
||||||
& 4^10 = 1,048,576 \
|
|
||||||
& => 3 "numbers that are both" x^2 "and" x^5 "exist"
|
|
||||||
$$
|
|
||||||
#### Final calculation:
|
|
||||||
Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
|
|
||||||
$==> 1,000 + 15 - 3 = 1,012$
|
|
||||||
|
|
||||||
### For two sets
|
|
||||||
$|"Total possibilities"| - |"Avoid 1"| - |"Avoid 2"| + |"Avoid Both"|$
|
|
||||||
Where
|
|
||||||
- $"Avoid 1"$ are all elements *not matching* condition 1
|
|
||||||
- $"Avoid 2"$ are all elements *not matching* condition 2
|
|
||||||
- $"Avoid Both"$ are all elements *not matching* condition 1 **and** 2
|
|
||||||
|
|
||||||
## Type of Task: Boolean Lattice
|
|
||||||
Find the amount of upper bounds in ${0, 1}^5$ for the set of vectors $S = {x, y, z}$
|
|
||||||
$$
|
|
||||||
& x = (0, 1, 0, 0, 0) \
|
|
||||||
& y = (0, 0, 1, 0, 1) \
|
|
||||||
& z = (0, 1, 1, 0, 0) \
|
|
||||||
$$
|
|
||||||
> [!INFO]
|
|
||||||
> Method used is called **All-Zero Column Method**
|
|
||||||
|
|
||||||
We look for columns that have $0$'s in all three vectors: Column $1$ and $4$, that's $2$ columns, so we define $k = 2$.
|
|
||||||
To get our result we calculate $2^k$ since there are only two possible states for each value $(0, 1)$:
|
|
||||||
$2^2 = 4$.
|
|
||||||
So we have **4** upper bounds in total.
|
|
||||||
|
|
||||||
> [!INFO]
|
|
||||||
> For the amount of lower bounds we'd check for all-ones columns
|
|
||||||
|
|
||||||
---
|
|
||||||
|
|
||||||
# Functions
|
|
||||||
A function takes in an element from its **domain**, transforms it in someway and outputs that transformed element, which is part of the **co-domain**.
|
|
||||||
Every element of the **domain** must map to a value in the **co-domain**, all values of the **co-domain** that are mapped to form the functions **image**.
|
|
||||||
A function must be **deterministic** - one input can only map to a single output.
|
|
||||||
|
|
||||||
## Notation
|
|
||||||
General function notation: $f: X -> Y$
|
|
||||||
|
|
||||||
> [!INFO]
|
|
||||||
> $f$: name of the function
|
|
||||||
> $X$: Domain
|
|
||||||
> $Y$: Co-domain
|
|
||||||
> $f(x)$: Image of $f$
|
|
||||||
> $X$, $f(x)$ and $Y$ are [[Set Theory | Set]]
|
|
||||||
|
|
||||||
For any $x in X$ the output $f(x)$ is an element of $Y$.
|
|
||||||
|
|
||||||
## Mapping Properties
|
|
||||||
### Injectivity
|
|
||||||
A function is _injective_ if every element in $y in f(x)$ has _at most_ one matching $x in X$.
|
|
||||||
- $forall y in Y,exists excl x in X : f(x) = y$
|
|
||||||
### Surjectivity
|
|
||||||
A function is _surjective_ if every element $y in Y$ has _at minimum_ one matching $x in X$
|
|
||||||
- $forall y in Y, exists x in X : f(x) = y$
|
|
||||||
|
|
||||||
### Bijectivity
|
|
||||||
A function is _bijective_ if every element $y in Y$ has _exactly_ one matching $x in X$ (it is _injective_ and _surjective_)
|
|
||||||
- $forall y in Y, exists excl x in X : f(x) = y$
|
|
||||||
|
|
||||||
---
|
|
||||||
|
|
||||||
# Logic
|
|
||||||
|
|
||||||
## Operators
|
|
||||||
| Operation | Explanation | Notation |
|
|
||||||
| ----------------- | ------------------------------------ | --------- |
|
|
||||||
| **and**<br> | Both $p$ and $q$ must be true | $p and q$ |
|
|
||||||
| **or** | Either $p$ or $q$ (or both) are true | $p or q$ |
|
|
||||||
| **not** | Negates the statement | $not p$ |
|
|
||||||
| **Implication** | If $p$ then $q$ | $=>$ |
|
|
||||||
| **Biconditional** | $p$ if and _only_ if $q$ | $<=>$ |
|
|
||||||
| **xor** | Either $p$ or $q$ but not both | $xor$ |
|
|
||||||
|
|
||||||
### Implied Operators
|
|
||||||
| Operation | Explanion | Notation |
|
|
||||||
| --------- | --------------------------------------- | -------------- |
|
|
||||||
| **nand** | $p$ and $q$ are not both true | $not(p and q)$ |
|
|
||||||
| **nor** | neither of $p$ and $q$ are true | $not(p or q)$ |
|
|
||||||
| **xnor** | $p$ and $q$ are both false or both true | $not xor$ |
|
|
||||||
|
|
||||||
---
|
|
||||||
|
|
||||||
# Relations
|
|
||||||
|
|
||||||
## Types of Relations
|
|
||||||
|
|
||||||
| Relation | Explanation | Example |
|
|
||||||
| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
|
|
||||||
| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$ |
|
|
||||||
| *reflexive* | every element is related to itself with the given relation | $a <= a, 5 = 5$ |
|
|
||||||
| *anti-reflexive* | every element is *NOT* related to itself in the given relation | $a < a$ |
|
|
||||||
| *symmetric* | the given relation work both ways | $a = b => b = a$ |
|
|
||||||
| *antisymmetric* | the given relation only works both ways if $a$ and $b$ are the same | $a <= b, b <= a => a = b$ |
|
|
||||||
|
|
||||||
|
|
||||||
## Equivalence Relations
|
|
||||||
A relation $R$ is called _equivalence relation_ when it is _transitive, reflexive and symmetric_.
|
|
||||||
|
|
||||||
### Example:
|
|
||||||
**Question:** How many equivalence classes are there for the given equivalence relation?
|
|
||||||
$$
|
|
||||||
& ~ "on" {0, 1, 2, 3}^(2) \
|
|
||||||
& "defined by" (x_1, y_1) ~ (x_2, y_2) <==> x_1 + y_1 = x_2 + y_2
|
|
||||||
$$
|
|
||||||
> [!INFO]
|
|
||||||
> Meaning:
|
|
||||||
> The pairs $(x_1, y_1)$ and $(x_2, y_2)$ are equivalent to each other when the components of the pair added up have the same result.
|
|
||||||
|
|
||||||
Solving:
|
|
||||||
- Smallest possible sum: $(0 + 0) = 0$
|
|
||||||
- Biggest possible sum: $(3 + 3) = 6$
|
|
||||||
- All possible sums: $0, 1, 2, 3, 4, 5, 6$
|
|
||||||
|
|
||||||
Each possible sum creates it's own equivalence class. So there are $7$ equivalence classes.
|
|
||||||
|
|
||||||
> [!NOTE]
|
|
||||||
> All equivalence classes:
|
|
||||||
> $[0]_(~) = {(0, 0)}$
|
|
||||||
> $[1]_(~) = {(0, 1), (1, 0)}$
|
|
||||||
> $[2]_(~) = {(0, 2), (1, 1), (2, 0)}$
|
|
||||||
> $[3]_(~) = {(0, 3), (1, 2), (2, 1), (3, 0)}$
|
|
||||||
> $[4]_(~) = {(1, 3), (2, 2), (3, 1)}$
|
|
||||||
> $[5]_(~) = {(2, 3), (3, 2)}$
|
|
||||||
> $[6]_(~) = {(3, 3)}$
|
|
||||||
|
|
||||||
## Binary Relation
|
|
||||||
A binary relation is a relation $R$ between _exactly two_ elements $a in R$ and $b in R$. An example for a binary relation is $a <= b$
|
|
||||||
|
|
||||||
## Converse Relation
|
|
||||||
$C^top$ or $C^(-1)$ is the relation that occurs if the elements of a _binary relation_ are switched.
|
|
||||||
|
|
||||||
## Composition of Relations - Example
|
|
||||||
$$
|
|
||||||
"Compute" Q^top compose R "with:"\
|
|
||||||
Q = {(2, 2), (3, 3), (2, 1)} \
|
|
||||||
R = {(1, 2), (3, 3), (3, 1)} \
|
|
||||||
$$
|
|
||||||
### 1. Apply converse to $Q$:
|
|
||||||
$$
|
|
||||||
Q^top = {(2, 2), (3, 3), (1, 2)}
|
|
||||||
$$
|
|
||||||
### 2. Perform Composition:
|
|
||||||
Look at each pair in $R$, check if $Q^top$ has a pair starting with se second element in that pair:
|
|
||||||
|
|
||||||
$$
|
|
||||||
(1, 2) -> (2, 2) => (1, 2) \
|
|
||||||
(3, 3) -> (3, 3) => (3, 3) \
|
|
||||||
(3, 1) -> (1, 2) => (3, 2)
|
|
||||||
$$
|
|
||||||
### 3. Result:
|
|
||||||
$$Q^top compose R = {(1, 2), (3, 2), (3, 3)}$$
|
|
||||||
|
|
||||||
## Orders
|
|
||||||
An **Order** is a mathematical way to sort, rank or compare elements within a set, where some elements come "before" and "after" others.
|
|
||||||
|
|
||||||
A _binary relation_ is called an order if it is...
|
|
||||||
- [?] a *reflexive relation*
|
|
||||||
- [?] a *antisymmetric relation*
|
|
||||||
- [?] a *transitive relation*
|
|
||||||
|
|
||||||
---
|
|
||||||
|
|
||||||
# Set Theory
|
|
||||||
A set is a collection of _unordered_ elements.
|
|
||||||
A set cannot contain duplicates.
|
|
||||||
|
|
||||||
## Notation
|
|
||||||
### Set Notation
|
|
||||||
Declaration of a set $A$ with elements $a$, $b$, $c$:
|
|
||||||
$$A := {a, b, c}$$
|
|
||||||
|
|
||||||
### Cardinality
|
|
||||||
Amount of Elements in a set $A$
|
|
||||||
Notation: $|A|$
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3, 4} \
|
|
||||||
|A| = 4
|
|
||||||
$$
|
|
||||||
|
|
||||||
### Well-Known Sets
|
|
||||||
- Empty Set: $emptyset = {}$
|
|
||||||
- Natural Numbers: $N = {1, 2, 3, ...}$
|
|
||||||
- Integers: $ZZ = {-2, -1, 0, 1, 2}$
|
|
||||||
- Rational Numbers: $QQ = {1/2, 22/7 }$
|
|
||||||
- Real Numbers: $RR = {1, pi, sqrt(2)}$
|
|
||||||
- Complex Numbers: $CC = {i, pi, 1, sqrt(-1)}$
|
|
||||||
|
|
||||||
### Set-Builder Notation
|
|
||||||
Common form of notation to create sets without explicitly specifying elements.
|
|
||||||
$$
|
|
||||||
A := {x in N | 0 <= x <= 5} \
|
|
||||||
A = {1, 2, 3, 4, 5}
|
|
||||||
$$
|
|
||||||
|
|
||||||
### Member of
|
|
||||||
Denote whether $x$ is an element of the set $A$
|
|
||||||
Notation: $x in A$
|
|
||||||
Negation: $x in.not A$
|
|
||||||
|
|
||||||
### Subsets
|
|
||||||
| Type | Explanation | Notation |
|
|
||||||
| ------------------------ | ---------------------------------------------------------------------- | ------------------- |
|
|
||||||
| **Subset** | Every element of $A$ is in $B$ | $A subset B$ |
|
|
||||||
| **Subset or equal to** | Every element of $A$ is in $B$, or they are the exactly same set | $A subset.eq B$ |
|
|
||||||
| **Proper subset** | Every element of $A$ is in $B$, but $A$ is definitely smaller than $B$ | $A subset.sq B$<br> |
|
|
||||||
| **Superset**<br> | $A$ contains everything that is in $B$ | $A supset B$ |
|
|
||||||
| **Superset or equal to** | $A$ contains everything that is in $B$, or they are identical | $A supset.eq B$ |
|
|
||||||
|
|
||||||
## Operations
|
|
||||||
### Union
|
|
||||||
Notation: $A union B$
|
|
||||||
Definition: all elements from both sets _without adding duplicates_
|
|
||||||
$$A := {1, 2, 3}\ B := {3, 4, 5}\ A union B = {1, 2, 3, 4, 5}$$
|
|
||||||
|
|
||||||
### Intersection
|
|
||||||
Notation:$A inter B$
|
|
||||||
Definition: all elements _contained in both sets_
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3} \
|
|
||||||
B := {2, 3, 4} \
|
|
||||||
A inter B = {2, 3}
|
|
||||||
$$
|
|
||||||
|
|
||||||
### Difference
|
|
||||||
Notation: $A backslash B$
|
|
||||||
Definition: all elements _in $A$ that are not in $B$_
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3} \
|
|
||||||
B := {3, 4, 5} \
|
|
||||||
A backslash B = {1, 2}
|
|
||||||
$$
|
|
||||||
|
|
||||||
### Symmetric Difference
|
|
||||||
Notation: $A Delta B$
|
|
||||||
Definition: all elements _only in $A$ or only in $B$_
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3} \
|
|
||||||
B := {2, 3, 4} \
|
|
||||||
A Delta B = {1, 4}
|
|
||||||
$$
|
|
||||||
|
|
||||||
### Cartesian Product
|
|
||||||
Notation: $A times B$
|
|
||||||
Definition: all pairs of all elements in $A$ and $B$
|
|
||||||
$$
|
|
||||||
A := {1, 2} \
|
|
||||||
B := {3, 4, 5} \
|
|
||||||
A times B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
|
|
||||||
$$
|
|
||||||
|
|
||||||
### Powerset
|
|
||||||
Notation: $cal(P)(A)$
|
|
||||||
Definition: all possible _Subsets of A_
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3} \
|
|
||||||
cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
|
|
||||||
$$
|
|
||||||
@@ -1,5 +0,0 @@
|
|||||||
# Notation
|
|
||||||
|
|
||||||
## 1. Sets and Logic
|
|
||||||
$|A|$: The *cardinality* (size) of finite set $A$.
|
|
||||||
$script(p)$
|
|
||||||
284
.trash/merge.md
284
.trash/merge.md
@@ -1,284 +0,0 @@
|
|||||||
## Asymptotic Equivalence Classes (Big-O)
|
|
||||||
|
|
||||||
The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
|
|
||||||
|
|
||||||
$f asymp g <==> f in O(g) and g in O(f)$
|
|
||||||
|
|
||||||
Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
|
|
||||||
|
|
||||||
### The "Dominant Term" Rule
|
|
||||||
To find which class a function belongs to, simplify it to its core growth rate:
|
|
||||||
1. **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
|
|
||||||
2. **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
|
|
||||||
3. **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
|
|
||||||
|
|
||||||
## Euclidian Algorithm
|
|
||||||
Purpose is to find the **GCD** (Greatest Common Divisor).
|
|
||||||
|
|
||||||
### Core Rule
|
|
||||||
Fill out this formula:
|
|
||||||
$$"Dividend" = ("Quotient" * "Divisor") + "Remainder"$$
|
|
||||||
1. **Divide** the bigger number by the smaller one
|
|
||||||
2. **How many times** does it fit -> $"Quotient"$
|
|
||||||
3. Find out **whats leftover** -> $"Remainder"$
|
|
||||||
4. **Shift to left** and repeat ($"Old Divisor" -> "Dividend"$, $"Remainder" -> "Divisor"$)
|
|
||||||
|
|
||||||
Result is the last $"Remainder"$ that is not $0$.
|
|
||||||
|
|
||||||
### Bezout Coefficients
|
|
||||||
**Goal:** find a $x$ and $y$ so that $"Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")$
|
|
||||||
|
|
||||||
1. **Rewrite Euclid (above) equations** to solve for remainder ($"Remainder" = "Old Remainder" - "Dividend" * "Divisor"$)
|
|
||||||
2. **Substitute remainders** -> ## Inclusion-Exclusion Principle
|
|
||||||
Principle that dictates that when combining / overlapping sets, you have to make sure to not include elements that occur in multiple sets multiple times.
|
|
||||||
|
|
||||||
### Example:
|
|
||||||
How many integers between $1$ and $10^6$ are of the form $x^2$ or $x^5$ for some $x in NN$?
|
|
||||||
#### How many $x^2$?
|
|
||||||
$sqrt(10^6) = 10^3 = 1.000$
|
|
||||||
#### How many $x^5$?
|
|
||||||
By estimation:
|
|
||||||
$$
|
|
||||||
&15^5 = 759,375 && "--- in the range / below" 10^6 \
|
|
||||||
&16^5 = 1,048,576 && "--- outside the range / above" 10^6 \
|
|
||||||
&=> 15 "numbers in the form "x^5"exist" &&
|
|
||||||
$$
|
|
||||||
> [!warning]
|
|
||||||
> Now, we can't add $1,000$ and $15$, since there are numbers that match both, so we need to subtract these duplicates.
|
|
||||||
|
|
||||||
#### How many $x^2$ and $x^5$ / $x^10$?
|
|
||||||
$$
|
|
||||||
& 3^10 = 59,049 \
|
|
||||||
& 4^10 = 1,048,576 \
|
|
||||||
& => 3 "numbers that are both" x^2 "and" x^5 "exist"
|
|
||||||
$$
|
|
||||||
#### Final calculation:
|
|
||||||
Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
|
|
||||||
$==> 1,000 + 15 - 3 = 1,012$
|
|
||||||
|
|
||||||
### For two sets
|
|
||||||
$|"Total possibilities"| - |"Avoid 1"| - |"Avoid 2"| + |"Avoid Both"|$
|
|
||||||
Where
|
|
||||||
- $"Avoid 1"$ are all elements *not matching* condition 1
|
|
||||||
- $"Avoid 2"$ are all elements *not matching* condition 2
|
|
||||||
- $"Avoid Both"$ are all elements *not matching* condition 1 **and** 2
|
|
||||||
|
|
||||||
## Type of Task: Boolean Lattice
|
|
||||||
Find the amount of upper bounds in ${0, 1}^5$ for the set of vectors $S = {x, y, z}$
|
|
||||||
$$
|
|
||||||
& x = (0, 1, 0, 0, 0) \
|
|
||||||
& y = (0, 0, 1, 0, 1) \
|
|
||||||
& z = (0, 1, 1, 0, 0) \
|
|
||||||
$$
|
|
||||||
> [!INFO]
|
|
||||||
> Method used is called **All-Zero Column Method**
|
|
||||||
|
|
||||||
We look for columns that have $0$'s in all three vectors: Column $1$ and $4$, that's $2$ columns, so we define $k = 2$.
|
|
||||||
To get our result we calculate $2^k$ since there are only two possible states for each value $(0, 1)$:
|
|
||||||
$2^2 = 4$.
|
|
||||||
So we have **4** upper bounds in total.
|
|
||||||
|
|
||||||
> [!INFO]
|
|
||||||
> For the amount of lower bounds we'd check for all-ones columns
|
|
||||||
A function takes in an element from its **domain**, transforms it in someway and outputs that transformed element, which is part of the **co-domain**.
|
|
||||||
Every element of the **domain** must map to a value in the **co-domain**, all values of the **co-domain** that are mapped to form the functions **image**.
|
|
||||||
A function must be **deterministic** - one input can only map to a single output.
|
|
||||||
|
|
||||||
## Notation
|
|
||||||
General function notation: $f: X -> Y$
|
|
||||||
|
|
||||||
> [!INFO]
|
|
||||||
> $f$: name of the function
|
|
||||||
> $X$: Domain
|
|
||||||
> $Y$: Co-domain
|
|
||||||
> $f(x)$: Image of $f$
|
|
||||||
> $X$, $f(x)$ and $Y$ are [[Set Theory | Set]]
|
|
||||||
|
|
||||||
For any $x in X$ the output $f(x)$ is an element of $Y$.
|
|
||||||
|
|
||||||
## Mapping Properties
|
|
||||||
### Injectivity
|
|
||||||
A function is _injective_ if every element in $y in f(x)$ has _at most_ one matching $x in X$.
|
|
||||||
- $forall y in Y,exists excl x in X : f(x) = y$
|
|
||||||
### Surjectivity
|
|
||||||
A function is _surjective_ if every element $y in Y$ has _at minimum_ one matching $x in X$
|
|
||||||
- $forall y in Y, exists x in X : f(x) = y$
|
|
||||||
|
|
||||||
### Bijectivity
|
|
||||||
A function is _bijective_ if every element $y in Y$ has _exactly_ one matching $x in X$ (it is _injective_ and _surjective_)
|
|
||||||
- $forall y in Y, exists excl x in X : f(x) = y$## Operators
|
|
||||||
| Operation | Explanation | Notation |
|
|
||||||
| ----------------- | ------------------------------------ | --------- |
|
|
||||||
| **and**<br> | Both $p$ and $q$ must be true | $p and q$ |
|
|
||||||
| **or** | Either $p$ or $q$ (or both) are true | $p or q$ |
|
|
||||||
| **not** | Negates the statement | $not p$ |
|
|
||||||
| **Implication** | If $p$ then $q$ | $=>$ |
|
|
||||||
| **Biconditional** | $p$ if and _only_ if $q$ | $<=>$ |
|
|
||||||
| **xor** | Either $p$ or $q$ but not both | $xor$ |
|
|
||||||
### Implied Operators
|
|
||||||
| Operation | Explanion | Notation |
|
|
||||||
| --------- | --------------------------------------- | -------------- |
|
|
||||||
| **nand** | $p$ and $q$ are not both true | $not(p and q)$ |
|
|
||||||
| **nor** | neither of $p$ and $q$ are true | $not(p or q)$ |
|
|
||||||
| **xnor** | $p$ and $q$ are both false or both true | $not xor$ |
|
|
||||||
## Types of Relations
|
|
||||||
|
|
||||||
| Relation | Explanation | Example |
|
|
||||||
| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
|
|
||||||
| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$ |
|
|
||||||
| *reflexive* | every element is related to itself with the given relation | $a <= a, 5 = 5$ |
|
|
||||||
| *anti-reflexive* | every element is *NOT* related to itself in the given relation | $a < a$ |
|
|
||||||
| *symmetric* | the given relation work both ways | $a = b => b = a$ |
|
|
||||||
| *antisymmetric* | the given relation only works both ways if $a$ and $b$ are the same | $a <= b, b <= a => a = b$ |
|
|
||||||
|
|
||||||
|
|
||||||
## Equivalence Relations
|
|
||||||
A relation $R$ is called _equivalence relation_ when it is _transitive, reflexive and symmetric_.
|
|
||||||
|
|
||||||
### Example:
|
|
||||||
**Question:** How many equivalence classes are there for the given equivalence relation?
|
|
||||||
$$
|
|
||||||
& ~ "on" {0, 1, 2, 3}^(2) \
|
|
||||||
& "defined by" (x_1, y_1) ~ (x_2, y_2) <==> x_1 + y_1 = x_2 + y_2
|
|
||||||
$$
|
|
||||||
> [!INFO]
|
|
||||||
> Meaning:
|
|
||||||
> The pairs $(x_1, y_1)$ and $(x_2, y_2)$ are equivalent to each other when the components of the pair added up have the same result.
|
|
||||||
|
|
||||||
Solving:
|
|
||||||
- Smallest possible sum: $(0 + 0) = 0$
|
|
||||||
- Biggest possible sum: $(3 + 3) = 6$
|
|
||||||
- All possible sums: $0, 1, 2, 3, 4, 5, 6$
|
|
||||||
|
|
||||||
Each possible sum creates it's own equivalence class. So there are $7$ equivalence classes.
|
|
||||||
|
|
||||||
> [!NOTE]
|
|
||||||
> All equivalence classes:
|
|
||||||
> $[0]_(~) = {(0, 0)}$
|
|
||||||
> $[1]_(~) = {(0, 1), (1, 0)}$
|
|
||||||
> $[2]_(~) = {(0, 2), (1, 1), (2, 0)}$
|
|
||||||
>$[3]_(~) = {(0, 3), (1, 2), (2, 1), (3, 0)}$
|
|
||||||
>$[4]_(~) = {(1, 3), (2, 2), (3, 1)}$
|
|
||||||
> $[5]_(~) = {(2, 3), (3, 2)}$
|
|
||||||
> $[6]_(~) = {(3, 3)}$
|
|
||||||
|
|
||||||
## Binary Relation
|
|
||||||
A binary relation is a relation $R$ between _exactly two_ elements $a in R$ and $b in R$. An example for a binary relation is $a <= b$
|
|
||||||
|
|
||||||
## Converse Relation
|
|
||||||
$C^top$ or $C^(-1)$ is the relation that occurs if the elements of a _binary relation_ are switched.
|
|
||||||
|
|
||||||
## Composition of Relations - Example
|
|
||||||
$$
|
|
||||||
"Compute" Q^top compose R "with:"\
|
|
||||||
Q = {(2, 2), (3, 3), (2, 1)} \
|
|
||||||
R = {(1, 2), (3, 3), (3, 1)} \
|
|
||||||
$$
|
|
||||||
### 1. Apply converse to $Q$:
|
|
||||||
$$
|
|
||||||
Q^top = {(2, 2), (3, 3), (1, 2)}
|
|
||||||
$$
|
|
||||||
### 2. Perform Composition:
|
|
||||||
Look at each pair in $R$, check if $Q^top$ has a pair starting with se second element in that pair:
|
|
||||||
|
|
||||||
$$
|
|
||||||
(1, 2) -> (2, 2) => (1, 2) \
|
|
||||||
(3, 3) -> (3, 3) => (3, 3) \
|
|
||||||
(3, 1) -> (1, 2) => (3, 2)
|
|
||||||
$$
|
|
||||||
### 3. Result:
|
|
||||||
$$ Q^top compose R = {(1, 2), (3, 2), (3, 3)} $$
|
|
||||||
|
|
||||||
## Orders
|
|
||||||
An **Order** is a mathematical way to sort, rank or compare elements within a set, where some elements come "before" and "after" others.
|
|
||||||
|
|
||||||
A _binary relation_ is called an order if it is...
|
|
||||||
- [?] a *reflexive relation*
|
|
||||||
- [?] a *antisymmetric relation*
|
|
||||||
- [?] a *transitive relation*
|
|
||||||
A set is a collection of _unordered_ elements.
|
|
||||||
A set cannot contain duplicates.
|
|
||||||
## Notation
|
|
||||||
### Set Notation
|
|
||||||
Declaration of a set $A$ with elements $a$, $b$, $c$:
|
|
||||||
$$ A := {a, b, c} $$
|
|
||||||
### Cardinality
|
|
||||||
Amount of Elements in a set $A$
|
|
||||||
Notation: $|A|$
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3, 4} \
|
|
||||||
|A| = 4
|
|
||||||
$$
|
|
||||||
### Well-Known Sets
|
|
||||||
- Empty Set: $emptyset = {}$
|
|
||||||
- Natural Numbers: $N = {1, 2, 3, ...}$
|
|
||||||
- Integers: $ZZ = {-2, -1, 0, 1, 2}$
|
|
||||||
- Rational Numbers: $QQ = {1/2, 22/7 }$
|
|
||||||
- Real Numbers: $RR = {1, pi, sqrt(2)}$
|
|
||||||
- Complex Numbers: $CC = {i, pi, 1, sqrt(-1)}$
|
|
||||||
|
|
||||||
### Set-Builder Notation
|
|
||||||
Common form of notation to create sets without explicitly specifying elements.
|
|
||||||
$$
|
|
||||||
A := {x in N | 0 <= x <= 5} \
|
|
||||||
A = {1, 2, 3, 4, 5}
|
|
||||||
$$
|
|
||||||
### Member of
|
|
||||||
Denote whether $x$ is an element of the set $A$
|
|
||||||
Notation: $x in A$
|
|
||||||
Negation: $x in.not A$
|
|
||||||
|
|
||||||
### Subsets
|
|
||||||
| Type | Explanation | Notation |
|
|
||||||
| ------------------------ | ---------------------------------------------------------------------- | ------------------- |
|
|
||||||
| **Subset** | Every element of $A$ is in $B$ | $A subset B$ |
|
|
||||||
| **Subset or equal to** | Every element of $A$ is in $B$, or they are the exactly same set | $A subset.eq B$ |
|
|
||||||
| **Proper subset** | Every element of $A$ is in $B$, but $A$ is definitely smaller than $B$ | $A subset.sq B$<br> |
|
|
||||||
| **Superset**<br> | $A$ contains everything that is in $B$ | $A supset B$ |
|
|
||||||
| **Superset or equal to** | $A$ contains everything that is in $B$, or they are identical | $A supset.eq B$ |
|
|
||||||
|
|
||||||
## Operations
|
|
||||||
### Union
|
|
||||||
Notation: $A union B$
|
|
||||||
Definition: all elements from both sets _without adding duplicates_
|
|
||||||
$$ A := {1, 2, 3}\ B := {3, 4, 5}\ A union B = {1, 2, 3, 4, 5} $$
|
|
||||||
### Intersection
|
|
||||||
Notation:$A inter B$
|
|
||||||
Definition: all elements _contained in both sets_
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3} \
|
|
||||||
B := {2, 3, 4} \
|
|
||||||
A inter B = {2, 3}
|
|
||||||
$$
|
|
||||||
### Difference
|
|
||||||
Notation: $A backslash B$
|
|
||||||
Definition: all elements _in $A$ that are not in $B$_
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3} \
|
|
||||||
B := {3, 4, 5} \
|
|
||||||
A backslash B = {1, 2}
|
|
||||||
$$
|
|
||||||
### Symmetric Difference
|
|
||||||
Notation: $A Delta B$
|
|
||||||
Definition: all elements _only in $A$ or only in $B$_
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3} \
|
|
||||||
B := {2, 3, 4} \
|
|
||||||
A Delta B = {1, 4}
|
|
||||||
$$
|
|
||||||
### Cartesian Product
|
|
||||||
Notation: $A times B$
|
|
||||||
Definition: all pairs of all elements in $A$ and $B$
|
|
||||||
$$
|
|
||||||
A := {1, 2} \
|
|
||||||
B := {3, 4, 5} \
|
|
||||||
A times B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
|
|
||||||
$$
|
|
||||||
### Powerset
|
|
||||||
Notation: $cal(P)(A)$
|
|
||||||
Definition: all possible _Subsets of A_
|
|
||||||
$$
|
|
||||||
A := {1, 2, 3} \
|
|
||||||
cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
|
|
||||||
$$
|
|
||||||
|
|
||||||
@@ -0,0 +1,7 @@
|
|||||||
|
---
|
||||||
|
created: 2026-04-08 08:50
|
||||||
|
course: "[[29593850 - Automationtheory]]"
|
||||||
|
type: overview
|
||||||
|
tags:
|
||||||
|
- university
|
||||||
|
---
|
||||||
71
10 Courses/02 - SoSe 2026/AT/29593852 - Strings.md
Normal file
71
10 Courses/02 - SoSe 2026/AT/29593852 - Strings.md
Normal file
@@ -0,0 +1,71 @@
|
|||||||
|
---
|
||||||
|
created: 2026-04-08 08:52
|
||||||
|
course: "[[29593850 - Automationtheory]]"
|
||||||
|
topic: strings
|
||||||
|
related: "[[29593929 - Alphabets]]"
|
||||||
|
type: lecture
|
||||||
|
status: 🟢
|
||||||
|
tags:
|
||||||
|
- university
|
||||||
|
---
|
||||||
|
## 📌 Summary
|
||||||
|
|
||||||
|
> [!abstract]
|
||||||
|
> Overview of lecture 1 on `Wednesday, 2026/Apr/08`
|
||||||
|
|
||||||
|
---
|
||||||
|
|
||||||
|
## 📝 Content
|
||||||
|
|
||||||
|
A word (or _string_) is a finite sequence $w = a_1 a_2 ... a_n$ if characters from $Sigma$.
|
||||||
|
|
||||||
|
> [!CONVENTION]
|
||||||
|
> We will use small letters to describe strings that are part of a language.
|
||||||
|
|
||||||
|
> [!EXAMPLE]
|
||||||
|
> $"aa", "ab", "bba"$ and $"baab"$ are strings over $Sigma = {a, b}.
|
||||||
|
|
||||||
|
#### Length of a string
|
||||||
|
The _length_ $abs(x)$ of a string $x = a_1 ... a_n$ is its number $abs(x) = n$ of characters.
|
||||||
|
#### Empty String
|
||||||
|
The empty string is denoted by $epsilon$, this is the neutral element.
|
||||||
|
-> $abs(epsilon) = 0$
|
||||||
|
|
||||||
|
## String Operations
|
||||||
|
### Concatenation
|
||||||
|
String can be concatenated, where one string is appended to another.
|
||||||
|
For strings $x = a_1 ... a_n$ and $y = b_1 ... b_m$ over alphabets $Sigma_x$ and $Sigma_y$, their _concatenation_ over the alphabet $Sigma = Sigma_x union Sigma_y$ is the string
|
||||||
|
$$x circle.small y = x y = a_1 a_2 ... a_n b_1 b_2 ... b_m$$
|
||||||
|
> This string is of the length $abs(x y) = n + m$
|
||||||
|
|
||||||
|
> [!EXAMPLE]
|
||||||
|
> $x = "apple"$
|
||||||
|
> $y = "pie"$
|
||||||
|
> $x circle.small y = "applepie"$
|
||||||
|
|
||||||
|
Order of operations / Brackets do _not matter_. (Concatenation is associative but **not** commutative $x y eq.not y x$)
|
||||||
|
> $(x circle.small y) circle.small z = x circle.small (y circle.small z)$
|
||||||
|
|
||||||
|
Any string concatenated with the empty string $epsilon$ will result in itself.
|
||||||
|
> $x circle.small epsilon = x = epsilon circle.small x$
|
||||||
|
|
||||||
|
### Exponentiation
|
||||||
|
|
||||||
|
The $n^"th"$ power $x^n$ of a string $x$ is the $(n-1)$-fold concatenation of $x$ with itself.
|
||||||
|
> $x^0 := epsilon$
|
||||||
|
> $x^n := x^(n-1) circle.small x$ for $n in NN$
|
||||||
|
|
||||||
|
> [!Example]
|
||||||
|
> $x^4 = x x x x$
|
||||||
|
> $(a b)^3 = a b a b a b$
|
||||||
|
|
||||||
|
### Reversing / Mirroring
|
||||||
|
For a string $x = a_1 a_2 ... a_(n-1) a_n$ of length $n$, it's _mirrored string_ is given by
|
||||||
|
$$ x^("Rev") = a_n a_(n-1)...a_2 a_1$$
|
||||||
|
## Substrings
|
||||||
|
A string $x$ is a _substring_ of a string $y$ if $y = u x v$, where $u$ and $v$ can be arbitrary strings.
|
||||||
|
- If $u = epsilon$ then $x$ is a _prefix_ of $y$.
|
||||||
|
- If $v = epsilon$ then $x$ is a suffix of $y$.
|
||||||
|
|
||||||
|
For strings $x$ and $y$ the quantity $abs(y)_x$ is the number of times that $x$ is a substring of $y$.
|
||||||
|
|
||||||
26
10 Courses/02 - SoSe 2026/AT/29593929 - Alphabets.md
Normal file
26
10 Courses/02 - SoSe 2026/AT/29593929 - Alphabets.md
Normal file
@@ -0,0 +1,26 @@
|
|||||||
|
---
|
||||||
|
created: 2026-04-08 10:09
|
||||||
|
course: "[[29593850 - Automationtheory]]"
|
||||||
|
topic: alphabets, characters
|
||||||
|
related:
|
||||||
|
type: lecture
|
||||||
|
status: 🟢
|
||||||
|
tags:
|
||||||
|
- university
|
||||||
|
---
|
||||||
|
## 📌 Summary
|
||||||
|
|
||||||
|
> [!abstract]
|
||||||
|
> Definition and examples of alphabets.
|
||||||
|
|
||||||
|
---
|
||||||
|
|
||||||
|
## 📝 Content
|
||||||
|
|
||||||
|
Alphabets are formal, non-empty, finite, sets of characters (or _letters_ or _symbols_). They are denoted by $Sigma$.
|
||||||
|
|
||||||
|
$Sigma = {a, b}$
|
||||||
|
> Alphabet $Sigma$ contains the characters $a$ and $b$.
|
||||||
|
|
||||||
|
$Sigma = {a, ..., z, A, ..., Z, 0, ..., 9}$
|
||||||
|
> usual alphabet for writing text
|
||||||
@@ -0,0 +1,47 @@
|
|||||||
|
---
|
||||||
|
created: 2026-04-08 10:15
|
||||||
|
course: "[[29593850 - Automationtheory]]"
|
||||||
|
topic: kleen
|
||||||
|
related: "[[29593929 - Alphabets]]"
|
||||||
|
type: lecture
|
||||||
|
status: 🟢
|
||||||
|
tags:
|
||||||
|
- university
|
||||||
|
---
|
||||||
|
## 📌 Summary
|
||||||
|
|
||||||
|
> [!abstract]
|
||||||
|
> Definition and Examples of the Kleene Star, the Kleene Plus and the Lemma Group Structure
|
||||||
|
|
||||||
|
|
||||||
|
---
|
||||||
|
|
||||||
|
## 📝 Content
|
||||||
|
|
||||||
|
### Kleene Star
|
||||||
|
Denoted by $Sigma^*$. The Kleene Star (or _Kleene operator_ or _Kleene Closure_) gives an infinite amount of [[29593852 - Strings|strings]] made up of the characters of the [[29593929 - Alphabets|alphabets]] $Sigma ^ *$.
|
||||||
|
$Sigma^*$ is the set of all string that can be generated by arbitrary concatenation of its characters.
|
||||||
|
> $Sigma^* := union.big_(n>=0) A_n$
|
||||||
|
> where $A_n$ is the set of all string combinations of length $n$
|
||||||
|
|
||||||
|
#### Remarks
|
||||||
|
- The same character can be used multiple times.
|
||||||
|
- The empty string $epsilon$ is also part f $Sigma^*$.
|
||||||
|
|
||||||
|
> [!Example]
|
||||||
|
> $Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}$
|
||||||
|
|
||||||
|
> [!FACT]
|
||||||
|
> - The set $Sigma^*$ is infinite, since we defined $Sigma$ to be non-empty.
|
||||||
|
> - It is _countable_ and has the same cardinality as the set $NN$ of natural numbers
|
||||||
|
|
||||||
|
### Kleene Plus
|
||||||
|
The _Kleene Plus_ of an alphabet $Sigma$ is given by $Sigma^+ = Sigma^* backslash {epsilon}$
|
||||||
|
|
||||||
|
### Lemma group structure
|
||||||
|
The _Lemma group structure_ is induced by the [[29593935 - Kleene Star & Kleene Plus#Kleene Star|Kleene Star]] - it is a monoid, that is a semigroup with a neutral element.
|
||||||
|
|
||||||
|
> [!PROOF]
|
||||||
|
> - Associativity has been shown
|
||||||
|
> - Existence of a neutral element has been shown.
|
||||||
|
> - Closure under $circle.small$: Let $x in Sigma^*$ and $y in Sigma^*$ be two string over the alphabet $Sigma$. Then $x circle.small y = x y in Sigma^*$
|
||||||
54
10 Courses/02 - SoSe 2026/AT/29593940 - Formal Languages.md
Normal file
54
10 Courses/02 - SoSe 2026/AT/29593940 - Formal Languages.md
Normal file
@@ -0,0 +1,54 @@
|
|||||||
|
---
|
||||||
|
created: 2026-04-08 10:20
|
||||||
|
course: "[[29593850 - Automationtheory]]"
|
||||||
|
topic: languages
|
||||||
|
related:
|
||||||
|
type: lecture
|
||||||
|
status: 🟢
|
||||||
|
tags:
|
||||||
|
- university
|
||||||
|
---
|
||||||
|
## 📌 Summary
|
||||||
|
|
||||||
|
> [!abstract]
|
||||||
|
> Definition and example for formal languages
|
||||||
|
|
||||||
|
|
||||||
|
---
|
||||||
|
|
||||||
|
## 📝 Content
|
||||||
|
|
||||||
|
A formal _language_ of the alphabet $Sigma$ is a subset $L$ of $Sigma^*$.
|
||||||
|
|
||||||
|
> [!EXAMPLE]
|
||||||
|
> For alphabet $Sigma = {a, b}$, let $L_1$ be the set of all string starting with $b$, followed by an arbitrary number of $a$'s, and ending with $b$:
|
||||||
|
> $L_1 = {b a^n b | n in NN_0}$
|
||||||
|
> Then $b b in L_1, b a b in L_1$, etc.
|
||||||
|
>
|
||||||
|
>More in [[29593895 - atfl-st2026-l01-formal-languages-full.pdf#page=54]]
|
||||||
|
|
||||||
|
## Language Operations
|
||||||
|
|
||||||
|
### Concatenation of languages
|
||||||
|
For languages $X subset Sigma^*_X$ over alphabet $Sigma_X$ and $Y subset Sigma^*_Y$ over alphabet $Sigma_Y$, their _concatenation_ is
|
||||||
|
> $X circle.small Y = X Y = {x y bar x in X and y in Y}$
|
||||||
|
|
||||||
|
The concatenation of $X$ and $Y$ thus contains all string combinations where the prefix is a string from $X$ and the suffix is a string from $Y$.
|
||||||
|
|
||||||
|
> [!CONVENTION]
|
||||||
|
> Concatenation has a higher precedence than set operations ($union, inter$).
|
||||||
|
|
||||||
|
### Exponentiation of languages
|
||||||
|
The $n^"th"$ power of the language $L subset.eq Sigma^*$ over alphabet $Sigma$ is
|
||||||
|
- $L^0 := { epsilon }$
|
||||||
|
- $L^n := L^(n-1) L$ if $n > 0$
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
## Finite representation of languages
|
||||||
|
**Goal:** Represent a language using _finite_ information
|
||||||
|
### Using set notation
|
||||||
|
$S = {a^n b^m bar n, m >= 0} = {epsilon, a, b, "aa", "ab", ...}$
|
||||||
|
> This is limited in practice.
|
||||||
|
|
||||||
|
### Using regular expressions
|
||||||
24
10 Courses/02 - SoSe 2026/AT/29593952 - Regular Languages.md
Normal file
24
10 Courses/02 - SoSe 2026/AT/29593952 - Regular Languages.md
Normal file
@@ -0,0 +1,24 @@
|
|||||||
|
---
|
||||||
|
created: 2026-04-08 10:32
|
||||||
|
course: "[[29593850 - Automationtheory]]"
|
||||||
|
topic: languages
|
||||||
|
related: "[[29593940 - Formal Languages]]"
|
||||||
|
type: lecture
|
||||||
|
status: 🟢
|
||||||
|
tags:
|
||||||
|
- university
|
||||||
|
---
|
||||||
|
## 📌 Summary
|
||||||
|
|
||||||
|
> [!abstract]
|
||||||
|
>
|
||||||
|
|
||||||
|
---
|
||||||
|
|
||||||
|
## 📝 Content
|
||||||
|
|
||||||
|
A language $L$ that can be described by a [[29593975 - Regular Expressions|Regular Expression]] $r$ (i. e. $L(r) = L$) is called _regular_.
|
||||||
|
|
||||||
|
> [!WARNING]
|
||||||
|
> There a several important languages that **cannot** be described by a regular expression.
|
||||||
|
> <mark style="background: #CACFD9A6;">(proof later)</mark>
|
||||||
@@ -0,0 +1,25 @@
|
|||||||
|
---
|
||||||
|
created: 2026-04-08 10:38
|
||||||
|
course: "[[29593850 - Automationtheory]]"
|
||||||
|
topic: kleene
|
||||||
|
related: "[[29593940 - Formal Languages]]"
|
||||||
|
type: lecture
|
||||||
|
status: 🟢
|
||||||
|
tags:
|
||||||
|
- university
|
||||||
|
---
|
||||||
|
## 📌 Summary
|
||||||
|
|
||||||
|
> [!abstract]
|
||||||
|
>
|
||||||
|
|
||||||
|
---
|
||||||
|
|
||||||
|
## 📝 Content
|
||||||
|
|
||||||
|
The _Kleene star_ $L^*$ of a language $L subset Sigma^*$ is the set of all strings (including the empty string $epsilon$) that can be generated by arbitrary concatenation of strings in the language, that is
|
||||||
|
> $L^* := union.big_(n >= 0) L^n$
|
||||||
|
|
||||||
|
> [!EXAMPLE]
|
||||||
|
> For $L = {01}$ one has $L^* = {epsilon, 01, 0101, 010101, ...}$
|
||||||
|
> $= {(01)^n bar n >= 0}$
|
||||||
@@ -0,0 +1,34 @@
|
|||||||
|
---
|
||||||
|
created: 2026-04-08 10:55
|
||||||
|
course: "[[29593850 - Automationtheory]]"
|
||||||
|
topic: languages
|
||||||
|
related: "[[29593940 - Formal Languages#Finite representation of languages]]"
|
||||||
|
type: lecture
|
||||||
|
status: 🟢
|
||||||
|
tags:
|
||||||
|
- university
|
||||||
|
---
|
||||||
|
## 📌 Summary
|
||||||
|
|
||||||
|
> [!abstract]
|
||||||
|
>
|
||||||
|
|
||||||
|
---
|
||||||
|
|
||||||
|
## 📝 Content
|
||||||
|
|
||||||
|
A _regular expression_ $r$ over an alphabet $Sigma$ is defined recursively:
|
||||||
|
- $emptyset, epsilon$ and each $a in Sigma$ are regular expression, which represent the Languages $L(emptyset) = emptyset, L(epsilon) = {epsilon}$ and $L(a) = {a}$
|
||||||
|
- If $r$ and $s$ are regular expressions then these are also regular expressions:
|
||||||
|
- $(r + s)$ with $L(r + s) = L(r) union L(s)$
|
||||||
|
- $(r s)$ with $L(r s) = L(r)L(s)$
|
||||||
|
- $r^*$ with $L(r^*) = L(r)^*$
|
||||||
|
|
||||||
|
|
||||||
|
> [!EXAMPLE]
|
||||||
|
> The language $L$ over $Sigma = {a, b}$ containing the substring $a b$ is regular, since it can be expressed using the regular expression
|
||||||
|
> $r = (a +b)^* a b (a + b)^*$
|
||||||
|
|
||||||
|
## Equivalence of regular expressions
|
||||||
|
Two regular expressions $r$ and $s$ are _equivalent_ ($r eq.triple s$ or $r hat(eq) s$) if they generate the same language ($L(r) eq L(s)$).
|
||||||
|
|
||||||
7
20 Atlas/AT - Map of Content.md
Normal file
7
20 Atlas/AT - Map of Content.md
Normal file
@@ -0,0 +1,7 @@
|
|||||||
|
|
||||||
|
```dataview
|
||||||
|
TABLE topic AS "Concept", status AS "Review Status"
|
||||||
|
FROM "10 Courses/02 - SoSe 2026/AT"
|
||||||
|
WHERE type = "lecture" OR type = "concept"
|
||||||
|
SORT created ASC
|
||||||
|
```
|
||||||
BIN
30 Library/29593895 - atfl-st2026-l01-formal-languages-full.pdf
Normal file
BIN
30 Library/29593895 - atfl-st2026-l01-formal-languages-full.pdf
Normal file
Binary file not shown.
Binary file not shown.
BIN
30 Library/29595284 - v00.pdf
Normal file
BIN
30 Library/29595284 - v00.pdf
Normal file
Binary file not shown.
BIN
30 Library/29595284 - v01.pdf
Normal file
BIN
30 Library/29595284 - v01.pdf
Normal file
Binary file not shown.
36
30 Library/normalize_name.sh
Executable file
36
30 Library/normalize_name.sh
Executable file
@@ -0,0 +1,36 @@
|
|||||||
|
#!/bin/bash
|
||||||
|
|
||||||
|
# Check if a filename was provided
|
||||||
|
if [ -z "$1" ]; then
|
||||||
|
echo "Usage: $0 <filename>"
|
||||||
|
exit 1
|
||||||
|
fi
|
||||||
|
|
||||||
|
TARGET="$1"
|
||||||
|
|
||||||
|
# Check if the file actually exists
|
||||||
|
if [ ! -f "$TARGET" ]; then
|
||||||
|
echo "Error: File '$TARGET' not found."
|
||||||
|
exit 1
|
||||||
|
fi
|
||||||
|
|
||||||
|
# Get creation time (%W).
|
||||||
|
# Note: Returns 0 or '-' if the filesystem doesn't support birth time.
|
||||||
|
BTIME=$(stat -c %W "$TARGET")
|
||||||
|
|
||||||
|
# Fallback to last modification time (%Y) if birth time is unavailable
|
||||||
|
if [ "$BTIME" -eq 0 ] || [ "$BTIME" == "-" ]; then
|
||||||
|
BTIME=$(stat -c %Y "$TARGET")
|
||||||
|
fi
|
||||||
|
|
||||||
|
# Calculate the minute-based timestamp (equivalent to Math.floor(ms / 60000))
|
||||||
|
# Since stat returns seconds, we divide by 60.
|
||||||
|
FORMATTED_DATE=$(( BTIME / 60 ))
|
||||||
|
|
||||||
|
# Define the new name
|
||||||
|
NEW_NAME="${FORMATTED_DATE} - ${TARGET}"
|
||||||
|
|
||||||
|
# Perform the rename
|
||||||
|
mv "$TARGET" "$NEW_NAME"
|
||||||
|
|
||||||
|
echo "Renamed: '$TARGET' -> '$NEW_NAME'"
|
||||||
1
40 Extras/OOP_Lecture
Submodule
1
40 Extras/OOP_Lecture
Submodule
Submodule 40 Extras/OOP_Lecture added at 55aef6d92e
@@ -1,9 +1,3 @@
|
|||||||
<%*
|
|
||||||
// Move the file to the Inbox immediately upon creation
|
|
||||||
let date = Math.floor(Date.now() / 60000)
|
|
||||||
let newName = `${date} - ${tp.file.title}`
|
|
||||||
await tp.file.move("/00 Inbox/" + newName)
|
|
||||||
-%>
|
|
||||||
---
|
---
|
||||||
created: <% tp.date.now("YYYY-MM-DD HH:mm") %>
|
created: <% tp.date.now("YYYY-MM-DD HH:mm") %>
|
||||||
course:
|
course:
|
||||||
@@ -14,6 +8,12 @@ status: 🔴
|
|||||||
tags:
|
tags:
|
||||||
- university
|
- university
|
||||||
---
|
---
|
||||||
|
<%*
|
||||||
|
// Move the file to the Inbox immediately upon creation
|
||||||
|
let date = Math.floor(Date.now() / 60000);
|
||||||
|
let newName = `${date} - ${tp.file.title}`;
|
||||||
|
await tp.file.move("/00 Inbox/" + newName);
|
||||||
|
-%>
|
||||||
## 📌 Summary
|
## 📌 Summary
|
||||||
|
|
||||||
> [!abstract]
|
> [!abstract]
|
||||||
@@ -22,4 +22,3 @@ tags:
|
|||||||
---
|
---
|
||||||
|
|
||||||
## 📝 Content
|
## 📝 Content
|
||||||
|
|
||||||
|
|||||||
Reference in New Issue
Block a user