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4093cde50a
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| 4093cde50a | |||
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58d114d895 | ||
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a36d8b0c51 | ||
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a578c545e8 | ||
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042300ed1f | ||
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af0d1d060e | ||
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8aa363b825 |
@@ -29,15 +29,15 @@ jobs:
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- name: Compile Analysis1
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- name: Compile Analysis1
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continue-on-error: true
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continue-on-error: true
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run: typst compile --root src src/cheatsheets/Analysis1.typ build/Analysis1.pdf
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run: typst compile --root src src/cheatsheets/Analysis1.typ "build/Analysis 1.pdf"
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- name: Compile Schaltungstheorie
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- name: Compile Schaltungstheorie
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continue-on-error: true
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continue-on-error: true
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run: typst compile --root src src/cheatsheets/Schaltungstheorie.typ build/Schaltungstheorie.pdf
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run: typst compile --root src src/cheatsheets/Schaltungstheorie.typ "build/Schaltungstheorie.pdf"
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- name: Compile LinAlg
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- name: Compile LinAlg
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continue-on-error: true
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continue-on-error: true
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run: typst compile --root src src/cheatsheets/LinearAlgebra.typ build/LinearAlgebra.pdf
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run: typst compile --root src src/cheatsheets/LinearAlgebra.typ "build/Linear Algebra.pdf"
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- name: Create Gitea Release
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- name: Create Gitea Release
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continue-on-error: true
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continue-on-error: true
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@@ -1,6 +1,11 @@
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#import "../lib/common_rewrite.typ" : *
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#import "../lib/common_rewrite.typ" : *
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#import "@preview/mannot:0.3.1"
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#import "@preview/mannot:0.3.1"
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#show math.integral: it => math.limits(math.integral)
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#show math.sum: it => math.limits(math.sum)
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#set text(7pt)
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#set page(
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#set page(
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paper: "a4",
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paper: "a4",
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margin: (
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margin: (
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@@ -40,40 +45,28 @@
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#columns(4, gutter: 2mm)[
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#columns(4, gutter: 2mm)[
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#bgBlock(fill: colorAllgemein)[
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Allgemeins]
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#subHeading(fill: colorAllgemein)[Allgemeins]
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#grid(
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columns: (auto, auto),
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row-gutter: 2mm,
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column-gutter: 3mm,
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[Dreiecksungleichung], [
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$abs(x + y) <= abs(x) + abs(y)$ \
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$abs(abs(x) - abs(y)) <= abs(x - y)$
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],
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[Cauchy-Schwarz-Ungleichung], [
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$abs(x dot y) <= abs(abs(x) dot abs(y))$
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],
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[Geometrische Summenformel], [
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#MathAlignLeft($ limits(sum)_(k=1)^(n) k = (n(n+1))/2 $)
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],
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[Bernoulli-Ungleichung ], [
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$(1 + a)^n x in RR >= 1 + n a$
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],
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[Binomialkoeffizient], [
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$binom(n, k) = (n!)/(k!(n-k)!)$
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],
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[Binomische Formel], [
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#MathAlignLeft($ (a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $)
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],
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[Fakultäten], [$ 0! = 1! = 1 $],
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[Gausklammer], [
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*Dreiecksungleichung* \
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$abs(x + y) <= abs(x) + abs(y)$ \
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$abs(abs(x) - abs(y)) <= abs(x - y)$ \
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*Cauchy-Schwarz-Ungleichung*\
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$abs(x dot y) <= abs(abs(x) dot abs(y))$ \
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*Geometrische Summenformel*\
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$sum_(k=1)^(n) k = (n(n+1))/2$ \
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*Bernoulli-Ungleichung* \
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$(1 + a)^n x in RR >= 1 + n a$ \
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*Binomialkoeffizient* $binom(n, k) = (n!)/(k!(n-k)!)$
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*Binomische Formel*\
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$(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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*Fakultäten* $0! = 1! = 1$ \
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*Gaußklammer*: \
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$floor(x) = text("floor")(x)$ \
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$floor(x) = text("floor")(x)$ \
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$ceil(x) = text("ceil")(x)$
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$ceil(x) = text("ceil")(x)$ \
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],
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*Bekannte Werte* \
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[Bekannte Werte], [
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$e approx 2.71828$ ($2 < e < 3$) \
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$e approx 2.71828$ ($2 < e < 3$) \
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$pi approx 3.14159$ ($3 < pi < 4$)
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$pi approx 3.14159$ ($3 < pi < 4$)
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]
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)
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]
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]
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#bgBlock(fill: colorAllgemein)[
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#bgBlock(fill: colorAllgemein)[
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@@ -84,8 +77,20 @@
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#grid(
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#grid(
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columns: (1fr, 1fr),
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columns: (1fr, 1fr),
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row-gutter: 2mm,
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[$ sin(x) = (e^(i x) - e^(-i x))/(2i) $],
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[$ sin(x) = (e^(i x) - e^(-i x))/(2i) $],
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[$ cos(x) = (e^(i x) + e^(-i x))/(2) $]
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[$ cos(x) = (e^(i x) + e^(-i x))/(2) $],
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grid.cell(
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colspan: 2,
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align: center,
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$ tan(x) = 1/2i ln((1+i x)/(1-i x)) $
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),
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grid.cell(
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colspan: 2,
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align: center,
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$ arctan(x) = 1/2i ln((1+i x)/(1-i x)) $
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)
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)
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)
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#subHeading(fill: colorAllgemein)[Trigonmetrie]
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#subHeading(fill: colorAllgemein)[Trigonmetrie]
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*Additionstheorem* \
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*Additionstheorem* \
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@@ -93,6 +98,10 @@
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$cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$ \
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$cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$ \
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$tan(x) + tan(y) = (tan(a) + tan(b))/(1 - tan(a) tan(b))$ \
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$tan(x) + tan(y) = (tan(a) + tan(b))/(1 - tan(a) tan(b))$ \
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$arctan(x) + arctan(y) = arctan((x+y)/(1 - x y))$ \
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$arctan(x) + arctan(y) = arctan((x+y)/(1 - x y))$ \
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$arctan(1/x) + arctan(x) = cases(
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x > 0 : pi/2,
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x < 0 : -pi/2
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)$
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*Doppelwinkel Formel* \
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*Doppelwinkel Formel* \
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$cos(2x) = cos^2(x) - sin^2(x)$ \
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$cos(2x) = cos^2(x) - sin^2(x)$ \
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@@ -176,8 +185,10 @@
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$forall m,n >= n_epsilon : abs(a_n - a_m) < epsilon$ \
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$forall m,n >= n_epsilon : abs(a_n - a_m) < epsilon$ \
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Cauchyfolge $=>$ Konvergenz)
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Cauchyfolge $=>$ Konvergenz)
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- $a_n$ unbeschränkt $=>$ divergenz
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- $a_n$ unbeschränkt $=>$ divergenz
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]
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*Konvergent Grenzwert finden*
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[Folgen Konvergenz Strategien]
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- Von Bekannten Ausdrücken aufbauen
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- Von Bekannten Ausdrücken aufbauen
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- Fixpunk Gleichung: $a = f(a)$ \
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- Fixpunk Gleichung: $a = f(a)$ \
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für rekusive $a_(n+1) = f(a_n)$ (Zu erst machen!)
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für rekusive $a_(n+1) = f(a_n)$ (Zu erst machen!)
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@@ -189,6 +200,23 @@
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$b_n -> +infinity$: $c_n <= b_n $, wenn $a_n -> +infinity$
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$b_n -> +infinity$: $c_n <= b_n $, wenn $a_n -> +infinity$
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- Zwerlegen in Konvergente Teil folgen \
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- Zwerlegen in Konvergente Teil folgen \
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(Vorallem bei $(-1)^n dot a_n$)
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(Vorallem bei $(-1)^n dot a_n$)
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*L'Hospital*
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$x in (a,b): limits(lim)_(x->b)f(x)/g(x)$
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(Konvergenz gegen $b$, beliebiges $a$)
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Bendingungen:
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1. $limits(lim)_(x->b)f(x) = limits(lim)_(x->b)g(x)= 0 "oder" infinity$
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2. $g'(x) != 0, x in (a,b)$
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3. $limits(lim)_(x->b) (f'(x))/(g'(x))$ existiert
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$=> limits(lim)_(x->b) (f'(x))/(g'(x)) = limits(lim)_(x->b) (f(x))/(g(x))$
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Kann auch Reksuive angewendet werden!
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]
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]
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#bgBlock(fill: colorFolgen)[
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#bgBlock(fill: colorFolgen)[
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@@ -212,15 +240,14 @@
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#bgBlock(fill: colorFolgen)[
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[Bekannte Folgen]
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#subHeading(fill: colorFolgen)[Bekannte Folgen]
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#grid(
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#grid(
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columns: (auto, auto, auto),
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columns: (auto, auto),
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column-gutter: 4mm,
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column-gutter: 4mm,
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row-gutter: 2mm,
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row-gutter: 2mm,
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align: bottom,
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align: bottom,
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MathAlignLeft($ lim_(n->infinity) 1/n = 0 $),
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MathAlignLeft($ lim_(n->infinity) 1/n = 0 $),
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[],
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MathAlignLeft($ lim_(n->infinity) k = k, k in RR $),
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grid.cell(colspan: 2, MathAlignLeft($ exp(x) = e^x = lim_(n->infinity) (1 + x/n)^n $)),
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MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $),
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MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $),
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MathAlignLeft($ lim_(n->infinity) k = k, k in RR $),
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MathAlignLeft($ e^x = lim_(n->infinity) (1 + x/n)^n $),
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grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) q^n = cases(
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grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) q^n = cases(
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0 &abs(q),
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0 &abs(q),
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1 &q = 1,
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1 &q = 1,
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@@ -246,8 +273,6 @@
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- *Absolute Konvergenz* \
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- *Absolute Konvergenz* \
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$limits(sum)_(n=1)^infinity abs(a_n) = a => limits(sum)_(n=1)^infinity a_n$ konvergent
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$limits(sum)_(n=1)^infinity abs(a_n) = a => limits(sum)_(n=1)^infinity a_n$ konvergent
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- *Partialsummen* \
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- *Partialsummen* \
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ALLE Partialsummen von $limits(sum)_(k=1)^infinity abs(a)$ beschränkt\
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ALLE Partialsummen von $limits(sum)_(k=1)^infinity abs(a)$ beschränkt\
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$=>$ _Absolute Konvergent_
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$=>$ _Absolute Konvergent_
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@@ -278,19 +303,6 @@
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2. $rho = lim_(n -> infinity) root(n, abs(a_(n+1))) $ \
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2. $rho = lim_(n -> infinity) root(n, abs(a_(n+1))) $ \
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divergent: $rho > 1$, keine Aussage $rho = 1$, konvergent $rho < 1$
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divergent: $rho > 1$, keine Aussage $rho = 1$, konvergent $rho < 1$
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- *Geometrische Reihe*
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$limits(sum)_(n=0)^infinity q^n$
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- konvergent $abs(q) < 1$, divergent $abs(q) >= 1$
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- Grenzwert: (Muss $n=0$) $=1/(1-q)$
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- *Harmonische Reihe* $limits(sum)_(n=0)^infinity 1/n = +infinity$
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- *Reihendarstellungen*
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1. $e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$
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2. $ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$
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3. $sin(x) = limits(sum)_(n=0)^infinity $
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4. $cos(x) = limits(sum)_(n=0)^infinity $
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]
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]
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#bgBlock(fill: colorReihen)[
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#bgBlock(fill: colorReihen)[
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@@ -305,9 +317,16 @@
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*Harmonische Reihe:* $sum_(n=0)^infinity 1/n = +infinity$
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*Harmonische Reihe:* $sum_(n=0)^infinity 1/n = +infinity$
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*Andere*
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*Reihendarstellungen*
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- $e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$
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#grid(
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- $ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$
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columns: (1fr, 1fr),
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gutter: 3mm,
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row-gutter: 2mm,
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$e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$,
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$ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$,
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$sin(x) = limits(sum)_(n=0)^infinity $,
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$cos(x) = limits(sum)_(n=0)^infinity $
|
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|
)
|
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]
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]
|
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#colbreak()
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#colbreak()
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@@ -421,7 +440,7 @@
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{ color.hsl(180deg, 81.82%, 95.69%) },
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{ color.hsl(180deg, 81.82%, 95.69%) },
|
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[$1/(q + x) x^(q+1)$], [$x^q$], [$q x^(q-1)$],
|
[$1/(q + x) x^(q+1)$], [$x^q$], [$q x^(q-1)$],
|
||||||
[$ln abs(x)$], [$1/x$], [$-1/x^2$],
|
[$ln abs(x)$], [$1/x$], [$-1/x^2$],
|
||||||
[$x ln(a x) - x$], [$ln(a x)$], [$1 / x$],
|
[$x ln(a x) - x$], [$ln(a x)$], [$a / x$],
|
||||||
[$2/3 sqrt(a x^3)$], [$sqrt(a x)$], [$a/(2 sqrt(a x))$],
|
[$2/3 sqrt(a x^3)$], [$sqrt(a x)$], [$a/(2 sqrt(a x))$],
|
||||||
[$e^x$], [$e^x$], [$e^x$],
|
[$e^x$], [$e^x$], [$e^x$],
|
||||||
[$a^x/ln(a)$], [$a^x$], [$a^x ln(a)$],
|
[$a^x/ln(a)$], [$a^x$], [$a^x ln(a)$],
|
||||||
@@ -470,6 +489,50 @@
|
|||||||
3. $x$-kürzen sich weg
|
3. $x$-kürzen sich weg
|
||||||
])
|
])
|
||||||
|
|
||||||
|
#bgBlock(fill: colorIntegral, [
|
||||||
|
#subHeading(fill: colorIntegral, [Integral])
|
||||||
|
|
||||||
|
*Riemann Integral*\
|
||||||
|
$limits(sum)_(x=a)^(b) f(i)(x_())$
|
||||||
|
|
||||||
|
Summen: $integral f(x) + g(x) d x = integral f(x) d x + integral g(x)$
|
||||||
|
|
||||||
|
Vorfaktoren: $integral lambda f(x) d x = lambda f(x) d x$
|
||||||
|
|
||||||
|
*Integral Type*\
|
||||||
|
- Eigentliches Int.: $integral_a^b f(x) d x$
|
||||||
|
- Uneigentliches Int.: \
|
||||||
|
$limits(lim)_(epsilon -> 0) integral_a^(b + epsilon) f(x) d x$ \
|
||||||
|
$limits(lim)_(epsilon -> plus.minus infinity) integral_a^(epsilon) f(x) d x$
|
||||||
|
- Unbestimmtes Int.: $integral f(x) d x = F(x) + c, c in RR$- Uneigentliches Int.:
|
||||||
|
|
||||||
|
|
||||||
|
*Cauchy-Hauptwert*
|
||||||
|
|
||||||
|
$integral_(-infinity)^(+infinity) f(x)$ \
|
||||||
|
NUR konvergent wenn: \
|
||||||
|
$limits(lim)_(R -> -infinity) integral_(R)^(a) f(x) d x$ und $limits(lim)_(R -> infinity) integral_(a)^(R) f(x) d x$ konvergent für $a in RR$
|
||||||
|
|
||||||
|
$integral_(-infinity)^(infinity) f(x) d x$ existiert \
|
||||||
|
$=> lim_(M -> infinity) integral_(-M)^(M) f(x) d x = integral_(-infinity)^(infinity) f(x) d x$
|
||||||
|
|
||||||
|
*Partial Integration*
|
||||||
|
|
||||||
|
$integral u(x) dot v'(x) d x = u(x)v(x) - integral u'(x) dot v(x)$
|
||||||
|
|
||||||
|
*Subsitution*
|
||||||
|
|
||||||
|
$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot 1/(g'(x)) d x$
|
||||||
|
|
||||||
|
1. Ersetzung: $ d x := d t dot g'(x)$ und $t := g(x)$
|
||||||
|
2. Grenzen: $t_0 = g(x_0)$, $t_1 = g(x_1)$
|
||||||
|
3. $x$-kürzen sich weg
|
||||||
|
|
||||||
|
*Absolute "Konvergenz"* \
|
||||||
|
Wenn $g(x)$ konvergent,
|
||||||
|
$abs(f(x)) <= g(x) => $ $f(x)$ konvergent
|
||||||
|
])
|
||||||
|
|
||||||
]
|
]
|
||||||
|
|
||||||
#bgBlock(fill: colorAllgemein, [
|
#bgBlock(fill: colorAllgemein, [
|
||||||
@@ -524,23 +587,4 @@ Konvergenz Radius $R = [0, infinity)$$$
|
|||||||
)$
|
)$
|
||||||
|
|
||||||
$ R = limsup_(n -> infinity) $
|
$ R = limsup_(n -> infinity) $
|
||||||
#bgBlock(fill: colorIntegral, [
|
|
||||||
#subHeading(fill: colorIntegral, [Integral])
|
|
||||||
|
|
||||||
Summen: $integral f(x) + g(x) d x = integral f(x) d x + integral g(x)$
|
|
||||||
|
|
||||||
Vorfaktoren: $integral lambda f(x) d x = lambda f(x) d x$
|
|
||||||
|
|
||||||
*Partial Integration*
|
|
||||||
|
|
||||||
$integral u(x) dot v'(x) d x = u(x)v(x) - integral u'(x) dot v(x)$
|
|
||||||
|
|
||||||
*Subsitution*
|
|
||||||
|
|
||||||
$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot g'(x) d x$
|
|
||||||
|
|
||||||
1. Ersetzung: $ d x := d t dot 1/(g'(x))$ und $t := g(x)$
|
|
||||||
2. Grenzen: $t_0 = g(x_0)$, $t_1 = g(x_1)$
|
|
||||||
3. $x$-kürzen sich weg
|
|
||||||
])
|
|
||||||
|
|
||||||
|
|||||||
206
src/cheatsheets/Digitaltechnik.typ
Normal file
206
src/cheatsheets/Digitaltechnik.typ
Normal file
@@ -0,0 +1,206 @@
|
|||||||
|
#import "../lib/common_rewrite.typ" : *
|
||||||
|
#import "@preview/mannot:0.3.1"
|
||||||
|
|
||||||
|
#show math.integral: it => math.limits(math.integral)
|
||||||
|
#show math.sum: it => math.limits(math.sum)
|
||||||
|
|
||||||
|
#set page(
|
||||||
|
paper: "a4",
|
||||||
|
margin: (
|
||||||
|
bottom: 10mm,
|
||||||
|
top: 5mm,
|
||||||
|
left: 5mm,
|
||||||
|
right: 5mm
|
||||||
|
),
|
||||||
|
flipped:true,
|
||||||
|
footer: context [
|
||||||
|
#grid(
|
||||||
|
align: center,
|
||||||
|
columns: (1fr, 1fr, 1fr),
|
||||||
|
[#align(left, datetime.today().display("[day].[month].[year]"))],
|
||||||
|
[#align(center, counter(page).display("- 1 -"))],
|
||||||
|
[#align(right, image("../images/cc0.png", height: 5mm,))]
|
||||||
|
)
|
||||||
|
],
|
||||||
|
)
|
||||||
|
|
||||||
|
#place(top+center, scope: "parent", float: true, heading(
|
||||||
|
[Digitaltechnik]
|
||||||
|
))
|
||||||
|
|
||||||
|
#let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm))
|
||||||
|
#let MathAlignLeft(e) = {
|
||||||
|
align(left, block(e))
|
||||||
|
}
|
||||||
|
|
||||||
|
#let colorBoolscheLogic = color.hsl(105.13deg, 92.13%, 75.1%)
|
||||||
|
#let colorOptimierung = color.hsl(202.05deg, 92.13%, 75.1%)
|
||||||
|
#let colorRealsierung = color.hsl(280deg, 92.13%, 75.1%)
|
||||||
|
#let colorState = color.hsl(356.92deg, 92.13%, 75.1%)
|
||||||
|
//#let colorIntegral = color.hsl(34.87deg, 92.13%, 75.1%)
|
||||||
|
|
||||||
|
#let LNot(x) = math.op($overline(#x)$)
|
||||||
|
|
||||||
|
#columns(4, gutter: 2mm)[
|
||||||
|
#bgBlock(fill: colorBoolscheLogic)[
|
||||||
|
#subHeading(fill: colorBoolscheLogic)[Allgemein]
|
||||||
|
*Moorsches Gesetz:* 2x der Anzahl der Transistoren pro Fläche (in 2 Jahren)
|
||||||
|
|
||||||
|
Flächenskalierung eines Transistors: $1/sqrt(2)$
|
||||||
|
|
||||||
|
*Kombinatorisch:* Kein Gedächtnis
|
||||||
|
|
||||||
|
*(Synchrone) sequenentielle:* Mit Gedächtnis
|
||||||
|
|
||||||
|
*Fan-In:* Anzahl der Inputs eines Gatters
|
||||||
|
|
||||||
|
*Fan-Out:* Anzahl der Output Verbindungen eines Gatters
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorBoolscheLogic)[
|
||||||
|
#subHeading(fill: colorBoolscheLogic)[Boolsche Algebra]
|
||||||
|
|
||||||
|
*Dualität*
|
||||||
|
$LNot(0) = 1$, $LNot(1) = 0$
|
||||||
|
|
||||||
|
*Äquivalenz* $LNot((LNot(A)))=A$\
|
||||||
|
$A dot A = A$, $A + 0 = A$ \
|
||||||
|
|
||||||
|
*Konstanz*
|
||||||
|
$A dot 1 = A$ $A + 1 = 1$
|
||||||
|
|
||||||
|
*Komplementärgesetz* \
|
||||||
|
$A dot LNot(A) = 0$, $A + LNot(A) = 1$
|
||||||
|
|
||||||
|
*Kommutativgesetz* \
|
||||||
|
$A dot B = B dot A$, $A + B = B + A$
|
||||||
|
|
||||||
|
*Assoziativgesetz*\
|
||||||
|
$A dot (B dot C) = (A dot B) dot C$\
|
||||||
|
$A + (B + C) = (A + B) + C$
|
||||||
|
|
||||||
|
*Distributivgesetz*\
|
||||||
|
$A dot (B + C) = A dot B + A dot C$ \
|
||||||
|
$A + (B dot C) = (A + B) dot (A + C)$
|
||||||
|
|
||||||
|
*De Morgan*\
|
||||||
|
$LNot((A + B)) = LNot(A) dot LNot(B)$\
|
||||||
|
$LNot((A dot B)) = LNot(A) + LNot(B)$
|
||||||
|
|
||||||
|
*Absorptionsgesetz*\
|
||||||
|
$A + (A dot B) = A$\
|
||||||
|
$A dot (A + B) = A$
|
||||||
|
|
||||||
|
*Resolutionsgesetz (allgemein)*\
|
||||||
|
$X dot A + LNot(X) + B = X dot A + LNot(X) dot B + bold(A dot B)$
|
||||||
|
|
||||||
|
*Resolutionsgesetz (speziell)*\
|
||||||
|
$X dot A + LNot(X) dot A = A$\
|
||||||
|
$(X + A) dot (LNot(X) + A) = A$
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorBoolscheLogic)[
|
||||||
|
#subHeading(fill: colorBoolscheLogic)[Boolsche Funktionen]
|
||||||
|
|
||||||
|
$f: {0,1}^n -> {0,1}$
|
||||||
|
|
||||||
|
Variablenmenge: ${x_0, x_1, ..., x_n}$\
|
||||||
|
Literalmenge: ${x_0, ..., x_n, LNot(x_0), ... LNot(x_n)}$ \
|
||||||
|
Einsmenge: $F = {underline(v) in {0,1}^n | f(underline(v)) = 1}$
|
||||||
|
Nullmenge: $overline(F) = {underline(v) in {0,1}^n | f(underline(v)) = 0}$
|
||||||
|
Don't-Care-Set: ${underline(v) in {0,1}^n | f(underline(v)) = *}$
|
||||||
|
|
||||||
|
Funktionsbündel: $underline(y) = underline(f)(underline(x))$ \
|
||||||
|
$underline(f): {0,1}^n -> {0,1}^m$
|
||||||
|
|
||||||
|
*Kofaktoren* aka Bit $n$ fixen\
|
||||||
|
$x_i : f_x_i = f(x_1, ..., 1, ..., x_n)$\
|
||||||
|
$overline(x)_i : f_overline(x)_i = f(x_1, ..., 0, ..., x_n)$
|
||||||
|
|
||||||
|
*Substitutionsregel*
|
||||||
|
|
||||||
|
$x_i dot f = x_i dot f_x_i$
|
||||||
|
|
||||||
|
$overline(x)_i dot f = overline(x)_i dot f_overline(x)_i$
|
||||||
|
|
||||||
|
$x_i + f = x_i + f_overline(x)_i$
|
||||||
|
|
||||||
|
$overline(x)_i + f = overline(x)_i + f_x_i$
|
||||||
|
|
||||||
|
*Boolsche Expansion*\
|
||||||
|
$f(underline(x)) = x_i dot f_x_i + overline(x)_i dot f_overline(x)_i$
|
||||||
|
|
||||||
|
$f(underline(x)) = (x_i + f_overline(x)_i) dot (overline(x)_i + f_x_i)$
|
||||||
|
|
||||||
|
$overline(f(underline(x))) = overline(x)_i dot overline(f_overline(x)_i) + x_i dot overline(f_x_i)$
|
||||||
|
|
||||||
|
$overline(f(underline(x))) = (overline(x)_i + overline(f_x_i)) dot (x_i + overline(f_overline(x)_i)) $
|
||||||
|
|
||||||
|
*Eigentschaften:*
|
||||||
|
|
||||||
|
tautologisch: $f(underline(x)) = 1, forall underline(x) in {0,1}^n$\
|
||||||
|
kontradiktorisch: $f(underline(x)) = 0, forall underline(x) in {0,1}^n$\
|
||||||
|
unabhängig von $x_i <=> f_x_i = f_overline(x)_i$\
|
||||||
|
abhängig von $x_i <=> f_x_i != f_overline(x)_i$\
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorOptimierung)[
|
||||||
|
#subHeading(fill: colorOptimierung)[Hauptsatz der Schaltalgebra]
|
||||||
|
Jede $f(x_0, ...,x_n)$ kann als...
|
||||||
|
- *Minterme $m$:* $ = LNot(x)_0 dot x_1 dot ...$\
|
||||||
|
VerODERungen von VerUNDungen\
|
||||||
|
$f(underline(x)) = m_0 + m_1 + ... + m_n$
|
||||||
|
|
||||||
|
- *Maxterme $M$:* $ = LNot(x)_0 + x_1 ü ...$\
|
||||||
|
VerUNDungen von VerODERungen\
|
||||||
|
$f(underline(x)) = m_0 dot m_1 dot ... dot m_n$
|
||||||
|
|
||||||
|
... dargestellt werden
|
||||||
|
|
||||||
|
*DNF:* Disjunktive Normalform, *Minterme*
|
||||||
|
- Term $tilde.equiv$ $1$-Zeile
|
||||||
|
- $LNot(x)_0 dot x_1 + x_0 dot x_1 +...$\
|
||||||
|
- $1 tilde.equiv x_0$, $0 tilde.equiv overline(x_0)$
|
||||||
|
|
||||||
|
*KNF:* Konjunktive Normalform, *Maxterme*
|
||||||
|
- Term $tilde.equiv$ $0$-Zeile
|
||||||
|
- $(LNot(x)_0 + LNot(x)_1) dot (x_0 + x_1) dot...$\
|
||||||
|
- $1 tilde.equiv overline(x_0)$, $0 tilde.equiv x_0$
|
||||||
|
|
||||||
|
Kanonische: In jedem Term müssen alle enthalten sein.
|
||||||
|
|
||||||
|
*KDNF:* Kanonische DNF\
|
||||||
|
*KKNF:* Kanonische KNF
|
||||||
|
|
||||||
|
$f(underline(x)) -->$ *KKNF* / *KDNF* mit Boolsche Expansion
|
||||||
|
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorOptimierung)[
|
||||||
|
#subHeading(fill: colorOptimierung)[Quine McCluskey]
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorRealsierung)[
|
||||||
|
#subHeading(fill: colorRealsierung)[NMOS/PMOS]
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorRealsierung)[
|
||||||
|
#subHeading(fill: colorRealsierung)[CMOS]
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorState)[
|
||||||
|
#subHeading(fill: colorState)[Timing]
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorState)[
|
||||||
|
#subHeading(fill: colorState)[Latches und Register]
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorState)[
|
||||||
|
#subHeading(fill: colorState)[Pipeline/Parallele Verarbeitungseinheiten]
|
||||||
|
]
|
||||||
|
|
||||||
|
#bgBlock(fill: colorState)[
|
||||||
|
#subHeading(fill: colorState)[Zustandsautomaten]
|
||||||
|
]
|
||||||
|
]
|
||||||
@@ -2,7 +2,9 @@
|
|||||||
#import "@preview/mannot:0.3.1"
|
#import "@preview/mannot:0.3.1"
|
||||||
#import "@preview/zap:0.5.0"
|
#import "@preview/zap:0.5.0"
|
||||||
|
|
||||||
|
#set math.mat(delim: "[")
|
||||||
#show math.equation.where(block: true): it => math.inline(it)
|
#show math.equation.where(block: true): it => math.inline(it)
|
||||||
|
#set math.mat(delim: "[")
|
||||||
|
|
||||||
#set page(
|
#set page(
|
||||||
paper: "a4",
|
paper: "a4",
|
||||||
@@ -65,7 +67,7 @@
|
|||||||
|
|
||||||
Knotenzidenzmatrix $bold(A)$
|
Knotenzidenzmatrix $bold(A)$
|
||||||
|
|
||||||
$bold(A) : bold(i_k) -> text("Knotenstrombianz") = 0$ \
|
$bold(A) : bold(i_k) -> text("Knotenstrombilanz") = 0$ \
|
||||||
$bold(A^T) : bold(u_b)-> bold(u_k)$
|
$bold(A^T) : bold(u_b)-> bold(u_k)$
|
||||||
$
|
$
|
||||||
bold(A) = quad mannot.mark(mat(
|
bold(A) = quad mannot.mark(mat(
|
||||||
@@ -80,6 +82,9 @@
|
|||||||
|
|
||||||
a in {-1, 0, 1}
|
a in {-1, 0, 1}
|
||||||
$
|
$
|
||||||
|
|
||||||
|
$-1$: In Knoten rein \
|
||||||
|
$1$: Aus Knoten raus \
|
||||||
|
|
||||||
#line(length: 100%, stroke: (thickness: 0.2mm))
|
#line(length: 100%, stroke: (thickness: 0.2mm))
|
||||||
|
|
||||||
@@ -103,6 +108,9 @@
|
|||||||
b in {-1, 0, 1}
|
b in {-1, 0, 1}
|
||||||
$
|
$
|
||||||
|
|
||||||
|
$-1$: Gegen Maschenrichtung
|
||||||
|
$1$: In Maschenrichtung
|
||||||
|
|
||||||
#line(length: 100%, stroke: (thickness: 0.2mm))
|
#line(length: 100%, stroke: (thickness: 0.2mm))
|
||||||
|
|
||||||
*KCL und KVL* \
|
*KCL und KVL* \
|
||||||
@@ -120,6 +128,10 @@
|
|||||||
|
|
||||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||||
#subHeading(fill: colorAnalyseVerfahren)[Baumkonzept]
|
#subHeading(fill: colorAnalyseVerfahren)[Baumkonzept]
|
||||||
|
KCLs: $n-1$\
|
||||||
|
KVLs: $b-(n-1)$
|
||||||
|
|
||||||
|
Baum einzeichnen (Keine Schleifen!)
|
||||||
]
|
]
|
||||||
|
|
||||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||||
@@ -129,19 +141,16 @@
|
|||||||
#bgBlock(fill: colorAnalyseVerfahren)[
|
#bgBlock(fill: colorAnalyseVerfahren)[
|
||||||
#subHeading(fill: colorAnalyseVerfahren)[Reduzierte Knotenpotenzial-Analyse]
|
#subHeading(fill: colorAnalyseVerfahren)[Reduzierte Knotenpotenzial-Analyse]
|
||||||
]
|
]
|
||||||
|
|
||||||
|
|
||||||
]
|
]
|
||||||
|
|
||||||
#pagebreak()
|
#pagebreak()
|
||||||
#place(bottom+left, scope: "parent", float: true)[
|
#place(bottom+left, scope: "parent", float: true)[
|
||||||
#bgBlock(fill: colorZweiTore)[
|
#bgBlock(fill: colorZweiTore)[
|
||||||
#subHeading(fill: colorZweiTore)[Umrechnung Zweitormatrizen]
|
#subHeading(fill: colorZweiTore)[Umrechnung Zweitormatrizen]
|
||||||
#show table.cell: it => pad(),
|
|
||||||
|
|
||||||
#table(
|
#table(
|
||||||
columns: (auto, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr),
|
columns: (auto, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr),
|
||||||
align: center,
|
align: center,
|
||||||
|
inset: (bottom: 4mm, top: 4mm),
|
||||||
gutter: 0.1mm,
|
gutter: 0.1mm,
|
||||||
[In $->$], $bold(R)$, $bold(G)$, $bold(H)$, $bold(H')$, $bold(A)$, $bold(A')$,
|
[In $->$], $bold(R)$, $bold(G)$, $bold(H)$, $bold(H')$, $bold(A)$, $bold(A')$,
|
||||||
|
|
||||||
|
|||||||
Reference in New Issue
Block a user