Shared Hard Cards
Cards that are collectively hardest across all users
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940 cards
nid:1772046331702
IO r2
A&W
[Image Occlusion region 2]
11
lapses
3/4
users
192%
ease
nid:1772046331702
Cloze c2
Q: {{c3::image-occlusion:rect:left=.1591:top=.8923:width=.7185:height=.0742}}{{c2::image-occlusion:rect:left=.3252:top=.7428:width=.5272:height=.0923}}{{c1::image-occlusion:rect:left=.0549:top=.1782:width=.9041:height=.1203}}{{c4::image-occlusion:rect:left=.1645:top=.4824:width=.1234:height
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772046331702 |
6 | 130% | 22d | 27 |
| niklas | cid:1772209100380 |
4 | 215% | 12d | 14 |
| tomas | cid:1772090857647 |
1 | 230% | 1d | 6 |
nid:1766314094848
c1
DiskMat
A cyclic group of order \(n\) {{c1::is isomorphic to \(\lan...
9
lapses
3/4
users
210%
ease
nid:1766314094848
Cloze c1
Q: A cyclic group of order \(n\) {{c1::is isomorphic to \(\langle \mathbb{Z}_n,\oplus)\), and hence commutative.::has which useful property?}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094921 |
4 | 170% | 3d | 19 |
| lorenz | cid:1764867990841 |
4 | 170% | 95d | 19 |
| niklas | cid:1762856074705 |
1 | 290% | 63d | 11 |
nid:1772548090724
c1
A&W
ein perfektes Matching
8
lapses
2/4
users
162%
ease
nid:1772548090724
Cloze c1
Cloze answer: ein perfektes Matching
Q: Für alle \( k \) gilt: jeder \( k \)-reguläre bipartite Graph enthält {{c1::ein perfektes Matching}}.Theorem-name included
A: (Frobenius, 1917)Es gilt sogar: Der Graph ist die Vereinigung von perfekten Matchings.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772548090724 |
5 | 150% | 13d | 25 |
| niklas | cid:1772569386218 |
3 | 175% | 2d | 15 |
nid:1771973928570
c1
Analysis
e^a \cdot (\cos(b) + i \sin(b))
7
lapses
3/4
users
218%
ease
nid:1771973928570
Cloze c1
Cloze answer: e^a \cdot (\cos(b) + i \sin(b))
Q: Addition von komplexen Zahlen in Polarform: \(e^z = e^{a + ib} = {{c1:: e^a \cdot (\cos(b) + i \sin(b)) }}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928571 |
4 | 170% | 17d | 22 |
| niklas | cid:1771970299379 |
2 | 255% | 112d | 10 |
| tomas | cid:1772003104447 |
1 | 230% | 1d | 4 |
nid:1766314094781
c2
DiskMat
least upper bound
7
lapses
2/4
users
195%
ease
nid:1766314094781
Cloze c2
Cloze answer: least upper bound
Q: Consider the poset \((A;\preceq)\). If \(\{a,b\}\) has a {{c2::least upper bound}}, then it is called the {{c1::join of \(a\) and \(b\) (also denoted \(a \lor b\)).}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094832 |
4 | 170% | 9d | 14 |
| niklas | cid:1762856073631 |
3 | 220% | 16d | 11 |
nid:1772327995541
Analysis
Wie lautet die Bernoulli Ungleichung?
7
lapses
2/4
users
202%
ease
nid:1766314094913
c1
DiskMat
a^0
6
lapses
3/4
users
215%
ease
nid:1766314095056
DiskMat
What is the number of subgroups of \(\mathbb{Z}_n\)?
6
lapses
3/4
users
220%
ease
nid:1766314095056
Q: What is the number of subgroups of \(\mathbb{Z}_n\)?
A: The number of divisors of \(n\) (as the order of each subgroup divides the group order (which is n here) by Lagrange). If \(n\) is written \(n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}\) then it is \(\prod_{i=1}^k (e_i+1)\).Note: This only holds because \(\mathbb{Z}_n\) is cyclic and therefore the subgroups are unique.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095224 |
3 | 190% | 4d | 13 |
| lorenz | cid:1766229407421 |
2 | 210% | 90d | 12 |
| niklas | cid:1766000828774 |
1 | 260% | 15d | 10 |
nid:1771973928505
Analysis
Dreiecksungleichung (Subtraktion)
6
lapses
3/4
users
205%
ease
nid:1766314094777
c1
DiskMat
lower (upper) bound of \(S\)
6
lapses
2/4
users
198%
ease
nid:1766314094777
Cloze c1
Cloze answer: lower (upper) bound of \(S\)
Q: Consider the poset \((A; \preceq)\) and \( S \subseteq A\).\(a \in A\) is a {{c1::lower (upper) bound of \(S\)}} if {{c2::\(a \preceq b\) (\(a \succeq b) \) for all \(b \in S\)}}
A: Note that a is not necessarily in the subset S (difference to the least and greatest elements).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073624 |
5 | 180% | 11d | 20 |
| jonas | cid:1766314094825 |
1 | 215% | 24d | 13 |
nid:1766314094806
c2
DiskMat
commutative ring without zerodivisors (\( \forall a \ \foral...
6
lapses
2/4
users
190%
ease
nid:1766314094806
Cloze c2
Cloze answer: commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).
Q: An {{c1::integral domain}} is a {{c2::commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).}}
A: A domain of elements behaving like integers.Examples: \(\mathbb{Z}, \mathbb{R}\)Counterexample: \(\mathbb{Z}_m, m\) not prime
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073681 |
4 | 170% | 34d | 19 |
| jonas | cid:1766314094870 |
2 | 210% | 16d | 12 |
nid:1766314094941
DiskMat
State Corollary 5.11 about groups of prime order (what prope...
6
lapses
2/4
users
182%
ease
nid:1766314094941
Q: State Corollary 5.11 about groups of prime order (what property, what does each element satisfy). (Proof Included)
A: Corollary 5.11: Every group of prime order is cyclic, and in such a group every element except the neutral element is a generator.
Proof: Only \(1 \mid p\) and \(p \mid p\) for \(p\) prime. So for \(a \in G\), either \(\text{ord}(a) = 1\) (meaning \(a = e\)) or \(\text{ord}(a) = p\) (meaning \(a\) generates the whole group; Lagrange).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095075 |
3 | 175% | 8d | 11 |
| niklas | cid:1764859231418 |
3 | 190% | 12d | 18 |
nid:1766314094961
DiskMat
State Lemma 5.18 about the units of a ring and the property ...
6
lapses
2/4
users
190%
ease
nid:1766314094961
Q: State Lemma 5.18 about the units of a ring and the property their set satisfies? (Proof included)
A: Lemma 5.18: For a ring \(R\), \(R^*\) is a group (the multiplicative group of units of \(R\)).
Proof idea: Every element of \(R^*\) has an inverse by definition, so axiom G3 holds. The other group axioms (associativity, neutral element) are inherited from the ring.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991217 |
5 | 150% | 55d | 22 |
| jonas | cid:1766314095101 |
1 | 230% | 4d | 7 |
nid:1772045507878
c1
A&W
Minimalgrad \(\delta(G) \geq |V|/2\)
6
lapses
2/4
users
198%
ease
nid:1772045507878
Cloze c1
Cloze answer: Minimalgrad \(\delta(G) \geq |V|/2\)
Q: Jeder Graph \(G = (V, E)\) mit \(|V| \geq 3\) und {{c1::Minimalgrad \(\delta(G) \geq |V|/2\)}} enthält {{c2::einen Hamiltonkreis}}.
A: Satz von Dirac
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772045507878 |
4 | 170% | 6d | 22 |
| niklas | cid:1772209100360 |
2 | 225% | 22d | 10 |
nid:1772046468683
c1
A&W
Das Problem „Gegeben ein Graph \(G = (V, E)\), enthält \(G\)...
6
lapses
2/4
users
205%
ease
nid:1772046468683
Cloze c1
Q: Das Problem „Gegeben ein Graph \(G = (V, E)\), enthält \(G\) einen Hamiltonkreis?" kann man in Zeit {{c1::\(O(|V|^2 \cdot 2^{|V|})\) entscheiden und, falls ja, einen solchen finden}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772046468684 |
4 | 170% | 13d | 21 |
| niklas | cid:1772209100367 |
2 | 240% | 30d | 9 |
nid:1773311192739
c1
A&W
In jedem Subgraphen gibt es einen Knoten mit Grad \(\leq k\)...
6
lapses
2/4
users
190%
ease
nid:1773311192739
Cloze c1
Cloze answer: In jedem Subgraphen gibt es einen Knoten mit Grad \(\leq k\)
Q: Heuristik:\(v_n\) := Knoten vom kleinsten Grad. Lösche \(v_n\).\(v_{n-1}\) := Knoten vom kleinsten Grad im Restgraph. Lösche \(v_{n-1}\). Iteriere.Falls \(G=(V,E)\) erfüllt:{{c1::In jedem Subgraphen gibt es ein
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773311192740 |
5 | 150% | 12d | 25 |
| niklas | cid:1773420068144 |
1 | 230% | 4d | 5 |
nid:1771364277512
c1
PProg
CPU state (registers, program counter)
6
lapses
2/4
users
198%
ease
nid:1771364277512
Cloze c1
Cloze answer: CPU state (registers, program counter)
Q: Process context includes:{{c1::CPU state (registers, program counter)}}{{c2::program state (stack, heap, resource handles)}}{{c3::additional management information}}.
A: A thread also has a context, but it is typically much smaller.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955105 |
4 | 170% | 13d | 17 |
| niklas | cid:1771364277648 |
2 | 225% | 13d | 10 |
nid:1774631277043
c1
A&W
Sei \(X\) eine Zufallsvariable mit Wertebereich \(W_X\subset...
6
lapses
1/4
users
130%
ease
nid:1774631277043
Cloze c1
Q: Sei \(X\) eine Zufallsvariable mit Wertebereich \(W_X\subseteq\mathbb{N}_0\).Dann gilt:\[ \mathbb{E}[X] = {{c1::\sum_{i=1}^{\infty}\Pr[X\ge i] :: \text{Schrankenform} }} \]Proof Included
A: Proof:\[ \mathbb{E}[X]=\sum_{i=0}^{\infty}i\cdot\Pr[X=i]=\sum_{i=0}^{\infty}\sum_{j=1}^{i}\Pr[X=i]=\sum_{j=1}^{\infty}\sum_{i=j}^{\infty}\Pr[X=i]=\sum_{j=1}^{\infty}\Pr[X\ge j].\quad\square \](Der Schlüsselschritt ist das Vertauschen der Summationsreihenfolge: Statt über \(i\) zu summieren und für jedes \(j\le i\) eine 1 zu zählen, wird über \(j\) summiert und alle \(i\ge j\) gezählt.)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277044 |
6 | 130% | 3d | 23 |
nid:1772547552647
c2
A&W
State of the Art Matching:\( O({{c1::|E|^{1+o(1)} }}) \) für...
6
lapses
1/4
users
130%
ease
nid:1772547552647
Cloze c2
Q: State of the Art Matching:\( O({{c1::|E|^{1+o(1)} }}) \) für bipartite Graphen \( O({{c2::|V|^{1/2} \cdot |E|}}) \) für generelle Graphen (Hopcroft-Karp)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772547552649 |
6 | 130% | 12d | 26 |
nid:1771973928521
c2
Analysis
Youngsche UngleichungFür jedes \(x, y \in \mathbb{R}\), \(\e...
6
lapses
1/4
users
130%
ease
nid:1771973928521
Cloze c2
Q: Youngsche UngleichungFür jedes \(x, y \in \mathbb{R}\), \(\epsilon > 0\) gilt: \[ {{c1:: 2|xy| }} \leq {{c2:: \epsilon x^2 + \frac{1}{\epsilon} y^2 }}\]Proof Included
A: Proof: Setze \(\gamma = \sqrt{\epsilon} > 0\). OBDA gelte \(x \cdot y \geq 0\). \[ 0 \leq (\gamma x - \frac{y}{\gamma})^2 = \gamma^2 x^2 - 2x\cdot y + \frac{1}{\gamma^2}y^2 \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928521 |
6 | 130% | 7d | 24 |
nid:1766314094728
DiskMat
Why is Bézout's identity useful for finding modular inverses...
5
lapses
3/4
users
227%
ease
nid:1766314094728
Q: Why is Bézout's identity useful for finding modular inverses?
A: If \(\text{gcd}(a, m) = 1\), then \(ua + vm = 1\) for some \(u, v\). This means \(ua = 1 - vm\), so \(ua \equiv_m 1\), making \(u\) the multiplicative inverse of \(a\) modulo \(m\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990469 |
3 | 190% | 88d | 19 |
| jonas | cid:1766314094750 |
1 | 230% | 20d | 8 |
| niklas | cid:1762106939359 |
1 | 260% | 50d | 8 |
nid:1766314094729
DiskMat
Why does \(ax \equiv_m 1\) have no solution when \(\text{gcd...
5
lapses
3/4
users
227%
ease
nid:1766314094729
Q: Why does \(ax \equiv_m 1\) have no solution when \(\text{gcd}(a, m) = d > 1\)?
A: We can rewrite \(ax \equiv_m 1\) as \(ax - 1 = km \Leftrightarrow ax - km = 1\). Now since, \(d \mid a\) and \(d \mid m\), then \(d \mid ax\) and \(d \mid km\) for any \(x\).Thus \(d \mid (ax - km)\), and \(ax - km = 1\).But \(d \nmid 1 \implies d \nmid (ax - km)\), which is a contradiction. Thus \(ax\) can never be congruent to \(1\) modulo \(m\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094751 |
2 | 195% | 8d | 12 |
| niklas | cid:1762106939361 |
2 | 255% | 6d | 11 |
| lorenz | cid:1764867990472 |
1 | 230% | 118d | 11 |
nid:1766314094970
DiskMat
State Lemma 5.20 about division in integral domains: (The qu...
5
lapses
3/4
users
232%
ease
nid:1766314094970
Q: State Lemma 5.20 about division in integral domains: (The quotient has what property?)
A: Lemma 5.20: In an integral domain, if \(a \mid b\) (i.e., \(b = ac\) for some \(c\)), then \(c\) is unique and is denoted by \(c = b/a\) (the quotient).
Explanation: If \(b = ac_1\) and \(b = ac_2\), then \(a(c_1 - c_2) = 0\). Since \(a \neq 0\) in an integral domain, we must have \(c_1 - c_2 = 0\)\(\implies c_1 = c_2\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095113 |
2 | 210% | 3d | 10 |
| niklas | cid:1764859231486 |
2 | 255% | 4d | 14 |
| lorenz | cid:1764867991246 |
1 | 230% | 135d | 12 |
nid:1771526451947
c1
A&W
Formale Definition der low-Werte:\(low[v] = {{c1::\min \left...
5
lapses
3/4
users
242%
ease
nid:1771526451947
Cloze c1
Q: Formale Definition der low-Werte:\(low[v] = {{c1::\min \left( dfs[v], \min_{(v,w) \in E} \begin{cases} dfs[w], & \text{falls } (v,w) \text{ Restkante} \\ low[w], & \text{falls } (v,w) \text{ Baumkante} \end{cases} \right)}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771535790938 |
3 | 265% | 11d | 15 |
| lorenz | cid:1771526451947 |
1 | 230% | 39d | 8 |
| tomas | cid:1771530245016 |
1 | 230% | 2d | 8 |
nid:1766314095018
DiskMat
What property does every finite field \(\text{GF}(q)\) have ...
5
lapses
2/4
users
178%
ease
nid:1766314095018
Q: What property does every finite field \(\text{GF}(q)\) have (and what does \(q\) satisfy)?
A: Theorem 5.40: The multiplicative group of every finite field \(\text{GF}(q)\) is cyclic (as \(q\) is a power of a prime, if \(\text{GF}(q)\) is cyclic).
This group has order \(q - 1\) and \(\varphi(q-1)\) generators.Note that even though q is not prime thus not every integer is coprime, GF(q) is not Z_q.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095172 |
3 | 145% | 4d | 17 |
| lorenz | cid:1764867991401 |
2 | 210% | 76d | 17 |
nid:1766940295685
c2
DiskMat
all the atomic formulas
5
lapses
2/4
users
215%
ease
nid:1767089600366
DiskMat
Number of subgroups of \(\langle \mathbb{Z}_m \times \mathbb...
5
lapses
2/4
users
200%
ease
nid:1764867989741
c2
A&D
equivalence class of the relation defined as follows: \(u = ...
5
lapses
2/4
users
200%
ease
nid:1764867989741
Cloze c2
Cloze answer: equivalence class of the relation defined as follows: \(u = v\) if \(u\) reaches \(v\)
Q: A {{c1::connected component}} of \(G\) is a {{c2::equivalence class of the relation defined as follows: \(u = v\) if \(u\) reaches \(v\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1763363268619 |
3 | 190% | 2d | 14 |
| lorenz | cid:1764867989742 |
2 | 210% | 129d | 12 |
nid:1764867990714
c2
DiskMat
A {{c1::field (Körper)}} is {{c2::a nontrivial commutative r...
5
lapses
2/4
users
208%
ease
nid:1764867990714
Cloze c2
Q: A {{c1::field (Körper)}} is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}}
A: Example: \(\mathbb{R}\), but not \(\mathbb{Z}\)Non-trivial: {0} is not a field. In particular, 1 = 0 (neutral element of mult. = neutral element of add.) causes trouble.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073684 |
4 | 185% | 3d | 17 |
| lorenz | cid:1764867990715 |
1 | 230% | 76d | 8 |
nid:1771361604906
c1
A&W
Jeder \(u\)-\(v\)-Knotenseparator hat Grösse mindestens \(k ...
5
lapses
2/4
users
230%
ease
nid:1771361604906
Cloze c1
Cloze answer: Jeder \(u\)-\(v\)-Knotenseparator hat Grösse mindestens \(k \); Jeder \(u\)-\(v\)-Kantenseparator hat Grösse mindestens \(k\)
Q: Sei \(G = (V, E)\) ein Graph und \(u, v \in V, u \neq v\). Dann gilt:
{{c1::Jeder \(u\)-\(v\)-Knotenseparator hat Grösse mindestens \(k \)}}\(\iff\){{c2::Es gibt mindestens \(k\) intern-knotendisjunkte \(u\)-\(v\)-Pfade.}}{{c1::Jeder \(u\)-\(v\)-Kantenseparator hat Grösse mindes
A: Satz von Karl Menger (Sohn vom sehr baseden Carl Menger)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771366536202 |
3 | 250% | 16d | 22 |
| lorenz | cid:1771361604907 |
2 | 210% | 21d | 15 |
nid:1772928333353
c1
Analysis
\[ \cos\!\left(\frac{\pi}{3}\right) = {{c1::\frac{1}{2} }} \...
5
lapses
2/4
users
208%
ease
nid:1766580142830
A&D
Explain how union works in the optimised Union-Find:
5
lapses
1/4
users
150%
ease
nid:1766580142830
Q: Explain how union works in the optimised Union-Find:
A: Arrays:rep, where rep[v] gives the representative of \(v\).members, where members[rep[v]] which contains all members of the ZHK of \(v\)rank, where rank[rep[v]] contains the size of the ZHK of \(v\).We always merge the smaller ZHK into the bigger to minimise updates.We update the reps, then the member
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580142830 |
5 | 150% | 76d | 19 |
nid:1772928333395
c1
Analysis
\[ \cos\!\left(\frac{7\pi}{4}\right) = {{c1::\frac{\sqrt{2} ...
5
lapses
1/4
users
150%
ease
nid:1772928333395
Cloze c1
Q: \[ \cos\!\left(\frac{7\pi}{4}\right) = {{c1::\frac{\sqrt{2} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333395 |
5 | 150% | 20d | 22 |
nid:1764859231444
DiskMat
State Fermat's Little Theorem (Corollary 5.14) (both totient...
5
lapses
1/4
users
165%
ease
nid:1764859231444
Q: State Fermat's Little Theorem (Corollary 5.14) (both totient and prime):
A: Corollary 5.14 (Fermat's Little Theorem): For all \(m \geq 2\) and all \(a\) with \(\gcd(a, m) = 1\): \[a^{\varphi(m)} \equiv_m 1\]
In particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \[a^{p-1} \equiv_p 1\]
Proof: This follows from Corollary 5.10 (\(a^{|G|} = e\)). Since \(\gcd(a, m)=1\), it is an element of \(\mathbb{Z}_m^*\) and thus an element of a group. \(\langle a \rangle\) there
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231445 |
5 | 165% | 4d | 17 |
nid:1771366536192
c2
A&W
\(|V| \geq k + 1\) und für alle Teilmengen \(X \subseteq V\)...
5
lapses
1/4
users
210%
ease
nid:1771366536192
Cloze c2
Cloze answer: \(|V| \geq k + 1\) und für alle Teilmengen \(X \subseteq V\) mit \(|X| < k\) gilt: Der Graph \(G[V \setminus X]\) ist zusammenhängend
Q: Ein Graph \(G = (V, E)\) heisst {{c1::\(k\)-zusammenhängend}}, falls {{c2::\(|V| \geq k + 1\) und für alle Teilmengen \(X \subseteq V\) mit \(|X| < k\) gilt: Der Graph \(G[V \setminus X]\) ist zusammenhängend}}.
A: Man muss mindestens \(k\)-Knoten (und die inzidenten Kanten) löschen, um den Zusammenhang zu zerstören.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771366536201 |
5 | 210% | 13d | 23 |
nid:1766314094712
DiskMat
State Bézout's identity (Corollary 4.5).
4
lapses
3/4
users
243%
ease
nid:1766314094712
Q: State Bézout's identity (Corollary 4.5).
A: For \(a, b \in \mathbb{Z}\) (not both 0), there exist \(u, v \in \mathbb{Z}\) such that:
\[\text{gcd}(a, b) = ua + vb\]
The GCD can be expressed as an integer linear combination.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990414 |
2 | 210% | 63d | 11 |
| jonas | cid:1766314094733 |
1 | 230% | 14d | 8 |
| niklas | cid:1762106939325 |
1 | 290% | 119d | 10 |
nid:1766314094927
DiskMat
Which elements generate \(\mathbb{Z}_n\)? How can this be pr...
4
lapses
3/4
users
233%
ease
nid:1766314094927
Q: Which elements generate \(\mathbb{Z}_n\)? How can this be proven?
A: \(\mathbb{Z}_n\) is generated by all \(a \in \mathbb{Z}_n\) for which \(\gcd(a, n) = 1\) (all elements coprime to \(n\)).
Proof:\(a\) generator \(\implies\)\(\gcd(a, n) = 1\)\(\mathbb{Z}_n = \langle a \rangle\)\(\implies\)\(1 \in \langle a \rangle\)\(\implies\)\(a^u = au \equiv_n 1\) for some \(u\)\(\implies\)\(\gcd(a, n) = 1\) (\(\gcd\) must divide both \(au-qn\) and 1).\(\gcd(a, n) = 1 \implies
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231381 |
2 | 255% | 4d | 14 |
| jonas | cid:1766314095054 |
1 | 215% | 7d | 10 |
| lorenz | cid:1764867991106 |
1 | 230% | 81d | 11 |
nid:1771973928567
c1
Analysis
Eulersche Formel:\[ \cos(t) = {{c1:: \frac{e^{it} + e^{-it} ...
4
lapses
3/4
users
228%
ease
nid:1766314094853
c1
DiskMat
the order of \(1\) in the additive group if it is finite, an...
4
lapses
2/4
users
232%
ease
nid:1766314094853
Cloze c1
Cloze answer: the order of \(1\) in the additive group if it is finite, and 0 if it is infinite.
Q: The characteristic of a ring is {{c1::the order of \(1\) in the additive group if it is finite, and 0 if it is infinite.}}
A: Example: the characteristic of \(\langle \mathbb{Z}_m;\oplus,\ominus,0,\odot,1\rangle\)is \(m\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094926 |
3 | 190% | 8d | 13 |
| niklas | cid:1762856074719 |
1 | 275% | 10d | 10 |
nid:1767089604935
c1
LinAlg
\(x = 0\) is the only vector for which \(Ax = 0\)
4
lapses
2/4
users
210%
ease
nid:1767089604935
Cloze c1
Cloze answer: \(x = 0\) is the only vector for which \(Ax = 0\)
Q: The columns of \(A\) are independent if and only if {{c1::\(x = 0\) is the only vector for which \(Ax = 0\)::Linear combination view}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1767089604936 |
2 | 210% | 4d | 10 |
| lorenz | cid:1767105283315 |
2 | 210% | 71d | 15 |
nid:1766580143526
A&D
Kruskal's Algorithm
4
lapses
2/4
users
218%
ease
nid:1766580143526
Q: Kruskal's Algorithm
A: \(O(|E| \log |E| + |V| \log |V|)\)Outer loop: Iterate \(|E|\) times at most:Inner loop: find and union take \(O(\log |V|)\) per call amortised, thus \(O(|V| \log |V|)\) total.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580143526 |
3 | 190% | 126d | 16 |
| niklas | cid:1766568909602 |
1 | 245% | 19d | 5 |
nid:1765372936281
c2
A&D
{{c1:: \(\sum_{i = 1}^{n} i^3\)::Sum}} \(=\) {{c2::\(\frac{...
4
lapses
2/4
users
210%
ease
nid:1766448533056
c1
DiskMat
a field.
4
lapses
2/4
users
210%
ease
nid:1764867991445
DiskMat
What is the minimum distance of two codewords in a polynomia...
4
lapses
2/4
users
218%
ease
nid:1764867989947
DiskMat
What is the modus ponens logical rule?
4
lapses
2/4
users
232%
ease
nid:1764867990087
DiskMat
When is a relation \(\rho\) on set \(A\) irreflexive?
4
lapses
2/4
users
210%
ease
nid:1764867990087
Q: When is a relation \(\rho\) on set \(A\) irreflexive?
A: When \(a \ \not\rho \ a\) is true for all \(a \in A\), i.e., \(\rho \cap \text{id} = \emptyset\).Note that irreflexive is NOT the negation of reflexive!
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477382 |
3 | 190% | 23d | 17 |
| lorenz | cid:1764867990087 |
1 | 230% | 145d | 9 |
nid:1768182518186
c1
LinAlg
at the same indices; rank
4
lapses
2/4
users
218%
ease
nid:1768182518186
Cloze c1
Cloze answer: at the same indices; rank
Q: For \(A\) and \(MA\) (\(M\) invertible) they have:the independent columns {{c1:: at the same indices}}the same {{c1::rank}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182518186 |
3 | 190% | 69d | 14 |
| niklas | cid:1768139535123 |
1 | 245% | 21d | 7 |
nid:1768182518580
LinAlg
Prove that the row space of \(A\) and \(MA\) is the same for...
4
lapses
2/4
users
210%
ease
nid:1768182518580
Q: Prove that the row space of \(A\) and \(MA\) is the same for \(M\) invertible!
A: \(\textbf{R}(A) = \textbf{C}(A^\top) \overset{!}{=} \textbf{C}(A^\top M^\top) = \textbf{C}((MA)^\top) = \textbf{R}(MA)\)where ! holds because:
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182518580 |
3 | 190% | 70d | 16 |
| niklas | cid:1768148472221 |
1 | 230% | 2d | 4 |
nid:1765553400194
LinAlg
What is the 1-norm of a vector?
4
lapses
2/4
users
225%
ease
nid:1772045795752
A&W
Wie kann man mit der Siebformel die Zahl der Hamiltonkreise ...
4
lapses
2/4
users
210%
ease
nid:1773310950541
c1
A&W
|V|/2
4
lapses
2/4
users
218%
ease
nid:1773310950541
Cloze c1
Cloze answer: |V|/2
Q: Es gibt bipartite Graphen und eine Reihenfolge \(V = \{v_1, \ldots, v_n\}\) der Knoten, für die der Greedy-Algorithmus \({{c1::|V|/2}}\) viele Farben benötigt.
A: Vollständig bipartiter Graph ohne ein perfektes Matching
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773310950541 |
3 | 190% | 19d | 17 |
| niklas | cid:1773420068142 |
1 | 245% | 4d | 8 |
nid:1772547451587
c1
A&W
|V| \cdot |E|
4
lapses
2/4
users
218%
ease
nid:1772545721385
c1
A&W
Für jede Kante in \( M_{\text{max}} \) gilt: {{c1::Mindesten...
4
lapses
2/4
users
232%
ease
nid:1772545721385
Cloze c1
Q: Für jede Kante in \( M_{\text{max}} \) gilt: {{c1::Mindestens einer der beiden Endpunkte wird von einer Kante aus \( M_{\text{Greedy} } \) überdeckt}}
A: (Denn sonst könnten wir die Kante zu \( M_{\text{Greedy}} \) hinzufügen.)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386195 |
3 | 235% | 5d | 14 |
| lorenz | cid:1772545721385 |
1 | 230% | 40d | 8 |
nid:1772046331702
IO r3
A&W
[Image Occlusion region 3]
4
lapses
2/4
users
210%
ease
nid:1772046331702
Cloze c3
Q: {{c3::image-occlusion:rect:left=.1591:top=.8923:width=.7185:height=.0742}}{{c2::image-occlusion:rect:left=.3252:top=.7428:width=.5272:height=.0923}}{{c1::image-occlusion:rect:left=.0549:top=.1782:width=.9041:height=.1203}}{{c4::image-occlusion:rect:left=.1645:top=.4824:width=.1234:height
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772046331703 |
2 | 210% | 32d | 15 |
| niklas | cid:1772209100381 |
2 | 210% | 1d | 9 |
nid:1771973928588
c3
Analysis
Beweis: Für alle \(a < b\) in \(\mathbb{R}\) existiert ein \...
4
lapses
2/4
users
225%
ease
nid:1771973928588
Cloze c3
Q: Beweis: Für alle \(a < b\) in \(\mathbb{R}\) existiert ein \(\mathbb{Q}\) mit \(a < q < b\){{c1:: Wähle nach Archimedischem Prinzip \(n \in \mathbb{N}\) so dass \(\frac{1}{n} < b - a\).}}{{c2:: \(\frac{m}{n} \mid m \in \mathbb{Z}\) diese
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771969342906 |
3 | 220% | 69d | 10 |
| lorenz | cid:1771973928589 |
1 | 230% | 40d | 7 |
nid:1772928333487
c1
Analysis
\[ \tan\!\left(\frac{\pi}{3}\right) = {{c1::\sqrt{3} }} \]
4
lapses
2/4
users
225%
ease
nid:1772496585520
c1
Analysis
Es sei \((a_n)_{n \in \mathbb{N}_0}\) eine Folge in \(\mathb...
4
lapses
2/4
users
240%
ease
nid:1772496585520
Cloze c1
Q: Es sei \((a_n)_{n \in \mathbb{N}_0}\) eine Folge in \(\mathbb{R}\).Eine Teilfolge ist eine Folge der Form \(({a_n}_k)_{k \in \mathbb{N}_0}\) wobei \((n_k)_{k \in \mathbb{N}_0}\) eine {{c1:: Folge nicht-negativer ganze
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772496585520 |
2 | 210% | 34d | 13 |
| niklas | cid:1772520282865 |
2 | 270% | 205d | 13 |
nid:1762856073654
c1
A&D
closed walk without repeated vertices; at least three vertic...
4
lapses
2/4
users
210%
ease
nid:1762856073654
Cloze c1
Cloze answer: closed walk without repeated vertices; at least three vertices
Q: In graph theory, a {{c2::cycle (Kreis)}} is a {{c1::closed walk without repeated vertices}} and {{c1::at least three vertices}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073667 |
2 | 210% | 121d | 16 |
| tomas | cid:1765551666552 |
2 | 210% | 50d | 12 |
nid:1771872607286
DDCA
How can we build NOR from NOT and AND?
4
lapses
2/4
users
225%
ease
nid:1766580144345
c1
A&D
the values of the vertices in the priority queue (see line d...
4
lapses
1/4
users
170%
ease
nid:1766580144345
Cloze c1
Cloze answer: the values of the vertices in the priority queue (see line decrease_key(H, v, d[v]))
Q: Prim's Algorithm Invariants:The distances "d[.] = " in the distance array are {{c1::the values of the vertices in the priority queue (see line decrease_key(H, v, d[v]))}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580144345 |
4 | 170% | 87d | 18 |
nid:1764867991302
DiskMat
How do you perform polynomial division when the divisor is n...
4
lapses
1/4
users
170%
ease
nid:1764867991302
Q: How do you perform polynomial division when the divisor is not monic (e.g., in \(\text{GF}(7)[x]\))?
A: If dividing by a non-monic polynomial like \(4x + 2\) in \(\text{GF}(7)[x]\):
Find the multiplicative inverse of the leading coefficient in the field
For \(4\) in \(\text{GF}(7)\): \(4 \cdot 2 \equiv_7 1\), so \(4^{-1} = 2\)
Multiply the polynomial by this inverse to make it monic
\(2 \cdot (4x + 2) = 8x + 4 \equiv_7 x + 4\)
Now divide by the monic polynomial
Example: \(3x^2 + 6x + 5\) divided by \(4x + 2\) become
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991302 |
4 | 170% | 100d | 17 |
nid:1774631277033
A&W
In einer Gruppe von \(m\) Personen (mit \(n=365\) Tagen), wi...
4
lapses
1/4
users
170%
ease
nid:1774631277033
Q: In einer Gruppe von \(m\) Personen (mit \(n=365\) Tagen), wie gross ist die Wahrscheinlichkeit, dass alle Geburtstage verschieden sind? Leite die Formel her.Proof Included
A: (Geburtstagsproblem) Modell: Werfe \(m\) Bälle gleichverteilt in \(n\) Urnen. Sei \(A_j\) = "Ball \(j\) landet in einer leeren Urne."Mit dem Multiplikationssatz:\[ \Pr\!\left[\bigcap_{j=1}^m A_j\right] = \prod_{j=2}^{m} \frac{n-(j-1)}{n} = \prod_{j=2}^{m}\!\left(1-\frac{j-1}{n}\right). \]Obere Schranke mit \(1-x \le e^{-x}\):\[ \Pr[\text{alle verschieden}] \le \prod_{j=2}^{m} e^{-(j-1)/n} = e^{-m(m-1)/(2n)}. \]Also ist die Wahrscheinlichkeit für mindes
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277034 |
4 | 170% | 6d | 19 |
nid:1774487165282
Analysis
Wie beweist man \(\exp(z+w) = \exp(z) \cdot \exp(w)\)?
4
lapses
1/4
users
170%
ease
nid:1774487165282
Q: Wie beweist man \(\exp(z+w) = \exp(z) \cdot \exp(w)\)?
A: Beide Reihen \(\sum z^n/n!\) und \(\sum w^n/n!\) sind absolut konvergent.Das Cauchy-Produkt liefert:\[c_n = \sum_{k=0}^n \frac{z^{n-k}}{(n-k)!} \cdot \frac{w^k}{k!} = \frac{1}{n!}\sum_{k=0}^n \binom{n}{k} z^{n-k} w^k = \frac{(z+w)^n}{n!}\]Also \(\exp(z)\exp(w) = \sum c_n = \sum \frac{(z+w)^n}{n!} = \exp(z+w)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165282 |
4 | 170% | 1d | 18 |
nid:1774917595550
c1
Analysis
0 < |x - x_0| < \delta \implies |f(x) - L| < \varepsilon
4
lapses
1/4
users
170%
ease
nid:1774917595550
Cloze c1
Cloze answer: 0 < |x - x_0| < \delta \implies |f(x) - L| < \varepsilon
Q: Die alternative Grenzwertdefinition schliesst \(x_0\) selbst aus:\[\begin{gathered}\forall \varepsilon > 0\;\exists \delta > 0 \text{ so dass für alle } x \in \mathbb{D}(f) \\ {{c1:: 0 < |x - x_0| < \delta \implies |f(x) - L| < \varepsilon}}\end{gathered}\]
A: Durch \(0 < |x - x_0|\) ist der Grenzwert unabhängig vom Funktionswert bei \(x_0\) - selbst wenn \(f(x_0)\) nicht definiert ist.Da gilt \(0 < |x - x_0|\) kann \(x\) nicht den Wert \(x_0\) annehmen.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917595550 |
4 | 170% | 1d | 18 |
nid:1774917595090
c1
Analysis
Es gelte \(\mathbb{D}(f) \cap [x_0,\, x_0 + \delta) \neq \em...
4
lapses
1/4
users
170%
ease
nid:1774917595090
Cloze c1
Q: Es gelte \(\mathbb{D}(f) \cap [x_0,\, x_0 + \delta) \neq \emptyset \;\forall \delta > 0\).Falls gilt \(\forall \varepsilon > 0 \;\exists \delta > 0\): \[{{c1::x \in \mathbb{D}(f) \cap [x_0,\, x_0 + \delta) \;\Rightarrow\; |f(x) - L| < \varepsilon }},\] hat \(f\) in \(x_
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917595090 |
4 | 170% | 4d | 19 |
nid:1774487165385
c1
Analysis
Für die geometrische Reihe \(\sum_{n=0}^\infty q^n\) gilt \(...
4
lapses
1/4
users
170%
ease
nid:1774487165385
Cloze c1
Q: Für die geometrische Reihe \(\sum_{n=0}^\infty q^n\) gilt \(S_n = {{c1:: \frac{1 - q^{n + 1} }{1 - q} }}\)
A: \[ \begin{align} qS_n &= q + q^2 + \dots + q^{n + 1} \\ S_n - qS_n &= 1 - q^{n + 1} \\ S_n &= \frac{1 - q^{n + 1}}{1 - q} \end{align} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165385 |
4 | 170% | 8d | 20 |
nid:1772928333323
c1
Analysis
streng monoton steigend
4
lapses
1/4
users
170%
ease
nid:1772928333323
Cloze c1
Cloze answer: streng monoton steigend
Q: Der Wertebereich von \(\arctan\) ist \({{c1::\left(-\frac{\pi}{2},\, \frac{\pi}{2}\right)}}\), und die Funktion ist {{c1::streng monoton steigend::Wachstumsverhalten}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333323 |
4 | 170% | 15d | 19 |
nid:1771973928491
c1
Analysis
Sei \(n \in \mathbb{N}\), \(n \ge 1\). Dann hat die Gleichun...
4
lapses
1/4
users
170%
ease
nid:1771973928491
Cloze c1
Q: Sei \(n \in \mathbb{N}\), \(n \ge 1\). Dann hat die Gleichung \(z^n = 1\) genau \(n\) Lösungen in \(\mathbb{C}\): \(z_1, z_2, \dots, z_n\) wobei: \[ z_j = {{c1:: \cos \frac{2\pi j}{n} + i \cdot \sin \frac{2 \pi j}{n} }}, \quad 1 \le j \le n \]
A: Die Lösungen liegen alle auf einem Kreis mit Radius 1 und sind gleichmäßig verteilt (formen ein n-Eck). Beispiel: Für \(w = R \cdot e^{i \varphi}\) sind die Lösungen von \(z^n = w\) gleich der \(n\) komplexen Zahlen mit Betrag \(\sqrt[^n]{R}\) und Winkeln \(\varphi_k = \frac{\varphi}{n} + k \cdot \frac{2 \pi}{n}\) für \(k = 0, \dots, n - 1\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928491 |
4 | 170% | 10d | 25 |
nid:1772928333414
c1
Analysis
\[ \sin\!\left(\frac{\pi}{4}\right) = {{c1::\frac{\sqrt{2} }...
4
lapses
1/4
users
170%
ease
nid:1772928333414
Cloze c1
Q: \[ \sin\!\left(\frac{\pi}{4}\right) = {{c1::\frac{\sqrt{2} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333414 |
4 | 170% | 17d | 18 |
nid:1761028602734
LinAlg
An linear combination of \(\lambda_1\textbf{v}_1 + \lambda_...
4
lapses
1/4
users
215%
ease
nid:1761028602734
Q: An linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is convex if
A: it is both affine and conic
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761028602734 |
4 | 215% | 89d | 15 |
nid:1761491477391
DiskMat
How does antisymmetry appear in graph representation?
4
lapses
1/4
users
215%
ease
nid:1761491477391
Q: How does antisymmetry appear in graph representation?
A: There is not a single cycle of length 2 (no edge from \(a\) to \(b\) AND from \(b\) to \(a\)).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477392 |
4 | 215% | 11d | 17 |
nid:1761491477501
DiskMat
What is the set \(\{0, 1\}^{\infty}\)?
4
lapses
1/4
users
245%
ease
nid:1761491477501
Q: What is the set \(\{0, 1\}^{\infty}\)?
A: The set of semi-infinite binary sequences, or equivalently, the set of functions \(\mathbb{N} \to \{0,1\}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477502 |
4 | 245% | 32d | 21 |
nid:1765296057127
c1
A&D
O(n)
4
lapses
1/4
users
200%
ease
nid:1765296057127
Cloze c1
Cloze answer: O(n)
Q: Choose a tight bound!\({{c1::O(n)}} \leq {{c2::O(\log(n!))}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765296057127 |
4 | 200% | 8d | 12 |
nid:1772520447039
c2
A&W
\(\forall u, v \in V, u \neq v\) gibt mindestens \(k\) inter...
4
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nid:1772520447039
Cloze c2
Cloze answer: \(\forall u, v \in V, u \neq v\) gibt mindestens \(k\) intern-knotendisjunkte \(u\)-\(v\)-Pfade.; \(\forall u, v \in V, u \neq v\) gibt mindestens \(k\) kantendisjunkte \(u\)-\(v\)-Pfade.
Q: Sei \(G = (V, E)\) ein Graph. Dann gilt:
{{c1::\(G\) is \(k\)-knoten-zusammenhängend}}\(\iff\){{c2::\(\forall u, v \in V, u \neq v\) gibt mindestens \(k\) intern-knotendisjunkte \(u\)-\(v\)-Pfade.}}{{c1::\(G\) ist \(k\)-kanten-zusammenhängend}}\(\if
A: Satz von Karl Menger V2 (Sohn vom sehr baseden Carl Menger)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772520447040 |
4 | 200% | 9d | 14 |
nid:1766314094576
DiskMat
How are ordered pairs \((a, b)\) formally defined in set the...
3
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nid:1766314094825
c1
DiskMat
subset; \(A\times B\).; a relation on \(A\).
3
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nid:1766314094825
Cloze c1
Cloze answer: subset; \(A\times B\).; a relation on \(A\).
Q: A relation \(\rho\) from a set \(A\) to a set \(B\) (also called an \((A,B)\)-relation) is a {{c1::subset}} of {{c1::\(A\times B\).}} If \(A = B\), then \(\rho\) is called {{c1::a relation on \(A\).}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094897 |
1 | 230% | 4d | 5 |
| lorenz | cid:1764867990770 |
1 | 230% | 79d | 12 |
| niklas | cid:1762856074679 |
1 | 245% | 36d | 7 |
nid:1766940295781
DiskMat
In a finite group of order \(|G|\), for \(x^e = y\), \(d\) i...
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nid:1766940295781
Q: In a finite group of order \(|G|\), for \(x^e = y\), \(d\) is the inverse such that \(y^d = x\) iff: (Proof included)
A: \(ed \equiv_{|G|} 1\), i.e. \(d\) is the multiplicative inverse of \(e\) modulo \(|G|\).Proof\(ed = k \cdot |G| + 1\) (multiplicative inverse)\((x^e)^d = x^{ed} = x^{k\cdot |G| + 1}\)\((x^{|G|})^k \cdot x = 1^k \cdot x = x\)Thus this returns \(x\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295939 |
1 | 230% | 4d | 6 |
| lorenz | cid:1766448532968 |
1 | 230% | 103d | 11 |
| niklas | cid:1766318730351 |
1 | 245% | 10d | 5 |
nid:1764867991514
LinAlg
The scalar product of \(\textbf{v} \cdot \textbf{v}\) is \(\...
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nid:1764867991514
Q: The scalar product of \(\textbf{v} \cdot \textbf{v}\) is \(\leq or \geq\) to what?
A: \(\textbf{v} \cdot \textbf{v} \geq 0\) with equality exactly if \(\textbf{v} = \textbf{0}\).This is because we essentially square the entries and thus can't get negatives.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991514 |
1 | 230% | 119d | 11 |
| niklas | cid:1761028945284 |
1 | 245% | 43d | 8 |
| tomas | cid:1765551644273 |
1 | 245% | 11d | 4 |
nid:1766314094559
c1
DiskMat
\(F \rightarrow G\)
3
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nid:1766314094620
DiskMat
What important property do equivalence classes have?
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nid:1766314094620
Q: What important property do equivalence classes have?
A: The set \(A / \theta\) of equivalence classes of an equivalence relation \(\theta\) on \(A\) is a partition of \(A\).
(Equivalence classes are disjoint and cover the entire set)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094628 |
2 | 180% | 9d | 12 |
| niklas | cid:1761491477412 |
1 | 260% | 40d | 9 |
nid:1766314094637
DiskMat
What is the meet of elements \(a\) and \(b\) in a poset?
3
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nid:1766314094711
DiskMat
How is the GCD related to ideals? (Lemma 4.4)
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nid:1766314094711
Q: How is the GCD related to ideals? (Lemma 4.4)
A: Let \(a, b \in \mathbb{Z}\) (not both 0). If \((a, b) = (d)\), then \(d\) is a greatest common divisor of \(a\) and \(b\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990411 |
2 | 210% | 62d | 17 |
| jonas | cid:1766314094732 |
1 | 230% | 5d | 8 |
nid:1766314094928
c1
DiskMat
cyclic
3
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nid:1766314094985
DiskMat
If \(b(x)\) divides \(a(x)\), then so does:
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nid:1766314094985
Q: If \(b(x)\) divides \(a(x)\), then so does:
A: \(v \cdot b(x)\) for any nonzero \(v \in F\).
This holds because if \(a(x) = b(x) \cdot c(x)\), then \(a(x) = vb(x) \cdot (v^{-1} c(x))\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991292 |
2 | 210% | 85d | 12 |
| jonas | cid:1766314095135 |
1 | 230% | 4d | 8 |
nid:1766314094989
DiskMat
What does polynomial evaluation preserve?
3
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nid:1766314094989
Q: What does polynomial evaluation preserve?
A: Lemma 5.28: Polynomial evaluation is compatible with the ring operations:
- If \(c(x) = a(x) + b(x)\) then \(c(\alpha) = a(\alpha) + b(\alpha)\) for any \(\alpha\)
- If \(c(x) = a(x) \cdot b(x)\) then \(c(\alpha) = a(\alpha) \cdot b(\alpha)\) for any \(\alpha\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991309 |
2 | 210% | 101d | 12 |
| jonas | cid:1766314095139 |
1 | 185% | 8d | 10 |
nid:1766314094994
DiskMat
If we want to use roots to check that a polynomial is irredu...
3
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nid:1766314094994
Q: If we want to use roots to check that a polynomial is irreducible, it has to have?
A: Degree \(2\) or \(3\).
Important: This doesn't work for polynomials of higher degrees! A degree \(4\) polynomial might be the product of two irreducible degree \(2\) polynomials, each with no roots.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231542 |
2 | 240% | 18d | 8 |
| jonas | cid:1766314095144 |
1 | 215% | 8d | 9 |
nid:1766314095000
c1
DiskMat
A ring \(R\) is a field if and only if {{c1:: \(\langle R \s...
3
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nid:1766314095016
DiskMat
When does an irreducible polynomial exist in \(\text{GF}(p)[...
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nid:1766314095016
Q: When does an irreducible polynomial exist in \(\text{GF}(p)[x]\)?
A: For every prime \(p\) and every \(d > 1\), there exists an irreducible polynomial of degree \(d\) in \(\text{GF}(p)[x]\).
Result: we can construct a finite field with \(p^d\) elements by using an irreducible polynomial of degree \(d\) to cap the number of coefficients at \(d\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231591 |
2 | 210% | 4d | 14 |
| jonas | cid:1766314095170 |
1 | 230% | 9d | 9 |
nid:1766314111354
LinAlg
A linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2...
3
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nid:1766314111354
Q: A linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is conic if
A: \(\lambda_j \geq 0\) for \(j = 1, 2, \dots, n\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314111354 |
2 | 210% | 3d | 7 |
| lorenz | cid:1764867991507 |
1 | 230% | 133d | 12 |
nid:1766314111376
LinAlg
Was ist der rank einer full rank matrix \(A \in \mathbb{R}^{...
3
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users
235%
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nid:1767089600236
c1
DiskMat
factored uniquely into irreducible elements (up to associate...
3
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nid:1767089600236
Cloze c1
Cloze answer: factored uniquely into irreducible elements (up to associates)
Q: In a Euclidean domain every element can be {{c1:: factored uniquely into irreducible elements (up to associates)}}
A: \(a, b\) associates (\(a \sim b\)) if \(a = ub\) for some unit \(u\).Proof sketch:
Consider a nonzero, nonunit \(a \in R\).
If a is irreducible, we are done.
Otherwise, \(a = bc\) with both \(b,c\) nonunits.
By the Euclidean property, we may assume \(\delta(b), \delta(c) < \delta(a)\).
If either factor is reducible, factor it
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1767089600236 |
2 | 210% | 1d | 9 |
| niklas | cid:1767082205233 |
1 | 245% | 5d | 7 |
nid:1768160640380
LinAlg
How do we find a basis for the row space \(R(A) = C(A^\top)\...
3
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nid:1768160640380
Q: How do we find a basis for the row space \(R(A) = C(A^\top)\)?
A: The first \(r\) columns of \(R^\top\) where \(R\) is the RREF of \(A\) form a basis of the row space (the non-zero rows). In particular \(\dim(\textbf{R}(A)) = r\)This works because as noted before, multiplying by an invertible matrix \(M\) does not change the row-space of \(MA\) on the left.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182518113 |
2 | 210% | 84d | 11 |
| jonas | cid:1768160640388 |
1 | 230% | 1d | 4 |
nid:1766531635629
A&D
Bellman-Ford
3
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nid:1766531635629
Q: Bellman-Ford
A: \(O(|V| \cdot |E|)\) (uses DP)We iterate over all edges in the "relaxation" thus the time complexity of that step is \(O(m)\) (the actual check is \(O(1)\)).As we relax \(n - 1\) (or \(n\) for negative cycle check) times, the total runtime is \(O(n \cdot m)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635631 |
2 | 210% | 60d | 13 |
| tomas | cid:1766576739763 |
1 | 230% | 11d | 5 |
nid:1766531635563
c1
A&D
\(\exists\) back edge
3
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users
242%
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nid:1766580143624
A&D
Boruvka's Algorithm
3
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nid:1766580143624
Q: Boruvka's Algorithm
A: \(O((|V| + |E|) \cdot \log |V|)\)During each iteration, we examine all edges to find the cheapest one: \(O(|V| + |E|)\):Run DFS to find the connected components: \(O(|V| + |E|)\)Find the cheapest one \(O(|E|)\)We iterate a total of \(\log_2 |V|\) times as each iteration halves the number of connected components.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580143624 |
2 | 210% | 70d | 13 |
| niklas | cid:1766567785295 |
1 | 245% | 16d | 6 |
nid:1765372936212
c1
A&D
O(\log(n!))
3
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users
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nid:1766580143735
c1
A&D
\(O(|V| \log |V|)\)
3
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nid:1766580143735
Cloze c1
Cloze answer: \(O(|V| \log |V|)\)
Q: The amortised runtime of union in the Union-Find datastructure is {{c1:: \(O(|V| \log |V|)\)}}.
A: Union takes \(\Theta(\min \{ |ZHK(u)| , |ZHK(v)| \}\). In the worst case, the minimum is \(|V| / 2\) as both have the same size.Therefore over all loops, this would take \(O(|V| \log |V|)\) time, as on average we only take \(O(\log |V|)\) time.The graph stays worst case, this is the average of the calls in the worst case.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580143735 |
2 | 210% | 82d | 13 |
| niklas | cid:1766569467577 |
1 | 245% | 25d | 6 |
nid:1768344740183
c1
A&D
we know the graph is connected, i.e. \(m \geq n - 1\)
3
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2/4
users
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nid:1768344740183
Cloze c1
Cloze answer: we know the graph is connected, i.e. \(m \geq n - 1\)
Q: We can run DFS in \(O(m)\) if {{c1:: we know the graph is connected, i.e. \(m \geq n - 1\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768344740183 |
2 | 210% | 88d | 12 |
| tomas | cid:1768391364319 |
1 | 230% | 7d | 5 |
nid:1765372936179
A&D
When \(f = \Theta(g)\), this means?
3
lapses
2/4
users
242%
ease
nid:1765372936179
Q: When \(f = \Theta(g)\), this means?
A: \(\exists C_1,C_2 \ge 0 \quad \forall n \in \mathbb{N}\) \(C_1 \cdot g(n) \leq f(n) \leq C_2 \cdot g(n)\)\(f\) grows asymptotically the same as \(g\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936179 |
2 | 210% | 113d | 13 |
| niklas | cid:1765295553120 |
1 | 275% | 20d | 13 |
nid:1765372936170
A&D
What is l'Hôpital's Rule?
3
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nid:1765372936170
Q: What is l'Hôpital's Rule?
A: If \(\lim_{x \to \infty} f(x) = \lim_{x \to \infty} g(x) = \infty\) (or both \(=0\)), and \(\lim_{x \to \infty} \frac{f'(x)}{g'(x)}\) exists (or is \(\pm\infty\)), then:
\(\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{f'(x)}{g'(x)}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936170 |
2 | 210% | 117d | 15 |
| niklas | cid:1765295341110 |
1 | 245% | 17d | 13 |
nid:1766531635421
A&D
What is the tree condition for 2-3 Trees implementing a dict...
3
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2/4
users
228%
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nid:1766531635530
c1
A&D
it does not contain any cycles of odd length
3
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users
220%
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nid:1765372936194
c2
A&D
O(\log(n!))
3
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2/4
users
220%
ease
nid:1764867990542
c1
DiskMat
Find a suitable statement \( T\).
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nid:1764867990542
Cloze c1
Cloze answer: Find a suitable statement \( T\).
Q: Proof method: Proof by Contradiction1. {{c1:: Find a suitable statement \( T\).}}2. {{c2:: Prove that \( T \) is false.}}3. {{c3:: Assume that \( S \) is false and prove that \( T \) is true (-> contradiction).}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073567 |
2 | 240% | 43d | 11 |
| lorenz | cid:1764867990542 |
1 | 230% | 117d | 9 |
nid:1764867991070
c2
DiskMat
If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is \(|G...
3
lapses
2/4
users
235%
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nid:1764867991498
c1
DiskMat
4
3
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nid:1767918757756
IO r5
EProg
[Image Occlusion region 5]
3
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nid:1767918757756
Cloze c5
Q: {{c1::image-occlusion:rect:left=.2281:top=.3427:width=.0814:height=.2045:oi=1}}{{c2::image-occlusion:rect:left=.3053:top=.345:width=.1142:height=.2067:oi=1}}{{c3::image-occlusion:rect:left=.1625:top=.5221:width=.0693:height=.2181:oi=1}}{{c4::image-occlusion:rect:left=.1625:top=.713:width
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1767888505029 |
2 | 240% | 13d | 13 |
| lorenz | cid:1767918757756 |
1 | 230% | 117d | 10 |
nid:1768263610799
c3
LinAlg
Assume \(Q\) is orthogonal and square. Then:{{c1::\(QQ^\top ...
3
lapses
2/4
users
220%
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nid:1768263610799
Cloze c3
Q: Assume \(Q\) is orthogonal and square. Then:{{c1::\(QQ^\top = I\)}}{{c2::\(Q^{-1} = Q^\top\)}}{{c3::The columns form an orthonormal basis for \(\mathbb{R}^n\).}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768214114072 |
2 | 210% | 2d | 6 |
| lorenz | cid:1768263610801 |
1 | 230% | 72d | 11 |
nid:1768263610411
c1
LinAlg
Let \(A \in \mathbb{R}^{m \times n}\). Then \(N(A) = {{c1::N...
3
lapses
2/4
users
228%
ease
nid:1768263610411
Cloze c1
Q: Let \(A \in \mathbb{R}^{m \times n}\). Then \(N(A) = {{c1::N(A^\top A)::\text{another nullspace} }}\). Proof Included
A: \(N(A) = N(A^\top A)\) holds because:if \(x \in N(A)\) then \(Ax = 0 \implies A^\top Ax = A \cdot 0 \implies A^\top A x = 0\).if \(x \in N(A^\top A)\) then \(A^\top A x = 0\), which means \[ 0 = x^\top 0 = x^\top A^\top Ax = (Ax)^\top(Ax) = ||Ax||^2 \implies Ax = 0 \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768210666216 |
2 | 225% | 2d | 9 |
| lorenz | cid:1768263610411 |
1 | 230% | 80d | 11 |
nid:1768182517405
c1
LinAlg
unique
3
lapses
2/4
users
228%
ease
nid:1768344745873
c1
LinAlg
\(C(A^\top)\)
3
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2/4
users
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nid:1768344745873
Cloze c1
Cloze answer: \(C(A^\top)\)
Q: For a full row rank matrix \(A\), the unique solution to\[{{c1:: \min_{x \in \mathbb{R}^n} ||x||^2 \text{ s.t. } Ax = b}}\] is given by the vector \(\hat{x} = A^\dagger b\). This \(\hat{x}\) is in {{c1:: \(C(A^\top)\)}}. Proof Included
A: Proof By Lemma 6.4.5 we only need to show that \(\hat{x} = A^\dagger b\) satisfies \(A \hat{x} = b\) and that \(\hat{x} \in C(A^\top)\).\(A\hat{x} = AA^\dagger b = AA^\top (AA^\top)^{-1}b = b\) \(\hat{x} = A^\dagger b = A^\top ((AA^\top)^{-1} b) = A^\top y\) for some \(y\) thus \(x \in C(A^\top)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768344745873 |
2 | 210% | 122d | 14 |
| niklas | cid:1768302903149 |
1 | 230% | 2d | 6 |
nid:1772546471834
c1
A&W
gibt es mindestens einen Pfad mit Start- und Endkante in \( ...
3
lapses
2/4
users
235%
ease
nid:1772546471834
Cloze c1
Cloze answer: gibt es mindestens einen Pfad mit Start- und Endkante in \( M' \)
Q: Seien \( M \), \( M' \) beliebige Matchings.Betrachte den Teilgraphen mit Kantenmenge \( M \oplus M' \).Falls \( |M| < |M'| \), so {{c1::gibt es mindestens einen Pfad mit Start- und Endkante in \( M' \)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772546471834 |
2 | 210% | 19d | 15 |
| niklas | cid:1772569386186 |
1 | 260% | 20d | 8 |
nid:1773307908373
IO r1
A&W
[Image Occlusion region 1]
3
lapses
2/4
users
228%
ease
nid:1773307908373
Cloze c1
Q: {{c1::image-occlusion:rect:left=.1376:top=.5345:width=.6408:height=.0783}}{{c2::image-occlusion:rect:left=.0886:top=.6098:width=.903:height=.2198}}{{c3::image-occlusion:rect:left=.2343:top=.9079:width=.0768:height=.0783}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773307908375 |
2 | 210% | 23d | 14 |
| niklas | cid:1773420068090 |
1 | 245% | 5d | 6 |
nid:1771526674685
c1
A&W
\(v \neq root\), und \(v\) hat ein Kind \(u\) im DFS-Baum mi...
3
lapses
2/4
users
242%
ease
nid:1771526674685
Cloze c1
Cloze answer: \(v \neq root\), und \(v\) hat ein Kind \(u\) im DFS-Baum mit \(low[u] \geq dfs[v]\)
Q: \(v\) ist genau dann Artikulationsknoten, wenn:{{c1::\(v \neq root\), und \(v\) hat ein Kind \(u\) im DFS-Baum mit \(low[u] \geq dfs[v]\)}} oder {{c2::\(v = root\), und \(v\) hat mindestens zwei Kinder im DFS-Baum.}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771535790933 |
2 | 255% | 13d | 14 |
| lorenz | cid:1771526674686 |
1 | 230% | 30d | 12 |
nid:1771362440456
c1
A&W
einen Knoten mit Grad < \(k\)
3
lapses
2/4
users
250%
ease
nid:1771973928588
c1
Analysis
Beweis: Für alle \(a < b\) in \(\mathbb{R}\) existiert ein \...
3
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2/4
users
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nid:1771973928588
Cloze c1
Q: Beweis: Für alle \(a < b\) in \(\mathbb{R}\) existiert ein \(\mathbb{Q}\) mit \(a < q < b\){{c1:: Wähle nach Archimedischem Prinzip \(n \in \mathbb{N}\) so dass \(\frac{1}{n} < b - a\).}}{{c2:: \(\frac{m}{n} \mid m \in \mathbb{Z}\) diese
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928591 |
2 | 210% | 6d | 11 |
| niklas | cid:1771969342907 |
1 | 275% | 42d | 9 |
nid:1772496585510
Analysis
Was ist ein Häufungspunkt?
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nid:1772496585510
Q: Was ist ein Häufungspunkt?
A: Grenzwert einer Teilfolge (Punkt, an den eine Folge immer wieder beliebig nahe herankommt)\[\forall \varepsilon > 0 \forall N \in \mathbb{N}_0 \exists n \geq N \text{ so dass } | a_n - A | < \varepsilon\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772520282861 |
2 | 225% | 90d | 15 |
| lorenz | cid:1772496585510 |
1 | 230% | 33d | 8 |
nid:1771973928498
c1
Analysis
Der Abstand zwischen zwei komplexen Zahlen \(z_1, z_2\) ist ...
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nid:1771973928498
Cloze c1
Q: Der Abstand zwischen zwei komplexen Zahlen \(z_1, z_2\) ist \( d = {{c1:: |z_2 - z_1 | = |z_1 - z_2| ::\text{Beide Formen} }}\).
A: Hier gilt wieder die Dreiecksungleichung: \(|z + w| \leq |z| + |w|\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928498 |
2 | 210% | 9d | 16 |
| niklas | cid:1771970006360 |
1 | 245% | 45d | 5 |
nid:1771973928515
c1
Analysis
Division im Komplexen:\[ \frac{z}{w} = {{c1:: \frac{z \cdot ...
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nid:1774138446805
c1
Analysis
beschränkte Folge reeller Zahlen
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nid:1774138446805
Cloze c1
Cloze answer: beschränkte Folge reeller Zahlen
Q: Jede {{c1::beschränkte Folge reeller Zahlen}} hat {{c2::einen Häufungspunkt und eine konvergente Teilfolge}}.Proof idea included
A: (Bolzano-Weierstrass)Beachte: Dies gilt nur für die 1-norm!Proof Idea: Nested Intervals. Always bisect the interval. Since the sequence is infinite, at least one of the intervals must contain an infinite amount of terms.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1774006423271 |
2 | 225% | 1d | 11 |
| lorenz | cid:1774138446805 |
1 | 230% | 28d | 11 |
nid:1771973928615
c1
Analysis
Ordnungsvollständigkeit:Seien \(A, B \subseteq \mathbb{R}\),...
3
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users
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nid:1771973928615
Cloze c1
Q: Ordnungsvollständigkeit:Seien \(A, B \subseteq \mathbb{R}\), sodass
{{c2:: \(A \neq \emptyset\), \(B \neq \emptyset\)}}
{{c2:: \(\forall a \in A \ \forall b \in B \ : \ a \leq b\)}}
Dann {{c1:: gibt es ein \(c \in \mathbb{R}\), sodass \[ \foral
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771974617624 |
2 | 225% | 22d | 8 |
| lorenz | cid:1772327995619 |
1 | 230% | 33d | 12 |
nid:1771973928518
c1
Analysis
Für \(z \in \mathbb{C}\) gilt: \(z + \bar{z} = {{c1:: 2 \te...
3
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users
242%
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nid:1771777467576
DDCA
How can we convert the expansion of \(F\) to the expansion o...
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ease
nid:1771777467576
Q: How can we convert the expansion of \(F\) to the expansion of \(\overline F\)?
A: \[\begin{array}{r l c r l}
\text{E.g., } F(A,B,C) & = \sum m(3,4,5,6,7) & \longrightarrow & \overline{F}(A,B,C) & = \sum m(0,1,2) \\
& = \prod M(0,1,2) & \longrightarrow & & = \prod M(3,4,5,6,7)
\end{array}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771777467576 |
2 | 210% | 19d | 14 |
| niklas | cid:1771872607280 |
1 | 230% | 3d | 5 |
nid:1772117145754
DDCA
How do we guarantee correct operation of an R-S Latch?
3
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users
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ease
nid:1772117145754
Q: How do we guarantee correct operation of an R-S Latch?
A: We add two more NAND gates.\(Q\) takes the value of \(D\), when write enable (WE) is set to 1.\(S\) and \(R\) can never be 0 at the same time!
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772117145754 |
2 | 210% | 18d | 16 |
| niklas | cid:1772209100527 |
1 | 245% | 6d | 6 |
nid:1766498257927
c1
A&D
it must lie on a cycle
3
lapses
2/4
users
212%
ease
nid:1771364277451
c1
PProg
Scheduling overhead
3
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users
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ease
nid:1771364277451
Cloze c1
Cloze answer: Scheduling overhead
Q: {{c1::Scheduling overhead}} is the {{c2::extra time spent by the system or the algorithm}} to distribute work on {{c3::multiple threads/tasks}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955146 |
2 | 210% | 9d | 10 |
| niklas | cid:1771364277454 |
1 | 245% | 20d | 7 |
nid:1766314094859
DiskMat
For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15)
3
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users
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nid:1766314094859
Q: For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15)
A: The group \(\mathbb{Z}^*_m\) is cyclic if and only if:• \(m = 2\)• \(m = 4\)• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\)) Example: Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime). 2 is a generator.Powers of 2: 2, 4, 8, 16, 13, 7, 14
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094933 |
3 | 175% | 8d | 18 |
nid:1766940295689
c1
DiskMat
empty clause \(\emptyset\) (formula with no literals)
3
lapses
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users
175%
ease
nid:1766940295689
Cloze c1
Cloze answer: empty clause \(\emptyset\) (formula with no literals)
Q: The {{c1::empty clause \(\emptyset\) (formula with no literals)}} corresponds to an {{c2::unsatisfiable formula}}.
A: A disjunction with no disjuncts is false.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295781 |
3 | 175% | 3d | 11 |
nid:1766531635539
c1
A&D
\(\exists\) toposort
3
lapses
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users
190%
ease
nid:1766531635539
Cloze c1
Cloze answer: \(\exists\) toposort
Q: {{c1:: \(\exists\) toposort}} \(\Longleftrightarrow\) {{c2:: \(\lnot \exists\) directed closed walk}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635540 |
3 | 190% | 63d | 14 |
nid:1765372936281
c1
A&D
{{c1:: \(\sum_{i = 1}^{n} i^3\)::Sum}} \(=\) {{c2::\(\frac{...
3
lapses
1/4
users
190%
ease
nid:1765372936281
Cloze c1
Q: {{c1:: \(\sum_{i = 1}^{n} i^3\)::Sum}} \(=\) {{c2::\(\frac{n^2(n + 1)^2}{4}\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936281 |
3 | 190% | 87d | 17 |
nid:1765372936266
c1
A&D
{{c1:: \(\sum_{i = 1}^{n} \sum_{k = 1}^{\textbf{i} } 1\)::Su...
3
lapses
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users
190%
ease
nid:1765372936266
Cloze c1
Q: {{c1:: \(\sum_{i = 1}^{n} \sum_{k = 1}^{\textbf{i} } 1\)::Sum}} \(=\) {{c2:: \(\sum_{i = 1}^n i = \frac{n(n + 1)}{2}\)}}
A: inner loop depends on outer
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936267 |
3 | 190% | 97d | 16 |
nid:1765372936203
c1
A&D
O(k^n)
3
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users
190%
ease
nid:1765372936203
Cloze c1
Cloze answer: O(k^n)
Q: Choose a tight bound!\({{c1::O(k^n)}} \leq {{c2::O(n!)}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936203 |
3 | 190% | 108d | 17 |
nid:1769211470058
A&D
How can you find the upper bound of a geometric series like ...
3
lapses
1/4
users
190%
ease
nid:1769211470058
Q: How can you find the upper bound of a geometric series like \(T = 7^1, 7^2, \ldots, 7^n\)?
A: Use the multiply-subract trick.Mutliply the series by its base: \(7T\)Subtract: \(7T - T = 7^{n+1} - 7^1\) (middle terms cancel)Factor: \(T(7-1) = 7^{n+1} - 7^1\)Divide: \(T = \frac{7^{n+1} - 7^1}{6}\)This trick works even if every term has a constant coefficient.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1769211470058 |
3 | 190% | 98d | 15 |
nid:1766448532960
c2
DiskMat
\(e\) coprime to \(|G|\)
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lapses
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users
190%
ease
nid:1766448532960
Cloze c2
Cloze answer: \(e\) coprime to \(|G|\)
Q: In a finite group the function \(x \rightarrow x^e\) is {{c1:: a bijection}} if {{c2::\(e\) coprime to \(|G|\)}}.For \(x^e = y\), the inverse of \(y\) is {{c3:: the unique \(e\)-th root \(x = y^d\), with \(de \equiv_{|G|} 1\)}}.
A: Proof:We have \(ed = k \cdot |G| + 1\) for some \(k\). Thus, for any \(x \in G\) we have\[(x^e)^d = x^{ed} = x^{k \cdot |G| + 1} = \underbrace{(x^{|G|})^k}_{=1} \cdot x = x\]which means that the function \(y \mapsto y^d\) is the inverse function of the function \(x \mapsto x^e\) (which is hence a bijection). The under-braced term is equal to 1 because the order of \(x\) must divide the order of \(G\) (Lagrange).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448532960 |
3 | 190% | 62d | 17 |
nid:1764867990499
DiskMat
How can we use the CRT to decompose remainders like \(R_{77}...
3
lapses
1/4
users
190%
ease
nid:1764867990499
Q: How can we use the CRT to decompose remainders like \(R_{77}(n)\)?
A: We can decompose \(77 = 11 \cdot 7\) and then calculate:\(R_7(n) = 3\)\(R_{11}(n) = 5\)Then to find the result mod 77, we use the CRT.Find \(11^{-1} \pmod{7} = 2\) (since \(11 \cdot 2 = 22 \equiv 1 \pmod{7}\))Find \(7^{-1} \pmod{11} = 8\) (since \(7 \cdot 8 = 56 \equiv 1 \pmod{11}\))Calculate: \(x = 3 \cdot 11 \cdot 2 + 5 \cdot 7 \cdot 8 = 66 + 280 = 346 \equiv 38 \pmod{77}\)Therefo
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990499 |
3 | 190% | 72d | 18 |
nid:1767105269557
DiskMat
What's the definition of an Euclidean domain?
3
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1/4
users
190%
ease
nid:1767105269557
Q: What's the definition of an Euclidean domain?
A: A euclidean domain is an integral domain \(D\) together with a degree function \(d: D \setminus {0} \rightarrow \mathbb{N}\) such that:For every \(a\) and \(b \neq 0\) in \(D\) there exist \(q\) and \(r\) such that \(a = bq + r\) and \(d(r) < d(b)\) or \(r = 0\)For all nonzero \(a\) and \(b\) in \(D\), \(d(a) \leq d(ab)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1767105269557 |
3 | 190% | 93d | 18 |
nid:1768182517631
c3
LinAlg
one unique inverse \(-v\) for all \(v\)
3
lapses
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users
190%
ease
nid:1768182517631
Cloze c3
Cloze answer: one unique inverse \(-v\) for all \(v\)
Q: In a vector space \(V\) three important properties hold:{{c1::\(0v = 0\) for all \(v\)}}{{c2:: there is only one \(0\)}}{{c3:: one unique inverse \(-v\) for all \(v\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182517632 |
3 | 190% | 63d | 16 |
nid:1765553400173
LinAlg
What is a property that always holds for linear transformati...
3
lapses
1/4
users
190%
ease
nid:1765553400173
Q: What is a property that always holds for linear transformations?
A: \(T(0) = 0\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765553400173 |
3 | 190% | 76d | 14 |
nid:1768182517848
c2
LinAlg
\(b \neq 0\)
3
lapses
1/4
users
190%
ease
nid:1768182517848
Cloze c2
Cloze answer: \(b \neq 0\)
Q: If {{c2::\(b \neq 0\)}}, \(\textbf{Sol}(A, b)\) is {{c1::not a subspace of \(\mathbb{R}^n\)}}.
A: Because it doesn't contain the zero vector!If \(b \neq 0\), the the solution space is "shifted" off the origin:
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182517849 |
3 | 190% | 76d | 17 |
nid:1768182518360
LinAlg
How do we find a basis for the nullspace of \(A\)?
3
lapses
1/4
users
190%
ease
nid:1768182518360
Q: How do we find a basis for the nullspace of \(A\)?
A: Compute the RREF form \(R\) of \(A\) (\(MA\) has the same nullspace as \(A\): \(\textbf{N}(A) = \textbf{N}(MA)\))Remove any zero rows (because \(0^\top x = 0\) regardless of \(x\))Solve for \(Rx = 0\):We seperate the matrix into the identity and the "rest". Note that for this we take colu
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182518361 |
3 | 190% | 74d | 16 |
nid:1768182517987
LinAlg
Express \(\text{Sol}(A, b)\) in standard form:
3
lapses
1/4
users
190%
ease
nid:1768182517987
Q: Express \(\text{Sol}(A, b)\) in standard form:
A: \(\textbf{Sol}(A, 0) = \textbf{N}(A)\) as we search for the zeros. We thus first find the nullspace, and then shift it by an arbitrary solution of \(Ax = b\).Let \(s\) be some solution of \(Ax = b\). Then \[ \textbf{Sol}(A, b) = \{s + x : x \in \textbf{N}(A)\} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182517988 |
3 | 190% | 78d | 16 |
nid:1768182518324
c1
LinAlg
unique
3
lapses
1/4
users
190%
ease
nid:1768182518324
Cloze c1
Cloze answer: unique
Q: For \(A\) written in CR-Decomposition \(A = CR'\), \(R'\) is {{c1:: unique::property? and why proof?}}.
A: \(R'\) is unique because the \(C\) is linearly independent and there's only one way to write a vector (the columns of \(A\)) as the linear combination of independent vectors.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182518324 |
3 | 190% | 125d | 18 |
nid:1774487165116
c2
A&W
minimales (gewichtsminimales) perfektes Matching
3
lapses
1/4
users
190%
ease
nid:1774487165116
Cloze c2
Cloze answer: minimales (gewichtsminimales) perfektes Matching
Q: Für \(n\) gerade und \(\ell : \binom{[n]}{2} \to \mathbb{N}_0\) kann man in Zeit \(O({{c1::n^3}})\) ein {{c2::minimales (gewichtsminimales) perfektes Matching}} in \(K_n\) finden.
A: Das ist der Blossom-Algorithmus.Dies wird im Christofides-Algorithmus für das metrische TSP benötigt.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165117 |
3 | 190% | 3d | 14 |
nid:1776171249095
c1
A&W
Eine Zufallsvariable \(X\) mit Dichte\[f_X(i) = \begin{cases...
3
lapses
1/4
users
190%
ease
nid:1776171249095
Cloze c1
Q: Eine Zufallsvariable \(X\) mit Dichte\[f_X(i) = \begin{cases}{{c1:: \frac{e^{-\lambda} \lambda^i}{i!} }} & \text{für } i \in \mathbb{N}_0 \\ 0 & \text{sonst} \end{cases}\]heisst {{c2::poisson-verteilt}} mit Parameter \(\lambda\).Man schreibt das auch als \({{c2::X \sim \text{Po}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776171249097 |
3 | 190% | 2d | 14 |
nid:1776332605880
c1
A&W
Seien \(\delta, \varepsilon > 0\). Falls \({{c1::N \geq 3\,\...
3
lapses
1/4
users
190%
ease
nid:1776332605880
Cloze c1
Q: Seien \(\delta, \varepsilon > 0\). Falls \({{c1::N \geq 3\,\frac{|U|}{|S|} \cdot \frac{1}{\varepsilon^2} \cdot \ln(\tfrac{2}{\delta})}}\), ist die Ausgabe \(Y\) von Target-Shooting mit Wahrscheinlichkeit mindestens \(1 - \delta\) im Intervall \[{
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776332605882 |
3 | 190% | 4d | 13 |
nid:1774358596854
c1
A&W
Seien \(A_1, \ldots, A_n\) paarweise disjunkte Ereignisse un...
3
lapses
1/4
users
190%
ease
nid:1774358596854
Cloze c1
Q: Seien \(A_1, \ldots, A_n\) paarweise disjunkte Ereignisse und sei \(B \subseteq A_1 \cup \cdots \cup A_n\).
Dann gilt: \[\Pr[B] = {{c1::\sum_{i=1}^{n} \Pr[B\mid A_i] \cdot \Pr[A_i]}}.\]
A: Satz von der totalen WahrscheinlichkeitBeispiel: Ziegenproblem
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774358596854 |
3 | 190% | 8d | 17 |
nid:1774917593197
A&W
Sei \(X=\) Anzahl Würfe bis zum ersten Kopf mit \(\Pr[\text{...
3
lapses
1/4
users
190%
ease
nid:1774917593197
Q: Sei \(X=\) Anzahl Würfe bis zum ersten Kopf mit \(\Pr[\text{Kopf}] = p\). Welche Methode verwenden wir bei einem gedächtnislosen Problem wie diesem?
A: Definiere \(K_1\) = "erster Wurf ist Kopf." Wende totale Erwartung bedingt auf \(K_1\) an:\(\mathbb{E}[X \mid K_1] = 1\) (sofort fertig)\(\mathbb{E}[X \mid \overline{K}_1] = 1 + \mathbb{E}[X]\) (gedächtnislos: nach Zahl startet der Prozess identisch neu, plus der eine verbrauchte Wurf)Einsetzen in \(\mathbb{E}[X] = 1 \cdot p + (1 + \mathbb{E}[X])(1-p)\) und Auflösen ergibt \(\mathbb{E}[X] = 1/p\). Vermeidet die direkte Berechnung von \(\sum k \cdot (1-p)^{k-1} p\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917593197 |
3 | 190% | 5d | 17 |
nid:1776174687031
c1
A&W
\(N\) und \(X\) seien zwei unabhängige Zufallsvariablen mit ...
3
lapses
1/4
users
190%
ease
nid:1776174687031
Cloze c1
Q: \(N\) und \(X\) seien zwei unabhängige Zufallsvariablen mit \(W_N \subseteq \mathbb{N}\). Weiter sei\[Z := \sum_{i=1}^{N} X_i,\]wobei \(X_1, X_2, \ldots\) unabhängige Kopien von \(X\) sind.Dann gilt:\[\mathbb{E}[Z] = {{c1::\mathbb{E}[N] \cdot \mathbb{E}[X]}}.\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776174687031 |
3 | 190% | 6d | 12 |
nid:1773310695996
c1
A&W
chromatischer Zahl \(\geq r\)
3
lapses
1/4
users
190%
ease
nid:1773310695996
Cloze c1
Cloze answer: chromatischer Zahl \(\geq r\)
Q: \(\forall k \in \mathbb{N},\ \forall r \in \mathbb{N}\): Es gibt Graphen ohne einen Kreis mit Länge \(\leq k\), aber mit {{c1::chromatischer Zahl \(\geq r\)}}.
A: Lokal sieht der Graph aus wie ein Baum (alle Knoten, die man von einem \(v\) aus in \(k/2\) Schritten erreichen kann).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773310695996 |
3 | 190% | 8d | 18 |
nid:1774917593418
c1
A&W
Für eine beliebige Zufallsvariable \(X\) und \(a, b \in \mat...
3
lapses
1/4
users
190%
ease
nid:1774917593418
Cloze c1
Q: Für eine beliebige Zufallsvariable \(X\) und \(a, b \in \mathbb{R}\) gilt: \[\operatorname{Var}[a \cdot X + b] = {{c1::a^2 \cdot \operatorname{Var}[X]}}\]Proof Included
A: Beweis:\(\operatorname{Var}[X + b] = \mathbb{E}[(X + b - \mathbb{E}[X + b])^2]\) \(= \mathbb{E}[(X - \mathbb{E}[X])^2]\) \(= \operatorname{Var}[X]\) Mit Hilfe von \(\text{Var}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^2\) erhalten wir \(\operatorname{Var}[a \cdot X] = \mathbb{E}[(aX)^2] - \mathbb{E}[aX]^2\) \(= a^2 \mathbb{E}[X^2] - (a\mathbb{E}[X])^2 = a^2 \cdot \operatorname{Var}[X]\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917593418 |
3 | 190% | 8d | 14 |
nid:1772046206585
A&W
Was ist der Speicherbedarf von Hamiltonkreise mit DP?
3
lapses
1/4
users
190%
ease
nid:1772046206585
Q: Was ist der Speicherbedarf von Hamiltonkreise mit DP?
A: \(n\cdot2^n\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772046206585 |
3 | 190% | 17d | 20 |
nid:1772928333399
c1
Analysis
\[ \cos\!\left(\frac{11\pi}{6}\right) = {{c1::\frac{\sqrt{3}...
3
lapses
1/4
users
190%
ease
nid:1772928333399
Cloze c1
Q: \[ \cos\!\left(\frac{11\pi}{6}\right) = {{c1::\frac{\sqrt{3} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333399 |
3 | 190% | 15d | 18 |
nid:1774487165205
Analysis
Was gilt auf dem Rand des Konvergenzkreises \(|x - a| = R\) ...
3
lapses
1/4
users
190%
ease
nid:1774487165205
Q: Was gilt auf dem Rand des Konvergenzkreises \(|x - a| = R\) einer Potenzreihe?
A: Keine allgemeine Aussage - kommt auf den Einzelfall an:\(\sum \frac{x^n}{n}\): divergiert für \(x = 1\), konvergiert für \(x = -1\) (Leibniz)\(\sum \frac{x^n}{n^2}\): konvergiert für alle \(|x| = 1\) (absolut)\(\sum x^n\): divergiert für alle \(|x| = 1\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165206 |
3 | 190% | 1d | 13 |
nid:1774917594689
c1
Analysis
Es sei \(f : \mathbb{D}(f) \to \mathbb{R}\), es sei \(x_0 \i...
3
lapses
1/4
users
190%
ease
nid:1774917594689
Cloze c1
Q: Es sei \(f : \mathbb{D}(f) \to \mathbb{R}\), es sei \(x_0 \in \mathbb{R}\) und es gelte \[{{c1::\mathbb{D}(f) \cap (x_0 - \delta,\, x_0 + \delta) \neq \emptyset \quad \forall \delta > 0}}\]Dann ist \(L \in \mathbb{R}\) der Grenzwert/Limes von \(f(x)\) an der Stelle \(x_0\), falls gilt \[{{c2::\be
A: Beachte, dass die Funktion nicht unbedingt an der Stelle \(x_0\) des Grenzwerts definiert sein muss (siehe Sprungstelle, Definitionslücke).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917594689 |
3 | 190% | 1d | 14 |
nid:1774917595188
c1
Analysis
Unstetigkeitsstelle
3
lapses
1/4
users
190%
ease
nid:1774917595188
Cloze c1
Cloze answer: Unstetigkeitsstelle
Q: Dieser Graph hat eine {{c1::Unstetigkeitsstelle}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917595188 |
3 | 190% | 6d | 16 |
nid:1774138448037
Analysis
Trick: FixpunktSei eine Folge rekursiv definiert durch \(a_1...
3
lapses
1/4
users
190%
ease
nid:1774138448037
Q: Trick: FixpunktSei eine Folge rekursiv definiert durch \(a_1 = c\) und \(a_{n+1} = f(a_n)\).
A: Falls \((a_n)\) konvergiert (z.B. nach Weierstrass), setzt man \(l = \lim_{n \to \infty} a_n \) \(= \lim_{n \to \infty} a_{n+1}\) und erhält die Fixpunktgleichung: \(l = f(l)\) Man löst diese Gleichung nach \(l\) auf und schließt anhand der Eigenschaften der Folge (Vorzeichen, Monotonie, Beschränktheit) aus, welcher Kandidat der tatsächliche Grenzwert ist.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774138448037 |
3 | 190% | 2d | 16 |
nid:1774917594689
c2
Analysis
Es sei \(f : \mathbb{D}(f) \to \mathbb{R}\), es sei \(x_0 \i...
3
lapses
1/4
users
190%
ease
nid:1774917594689
Cloze c2
Q: Es sei \(f : \mathbb{D}(f) \to \mathbb{R}\), es sei \(x_0 \in \mathbb{R}\) und es gelte \[{{c1::\mathbb{D}(f) \cap (x_0 - \delta,\, x_0 + \delta) \neq \emptyset \quad \forall \delta > 0}}\]Dann ist \(L \in \mathbb{R}\) der Grenzwert/Limes von \(f(x)\) an der Stelle \(x_0\), falls gilt \[{{c2::\be
A: Beachte, dass die Funktion nicht unbedingt an der Stelle \(x_0\) des Grenzwerts definiert sein muss (siehe Sprungstelle, Definitionslücke).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917594690 |
3 | 190% | 8d | 16 |
nid:1774487166307
c1
Analysis
Exponentialreihe:\[\exp(z) = {{c1:: \sum_{n=0}^\infty \frac{...
3
lapses
1/4
users
190%
ease
nid:1774487166307
Cloze c1
Q: Exponentialreihe:\[\exp(z) = {{c1:: \sum_{n=0}^\infty \frac{z^n}{n!} }}\]Diese Reihe konvergiert {{c2::absolut für alle \(z \in \mathbb{C}\)::Konvergenztyp}}.
A: (Konvergenzradius \(R = \infty\))
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487166308 |
3 | 190% | 8d | 17 |
nid:1774487166317
c1
Analysis
Die Riemansche-Zeta Funktion Reihe \(\displaystyle\zeta(s) =...
3
lapses
1/4
users
190%
ease
nid:1774487166317
Cloze c1
Q: Die Riemansche-Zeta Funktion Reihe \(\displaystyle\zeta(s) = {{c1:: \sum_{n=1}^\infty \frac{1}{n^s} }}\) konvergiert für {{c2::\(s > 1\)}} und divergiert für {{c2::\(s\leq1\)}}.
A: Oft als Referenzreihe im Vergleichssatz nützlich (wenn Wurzel/Quotient versagen).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487166318 |
3 | 190% | 8d | 17 |
nid:1771973928588
c2
Analysis
Beweis: Für alle \(a < b\) in \(\mathbb{R}\) existiert ein \...
3
lapses
1/4
users
190%
ease
nid:1771973928588
Cloze c2
Q: Beweis: Für alle \(a < b\) in \(\mathbb{R}\) existiert ein \(\mathbb{Q}\) mit \(a < q < b\){{c1:: Wähle nach Archimedischem Prinzip \(n \in \mathbb{N}\) so dass \(\frac{1}{n} < b - a\).}}{{c2:: \(\frac{m}{n} \mid m \in \mathbb{Z}\) diese
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928590 |
3 | 190% | 25d | 17 |
nid:1771973928592
c1
Analysis
Eulersche Formel:\[ \sin(t) = {{c1:: \frac{e^{it} - e^{-it} ...
3
lapses
1/4
users
190%
ease
nid:1771973928592
Cloze c1
Q: Eulersche Formel:\[ \sin(t) = {{c1:: \frac{e^{it} - e^{-it} }{2i} ::\text{Exponentialform} }}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928592 |
3 | 190% | 20d | 20 |
nid:1771778752681
c1
DDCA
either the input A or the input B
3
lapses
1/4
users
190%
ease
nid:1771778752681
Cloze c1
Cloze answer: either the input A or the input B
Q: The output C of a MUX is always connected to {{c1::either the input A or the input B}}.
A: Output value depends on the value of the select line S.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771778752681 |
3 | 190% | 14d | 20 |
nid:1773754343154
c1
PProg
Overhead and synchronization barriers
3
lapses
1/4
users
190%
ease
nid:1773754343154
Cloze c1
Cloze answer: Overhead and synchronization barriers
Q: What factors limit scalability?Sequential part of the program (Amdahl's law)Data structures and algorithmsWork distribution strategyWork scheduling strategy{{c1::Overhead and synchronization barriers}}Memory access and caches
A: How much time is spent on synchronization, locking, context switching?Frequent context switches introduce delays that degrade parallel performance.High contention for shared resources or excessive synchronization barriers create bottlenecks that limit parallel efficiency.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773754343155 |
3 | 190% | 7d | 18 |
nid:1761491477383
DiskMat
When is a relation \(\rho\) on set \(A\) symmetric?
3
lapses
1/4
users
235%
ease
nid:1761491477383
Q: When is a relation \(\rho\) on set \(A\) symmetric?
A: When \(a \ \rho \ b \Longleftrightarrow b \ \rho \ a\) for all \(a, b \in A\), i.e., \(\rho = \hat{\rho}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477384 |
3 | 235% | 65d | 19 |
nid:1762856073615
c1
DiskMat
least (greatest) element of \(A\)
3
lapses
1/4
users
220%
ease
nid:1762856073615
Cloze c1
Cloze answer: least (greatest) element of \(A\)
Q: Consider the poset \((A; \preceq)\).\(a \in A\) is the {{c1::least (greatest) element of \(A\)}} if {{c2::\(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)}}
A: Note that a least or a greatest element need not exist. However, there can be at most one least element, as suggested by the word “the” in the definition. This follows directly from the antisymmetry of \(\preceq\). If there were two least elements, they would be mutually comparable, and hence must be equal.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073623 |
3 | 220% | 4d | 11 |
nid:1765198200589
A&D
How can one get a lower bound for the function \(n!\) ?
3
lapses
1/4
users
250%
ease
nid:1765198200589
Q: How can one get a lower bound for the function \(n!\) ?
A: One could simply take only the largest 90% of elements: \(n! \geq 1 \cdot 2 \cdot ... \cdot n \geq n/10 \cdot ... \cdot n\)\(\geq (n/10)^{0.9n}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765198200589 |
3 | 250% | 27d | 16 |
nid:1765296364773
c1
A&D
O(\log(n))
3
lapses
1/4
users
235%
ease
nid:1765296364773
Cloze c1
Cloze answer: O(\log(n))
Q: Choose a tight bound!\({{c1::O(\log(n))}}\leq {{c2::O(n)}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765296364773 |
3 | 235% | 10d | 17 |
nid:1766000828772
DiskMat
What is the number of generators of \(\mathbb{Z}_n^*\)?
3
lapses
1/4
users
205%
ease
nid:1766000828772
Q: What is the number of generators of \(\mathbb{Z}_n^*\)?
A: 1. Verify that \(\mathbb{Z}_n^*\)is cyclic (iff n = 2, 4, \(p^e\), \(2p^e\), with \(e \ge 1\) and \(p\) is an odd prime)2. If \(\mathbb{Z}_n^*\) is cyclic then it is isomorphic to \(\mathbb{Z}_{\varphi(n)}^+\) (by Lemma) 3. The number of generators of \(\mathbb{Z}_{\varphi(n)}^+\) is \(\varphi(\varphi(n))\) as it is the number of elements coprime to the group order.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766000828772 |
3 | 205% | 3d | 9 |
nid:1766488260288
A&D
Jump Game
3
lapses
1/4
users
205%
ease
nid:1766488260288
Q: Jump Game
A: \(O(n)\) (hyper-optimised version)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766488260289 |
3 | 205% | 1d | 12 |
nid:1766522811173
c1
A&D
a path
3
lapses
1/4
users
205%
ease
nid:1766522811173
Cloze c1
Cloze answer: a path
Q: The shortest walk is always {{c1::a path}}.
A: This is due to the triangle inequality, given that no negative cycles exist.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766522811173 |
3 | 205% | 1d | 9 |
nid:1766580201542
A&D
Cut and Paste Proof of Cut-Property:
3
lapses
1/4
users
190%
ease
nid:1766580201542
Q: Cut and Paste Proof of Cut-Property:
A: Let \((S, V \setminus S)\) be any cut of a graph \(G\).Let \(e = (u,v)\) be the minimal edge crossing this cut. We want to show that \(e \in T\). Assume \(e \not \in T\) for contradiction.Since \(T\) is a spanning tree, \(T \cup {e}\) contains a cycle, crossing the cut at least twice (once via \(e\) and once via another edge \(e’\).)W
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766580201542 |
3 | 190% | 1d | 11 |
nid:1769377096401
c1
EProg
(Husky) dog; Casting further down than dynamic type
3
lapses
1/4
users
190%
ease
nid:1769377096401
Cloze c1
Cloze answer: (Husky) dog; Casting further down than dynamic type
Q: Runtime Errors for Casting:
{{c1::(Husky) dog; Casting further down than dynamic type}}
{{c2:: (Cat) dog; Casting into sibling type}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1769377096402 |
3 | 190% | 2d | 9 |
nid:1771364277486
c2
PProg
during any possible execution, a memory location could be wr...
3
lapses
1/4
users
205%
ease
nid:1771364277486
Cloze c2
Cloze answer: during any possible execution, a memory location could be written from one thread, while concurrently being read or written from another thread.
Q: A program has a {{c1::data race}} if, {{c2::during any possible execution, a memory location could be written from one thread, while concurrently being read or written from another thread.}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277571 |
3 | 205% | 22d | 12 |
nid:1771366536198
c2
A&W
für alle Teilmengen \(X \subseteq E\) mit \(|X| < k\) gilt: ...
3
lapses
1/4
users
250%
ease
nid:1771366536198
Cloze c2
Cloze answer: für alle Teilmengen \(X \subseteq E\) mit \(|X| < k\) gilt: Der Graph \((V, E \setminus X)\) ist zusammenhängend
Q: Ein Graph \(G = (V, E)\) heisst {{c1::\(k\)-kanten-zusammenhängend}}, falls {{c2::für alle Teilmengen \(X \subseteq E\) mit \(|X| < k\) gilt: Der Graph \((V, E \setminus X)\) ist zusammenhängend}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771366536213 |
3 | 250% | 34d | 13 |
nid:1771968911590
Analysis
Archimedisches Prinzip
3
lapses
1/4
users
235%
ease
nid:1771968911590
Q: Archimedisches Prinzip
A: Für \(x \in \mathbb{R}\) und \(y > 0\) existiert \(n \in \mathbb{N}\) mit \(n \cdot y > x\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771968911590 |
3 | 235% | 107d | 17 |
nid:1772569386221
c1
A&W
|E|
3
lapses
1/4
users
235%
ease
nid:1772569386221
Cloze c1
Cloze answer: |E|
Q: In \( k \)-regulären bipartiten Graphen kann man in Zeit \( O({{c1::|E|}}) \) ein perfektes Matching bestimmen.
A: Perfektes Matching in \(k\)-regulären bipartiten GraphenDas Skript erwähnt, dass es einen Algorithmus gibt, der in Zeit \(O(|E|)\) ein perfektes Matching in \(k\)-regulären bipartiten Graphen findet, sagt aber explizit: „Der allgemeine Fall ist deutlich schwieriger."Bewiesen wird im Skript nur der Spezialfall \(k = 2^k\) (Satz 1.54).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386221 |
3 | 235% | 4d | 11 |
nid:1772788241867
c1
Analysis
\[ \tan\!\left(\frac{5\pi}{3}\right) = {{c1::-\sqrt{3} }} \]
3
lapses
1/4
users
175%
ease
nid:1772788241867
Cloze c1
Q: \[ \tan\!\left(\frac{5\pi}{3}\right) = {{c1::-\sqrt{3} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772788241867 |
3 | 175% | 7d | 13 |
nid:1774006743986
c2
Analysis
Eine Folge {{c1::konvergiert}} \(\Longleftrightarrow\) Sie i...
3
lapses
1/4
users
190%
ease
nid:1774006743986
Cloze c2
Q: Eine Folge {{c1::konvergiert}} \(\Longleftrightarrow\) Sie ist {{c2:: eine Cauchy-Folge (für Folgen in \(\mathbb{R}\) und \(\mathbb{C}\))}}.
A: Dies gilt nicht für Folgen in \(\mathbb{Q}\), da sie zum Beispiel auf \(\sqrt{2}\) konvergieren können, was jedoch nicht in \(\mathbb{Q}\) liegt -> ergo konvergiert nie.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1774006743987 |
3 | 190% | 2d | 13 |
nid:1766314077300
c2
A&D
Eulerian walk (Eulerweg)
2
lapses
2/4
users
260%
ease
nid:1766314077300
Cloze c2
Cloze answer: Eulerian walk (Eulerweg)
Q: In graph theory, an {{c2::Eulerian walk (Eulerweg)}} is a {{c1::walk that contains every edge of the graph exactly once}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314077304 |
1 | 245% | 14d | 8 |
| niklas | cid:1762856073668 |
1 | 275% | 39d | 7 |
nid:1766314077314
c1
A&D
take the first element from the unsorted input and place it ...
2
lapses
2/4
users
238%
ease
nid:1766314077314
Cloze c1
Cloze answer: take the first element from the unsorted input and place it correctly in the sorted output
Q: In every iteration of insertion sort, we {{c1::take the first element from the unsorted input and place it correctly in the sorted output}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314077321 |
1 | 245% | 22d | 6 |
| lorenz | cid:1764867989687 |
1 | 230% | 102d | 11 |
nid:1766314094565
DiskMat
What is the Pigeonhole Principle?
2
lapses
2/4
users
230%
ease
nid:1766314094565
Q: What is the Pigeonhole Principle?
A: If a set of \(n\) objects is partitioned into \(k < n\) sets, then at least one of those sets contains at least \(\lceil \frac{n}{k} \rceil\) objects.
(If you have more pigeons than holes, at least one hole must contain multiple pigeons)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094572 |
1 | 230% | 16d | 8 |
| lorenz | cid:1764867989956 |
1 | 230% | 130d | 9 |
nid:1766314094621
DiskMat
How are the rational numbers \(\mathbb{Q}\) defined using eq...
2
lapses
2/4
users
230%
ease
nid:1766314094621
Q: How are the rational numbers \(\mathbb{Q}\) defined using equivalence relations?
A: Let \(A = \mathbb{Z} \times (\mathbb{Z} \setminus \{0\})\) and \((a, b) \sim (c,d) \overset{\text{def}}{\Longleftrightarrow} ad = bc\)
Then: \(\mathbb{Q} \overset{\text{def}}{=} (\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})) / \sim\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094629 |
1 | 230% | 23d | 7 |
| lorenz | cid:1764867990134 |
1 | 230% | 91d | 8 |
nid:1766314094625
DiskMat
When is a poset \((A; \preceq)\) totally ordered (linearly o...
2
lapses
2/4
users
230%
ease
nid:1766314094635
DiskMat
When is a poset \((A; \preceq)\) well-ordered?
2
lapses
2/4
users
215%
ease
nid:1766314094664
c1
DiskMat
\(A^n\) (\(n\)-tuples) is countable
2
lapses
2/4
users
230%
ease
nid:1766314094664
Cloze c1
Cloze answer: \(A^n\) (\(n\)-tuples) is countable
Q: Which operations preserve countability?Let \(A\) and \(A_i\) for \(i \in \mathbb{N}\) be countable sets. Then: {{c1::\(A^n\) (\(n\)-tuples) is countable }}{{c2::\(\bigcup_{i\in \mathbb{N} } A_i\) (countable union) is countabl
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094677 |
1 | 230% | 16d | 9 |
| lorenz | cid:1764867990269 |
1 | 230% | 76d | 9 |
nid:1766314094710
DiskMat
What important property do ideals in \(\mathbb{Z}\) have? (L...
2
lapses
2/4
users
230%
ease
nid:1766314094710
Q: What important property do ideals in \(\mathbb{Z}\) have? (Lemma 4.3)
A: For \(a, b \in \mathbb{Z}\), there exists \(d \in \mathbb{Z}\) such that \((a, b) = (d)\).
Every ideal can be generated by a single integer.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094731 |
1 | 230% | 15d | 9 |
| lorenz | cid:1764867990409 |
1 | 230% | 97d | 9 |
nid:1766314094737
DiskMat
Does \( p \mid a \land q \mid a \land \gcd(p, q) = 1 \implie...
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nid:1766314094737
Q: Does \( p \mid a \land q \mid a \land \gcd(p, q) = 1 \implies pq \mid a \) hold? (Proof included)
A: Yes, but this has to be reproven before using.The proof technique is important. Replacing a neutral element by something it's equal to often is a smart move.
Proof: This is an important result for the exam:
\[p \mid a \land q \mid a \land \gcd(p, q) = 1 \implies pq \mid a\]
Which is the same as saying \(\exists k \in \mathbb{Z}\) such that \(a = pq \cdot k\).
Since \(p \mid a\) and \(q \mid a\), we have:
\[\exists k, k' \in \mathbb{Z} \text{ such
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094761 |
1 | 185% | 9d | 9 |
| niklas | cid:1762453251142 |
1 | 260% | 37d | 11 |
nid:1766314094897
c1
DiskMat
neutral to neutral: \(\psi(e_G) = e_h\)
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nid:1766314094897
Cloze c1
Cloze answer: neutral to neutral: \(\psi(e_G) = e_h\)
Q: Lemma 5.5(ii): A group homomorphism \(\psi: G \rightarrow H\) maps {{c1::inverses to inverses: \(\psi(\widehat{a}) = \widetilde{\psi(a)}\)}} for all \(a\).{{c1::neutral to neutral: \(\psi(e_G) = e_h\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095008 |
1 | 230% | 9d | 8 |
| lorenz | cid:1764867991002 |
1 | 230% | 79d | 10 |
nid:1766314094909
c1
DiskMat
For \(H\) to be a subgroup, it must have closure under {{c1:...
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nid:1766314094983
DiskMat
How can you check if a polynomial of degree \(d\) is irreduc...
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nid:1766314094983
Q: How can you check if a polynomial of degree \(d\) is irreducible?
A: To check if a polynomial of degree \(d\) is irreducible, check all monic irreducible polynomials of degree \(\leq d/2\) as possible divisors.
Why \(d/2\)? If \(a(x) = b(x) \cdot c(x)\) where \(b\) and \(c\) are non-constant, then \(\deg(b) + \deg(c) = \deg(a) = d\). So at least one of \(b\) or \(c\) has degree \(\leq d/2\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095133 |
1 | 230% | 3d | 8 |
| niklas | cid:1764859231520 |
1 | 260% | 21d | 11 |
nid:1766314095043
c1
DiskMat
\(a \ | \ bc\)
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nid:1766940295685
c1
DiskMat
free symbols of a formula
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nid:1766940295760
DiskMat
What does the semantics of a logic define?
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nid:1766940295760
Q: What does the semantics of a logic define?
A: The semantics defines:1. A function \(free\) that assigns to each formula which symbols occur free2. A function \(\sigma\) that assigns truth values to formulas under interpretations3. The meaning and behavior of logical operators
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295901 |
1 | 200% | 3d | 8 |
| niklas | cid:1766418002707 |
1 | 245% | 9d | 10 |
nid:1766940295779
c1
DiskMat
restricted to a certain type of mathematical statement
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nid:1766940295779
Cloze c1
Cloze answer: restricted to a certain type of mathematical statement
Q: A proof system is always {{c1::restricted to a certain type of mathematical statement}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295936 |
1 | 215% | 6d | 9 |
| lorenz | cid:1766448533176 |
1 | 230% | 66d | 8 |
nid:1767089604933
LinAlg
Let \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\) be a linea...
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nid:1767089604933
Q: Let \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\) be a linear transformation. There is a?
A: There is a unique \(m \times n\) matrix A such that \(T = T_A\) meaning that \(T(x) = T_A(x) = Ax\) for all \(x \in \mathbb{R}^n\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1767089604934 |
1 | 230% | 3d | 8 |
| lorenz | cid:1767105283299 |
1 | 230% | 73d | 8 |
nid:1766580142755
A&D
Johnson's Algorithm
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nid:1766580142755
A&D
Johnson's Algorithm
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nid:1766531635612
A&D
What is the optimal substructure property of shortest paths?
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nid:1766531635612
Q: What is the optimal substructure property of shortest paths?
A: Any subpath of a shortest path is itself the shortest path between its endpoints (requires no negative cycles).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635612 |
1 | 230% | 90d | 10 |
| niklas | cid:1766523798264 |
1 | 245% | 40d | 6 |
nid:1765198542527
A&D
Runtime to determine whether an Eulerian walk exists?
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nid:1765372936263
c1
A&D
{{c1:: \(\sum_{j = 1}^{n} \sum_{k = \textbf{j} }^{n} 1\)::Su...
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nid:1765372936263
Cloze c1
Q: {{c1:: \(\sum_{j = 1}^{n} \sum_{k = \textbf{j} }^{n} 1\)::Sum}} \(=\) {{c2:: \(\sum_{j = 1}^n (n - j + 1) = \frac{n(n + 1)}{2}\) }}
A: inner loop depends on outer
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|---|---|---|---|---|---|
| lorenz | cid:1765372936263 |
1 | 230% | 142d | 12 |
| niklas | cid:1765297991538 |
1 | 260% | 68d | 9 |
nid:1766531635566
c1
A&D
\(\geq\)
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nid:1766531635569
A&D
How do we get a topological sorting from DFS?
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nid:1765372936167
c2
A&D
What are the prerequisites for \(f\) and \(g\) to apply l'Hô...
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nid:1765372936167
Cloze c2
Q: What are the prerequisites for \(f\) and \(g\) to apply l'Hôpital's?{{c1::\(f, g\) are differentiable (for sufficiently large \(x\))}}{{c2::\(\lim_{x \to \infty} f(x) = \lim_{x \to \infty} g(x) = \infty\) (or both \(= 0\))}}{{c3::\(g'(x
A: Then: \(\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{f'(x)}{g'(x)}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766535228866 |
1 | 230% | 200d | 9 |
| niklas | cid:1766567318280 |
1 | 230% | 3d | 2 |
nid:1766448532960
c1
DiskMat
a bijection
2
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nid:1766448532960
Cloze c1
Cloze answer: a bijection
Q: In a finite group the function \(x \rightarrow x^e\) is {{c1:: a bijection}} if {{c2::\(e\) coprime to \(|G|\)}}.For \(x^e = y\), the inverse of \(y\) is {{c3:: the unique \(e\)-th root \(x = y^d\), with \(de \equiv_{|G|} 1\)}}.
A: Proof:We have \(ed = k \cdot |G| + 1\) for some \(k\). Thus, for any \(x \in G\) we have\[(x^e)^d = x^{ed} = x^{k \cdot |G| + 1} = \underbrace{(x^{|G|})^k}_{=1} \cdot x = x\]which means that the function \(y \mapsto y^d\) is the inverse function of the function \(x \mapsto x^e\) (which is hence a bijection). The under-braced term is equal to 1 because the order of \(x\) must divide the order of \(G\) (Lagrange).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448532962 |
1 | 230% | 78d | 8 |
| niklas | cid:1766318243105 |
1 | 245% | 6d | 6 |
nid:1764867990975
DiskMat
Give an example of a direct product of groups and explain it...
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nid:1764867990975
Q: Give an example of a direct product of groups and explain its structure.
A: The group \(\langle \mathbb{Z}_5; \oplus \rangle \times \langle \mathbb{Z}_7; \oplus \rangle\):
- Carrier: \(\mathbb{Z}_5 \times \mathbb{Z}_7\)
- Neutral element: \((0, 0)\)
- Operation is component-wise: \((a, b) \star (c, d) = (a \oplus_5 c, b \oplus_7 d)\)
By the Chinese Remainder Theorem, this group is isomorphic to \(\langle \mathbb{Z}_{35}; \oplus \rangle\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990975 |
1 | 230% | 84d | 8 |
| niklas | cid:1764859231274 |
1 | 245% | 23d | 9 |
nid:1764867990481
DiskMat
Why does the Chinese Remainder Theorem require \(m_1, \dots,...
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nid:1764867990481
Q: Why does the Chinese Remainder Theorem require \(m_1, \dots, m_r\) to be pairwise relatively prime?
A: If \(\text{gcd}(m_i, m_j) = d > 1\), then the system could be inconsistent (e.g., \(x \equiv 0 \pmod{6}\) and \(x \equiv 1 \pmod{4}\) has no solution) or have multiple solutions (destroying uniqueness).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990481 |
1 | 230% | 95d | 11 |
| niklas | cid:1762106939367 |
1 | 245% | 26d | 9 |
nid:1764867990386
DiskMat
Give the formal definition of the least common multiple \(\t...
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nid:1764867990386
Q: Give the formal definition of the least common multiple \(\text{lcm}(a, b)\).
A: \[a \mid l \land b \mid l \land \forall m \ ((a \mid m \land b \mid m) \rightarrow l \mid m)\]
\(l\) is a common multiple of \(a\) and \(b\) which divides every common multiple of \(a\) and \(b\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990386 |
1 | 230% | 102d | 8 |
| niklas | cid:1762106939307 |
1 | 260% | 30d | 7 |
nid:1764867989897
DiskMat
What's the difference between \(\equiv\), \(\leftrightarrow\...
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nid:1764867989897
Q: What's the difference between \(\equiv\), \(\leftrightarrow\), and \(\Leftrightarrow\)?
A: \(\equiv\): links formulas to statements (not part of PL itself)
\(\leftrightarrow\): formula → formula (part of PL)
\(\Leftrightarrow\): statement → statement
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867989897 |
1 | 230% | 108d | 9 |
| niklas | cid:1761491477262 |
1 | 260% | 38d | 7 |
nid:1764867990108
DiskMat
How can we test whether a relation is transitive using compo...
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nid:1764867990108
Q: How can we test whether a relation is transitive using composition?
A: A relation \(\rho\) is transitive if and only if \(\rho^2 \subseteq \rho\).
(If all two-step paths are already direct edges, the relation is transitive)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990108 |
1 | 230% | 134d | 11 |
| niklas | cid:1761491477396 |
1 | 230% | 30d | 9 |
nid:1764867990060
DiskMat
How many distinct relations are possible on a finite set \(A...
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nid:1764867990060
Q: How many distinct relations are possible on a finite set \(A\) with \(|A|\) elements?
A: \(2^{|A \times A|} = 2^{|A|^2}\) (because \(\rho \subseteq A \times A\))
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990060 |
1 | 230% | 135d | 9 |
| niklas | cid:1761491477366 |
1 | 260% | 67d | 10 |
nid:1764867991256
c1
DiskMat
\(0\) (all \(a_i\) are \(0\))
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nid:1764867991083
DiskMat
What property do the orders of elements in finite groups hav...
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nid:1764867991083
Q: What property do the orders of elements in finite groups have?
A: Lemma 5.6: In a finite group \(G\), every element has a finite order.
(This doesn't hold for infinite groups - elements can have infinite order.)Proof: Since the order is finite, elements must repeat. That means, there exist \(m > n \geq 0\) s.t. \(g^m = g^n\)\(\implies g^{m-n} = e\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991083 |
1 | 230% | 146d | 9 |
| niklas | cid:1764859231366 |
1 | 275% | 24d | 9 |
nid:1764867990681
c1
DiskMat
The order of an element \(a\) in a group (denoted \(\text{or...
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nid:1764867990681
Cloze c1
Q: The order of an element \(a\) in a group (denoted \(\text{ord}(a)\)) is {{c1::the smallest \(m \ge 1\) such that \(a^m = e\). If such an \(m\) does not exist, \(\text{ord}(a) = \infty\)}}
A: \(\text{ord}(e) = 1\) in any group
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990681 |
1 | 230% | 162d | 9 |
| niklas | cid:1762856073654 |
1 | 215% | 27d | 9 |
nid:1764867991649
c1
LinAlg
row vector; tuple
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nid:1768182518428
c2
LinAlg
Let \(V\) be a finitely generated vector space, \(F \subsete...
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nid:1768182518428
Cloze c2
Q: Let \(V\) be a finitely generated vector space, \(F \subseteq V\) a finite set of linearly independent vectors (note that \(F\) does not need to span \(V\)) and \(G \subseteq V\) a finite set of vectors with \(\textbf{Span}(G) = V\) (but they don't all need to be independent). Then the followin
A: We can use the lemma to argue that there can't be more than \(n\) independent vectors in a space of dimension \(n\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182518428 |
1 | 230% | 93d | 11 |
| niklas | cid:1768146856907 |
1 | 230% | 20d | 5 |
nid:1768344745614
LinAlg
Rewrite \(A^\dagger = R^\dagger C^\dagger\) in terms of \(A\...
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nid:1768344745614
Q: Rewrite \(A^\dagger = R^\dagger C^\dagger\) in terms of \(A\), \(R\), \(C\):
A: \(\begin{aligned}
A^\dagger &= R^\top (RR^\top)^{-1} (C^\top C)^{-1} C^\top \\
&= R^\top (C^\top C R R^\top)^{-1} C^\top \\
&= R^\top (C^\top A R^\top)^{-1} C^\top
\end{aligned}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768344745614 |
1 | 230% | 124d | 13 |
| niklas | cid:1768303035591 |
1 | 230% | 3d | 5 |
nid:1768344745392
c1
LinAlg
a right inverse; A A^\dagger = I
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nid:1768344745392
Cloze c1
Cloze answer: a right inverse; A A^\dagger = I
Q: For \(A \in \mathbb{R}^{m \times n}\) with \(\text{rank}(A) = m\), the pseudo-inverse \(A^\dagger \in \mathbb{R}^{n \times m}\) is {{c1::a right inverse}} of \(A\): \[ {{c1:: A A^\dagger = I }}\]Proof Included
A: Proof Since \(A^\top\) has full column rank, \(((A^\top)^\top A^\top) = AA^\top\) is invertible: \(AA^\dagger = AA^\top(A A^\top)^{-1} = I\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768344745392 |
1 | 230% | 143d | 11 |
| niklas | cid:1768302430259 |
1 | 230% | 2d | 6 |
nid:1772046170351
A&W
Was ist die Laufzeit von Hamiltonkreise mit DP?
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nid:1772547552647
c1
A&W
State of the Art Matching:\( O({{c1::|E|^{1+o(1)} }}) \) für...
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users
230%
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nid:1772547552647
Cloze c1
Q: State of the Art Matching:\( O({{c1::|E|^{1+o(1)} }}) \) für bipartite Graphen \( O({{c2::|V|^{1/2} \cdot |E|}}) \) für generelle Graphen (Hopcroft-Karp)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772547552648 |
1 | 230% | 24d | 11 |
| niklas | cid:1772569386229 |
1 | 230% | 1d | 5 |
nid:1773311287370
c1
A&W
2
2
lapses
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users
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nid:1773311287370
Cloze c1
Cloze answer: 2
Q: Heuristik:\(v_n\) := Knoten vom kleinsten Grad. Lösche \(v_n\).\(v_{n-1}\) := Knoten vom kleinsten Grad im Restgraph. Lösche \(v_{n-1}\). Iteriere.Die Heuristik findet immer eine Färbung mit {{c1::2}} Farben für Bäume.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773311287370 |
1 | 230% | 23d | 8 |
| niklas | cid:1773420068155 |
1 | 230% | 2d | 3 |
nid:1772702804038
A&W
Wahr oder falsch?Jede Brücke in einem Graphen ist zu mindest...
2
lapses
2/4
users
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nid:1772046117792
c1
A&W
Hamiltonkreise mit DPFür alle \(S \subseteq [n]\) mit \(1 \i...
2
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2/4
users
230%
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nid:1772046117792
Cloze c1
Q: Hamiltonkreise mit DPFür alle \(S \subseteq [n]\) mit \(1 \in S\) und alle \(x \in S\) mit \(x \neq 1\):
\[P_{S,x} := {{c1::\begin{aligned} &\begin{cases} 1, & \text{es gibt einen 1-x-Pfad, der genau die Knoten aus } S \text{ enthält} \\ 0, &
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772046117792 |
1 | 230% | 41d | 12 |
| niklas | cid:1772209100374 |
1 | 230% | 19d | 7 |
nid:1772928333503
c1
Analysis
\[ \tan\!\left(\frac{5\pi}{6}\right) = {{c1::-\frac{\sqrt{3}...
2
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users
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nid:1774138446805
c2
Analysis
einen Häufungspunkt und eine konvergente Teilfolge
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users
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nid:1774138446805
Cloze c2
Cloze answer: einen Häufungspunkt und eine konvergente Teilfolge
Q: Jede {{c1::beschränkte Folge reeller Zahlen}} hat {{c2::einen Häufungspunkt und eine konvergente Teilfolge}}.Proof idea included
A: (Bolzano-Weierstrass)Beachte: Dies gilt nur für die 1-norm!Proof Idea: Nested Intervals. Always bisect the interval. Since the sequence is infinite, at least one of the intervals must contain an infinite amount of terms.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774138446806 |
1 | 230% | 23d | 8 |
| niklas | cid:1774006423272 |
1 | 230% | 1d | 7 |
nid:1772928333368
c1
Analysis
\[ \cos\!\left(\frac{5\pi}{6}\right) = {{c1::-\frac{\sqrt{3}...
2
lapses
2/4
users
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nid:1771973928629
c1
Analysis
Um zu beweisen, dass eine komplexe Zahl \(z\) pur imaginär i...
2
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users
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nid:1771776367272
DDCA
What is an implicant?
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nid:1772199267990
c1
DDCA
synchronous
2
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nid:1772199267990
Cloze c1
Cloze answer: synchronous
Q: Most modern computers are {{c1::synchronous}} "machines".
A: State transitions take place at fixed units of time (i.e., potentially delayed response to input, synchronized to an external signal).Controlled in part by a clock, as we will see soon.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772199267990 |
1 | 230% | 35d | 8 |
| niklas | cid:1772209100512 |
1 | 245% | 24d | 6 |
nid:1772113400108
DDCA
How do we implement a logic function in a PLA?
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nid:1772113400108
Q: How do we implement a logic function in a PLA?
A: Connect the output of an AND gate to the input of an OR gate if the corresponding minterm is included in the SOP.This is a simple programmable logic construct.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772113400108 |
1 | 230% | 38d | 8 |
| niklas | cid:1772209100560 |
1 | 230% | 4d | 3 |
nid:1761491477251
c1
DiskMat
F and G are equivalent
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nid:1761491477251
Cloze c1
Cloze answer: F and G are equivalent
Q: {{c2::\(F \equiv G\)}} means {{c1::F and G are equivalent}}, i.e., {{c3:: their truth values are equal for all truth assignments to the propositional symbols appearing in \(F\) or \(G\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477252 |
1 | 245% | 14d | 6 |
| tomas | cid:1765551656856 |
1 | 230% | 7d | 4 |
nid:1762856074477
c1
A&D
Eulerian walk (Eulerweg) that ends at the start vertex
2
lapses
2/4
users
260%
ease
nid:1762856074477
Cloze c1
Cloze answer: Eulerian walk (Eulerweg) that ends at the start vertex
Q: In graph theory, a {{c2::closed Eulerian walk (Eulerzyklus)}} is an {{c1::Eulerian walk (Eulerweg) that ends at the start vertex}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856074511 |
1 | 275% | 154d | 9 |
| tomas | cid:1765551666572 |
1 | 245% | 23d | 5 |
nid:1764745041020
A&D
What is the Cut-Property (Schnittprinzip)?
2
lapses
2/4
users
238%
ease
nid:1764745041020
Q: What is the Cut-Property (Schnittprinzip)?
A: To join a set of disjoint connected components, we need to use an edge to join two of their vertices. The idea is that the cheapest such edge is always a safe edge.This is true only for distinct edge weights!
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764745041020 |
1 | 245% | 67d | 13 |
| tomas | cid:1765551666639 |
1 | 230% | 19d | 6 |
nid:1766573228813
A&D
Johnson's Algorithm
2
lapses
2/4
users
238%
ease
nid:1769446026075
c1
A&D
\(O(n!)\)
2
lapses
2/4
users
238%
ease
nid:1771363637254
c1
Analysis
obere Schranke
2
lapses
2/4
users
238%
ease
nid:1771363637254
Cloze c1
Cloze answer: obere Schranke
Q: Eine {{c1::obere Schranke}} einer Teilmenge \(X \subset \mathbb{R}\) ist ein Element \(y \in \mathbb{R}\) mit der folgenden Eigenschaft: {{c2::\(\forall x \in X\) \(x \leq y\)}}.
A: Eine untere Schranke ist entsprechend mit \(\geq\) definiert.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771363637255 |
1 | 245% | 145d | 10 |
| tomas | cid:1771364083972 |
1 | 230% | 9d | 10 |
nid:1771364277466
c2
PProg
an independently running instance of a program/application, ...
2
lapses
2/4
users
238%
ease
nid:1771364277466
Cloze c2
Cloze answer: an independently running instance of a program/application, typically on the operating system level
Q: A {{c1::process}} is {{c2::an independently running instance of a program/application, typically on the operating system level}}.
A: Similar to a thread, but usually more heavy-weight (since a whole program) and encapsulated in memory.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277500 |
1 | 245% | 18d | 7 |
| tomas | cid:1771363955099 |
1 | 230% | 4d | 5 |
nid:1771364277472
c2
PProg
circular waiting/blocking (no instructions are executed/CPU ...
2
lapses
2/4
users
238%
ease
nid:1771364277472
Cloze c2
Cloze answer: circular waiting/blocking (no instructions are executed/CPU time is used) between threads, so that the system (union of all threads) cannot make any progress anymore
Q: {{c1::Deadlock}} is {{c2::circular waiting/blocking (no instructions are executed/CPU time is used) between threads, so that the system (union of all threads) cannot make any progress anymore}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277521 |
1 | 245% | 19d | 7 |
| tomas | cid:1771363955017 |
1 | 230% | 10d | 15 |
nid:1771872607303
c1
DDCA
the delay between inputs changing and outputs responding
2
lapses
2/4
users
245%
ease
nid:1771872607303
Cloze c1
Cloze answer: the delay between inputs changing and outputs responding
Q: Timing specification describes {{c1::the delay between inputs changing and outputs responding}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607304 |
1 | 260% | 36d | 6 |
| tomas | cid:1771780392238 |
1 | 230% | 2d | 6 |
nid:1771970403211
c3
Analysis
Argument ausrechnen:
\(\varphi = {{c1:: \arctan(\frac{y}{x})...
2
lapses
2/4
users
238%
ease
nid:1771970403211
Cloze c3
Q: Argument ausrechnen:
\(\varphi = {{c1:: \arctan(\frac{y}{x}) }}\) falls \(x > 0\).
\(\varphi = {{c1:: \arctan(\frac{y}{x}) + \pi }}\) falls \(x < 0\) und \(y \ge 0\)
\(\varphi = {{c1:: \arctan(\frac{y}{x}) - \pi }}\) falls \(x < 0\) und \(y < 0\).
A: Achtung: Bei der Umrechnung von Normal- in Polarform ist der Fall \(x=y=0\) ausgeschlossen.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771970403213 |
1 | 245% | 32d | 7 |
| tomas | cid:1772003104419 |
1 | 230% | 4d | 5 |
nid:1766314094616
c1
DiskMat
Complete relation \(A \times A\) → single equivalence class...
2
lapses
1/4
users
225%
ease
nid:1766314094616
Cloze c1
Cloze answer: Complete relation \(A \times A\) → single equivalence class \(A\)
Q: What are the two trivial equivalence relations on a set \(A\)?{{c1:: Complete relation \(A \times A\) → single equivalence class \(A\)}}{{c2:: Identity relation → equivalence classes are all singletons \(\{a\}\
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094623 |
2 | 225% | 7d | 10 |
nid:1766314094664
c2
DiskMat
Which operations preserve countability?Let \(A\) and \(A_i\)...
2
lapses
1/4
users
180%
ease
nid:1766314094664
Cloze c2
Q: Which operations preserve countability?Let \(A\) and \(A_i\) for \(i \in \mathbb{N}\) be countable sets. Then: {{c1::\(A^n\) (\(n\)-tuples) is countable }}{{c2::\(\bigcup_{i\in \mathbb{N} } A_i\) (countable union) is countabl
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094676 |
2 | 180% | 8d | 9 |
nid:1766314094748
DiskMat
Proof method: "Indirect Proof of an Implication"
2
lapses
1/4
users
195%
ease
nid:1766314094748
Q: Proof method: "Indirect Proof of an Implication"
A: Indirect proof of \( S \implies T \): Assume T is false, prove that S is false.Follows from \( (\neg B \to \neg A) \models (A \to B) \)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094774 |
2 | 195% | 8d | 11 |
nid:1766314094775
c2
DiskMat
there exists no \(b \in A\) with \(b \prec a\) (\(b \succ a ...
2
lapses
1/4
users
210%
ease
nid:1766314094775
Cloze c2
Cloze answer: there exists no \(b \in A\) with \(b \prec a\) (\(b \succ a \) )
Q: Consider the poset \((A; \preceq)\) and \( S \subseteq A\).\(a \in A\) is a {{c1::minimal (maximal) element of \(A\)}} if {{c2::there exists no \(b \in A\) with \(b \prec a\) (\(b \succ a \) )}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094821 |
2 | 210% | 3d | 10 |
nid:1766314094778
c2
DiskMat
\(a\) is the greatest (least) element of the set of all lowe...
2
lapses
1/4
users
165%
ease
nid:1766314094778
Cloze c2
Cloze answer: \(a\) is the greatest (least) element of the set of all lower (upper) bounds of \(S\).
Q: Consider the poset \((A; \preceq)\) and \( S \subseteq A\).\(a \in A\) is the {{c1::greatest lower (least upper) bound of \(S\)}} if {{c2::\(a\) is the greatest (least) element of the set of all lower (upper) bounds of \(S\). }}
A: Note that greatest (least) refers to the operation \(\preceq\) and not to order by \(>\) or \(<\) (smaller, bigger).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094827 |
2 | 165% | 9d | 13 |
nid:1766314094948
DiskMat
\(\mathbb{Z}_m^*\) is defined as?
2
lapses
1/4
users
195%
ease
nid:1766314094948
Q: \(\mathbb{Z}_m^*\) is defined as?
A: \[ \overset{\text{def}}{=} \ \{a \in \mathbb{Z}_m \ | \ \gcd(a, m) = 1\} \]This is the set of all elements in \(\mathbb{Z}_m\) that are coprime to \(m\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095082 |
2 | 195% | 1d | 11 |
nid:1766940295803
DiskMat
\(F[x]_{m(x)}^*\) is defined as:
2
lapses
1/4
users
195%
ease
nid:1766940295803
Q: \(F[x]_{m(x)}^*\) is defined as:
A: \[\{ a(x) \in F[x]_{m(x)} \ | \ \gcd(a(x), m(x)) = 1 \}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295973 |
2 | 195% | 2d | 8 |
nid:1765372936275
c2
A&D
{{c1:: \(\sum_{i = 1}^{n} i^2\)::Sum}} \(=\) {{c2::\(\frac{...
2
lapses
1/4
users
210%
ease
nid:1765372936275
Cloze c2
Q: {{c1:: \(\sum_{i = 1}^{n} i^2\)::Sum}} \(=\) {{c2::\(\frac{n(n + 1)(2n + 1)}{6}\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936275 |
2 | 210% | 77d | 15 |
nid:1765372936263
c2
A&D
{{c1:: \(\sum_{j = 1}^{n} \sum_{k = \textbf{j} }^{n} 1\)::Su...
2
lapses
1/4
users
210%
ease
nid:1765372936263
Cloze c2
Q: {{c1:: \(\sum_{j = 1}^{n} \sum_{k = \textbf{j} }^{n} 1\)::Sum}} \(=\) {{c2:: \(\sum_{j = 1}^n (n - j + 1) = \frac{n(n + 1)}{2}\) }}
A: inner loop depends on outer
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936264 |
2 | 210% | 91d | 12 |
nid:1765198542546
A&D
Let \(W\) be a walk and let \(u\) be a vertex, what is \(\te...
2
lapses
1/4
users
210%
ease
nid:1765198542546
Q: Let \(W\) be a walk and let \(u\) be a vertex, what is \(\text{deg}_W(u)\)? (generally)
A: The number of edges incident to \(u\) which are part of \(W\) but repetitions are included, therefore it is possible that \(\text{deg}(u) < \text{deg}_W(u)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765198542546 |
2 | 210% | 107d | 11 |
nid:1766531635457
c3
A&D
Order of calculation (what depends on what entries, what var...
2
lapses
1/4
users
210%
ease
nid:1766531635457
Cloze c3
Cloze answer: Order of calculation (what depends on what entries, what variable incremented first)
Q: Steps of giving a DP solution:{{c1::Define the DP table (dimensions, index, range; meaning of entry): ex: DP[1..n+1][1..k+1]}}{{c2::Computation of entries (Base case, recursive formula, pay attention to bounds!)}}{{c3::Order of calculation (what depends on w
A: SMIROST (Size, Meaning, Initialisation, Recursive Relation, Order, Solution, Time)Smiling Monkey In Red Overall S
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635458 |
2 | 210% | 109d | 11 |
nid:1766531635590
A&D
What is the handshake lemma in directed graphs?
2
lapses
1/4
users
210%
ease
nid:1766531635590
Q: What is the handshake lemma in directed graphs?
A: \[ \sum_{v \in V} \deg_{out}(v) = \sum_{v \in V} \deg_{in}(v) = |E| \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635590 |
2 | 210% | 105d | 14 |
nid:1764867989852
c1
A&D
Dijkstra's
2
lapses
1/4
users
210%
ease
nid:1764867989852
Cloze c1
Cloze answer: Dijkstra's
Q: Prim's Algorithm is similar to {{c1:: Dijkstra's}} with the difference that {{c1:: \(d[v] = \min \{d[v], w(v*, v)\}\) instead of \(d[v^*] + w(v^*, v)\) }}.
A: Dijkstra's find the shortest distance to each vertex, thus it tracks the total distance.Prim's needs to build the MST. Since we add vertex \(v\) to the MST in the loop, we now want to know the new least distance to the MST for each node.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867989852 |
2 | 210% | 111d | 14 |
nid:1766580143889
A&D
Prim's Algorithm
2
lapses
1/4
users
210%
ease
nid:1766580143889
Q: Prim's Algorithm
A: \(O((|V| + |E|) \log |V|)\) (Adjacency List, otherwise \(\Theta(|V|^2)\) like Dijkstra's)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580143891 |
2 | 210% | 123d | 14 |
nid:1764867989799
c1
A&D
complete
2
lapses
1/4
users
210%
ease
nid:1764867989799
Cloze c1
Cloze answer: complete
Q: A graph \(G\) is {{c1::complete}} when it's set of edges is {{c2::\(\{\{u, v\} \ | \ u, v \in V, u \neq v\}\) }}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867989799 |
2 | 210% | 133d | 12 |
nid:1766531635467
A&D
Maximum Subarray Sum
2
lapses
1/4
users
210%
ease
nid:1766531635467
Q: Maximum Subarray Sum
A: \(\Theta(n)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635468 |
2 | 210% | 132d | 15 |
nid:1765372936269
c1
A&D
{{c1:: \(\sum_{i = 1}^{n} i\)::Sum}} \(=\) {{c2::\(\frac{n(...
2
lapses
1/4
users
210%
ease
nid:1765372936269
Cloze c1
Q: {{c1:: \(\sum_{i = 1}^{n} i\)::Sum}} \(=\) {{c2::\(\frac{n(n + 1)}{2}\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936270 |
2 | 210% | 153d | 13 |
nid:1764867991265
c2
DiskMat
integral domain
2
lapses
1/4
users
210%
ease
nid:1764867991265
Cloze c2
Cloze answer: integral domain
Q: The degree of the product of two polynomials is {{c1::equal to the sum of their degrees}} if \(R\) is an {{c2::integral domain}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991265 |
2 | 210% | 56d | 12 |
nid:1764867991385
c2
DiskMat
\(F[x]_{m(x)} =\) {{c2::\(F[x]_{m(x)}^* \cup \{0\}\)}} iff {...
2
lapses
1/4
users
210%
ease
nid:1764867991385
Cloze c2
Q: \(F[x]_{m(x)} =\) {{c2::\(F[x]_{m(x)}^* \cup \{0\}\)}} iff {{c1:: \(m(x)\) is irreducible}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991386 |
2 | 210% | 64d | 12 |
nid:1764867990613
c2
DiskMat
\(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)
2
lapses
1/4
users
210%
ease
nid:1764867990613
Cloze c2
Cloze answer: \(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)
Q: Consider the poset \((A; \preceq)\).\(a \in A\) is the {{c1::least (greatest) element of \(A\)}} if {{c2::\(a \preceq b\) (\(a \succeq b) \) for all \(b \in A\)}}
A: Note that a least or a greatest element need not exist. However, there can be at most one least element, as suggested by the word “the” in the definition. This follows directly from the antisymmetry of \(\preceq\). If there were two least elements, they would be mutually comparable, and hence must be equal.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990614 |
2 | 210% | 74d | 12 |
nid:1764867991211
DiskMat
If \(uv = vu = 1\) for some \(v \in R\) (we write \(v = u^{-...
2
lapses
1/4
users
210%
ease
nid:1764867991211
Q: If \(uv = vu = 1\) for some \(v \in R\) (we write \(v = u^{-1}\)), then \(u\) is a?
A: Unit.
Example The units of \(\mathbb{Z}\) are \(-1\) and \(1\). Therefore \(\mathbb{Z}^* = \{-1, 1\}\). In contrast, \(\mathbb{R}^* = \mathbb{R} \backslash \{0\}\), as we can divide any two numbers.
The set of units of \(R\) is denoted by \(R^*\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991211 |
2 | 210% | 77d | 14 |
nid:1764867990128
DiskMat
What is the quotient set \(A / \theta\)?
2
lapses
1/4
users
210%
ease
nid:1764867990128
Q: What is the quotient set \(A / \theta\)?
A: \[A / \theta \overset{\text{def}}{=} \{[a]_{\theta} \ | \ a \in A\}\]
The set of all equivalence classes of \(\theta\) on \(A\) (also called "\(A\) modulo \(\theta\)" or "\(A\) mod \(\theta\)").
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990128 |
2 | 210% | 83d | 13 |
nid:1764867990878
c2
DiskMat
\(\gcd(a, n) = 1\), i.e. \(a\) and \(n\) are coprime
2
lapses
1/4
users
210%
ease
nid:1764867990878
Cloze c2
Cloze answer: \(\gcd(a, n) = 1\), i.e. \(a\) and \(n\) are coprime
Q: We can reduce the exponent \(a^m\) modulo \(n\) by {{c1::the \(\text{ord}(a)\)}} iff. {{c2::\(\gcd(a, n) = 1\), i.e. \(a\) and \(n\) are coprime}}.
A: \((a^{\operatorname{ord}(a)})^q \cdot a^r \equiv_n a^r\)This is because if \(\gcd(a, n) = 1\) then there exists an \(m\) for which \(a^m = e\) (same as for the mult. inverse since \(a^{m-1}\) is the inverse).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990879 |
2 | 210% | 88d | 15 |
nid:1766448532935
c2
DiskMat
a basis \(g\), which is then exponentiated
2
lapses
1/4
users
210%
ease
nid:1766448532935
Cloze c2
Cloze answer: a basis \(g\), which is then exponentiated
Q: The Diffie-Hellman Key-Agreement selects two public values:{{c1:: a large prime \(p\)}}{{c2:: a basis \(g\), which is then exponentiated}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448532936 |
2 | 210% | 83d | 13 |
nid:1764867991346
DiskMat
In a field, you can:
2
lapses
1/4
users
210%
ease
nid:1764867991346
Q: In a field, you can:
A: add
subtract
multiply
divide by any nonzero element.
You can divide, because in a field the multiplicative monoid is also a group (without \(0\), thus \(0\) cannot be divided by - no inverse).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991346 |
2 | 210% | 97d | 12 |
nid:1764867990871
DiskMat
How does one show the injectivity of a function?
2
lapses
1/4
users
210%
ease
nid:1764867990871
Q: How does one show the injectivity of a function?
A: Assume \(a \not= b\) and show that\(f(a) \neq f(b)\). Equivalently (by contrapositive), assume \(f(a) = f(b)\) and show that \(a = b\).Example: \(f(x) = 2x\), if \(f(a) = f(b)\), then \(2a = 2b\), which implies \(a = b\). Hence \(f\) is injective.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990871 |
2 | 210% | 108d | 15 |
nid:1764867991333
DiskMat
How is Lagrange interpolation for polynomials in a field def...
2
lapses
1/4
users
210%
ease
nid:1764867991333
Q: How is Lagrange interpolation for polynomials in a field defined?
A: Let \(\beta_i = a(\alpha_i)\) for \(i = 1, \dots, d+1\) where \(\alpha_i\) distinct for all \(i.\)\(a(x)\) is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial \(u_i(x)\) is: \[u_i(x) = \frac{(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}{(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}\]
Note tha
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765655178920 |
2 | 210% | 111d | 15 |
nid:1764867991070
c1
DiskMat
it has "volle Ordung"
2
lapses
1/4
users
210%
ease
nid:1764867991070
Cloze c1
Cloze answer: it has "volle Ordung"
Q: If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is \(|G|\)}}, {{c1:: it has "volle Ordung"}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991071 |
2 | 210% | 110d | 13 |
nid:1764867991398
DiskMat
When is there a finite field with \(q\) elements?
2
lapses
1/4
users
210%
ease
nid:1764867991398
Q: When is there a finite field with \(q\) elements?
A: \(\text{GF}(q)\) is only finite if and only if \(q\) is a power of a prime, i.e. \(q = p^k\) for \(p\) prime.
Any two fields of the same size \(q\) are isomorphic.Why: to construct an extension field, use \(\mathbb{Z}_p\) for coefficients. To be a field, \(p\) must be prime. In a polynomial with degree \(k-1\), each coefficient can take any of the \(p\) values from the coefficient field.
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|---|---|---|---|---|---|
| lorenz | cid:1764867991398 |
2 | 210% | 110d | 14 |
nid:1769307700918
c1
EProg
false since it implies everything
2
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Cloze c1
Cloze answer: false since it implies everything
Q: The strongest precondition is {{c1::false since it implies everything}}.
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|---|---|---|---|---|---|
| lorenz | cid:1769307700918 |
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nid:1768944601191
c1
LinAlg
diagonalisable; the EW \(1\) has algebraic multiplicity 2 bu...
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nid:1768944601191
Cloze c1
Cloze answer: diagonalisable; the EW \(1\) has algebraic multiplicity 2 but geometric multiplicity 1
Q: \(A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) is invertible but not {{c1::diagonalisable}} since {{c1::the EW \(1\) has algebraic multiplicity 2 but geometric multiplicity 1}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
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| lorenz | cid:1768944601191 |
2 | 210% | 61d | 12 |
nid:1768344745505
c1
LinAlg
For \(A \in \mathbb{R}^{m \times n}\) with \(\text{rank}(A) ...
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Cloze c1
Q: For \(A \in \mathbb{R}^{m \times n}\) with \(\text{rank}(A) = n\), we define the pseudo-inverse \(A^\dagger \in \mathbb{R}^{n \times m}\) as \[ A^\dagger = {{c1::(A^\top A)^{-1} A^\top }}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
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| lorenz | cid:1768344745505 |
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nid:1764867991551
LinAlg
What special conditions (other than the 3 basic conditions) ...
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nid:1764867991551
Q: What special conditions (other than the 3 basic conditions) make a set of vectors linearly dependent?
A: If:one of the vectors is 0one vector \(\textbf{v}\) is contained twice
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|---|---|---|---|---|---|
| lorenz | cid:1764867991551 |
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nid:1768182518208
LinAlg
Give an example of a non-finitely generated vector space:
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nid:1768182518208
Q: Give an example of a non-finitely generated vector space:
A: \(\mathbb{R}[x]\) is not finitely generated for example.
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|---|---|---|---|---|---|
| lorenz | cid:1768182518208 |
2 | 210% | 78d | 14 |
nid:1768182517485
c1
LinAlg
singular
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nid:1768182517485
Cloze c1
Cloze answer: singular
Q: A matrix \(A\) that is not invertible is called {{c1:: singular}}.
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| lorenz | cid:1768182517485 |
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nid:1768182517514
c2
LinAlg
A vector space \(V\) is called {{c1::finitely generated}} if...
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Cloze c2
Q: A vector space \(V\) is called {{c1::finitely generated}} if {{c2::there exists a finite subset \(G \subseteq V\) with \(\textbf{Span}(G) = V\)}}.
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| lorenz | cid:1768182517515 |
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nid:1768870077025
c1
LinAlg
a complete set of real eigenvectors if and only if \(B\) doe...
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Cloze c1
Cloze answer: a complete set of real eigenvectors if and only if \(B\) does
Q: \(A \in \mathbb{R}^{n \times n}\) and \(B \in \mathbb{R}^{n \times n}\) are similar matrices. The matrix \(A\) has {{c1::a complete set of real eigenvectors if and only if \(B\) does :: EVs}}. Proof Included
A: Proof \(\lambda, v\) EW, EV pair for matrix \(A\) iff \(Av = \lambda v \Leftrightarrow \lambda S^{-1}v = S^{-1}Av = S^{-1}ASS^{-1}v = B(S^{-1}v)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768870077026 |
2 | 210% | 76d | 13 |
nid:1768182517624
c1
LinAlg
Three equivalent statements:{{c1::\(T_A : \mathbb{R}^m \righ...
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Cloze c1
Q: Three equivalent statements:{{c1::\(T_A : \mathbb{R}^m \rightarrow \mathbb{R}^m\) is bijective.::Transformation}}{{c2::There is an \(m \times m\) matrix \(B\) such that \(BA = I\).}}{{c3::The columns of \(A\) are linearly independent.}}
A: The third one can be derived from the fact that if \(BA = I\), there is only a single \(x \in \mathbb{R}^m\) such that \(A \textbf{x} = 0\).It is also intuitively clear that if not all columns were linearly independent, we'd actually have a tall linear transformation and would be losing information.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182517626 |
2 | 210% | 103d | 14 |
nid:1768608740503
c1
LinAlg
imaginary (or zero) eigenvalues
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nid:1768608740503
Cloze c1
Cloze answer: imaginary (or zero) eigenvalues
Q: Real antisymmetric matrices always have {{c1::imaginary (or zero) eigenvalues}}.
A: Antisymmetric means \(A^T=-A\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768608740503 |
2 | 210% | 95d | 14 |
nid:1768608741704
c2
LinAlg
\(\det(A - \lambda I) = 0\)
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Cloze c2
Cloze answer: \(\det(A - \lambda I) = 0\)
Q: Let \(A \in \mathbb{R}^{n \times n}\).\(\lambda \in \mathbb{R}\) is a {{c1::real eigenvalue}} of \(A\) if and only if {{c2::\(\det(A - \lambda I) = 0\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768608741704 |
2 | 210% | 100d | 13 |
nid:1768263610432
c1
LinAlg
making sure that the sum of all the \(t_k = 0\), which can b...
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nid:1768263610432
Cloze c1
Cloze answer: making sure that the sum of all the \(t_k = 0\), which can be achieved by shifting the graph on the x-axis
Q: If the columns of \(A\) are pairwise orthogonal, we get \(A^\top A\) a diagonal matrix which is very easy to invert, i.e. makes Least Squares easier.We can convert any \(A\) to have orthogonal columns by {{c1:: making sure that the sum of all the \(t_k = 0\), which can
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|---|---|---|---|---|---|
| lorenz | cid:1768263610432 |
2 | 210% | 106d | 14 |
nid:1768608739788
LinAlg
What is special about the characteristic polynomial?
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nid:1768608739788
Q: What is special about the characteristic polynomial?
A: The characteristic polynomial is always monic.The polynomial \(\det(A - zI)\) has a leading \((-1)\) if the degree is odd. Therefore working with the characteristic one is easier.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768608739788 |
2 | 210% | 115d | 15 |
nid:1768263611378
LinAlg
Intuition on where the normal equations \(A^\top A\hat{x} = ...
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Q: Intuition on where the normal equations \(A^\top A\hat{x} = A^\top b\) come from:
A: In the previous case, we had \(\mathbf{e} = (\mathbf{b} - proj_S(\mathbf{b})) \ \bot \ \mathbf{a}\). Here, the same orthogonality condition holds for all columns of \(A\) (that we are projecting on).This is the same as stating \(A^\top (\mathbf{b} - proj_S(\mathbf{b})) = 0\) which by substituting \(proj_S(b) = \mathbf{p} = A \mathbf{\hat{x}}\) gives \(A^\top \mathbf{b} - A^\top A\mathbf{\hat{x}} = 0\) which we can restate as \(A^\top A \mathbf{\hat{x}} = A^\top \ma
| User | Card ID | Lapses | Ease | Interval | Reviews |
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| lorenz | cid:1768263611378 |
2 | 210% | 135d | 14 |
nid:1768263611201
c1
LinAlg
\(Ax\) to be the projection of \(b\) onto \(C(A)\)
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nid:1768263611201
Cloze c1
Cloze answer: \(Ax\) to be the projection of \(b\) onto \(C(A)\)
Q: When solving Least Squares (asking for a minimiser of \(||Ax - b||^2\)) we are asking {{c1::\(Ax\) to be the projection of \(b\) onto \(C(A)\)}}.
A: Least Squares is basically projection without multiplying by \(A\) at the end.It's also basically the Pseudoinverse.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768263611201 |
2 | 210% | 132d | 14 |
nid:1774487164708
c1
A&W
Die Anzahl der geordneten Auswahlen von \(k\) aus \(n\) Obje...
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nid:1774487164708
Cloze c1
Q: Die Anzahl der geordneten Auswahlen von \(k\) aus \(n\) Objektenohne Zurücklegen (Reihenfolge wichtig) ist:\[P(n, k) = {{c1::\frac{n!}{(n-k)!} = n \cdot (n-1) \cdots (n-k+1) }}\]
A: Beispiel: Wie viele 3-stellige PINs aus den Ziffern 0–9 ohne Wiederholung?\(P(10,3) = 10 \cdot 9 \cdot 8 = 720\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487164709 |
2 | 210% | 4d | 10 |
nid:1774631277013
c1
A&W
Seien \(A_1,\ldots,A_n\) paarweise disjunkt, \(B\subseteq\bi...
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nid:1774631277013
Cloze c1
Q: Seien \(A_1,\ldots,A_n\) paarweise disjunkt, \(B\subseteq\bigcup A_i\), \(\Pr[B]>0\). Dann gilt für jedes \(i\):\[ \Pr[A_i|B] = {{c1::\frac{\Pr[B|A_i]\cdot\Pr[A_i]}{\sum_{j=1}^{n}\Pr[B|A_j]\cdot\Pr[A_j]} }}. \]Proof Included
A: (Satz von Bayes)Proof: Nach Definition gilt \(\Pr[A_i|B]=\Pr[A_i\cap B]/\Pr[B]\). Zähler: \(\Pr[A_i\cap B]=\Pr[B|A_i]\cdot\Pr[A_i]\). Nenner: \(\Pr[B]=\sum_j\Pr[B|A_j]\Pr[A_j]\) (totale Wahrscheinlichkeit). \(\square\)Zentrale Anwendung: Die Konditionierungsrichtung "umkehren" - von \(\Pr[B|A_i]\) (leicht zu messen) zu \(\Pr[A_i|B]\) (was wir wissen wollen).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277013 |
2 | 210% | 7d | 11 |
nid:1774631277135
A&W
Falls \(A\) und \(B\) unabhängig sind, beweise, dass \(\bar{...
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nid:1774631277135
Q: Falls \(A\) und \(B\) unabhängig sind, beweise, dass \(\bar{A}\) und \(B\) ebenfalls unabhängig sind. Proof Included
A: Zu zeigen: \(\Pr[\bar{A}\cap B]=\Pr[\bar{A}]\cdot\Pr[B]\).\[\begin{gathered}\Pr[\bar{A}\cap B] = \Pr[B] - \Pr[A\cap B] \\ = \Pr[B] - \Pr[A]\Pr[B] \\ = (1-\Pr[A])\Pr[B] = \Pr[\bar{A}]\Pr[B]. \quad\square\end{gathered}\]Folgerung: Falls \(A_1,\ldots,A_n\) gemeinsam unabhängig sind, so auch jede Familie, die durch Ersetzen einiger \(A_i\) durch \(\bar{A}_i\) entsteht (Lemma 2.23).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277135 |
2 | 210% | 4d | 9 |
nid:1776171326030
c1
A&W
Eine Zufallsvariable \(X\) mit Dichte\[f_X(i) = \begin{cases...
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Cloze c1
Q: Eine Zufallsvariable \(X\) mit Dichte\[f_X(i) = \begin{cases} {{c1::p \cdot (1 - p)^{i-1} }} & \text{für } i \in \mathbb{N} \\ 0 & \text{sonst} \end{cases}\]heisst {{c2::geometrisch verteilt}} mit Erfolgswahrscheinlichkeit \(p\).Man schreibt das auch als \({{c2::X \sim \t
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776171326032 |
2 | 210% | 2d | 11 |
nid:1776175078408
c1
A&W
nicht-negative
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nid:1776175078408
Cloze c1
Cloze answer: nicht-negative
Q: Für jede {{c1::nicht-negative}} Zufallsvariable \(X\) und alle \(t > 0\), gilt\[\Pr\left[X \geq t\right] \leq {{c2::\frac{\mathbb{E}[X]}{t} }}.\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776175078408 |
2 | 210% | 6d | 9 |
nid:1773311655486
c1
A&W
|E|
2
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nid:1773311655486
Cloze c1
Cloze answer: |E|
Q: Jeder Graph kann in Zeit \(O({{c1::|E|}})\) mit \(\Delta(G)+1\) Farben gefärbt werden.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773311655486 |
2 | 210% | 16d | 12 |
nid:1773914249130
c1
A&W
Für Ereignisse \(A_1, \ldots, A_n\) (mit \(n \geq 2\)) gilt\...
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nid:1773914249130
Cloze c1
Q: Für Ereignisse \(A_1, \ldots, A_n\) (mit \(n \geq 2\)) gilt\[\Pr\left[\bigcup_{i=1}^{n} A_i\right] = {{c1::\sum_{\ell=1}^{n} (-1)^{\ell+1} \cdot \sum_{1 \leq i_1 < \cdots < i_\ell \leq n} \Pr[A_{i_1} \cap \cdots \cap A_{i_\ell}]}}\]
A: \[= \sum_{i=1}^{n} \Pr[A_i] - \sum_{1 \leq i_1 < i_2 \leq n} \Pr[A_{i_1} \cap A_{i_2}] + \ldots - \ldots + \ldots + (-1)^{n+1} \cdot \Pr[A_1 \cap \cdots \cap A_n].\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773914249130 |
2 | 210% | 6d | 14 |
nid:1774631269375
c3
A&W
Die Rekursionsformel des Pascalschen Dreiecks lautet: \[\bin...
2
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nid:1774631269375
Cloze c3
Q: Die Rekursionsformel des Pascalschen Dreiecks lautet: \[\binom{n}{k} = {{c3::\binom{n-1}{k-1} + \binom{n-1}{k} }}\]
A: Intuition: Fixiere Element \(x\).\(x\) dabei → noch \(k-1\) aus \(n-1\) wählen\(x\) nicht dabei → alle \(k\) aus \(n-1\) wählenPascalsches Dreieck (Eintrag in Zeile \(n\), Position \(k\) ist \(\binom{n}{k}\)):\[\begin{array}{ccccccccc} & & & & 1 \\ & & & 1 & & 1 \\ & & 1 & & 2 & & 1 \\ &
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631269375 |
2 | 210% | 4d | 10 |
nid:1774487164704
c1
A&W
Summe von Indikatoren
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nid:1774487164704
Cloze c1
Cloze answer: Summe von Indikatoren
Q: Um \(\mathbb{E}[X]\) zu berechnen, schreibe \(X\) als {{c1::Summe von Indikatoren}}:\[X = {{c1::X_{A_1} + X_{A_2} + \cdots + X_{A_n} }},\]dann gilt per Linearität:\[\mathbb{E}[X] = {{c2::\Pr[A_1] + \Pr[A_2] + \cdots + \Pr[A_n] }} \]
A: Unabhängigkeit nicht nötig!Beispiel: Erwartete Anzahl Fixpunkte einer zufälligen Permutation von \([n]\)?\(X_i = [i \text{ ist Fixpunkt}]\), \(\Pr[X_i = 1] = \frac{1}{n}\), also \(\mathbb{E}[X] = n \cdot \frac{1}{n} = 1\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487164705 |
2 | 210% | 10d | 14 |
nid:1774631277127
A&W
Bei \(m\) fairen Münzwürfen sei \(X\) = Anzahl (möglicherwei...
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nid:1774631277127
Q: Bei \(m\) fairen Münzwürfen sei \(X\) = Anzahl (möglicherweise überlappender) Vorkommen von "KKK" (drei aufeinanderfolgende Köpfe). Bestimme \(\mathbb{E}[X]\).
A: Ansatz: "KKK" kann an Positionen \(i=1,\ldots,m-2\) beginnen. Definiere:\[X_i = \begin{cases}1 & \text{Würfe } i,i+1,i+2 \text{ sind alle Kopf}\\ 0 & \text{sonst}\end{cases}.\]Dann ist \(X=X_1+\cdots+X_{m-2}\).Jeder Term: \(\mathbb{E}[X_i]=\Pr[X_i=1]=(1/2)^3=1/8\).Ergebnis: \(\mathbb{E}[X]=(m-2)\cdot\tfrac{1}{8}=\dfrac{m-2}{8}\).(Die Überlappungen spielen keine Rolle, Linearität des Erwartungswerts erledigt das automatisch.)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277128 |
2 | 210% | 11d | 13 |
nid:1776332392617
c3
A&W
Seien \(X_1, \ldots, X_n\) unabhängige Bernoulli-verteilte Z...
2
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users
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nid:1776332392617
Cloze c3
Q: Seien \(X_1, \ldots, X_n\) unabhängige Bernoulli-verteilte Zufallsvariablen mit \(\Pr[X_i = 1] = p_i\) und \(\Pr[X_i = 0] = 1 - p_i\).
Dann gilt für \(X = \sum_{i=1}^{n} X_i\)\(\Pr[X \geq (1+\delta)\,\mathbb{E}[X]] \;\leq\; {{c1::e^{-\frac{1}{3}\delta^2\,\mathbb{E}[X]} }}\) für alle
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776332392617 |
2 | 210% | 8d | 11 |
nid:1774487164866
c1
A&W
\Pr[A] \cdot \Pr[B \mid A] = \Pr[B] \cdot \Pr[A \mid B]
2
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nid:1774487164866
Cloze c1
Cloze answer: \Pr[A] \cdot \Pr[B \mid A] = \Pr[B] \cdot \Pr[A \mid B]
Q: Für Ereignisse \(A, B\) mit \(\Pr[A], \Pr[B] > 0\) gilt:\[\Pr[A \cap B] = {{c1::\Pr[A] \cdot \Pr[B \mid A] = \Pr[B] \cdot \Pr[A \mid B]}}\]
A: Umgestellt ergibt sich direkt der Satz von Bayes.Beide Seiten sind gleich, weil \(\Pr[A \cap B]\) symmetrisch in \(A\) und \(B\) ist.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487164866 |
2 | 210% | 15d | 13 |
nid:1772496585226
IO r2
A&W
[Image Occlusion region 2]
2
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nid:1772496585226
Cloze c2
Q: {{c1::image-occlusion:rect:left=.186:top=.2984:width=.5344:height=.2754}}{{c2::image-occlusion:rect:left=.183:top=.5891:width=.8119:height=.3672}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772496585228 |
2 | 210% | 23d | 16 |
nid:1773307783473
IO r4
A&W
[Image Occlusion region 4]
2
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users
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nid:1773307783473
Cloze c4
Q: {{c1::image-occlusion:polygon:left=.011:top=.2474:points=.0836,.2506 .4728,.2474 .4728,.3534 .011,.3566 .011,.3052 .0836,.3052}}{{c2::image-occlusion:rect:left=.0572:top=.4433:width=.1363:height=.045}}{{c2::image-occlusion:rect:left=.0924:top=.5815:width=.1869:height=.0514}}{{c3::image-o
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773307783483 |
2 | 210% | 17d | 15 |
nid:1774631277234
A&W
Zeige, dass die bedingten Wahrscheinlichkeiten \(\Pr[\cdot|B...
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users
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nid:1774631277234
Q: Zeige, dass die bedingten Wahrscheinlichkeiten \(\Pr[\cdot|B]\) für ein festes Ereignis \(B\) mit \(\Pr[B]>0\) einen gültigen Wahrscheinlichkeitsraum auf \(\Omega\) definieren.
A: Zu zeigen: \(\sum_{\omega\in\Omega}\Pr[\omega|B]=1\):\[ \sum_{\omega\in\Omega}\Pr[\omega|B] = \sum_{\omega\in\Omega}\frac{\Pr[\omega\cap B]}{\Pr[B]} = \sum_{\omega\in B}\frac{\Pr[\omega]}{\Pr[B]} = \frac{\Pr[B]}{\Pr[B]} = 1. \]Intuition: Bedingen setzt \(\Pr[\omega|B]=0\) für alle \(\omega\notin B\) und reskaliert die verbleibenden Wahrscheinlichkeiten mit \(1/\Pr[B]\), damit sie sich zu 1 summieren.Konsequenz: Alle Wahrscheinlichkeitsregeln (Komplem
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277234 |
2 | 210% | 13d | 13 |
nid:1774631277262
c1
A&W
\Pr[A]
2
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users
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nid:1774631277262
Cloze c1
Cloze answer: \Pr[A]
Q: Für die Indikatorvariable \(X_A\) eines Ereignisses \(A\) gilt:\[ \mathbb{E}[X_A] = {{c1::\Pr[A]}}. \]Proof Included
A: Proof: \(\mathbb{E}[X_A]=1\cdot\Pr[X_A=1]+0\cdot\Pr[X_A=0]=\Pr[A].\quad\square\)Das ist die Brücke zwischen Ereignissen (Wahrscheinlichkeit) und Zufallsvariablen (Erwartungswert): Die Wahrscheinlichkeit eines Ereignisses entspricht dem Erwartungswert seiner Indikatorvariable.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277262 |
2 | 210% | 12d | 11 |
nid:1771526288993
c3
A&W
ggf update, wenn Algorithmus während des backtracks zum Knot...
2
lapses
1/4
users
210%
ease
nid:1771526288993
Cloze c3
Cloze answer: ggf update, wenn Algorithmus während des backtracks zum Knoten zurückkehrt
Q: Berechnung der low-Werte:{{c1::Initialisierung mit dfs-Wert}}{{c2::ggf update, wenn Restkanten gefunden werden}}{{c3::ggf update, wenn Algorithmus während des backtracks zum Knoten zurückkehrt}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771526288995 |
2 | 210% | 23d | 13 |
nid:1772545892871
c2
A&W
abwechselnd Kanten aus \( M \) und nicht aus \( M \) enthält...
2
lapses
1/4
users
210%
ease
nid:1772545892871
Cloze c2
Cloze answer: abwechselnd Kanten aus \( M \) und nicht aus \( M \) enthält und der in von \( M \) nicht überdeckten Knoten beginnt und endet
Q: Ein {{c1::M-augmentierender Pfad}} ist ein Pfad, der {{c2::abwechselnd Kanten aus \( M \) und nicht aus \( M \) enthält und der in von \( M \) nicht überdeckten Knoten beginnt und endet}}.
A: \( \Rightarrow \) durch Tauschen entlang \( M \) können wir das Matching vergrössern
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772545892872 |
2 | 210% | 31d | 13 |
nid:1774487165288
c1
Analysis
Quotient- und Wurzelkriterium versagen bei Reihen vom Typ {{...
2
lapses
1/4
users
210%
ease
nid:1774487165288
Cloze c1
Q: Quotient- und Wurzelkriterium versagen bei Reihen vom Typ {{c1::\(\sum \frac{1}{n^s}\) (p-Reihen)}}, da aufeinanderfolgende Terme asymptotisch gleich schnell wachsen (\(\rho = 1\)).
A: In diesem Fall: Verdichtungssatz oder Grenzwertkriterium verwenden.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165288 |
2 | 210% | 3d | 13 |
nid:1772928333435
c1
Analysis
\[ \sin\!\left(\frac{5\pi}{6}\right) = {{c1::\frac{1}{2} }} ...
2
lapses
1/4
users
210%
ease
nid:1772928333435
Cloze c1
Q: \[ \sin\!\left(\frac{5\pi}{6}\right) = {{c1::\frac{1}{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333435 |
2 | 210% | 20d | 14 |
nid:1774487165306
c1
Analysis
der erste weggelassene Term
2
lapses
1/4
users
210%
ease
nid:1774487165306
Cloze c1
Cloze answer: der erste weggelassene Term
Q: Beim Leibniz-Kriterium gilt die Fehlerabschätzung:\[|S - S_n| \leq {{c1::a_{n+1} }}\]D.h. der Fehler ist höchstens so gross wie {{c1::der erste weggelassene Term}}.
A: Nützlich zur numerischen Approximation alternierender Reihen.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165307 |
2 | 210% | 12d | 13 |
nid:1774138446782
c2
Analysis
sie beschränkt ist
2
lapses
1/4
users
210%
ease
nid:1774138446782
Cloze c2
Cloze answer: sie beschränkt ist
Q: Für eine {{c1:: monotone Folge reeller Zahlen \((a_n)_{n \in \mathbb{N}_0}\)}} gilt: Sie konvergiert genau dann, wenn {{c2::sie beschränkt ist}}.
A: (Weierstrass)Falls die Folge monoton wachsend ist, gilt: \[ \lim_{n \rightarrow \infty} a_n = \sup \{a_n \mid n \in \mathbb{N}_0\} \]Falls die Folge monoton fallend ist, gilt:\[\lim_{n \rightarrow \infty} a_n = \inf \{ a_n \mid n \in \mathbb{N}_0\}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774138446782 |
2 | 210% | 12d | 11 |
nid:1774487165589
c5
Analysis
Eine Potenzreihe hat die Form \({{c5:: \displaystyle\sum_{k=...
2
lapses
1/4
users
210%
ease
nid:1774487165589
Cloze c5
Q: Eine Potenzreihe hat die Form \({{c5:: \displaystyle\sum_{k=0}^\infty c_k (x - a)^k }}\), wobei:\(a\) ist {{c1::der Entwicklungspunkt (Zentrum)}}\(c_0, c_1, \ldots\) sind {{c2::die Koeffizienten}}\(x\) ist {{c3::das Argument}}\((a - R,\, a + R)\) ist {{c
A: Spezialfall \(a = 0\): \(\sum c_k x^k\) - Entwicklungspunkt im Ursprung.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165590 |
2 | 210% | 9d | 11 |
nid:1774631277856
c1
Analysis
Eine Teleskopreihe \(\sum_{k=1}^\infty (b_k - b_{k-1})\) kon...
2
lapses
1/4
users
210%
ease
nid:1774631277856
Cloze c1
Q: Eine Teleskopreihe \(\sum_{k=1}^\infty (b_k - b_{k-1})\) konvergiert genau dann, wenn {{c1::\(\lim_{n\to\infty} b_n\) existiert}}.
A: In diesem Fall gilt \(\sum_{k=1}^\infty (b_k - b_{k-1}) = \lim_{n\to\infty} b_n - b_0\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277856 |
2 | 210% | 8d | 14 |
nid:1772928333345
c1
Analysis
\[ \cos\!\left(\frac{\pi}{6}\right) = {{c1::\frac{\sqrt{3} }...
2
lapses
1/4
users
210%
ease
nid:1772928333345
Cloze c1
Q: \[ \cos\!\left(\frac{\pi}{6}\right) = {{c1::\frac{\sqrt{3} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333346 |
2 | 210% | 21d | 14 |
nid:1772928333361
c1
Analysis
\[ \cos\!\left(\frac{2\pi}{3}\right) = {{c1::-\frac{1}{2} }}...
2
lapses
1/4
users
210%
ease
nid:1772928333361
Cloze c1
Q: \[ \cos\!\left(\frac{2\pi}{3}\right) = {{c1::-\frac{1}{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333361 |
2 | 210% | 28d | 12 |
nid:1772928333383
c1
Analysis
\[ \cos\!\left(\frac{4\pi}{3}\right) = {{c1::-\frac{1}{2} }}...
2
lapses
1/4
users
210%
ease
nid:1772928333383
Cloze c1
Q: \[ \cos\!\left(\frac{4\pi}{3}\right) = {{c1::-\frac{1}{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333383 |
2 | 210% | 20d | 16 |
nid:1772928333456
c1
Analysis
-1
2
lapses
1/4
users
210%
ease
nid:1772928333456
Cloze c1
Cloze answer: -1
Q: \[ \sin\!\left(\frac{3\pi}{2}\right) = {{c1::-1}} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333456 |
2 | 210% | 24d | 14 |
nid:1772928333200
c1
Analysis
\[\tan(x \pm y) = {{c1:: \frac{\tan x \pm \tan y}{1 \mp \tan...
2
lapses
1/4
users
210%
ease
nid:1772928333200
Cloze c1
Q: \[\tan(x \pm y) = {{c1:: \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y} }}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333200 |
2 | 210% | 14d | 16 |
nid:1772928333443
c1
Analysis
\[ \sin\!\left(\frac{7\pi}{6}\right) = {{c1::-\frac{1}{2} }}...
2
lapses
1/4
users
210%
ease
nid:1772928333443
Cloze c1
Q: \[ \sin\!\left(\frac{7\pi}{6}\right) = {{c1::-\frac{1}{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333443 |
2 | 210% | 28d | 12 |
nid:1772928333357
c1
Analysis
0
2
lapses
1/4
users
210%
ease
nid:1772928333357
Cloze c1
Cloze answer: 0
Q: \[ \cos\!\left(\frac{\pi}{2}\right) = {{c1::0}} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333357 |
2 | 210% | 28d | 12 |
nid:1772928333334
c1
Analysis
\cos\theta
2
lapses
1/4
users
210%
ease
nid:1772928333334
Cloze c1
Cloze answer: \cos\theta
Q: \[ \cos(-\theta) = {{c1::\cos\theta}} \]
A: \(\cos\) ist eine gerade Funktion.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333334 |
2 | 210% | 22d | 13 |
nid:1771973928646
Analysis
Wie lautet \(re^{i\varphi}\) ausgeschrieben mit \(\cos\) und...
2
lapses
1/4
users
210%
ease
nid:1771973928646
Q: Wie lautet \(re^{i\varphi}\) ausgeschrieben mit \(\cos\) und \(\sin\)?
A: \(re^{i \varphi} = r (\cos(\varphi) + i \sin(\varphi))\)Herleitung:\[ e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \dots = \sum_{k = 0}^\infty \frac{1}{k!}x^k \]Setzen wir in diese formel \(x = it\) ein, so erhalten wir \(e^{it} = \cos(t) + i \sin(t)\), \(t \in \mathbb{R}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928646 |
2 | 210% | 28d | 12 |
nid:1772928333452
c1
Analysis
\[ \sin\!\left(\frac{4\pi}{3}\right) = {{c1::-\frac{\sqrt{3}...
2
lapses
1/4
users
210%
ease
nid:1772928333452
Cloze c1
Q: \[ \sin\!\left(\frac{4\pi}{3}\right) = {{c1::-\frac{\sqrt{3} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333452 |
2 | 210% | 23d | 15 |
nid:1772496585317
Analysis
Wann konvergiert eine Folge \((a_n)_{n \in \mathbb{N_0}}\)?
2
lapses
1/4
users
210%
ease
nid:1772496585317
Q: Wann konvergiert eine Folge \((a_n)_{n \in \mathbb{N_0}}\)?
A: \[\text{Wenn }\forall \varepsilon > 0 \; \exists N > 0 \text{, so dass } \forall n > N : |a_n - L| < \varepsilon\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772496585317 |
2 | 210% | 22d | 15 |
nid:1772928333387
c1
Analysis
0
2
lapses
1/4
users
210%
ease
nid:1772928333387
Cloze c1
Cloze answer: 0
Q: \[ \cos\!\left(\frac{3\pi}{2}\right) = {{c1::0}} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333387 |
2 | 210% | 31d | 12 |
nid:1772113306847
DDCA
How do we determine the number of AND gates in a PLA?
2
lapses
1/4
users
210%
ease
nid:1772113306847
Q: How do we determine the number of AND gates in a PLA?
A: For an n-input logic function, we need a PLA with 2ⁿ n-input AND gates.Remember SOP: the number of possible minterms
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772113306847 |
2 | 210% | 26d | 14 |
nid:1772117784497
DDCA
Writing to MemoryWhat is \(D_i\) here?
2
lapses
1/4
users
210%
ease
nid:1772117784497
Q: Writing to MemoryWhat is \(D_i\) here?
A: Input.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772117784497 |
2 | 210% | 30d | 13 |
nid:1771779097113
c1
DDCA
OR gates
2
lapses
1/4
users
210%
ease
nid:1771779097113
Cloze c1
Cloze answer: OR gates
Q: Decoders can be combined with {{c1::OR gates}} to build logic functions.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771779097113 |
2 | 210% | 27d | 13 |
nid:1771777021996
DDCA
Convert this function to canonical form:
2
lapses
1/4
users
210%
ease
nid:1771777021996
Q: Convert this function to canonical form:
A: \(\begin{aligned} F(A,B,C) &= \sum m(3,4,5,6,7) \\ &= m3 + m4 + m5 + m6 + m7 \end{aligned}\)\(F = \overline{A}BC + A\overline{B}\overline{C} + A\overline{B}C + AB\overline{C} + ABC\)Note that this isn't minimal form! \(\Rightarrow F = A + BC\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771777021996 |
2 | 210% | 28d | 13 |
nid:1772114228305
c1
DDCA
gating of different signals onto a wire
2
lapses
1/4
users
210%
ease
nid:1772114228305
Cloze c1
Cloze answer: gating of different signals onto a wire
Q: A tri-state buffer enables {{c1::gating of different signals onto a wire}}.
A: It acts like a switch.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772114228305 |
2 | 210% | 35d | 12 |
nid:1774487167488
c3
PProg
Elastic; creates threads on demand, reuses idle ones.
2
lapses
1/4
users
210%
ease
nid:1774487167488
Cloze c3
Cloze answer: Elastic; creates threads on demand, reuses idle ones.
Q: The four standard ExecutorService pool types:newFixedThreadPool(n) - {{c1::Fixed n threads; excess tasks are queued.}}newSingleThreadExecutor() - {{c2::Exactly 1 thread; tasks execute sequentially.}}new
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631279571 |
2 | 210% | 4d | 10 |
nid:1774359509898
c1
PProg
decreasing span without increasing work too much
2
lapses
1/4
users
210%
ease
nid:1774359509898
Cloze c1
Cloze answer: decreasing span without increasing work too much
Q: Designing parallel algorithms is about {{c1::decreasing span without increasing work too much}}.
A: Amdahl's Law describes the limit of speedup due to sequential parts of a program
\(T_\infty\) (span) in the DAG is the practical representation of the "sequential fraction" in Amdahl's Law
\(T_\infty\) is the fundamental cause of the speedup limit - it represents the longest sequential dependency
If we reduce \(T_\infty\), we get closer to ideal speedup
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774359509898 |
2 | 210% | 4d | 10 |
nid:1761491477299
DiskMat
What is the logical rule for proof by contradiction?
2
lapses
1/4
users
240%
ease
nid:1761491477299
Q: What is the logical rule for proof by contradiction?
A: \((\lnot A \rightarrow B) \land \lnot B \models A\)
Alternative: \((A \lor B) \land \lnot B \models A\)
(If assuming \(\lnot A\) leads to something false, then \(A\) must be true)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477300 |
2 | 240% | 14d | 16 |
nid:1761491477305
DiskMat
How does an indirect proof of \(S \Rightarrow T\) work?
2
lapses
1/4
users
240%
ease
nid:1761491477305
Q: How does an indirect proof of \(S \Rightarrow T\) work?
A: An indirect proof assumes that \(T\) is false and proves that \(S\) is false under this assumption. This works because \(\lnot B \rightarrow \lnot A \models A \rightarrow B\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477306 |
2 | 240% | 15d | 16 |
nid:1761491477387
DiskMat
When is a relation \(\rho\) on set \(A\) antisymmetric?
2
lapses
1/4
users
240%
ease
nid:1761491477387
Q: When is a relation \(\rho\) on set \(A\) antisymmetric?
A: When \(a \ \rho \ b \land b \ \rho \ a \Longleftrightarrow a = b\) for all \(a, b \in A\), i.e., \(\rho \cap \hat{\rho} \subseteq \text{id}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477388 |
2 | 240% | 7d | 8 |
nid:1761491477545
DiskMat
List all types of symbols meaning equivalence:
2
lapses
1/4
users
270%
ease
nid:1761491477545
Q: List all types of symbols meaning equivalence:
A: Equivalences\(\equiv\) (formula→statement)\(\leftrightarrow\) (formula→formula)\(\Leftrightarrow\) (statement→statement)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477546 |
2 | 270% | 9d | 16 |
nid:1762856073655
c1
A&D
walk that contains every edge of the graph exactly once
2
lapses
1/4
users
240%
ease
nid:1762856073655
Cloze c1
Cloze answer: walk that contains every edge of the graph exactly once
Q: In graph theory, an {{c2::Eulerian walk (Eulerweg)}} is a {{c1::walk that contains every edge of the graph exactly once}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073669 |
2 | 240% | 90d | 11 |
nid:1762856074658
DiskMat
Sketch step-by-step how Cantor's diagonalization argument ca...
2
lapses
1/4
users
255%
ease
nid:1762856074658
Q: Sketch step-by-step how Cantor's diagonalization argument can be used to prove that the set \(\{0,1\}^\infty\) is uncountable.
A: Proof by contradiction: Assume a bijection to \(\mathbb{N}\) exists.That means there exists for each \(n\in \mathbb{N}\) a corresponding sequence of 0 and 1s, and vice-versa.We now construct a new sequence \(\alpha\) of 0s and 1s, by always taking the \(i\)-th bit from the \(i\)-th sequence, and inverting it.This new sequence does not agree with every existing sequence in at least one place.However, there is no&n
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856074690 |
2 | 255% | 32d | 13 |
nid:1763364140092
c2
A&D
the subgraph obtained after removing it and all it's inciden...
2
lapses
1/4
users
255%
ease
nid:1763364140092
Cloze c2
Cloze answer: the subgraph obtained after removing it and all it's incident edges is disconnected
Q: A vertex in a connected graph is a {{c1::cut vertex}} if {{c2::the subgraph obtained after removing it and all it's incident edges is disconnected}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1763364140092 |
2 | 255% | 70d | 9 |
nid:1764859231476
c2
DiskMat
(nontrivial, \(0 \neq 1\)) commutative ring
2
lapses
1/4
users
240%
ease
nid:1764859231476
Cloze c2
Cloze answer: (nontrivial, \(0 \neq 1\)) commutative ring
Q: An {{c1::integral domain \(D\)}} is a {{c2::(nontrivial, \(0 \neq 1\)) commutative ring}} without {{c3::zerodivisors (\(ab = 0 \implies a = 0 \lor b = 0\))}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231478 |
2 | 240% | 13d | 7 |
nid:1764859231569
DiskMat
What is \(F[x]_{m(x)}\)?
2
lapses
1/4
users
240%
ease
nid:1764859231569
Q: What is \(F[x]_{m(x)}\)?
A: Let \(m(x)\) be a polynomial of degree \(d\) over \(F\). Then: \[F[x]_{m(x)} \ \overset{\text{def}}{=} \ \{a(x) \in F[x] \ | \ \deg(a(x)) < d\}\]
This is the set of all polynomials over \(F\) with degree strictly less than \(d\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231570 |
2 | 240% | 6d | 14 |
nid:1764859231577
DiskMat
What is the cardinality of \(F[x]_{m(x)}\)?
2
lapses
1/4
users
255%
ease
nid:1764859231577
Q: What is the cardinality of \(F[x]_{m(x)}\)?
A: Lemma 5.34: Let \(F\) be a finite field with \(q\) elements and let \(m(x)\) be a polynomial of degree \(d\) over \(F\). Then: \[|F[x]_{m(x)}| = q^d\]
Explanation: Each polynomial of \(\deg d - 1\) has \(d\) coefficients (from \(0, \dots, d - 1\)), and each coefficient can be any of \(q\) elements from \(F\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231578 |
2 | 255% | 13d | 12 |
nid:1764860842101
c1
DiskMat
\(a \ | \ (b + c)\)
2
lapses
1/4
users
225%
ease
nid:1764860842101
Cloze c1
Cloze answer: \(a \ | \ (b + c)\)
Q: In any commutative ring, if \(a \ | \ b\) and \(a \ | \ c\), then {{c1:: \(a \ | \ (b + c)\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764860842101 |
2 | 225% | 4d | 6 |
nid:1765298196426
c2
A&D
{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) {{c2::\(\su...
2
lapses
1/4
users
240%
ease
nid:1765298196426
Cloze c2
Q: {{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) {{c2::\(\sum_{i = 1}^n n \log(n) = n^2 \log n\)::Sum}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765298196426 |
2 | 240% | 2d | 11 |
nid:1765300949586
A&D
Selection Sort
2
lapses
1/4
users
270%
ease
nid:1765300949586
Q: Selection Sort
A: Best Case: \(O(n^2)\)Worst Case: \(O(n^2)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765388611001 |
2 | 270% | 49d | 12 |
nid:1765301119701
A&D
Insertion Sort
2
lapses
1/4
users
225%
ease
nid:1765301119701
Q: Insertion Sort
A: Best Case: \(O(n \log n)\)Worst Case: \(O(n^2)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765388611006 |
2 | 225% | 2d | 5 |
nid:1766314818238
c2
DiskMat
sent over the network to their partner
2
lapses
1/4
users
240%
ease
nid:1766314818238
Cloze c2
Cloze answer: sent over the network to their partner
Q: For Diffie-Hellman key agreement, both Alice and Bob {{c1:: choose \(x_A, x_B\) (their private keys) at random}}.They then compute {{c2:: \(y_A := R_p(g^{x_A})\) and \(y_B\) analogously, which are their public keys}} which is {{c2:: sent over the network to thei
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766314818239 |
2 | 240% | 6d | 9 |
nid:1766485563842
c1
A&D
\(\leq \log_2(n)\)
2
lapses
1/4
users
225%
ease
nid:1766485563842
Cloze c1
Cloze answer: \(\leq \log_2(n)\)
Q: The height of a 2-3 Tree for \(n\) keys is {{c1::\(\leq \log_2(n)\)}} thus \(h={{c2::O(\log(n))::\textbf{O-notation} }}\).
A: Note that for the case \(n = 1\) the root has one leaf with the key.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766485563842 |
2 | 225% | 1d | 8 |
nid:1766488312297
A&D
Longest Common Subsequence
2
lapses
1/4
users
195%
ease
nid:1766488312297
Q: Longest Common Subsequence
A: \(\Theta(n \cdot m)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766488406684 |
2 | 195% | 8d | 12 |
nid:1766488967649
A&D
Edit Distance
2
lapses
1/4
users
210%
ease
nid:1766488967649
Q: Edit Distance
A: \(\Theta(n \cdot m)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766488967650 |
2 | 210% | 9d | 14 |
nid:1766499628105
c2
A&D
\(\exists\) directed closed walk
2
lapses
1/4
users
210%
ease
nid:1766499628105
Cloze c2
Cloze answer: \(\exists\) directed closed walk
Q: {{c1::\(\exists\) back edge}} \(\Longleftrightarrow\){{c2::\(\exists\) directed closed walk}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766499628106 |
2 | 210% | 6d | 9 |
nid:1768239387172
c2
EProg
Generics - type erasure means List<String> becomes just List...
2
lapses
1/4
users
210%
ease
nid:1768239387172
Cloze c2
Cloze answer: Generics - type erasure means List<String> becomes just List at runtime, so the check is impossible t instanceof List<String>
Q: The cases where instanceof causes a compile error:{{c1::Primitives - instanceof only works with reference types}}{{c2::Generics - type erasure means List<String> becomes just List at runtime, so the check is impossible t instanceof List<Str
A: However:Animal a = getanimal() could get a Dog which might implement List thus a instanceof List is not a compile error.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768239387173 |
2 | 210% | 4d | 10 |
nid:1768301518838
c1
A&D
a for loop over all unmarked nodes
2
lapses
1/4
users
225%
ease
nid:1768301518838
Cloze c1
Cloze answer: a for loop over all unmarked nodes
Q: DFS Pseudocode needs to include {{c1:: a for loop over all unmarked nodes}}, when we're not sure whether the graph is connected.
A: Otherwise we aren't visiting ZHKs that aren't connected to our chosen first node.DFS(g):
all vertices unmarked
for u unmarked:
visit(u)
visit(u):
mark u
for v adjacent to u:
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768301518838 |
2 | 225% | 12d | 10 |
nid:1769377807780
c1
EProg
attributes inside a subclass; shadowed
2
lapses
1/4
users
210%
ease
nid:1769377807780
Cloze c1
Cloze answer: attributes inside a subclass; shadowed
Q: We cannot override {{c1::attributes inside a subclass}}, they are {{c1::shadowed}}.
A: class Animal {
String name = "Animal";
String getName() { return "Animal"; }
}
class Dog extends Animal {
String name = "Dog"; // Shadows Animal.name (doesn't override it)
@Override
String getName() { return Dog"; } // Overrides Animal.getName()
}Animal a = new Dog();
System.out.println(a.name);
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1769377807780 |
2 | 210% | 1d | 8 |
nid:1771364277474
c2
PProg
it can never enter a/any critical section
2
lapses
1/4
users
240%
ease
nid:1771364277474
Cloze c2
Cloze answer: it can never enter a/any critical section
Q: A thread {{c1::starves}} if {{c2::it can never enter a/any critical section}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277526 |
2 | 240% | 14d | 7 |
nid:1771364277497
c1
PProg
Concurrency
2
lapses
1/4
users
255%
ease
nid:1771364277497
Cloze c1
Cloze answer: Concurrency
Q: {{c1::Concurrency}} means {{c2::dealing with multiple things at the same time}}.
A: (As opposed to parallelism: doing multiple things at the same time.)Involves managing shared resources and their interactions. Often used interchangeably with parallelism.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277596 |
2 | 255% | 39d | 9 |
nid:1771366536188
c1
A&W
(Knoten-)Zusammenhang
2
lapses
1/4
users
225%
ease
nid:1771366536188
Cloze c1
Cloze answer: (Knoten-)Zusammenhang
Q: Es gilt immer:{{c1::(Knoten-)Zusammenhang}} ≤ {{c2::Kanten-Zusammenhang}} ≤ {{c3::minimaler Grad}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771366536191 |
2 | 225% | 6d | 10 |
nid:1771535790927
c2
A&W
low[w] > dfs[v]
2
lapses
1/4
users
240%
ease
nid:1771535790927
Cloze c2
Cloze answer: low[w] > dfs[v]
Q: Eine Baumkante \(e = (v,w)\) (\(v\) Elternknoten, \(w\) Kindknoten) ist genau dann {{c1::eine Brücke}}, wenn \({{c2::low[w] > dfs[v]}}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771535790930 |
2 | 240% | 33d | 10 |
nid:1771535790935
c1
A&W
|V| + |E|
2
lapses
1/4
users
255%
ease
nid:1771535790935
Cloze c1
Cloze answer: |V| + |E|
Q: Alle low-Werte sind in Zeit \(O({{c1::|V| + |E|}})\) berechenbar.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771535790939 |
2 | 255% | 24d | 9 |
nid:1771872607305
DDCA
Which gate is this?
2
lapses
1/4
users
225%
ease
nid:1771872607305
Q: Which gate is this?
A: XOR
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607305 |
2 | 225% | 14d | 6 |
nid:1771872607315
DDCA
What is a maxterm?
2
lapses
1/4
users
210%
ease
nid:1771872607315
Q: What is a maxterm?
A: A sum (OR) that includes all input variables.\((A + \overline{B} + \overline{C}) \text{ , } (\overline{A} + B + \overline{C}) \text{ , } (A + B + \overline{C})\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607315 |
2 | 210% | 2d | 7 |
nid:1771969133128
c3
Analysis
Properties Absolutbetrag:
{{c1::\(|x| \geq 0\) für alle \(x\...
2
lapses
1/4
users
240%
ease
nid:1771969133128
Cloze c3
Q: Properties Absolutbetrag:
{{c1::\(|x| \geq 0\) für alle \(x\).::PSD}}
{{c2:: \(x \leq |x|, \forall x \in X\)::Vergleich}}
{{c3:: \(|xy| = |x||y| \forall x, y \in \mathbb{R}\).::Multiplikation}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771969133128 |
2 | 240% | 4d | 8 |
nid:1772209100534
c1
DDCA
Moore FSM; Mealy FSM
2
lapses
1/4
users
210%
ease
nid:1772209100534
Cloze c1
Cloze answer: Moore FSM; Mealy FSM
Q: Two types of finite state machines differ in the output logic:{{c1::Moore FSM}}: outputs depend only on the current state{{c1::Mealy FSM}}: outputs depend on the current state and the inputs
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772209100535 |
2 | 210% | 2d | 5 |
nid:1771872607447
c2
PProg
during any possible execution with the same inputs, its obse...
2
lapses
1/4
users
210%
ease
nid:1771872607447
Cloze c2
Cloze answer: during any possible execution with the same inputs, its observable behaviour (results, output, ...) may change if events happen in different order
Q: A program has a {{c1::race condition}} if, {{c2::during any possible execution with the same inputs, its observable behaviour (results, output, ...) may change if events happen in different order}}.
A: E.g. scheduler interactions causing different interleavings, changing network latency
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772532891703 |
2 | 210% | 2d | 5 |
nid:1772569386185
c1
A&W
wechselt ab zwischen Kanten aus \( M \) und \( M' \)
2
lapses
1/4
users
240%
ease
nid:1772569386185
Cloze c1
Cloze answer: wechselt ab zwischen Kanten aus \( M \) und \( M' \)
Q: Seien \( M \), \( M' \) beliebige Matchings.Betrachte den Teilgraphen mit Kantenmenge \( M \oplus M' \).Jeder Knoten hat Grad \( \leq 2 \) \( \Rightarrow \) Kollektion von Pfaden und Kreisen.Jeder
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386185 |
2 | 240% | 27d | 9 |
nid:1772788241826
c1
Analysis
\[ \sin\!\left(\frac{5\pi}{4}\right) = {{c1::-\frac{\sqrt{2}...
2
lapses
1/4
users
240%
ease
nid:1772788241826
Cloze c1
Q: \[ \sin\!\left(\frac{5\pi}{4}\right) = {{c1::-\frac{\sqrt{2} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772788241826 |
2 | 240% | 92d | 7 |
nid:1772788241864
c1
Analysis
1
2
lapses
1/4
users
240%
ease
nid:1772788241864
Cloze c1
Cloze answer: 1
Q: \[ \tan\!\left(\frac{5\pi}{4}\right) = {{c1::1}} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772788241864 |
2 | 240% | 2d | 11 |
nid:1773420068088
IO r2
A&W
[Image Occlusion region 2]
2
lapses
1/4
users
225%
ease
nid:1773420068088
Cloze c2
Q: {{c1::image-occlusion:rect:left=.1376:top=.5345:width=.6408:height=.0783}}{{c2::image-occlusion:rect:left=.0886:top=.6098:width=.903:height=.2198}}{{c3::image-occlusion:rect:left=.2343:top=.9079:width=.0768:height=.0783}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1773420068088 |
2 | 225% | 1d | 6 |
nid:1773420068121
A&W
Wahr oder falsch?Jeder \(k\)-reguläre bipartite Graph \(G = ...
2
lapses
1/4
users
210%
ease
nid:1773420068121
Q: Wahr oder falsch?Jeder \(k\)-reguläre bipartite Graph \(G = (A \cup B, E)\) für \(k \geq 1\) hat ein Matching der Größe \(|A|\).
A: WahrHall-Satz: Da \(G\) \(k\)-regulär ist, hat jeder Knoten in \(X\) Grad \(k\), jeder in \(N(X)\) Grad \(\leq k\). Weil in bipartiten Graphen die Gradsumme links gleich der Gradsumme rechts ist, folgt \(|N(X)| \geq |X|\). Damit ist Halls Bedingung erfüllt und ein Matching der Größe \(|A|\) existiert. Es gilt sogar: \(E\) lässt sich in \(k\) disjunkte perfekte Matchings partitionieren (iteratives Entfernen eines perfekten Matchings liefert jeweils einen \((k-1)\)-regu
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1773420068122 |
2 | 210% | 1d | 8 |
nid:1765551656886
DiskMat
Describe the three steps of a proof by contradiction of stat...
2
lapses
1/4
users
210%
ease
nid:1765551656886
Q: Describe the three steps of a proof by contradiction of statement \(S\).
A: 1. Find a suitable statement \(T\)
2. Prove that \(T\) is false
3. Assume \(S\) is false and prove that \(T\) is true (a contradiction)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551656886 |
2 | 210% | 1d | 6 |
nid:1766410023689
A&D
Runtime of operations in an adjacency matrix?
2
lapses
1/4
users
210%
ease
nid:1766410023689
Q: Runtime of operations in an adjacency matrix?
A: 1. Check if \(uv \in E\): \(O(1)\)2. Vertex \(u\) , find all adjacent vertices in: \(O(n)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766410023689 |
2 | 210% | 30d | 12 |
nid:1771363954980
c3
PProg
{{c1::Divide and conquer style parallelism (also called recu...
2
lapses
1/4
users
210%
ease
nid:1771363954980
Cloze c3
Q: {{c1::Divide and conquer style parallelism (also called recursive splitting)}} means: solve a problem by {{c2::recursively solving smaller sub-problems and combining their results}}.
A: Solve the sub-problems in separate threads to gain a speedup.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955012 |
2 | 210% | 4d | 7 |
nid:1771363955001
c1
PProg
parallelism
2
lapses
1/4
users
210%
ease
nid:1771363955001
Cloze c1
Cloze answer: parallelism
Q: The maximum possible speedup ({{c1::parallelism}}) is {{c2::\(\frac{T_1}{T_\infty} \)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955096 |
2 | 210% | 12d | 11 |
nid:1771363955028
c1
PProg
Work partitioning
2
lapses
1/4
users
210%
ease
nid:1771363955028
Cloze c1
Cloze answer: Work partitioning
Q: {{c1::Work partitioning}} is the {{c2::split-up of a program}} into smaller tasks that can be executed in {{c3::parallel}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955196 |
2 | 210% | 2d | 8 |
nid:1771578182870
c3
PProg
idle time due to task dependencies or waiting for data excha...
2
lapses
1/4
users
210%
ease
nid:1771578182870
Cloze c3
Cloze answer: idle time due to task dependencies or waiting for data exchange
Q: Parallel execution can introduce inefficiencies such as {{c1::communication overhead}}, {{c2::load imbalance}}, and {{c3::idle time due to task dependencies or waiting for data exchange}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771578182872 |
2 | 210% | 1d | 8 |
nid:1771616439344
c4
Advanced Finance
taking too many or too few risks
2
lapses
1/4
users
210%
ease
nid:1771616439344
Cloze c4
Cloze answer: taking too many or too few risks
Q: Agency problems include a manager:{{c1:: not putting in sufficient effort}}{{c2:: wasting money on personal benefits}}{{c3:: overinvesting in search of power or prestige}}{{c4:: taking too many or too few risks}}{{c5:: focusing on short-term results at
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771616439348 |
2 | 210% | 2d | 11 |
nid:1766314077328
c1
A&D
\(u\) reaches \(v\) (erreicht)
1
lapses
1/4
users
245%
ease
nid:1766314077328
Cloze c1
Cloze answer: \(u\) reaches \(v\) (erreicht)
Q: For \(u, v \in V\) we say that {{c1::\(u\) reaches \(v\) (erreicht)}} if {{c2::there is a walk with endpoints \(u\) and \(v\) (or a path)}}.
A: Reachability is an equivalence relation.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314077343 |
1 | 245% | 17d | 7 |
nid:1766314077330
c1
A&D
connected component
1
lapses
1/4
users
215%
ease
nid:1766314077330
Cloze c1
Cloze answer: connected component
Q: A {{c1::connected component}} of \(G\) is a {{c2::equivalence class of the relation defined as follows: \(u = v\) if \(u\) reaches \(v\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314077346 |
1 | 215% | 4d | 7 |
nid:1766314077346
c1
A&D
cut edge
1
lapses
1/4
users
230%
ease
nid:1766314077346
Cloze c1
Cloze answer: cut edge
Q: An edge in a connected graph is a {{c1::cut edge}} if {{c2::the subgraph obtained after removing it (keeping the vertices) is disconnected}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314077373 |
1 | 230% | 10d | 7 |
nid:1766314077348
c2
A&D
A graph \(G\) is {{c1::complete}} when it's set of edges is ...
1
lapses
1/4
users
215%
ease
nid:1766314077348
Cloze c2
Q: A graph \(G\) is {{c1::complete}} when it's set of edges is {{c2::\(\{\{u, v\} \ | \ u, v \in V, u \neq v\}\) }}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314077377 |
1 | 215% | 17d | 11 |
nid:1766314077369
c1
A&D
<
1
lapses
1/4
users
230%
ease
nid:1766314077369
Cloze c1
Cloze answer: <
Q: In BFS enter/leave ordering for all \(v\), enter[v] {{c1:: <}} leave[v].
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314077403 |
1 | 230% | 27d | 7 |
nid:1766314094599
DiskMat
How does the inverse of a relation appear in matrix and grap...
1
lapses
1/4
users
230%
ease
nid:1766314094599
Q: How does the inverse of a relation appear in matrix and graph representations?
A: Matrix: The transpose of the matrix
Graph: Reversing the direction of all edges
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094606 |
1 | 230% | 29d | 6 |
nid:1766314094602
DiskMat
Is composition of relations associative?
1
lapses
1/4
users
230%
ease
nid:1766314094602
Q: Is composition of relations associative?
A: Yes: \(\rho \circ (\sigma \circ \phi) = (\rho \circ \sigma) \circ \phi\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094609 |
1 | 230% | 28d | 6 |
nid:1766314094636
DiskMat
For what types of posets is well-ordering primarily of inter...
1
lapses
1/4
users
230%
ease
nid:1766314094636
Q: For what types of posets is well-ordering primarily of interest?
A: Infinite posets. (Every totally ordered finite poset is automatically well-ordered)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094646 |
1 | 230% | 9d | 9 |
nid:1766314094645
DiskMat
What is the preimage \(f^{-1}(T)\) of a subset \(T \subseteq...
1
lapses
1/4
users
230%
ease
nid:1766314094645
Q: What is the preimage \(f^{-1}(T)\) of a subset \(T \subseteq B\)?
A: \[f^{-1}(T) \overset{\text{def}}{=} \{a \in A \ | \ f(a) \in T\}\]
The set of values in \(A\) that map into \(T\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094656 |
1 | 230% | 14d | 8 |
nid:1766314094698
DiskMat
What is a composite number?
1
lapses
1/4
users
215%
ease
nid:1766314094698
Q: What is a composite number?
A: An integer greater than 1 that is not prime (i.e., it has divisors other than 1 and itself).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094719 |
1 | 215% | 16d | 10 |
nid:1766314094752
c3
DiskMat
Assume that \( S \) is false and prove that \( T \) is true...
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nid:1766314094752
Cloze c3
Cloze answer: Assume that \( S \) is false and prove that \( T \) is true (-> contradiction).
Q: Proof method: Proof by Contradiction1. {{c1:: Find a suitable statement \( T\).}}2. {{c2:: Prove that \( T \) is false.}}3. {{c3:: Assume that \( S \) is false and prove that \( T \) is true (-> contradiction).}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094781 |
1 | 230% | 13d | 10 |
nid:1766314094759
c2
DiskMat
\((a \ \rho \ b \wedge b \ \rho \ c) \implies a \ \rho \ c \...
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nid:1766314094759
Cloze c2
Cloze answer: \((a \ \rho \ b \wedge b \ \rho \ c) \implies a \ \rho \ c \) is true.
Q: A relation is {{c1::transitive}} if {{c2::\((a \ \rho \ b \wedge b \ \rho \ c) \implies a \ \rho \ c \) is true.}}
A: Examples: \( \le, \ge, <, |, \equiv_m\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094793 |
1 | 230% | 19d | 7 |
nid:1766314094762
c2
DiskMat
{{c1::Ein Körper}} ist eine Menge {{c1::\( \mathbb{K}\) mit ...
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nid:1766314094762
Cloze c2
Q: {{c1::Ein Körper}} ist eine Menge {{c1::\( \mathbb{K}\) mit Operationen \(+ , *\)}} mit folgenden Eigenschaften:{{c2::- \( (\mathbb{K}, +)\) ist eine abelsche Gruppe- \( (\mathbb{K} \backslash \{0\}, *)\) ist eine abelsche Gruppe- Distributivität:&
A: Beispiel: \( \mathbb{Q}, \mathbb{R}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094798 |
1 | 230% | 13d | 10 |
nid:1766314094763
c1
DiskMat
injective (or one-to-one)
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nid:1766314094763
Cloze c1
Cloze answer: injective (or one-to-one)
Q: A function is {{c1::injective (or one-to-one)}} if {{c2::for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions"}}.
A: Example: \(f(x) = x\), counterexample: \(f(x) = x^2, x \in \mathbb{R}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094801 |
1 | 230% | 12d | 7 |
nid:1766314094774
c2
DiskMat
totally ordered (also: linearly ordered) by \(\preceq\)
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nid:1766314094774
Cloze c2
Cloze answer: totally ordered (also: linearly ordered) by \(\preceq\)
Q: A poset \((A; \preceq)\) is called {{c2::totally ordered (also: linearly ordered) by \(\preceq\)}} if {{c1::any two elements of the poset are comparable.}}
A: Example: \((\mathbb{Z}; \ge)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094820 |
1 | 230% | 16d | 8 |
nid:1766314094788
DiskMat
Is the set \(\{0,1\}^*\) (finite binary sequences) countable...
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nid:1766314094788
Q: Is the set \(\{0,1\}^*\) (finite binary sequences) countable?
A: Yes. A possible injection to \(\mathbb{N}\) is to add a "1" at the beginning of each sequence and interpret it in binary.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094845 |
1 | 215% | 9d | 9 |
nid:1766314094807
c1
DiskMat
field (Körper)
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nid:1766314094807
Cloze c1
Cloze answer: field (Körper)
Q: A {{c1::field (Körper)}} is {{c2::a nontrivial commutative ring \(F\) in which every nonzero element is a unit, so \(F^* = F \backslash \{0\}\)}}
A: Example: \(\mathbb{R}\), but not \(\mathbb{Z}\)Non-trivial: {0} is not a field. In particular, 1 = 0 (neutral element of mult. = neutral element of add.) causes trouble.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094872 |
1 | 230% | 9d | 8 |
nid:1766314094832
c1
DiskMat
A partial function \(A \to B\) is a relation from \(A\) to \...
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nid:1766314094832
Cloze c1
Q: A partial function \(A \to B\) is a relation from \(A\) to \(B\) such that {{c1::\(\forall a \in A \; \forall b,b' \in B \; (a \mathop{f} b \land a\mathop{f} b' \to b = b')\) (well-defined).}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094904 |
1 | 215% | 5d | 8 |
nid:1766314094873
c1
DiskMat
right inverse element
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users
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nid:1766314094873
Cloze c1
Cloze answer: right inverse element
Q: A {{c1::right inverse element}} of \(a\) in \(\langle S; *, e \rangle\) is {{c2::an element \(b\) such that \(a * b = e\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094952 |
1 | 230% | 14d | 6 |
nid:1766314094875
DiskMat
Lemma about uniqueness of the inverse:
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nid:1766314094875
Q: Lemma about uniqueness of the inverse:
A: Lemma 5.2: In a monoid \(\langle M; *, e \rangle\), if \(a \in M\) has a left inverse and a right inverse, then they are equal. In particular, \(a\) has at most one inverse.Proof: \(L\) left inverse, \(R\) right inverse.\(L = L * e = L * (a * R) \) \(= (L * a) * R = e * R = R\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094959 |
1 | 230% | 23d | 8 |
nid:1766314094876
c3
DiskMat
G1 (associativity)
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nid:1766314094876
Cloze c3
Cloze answer: G1 (associativity)
Q: A {{c1::group}} is an algebra \(\langle G; *, \widehat{\ \ }, e \rangle\) satisfying {{c2::three}} axioms: {{c3::G1 (associativity)}}, {{c4::G2 (neutral element)}}, and {{c5::G3 (inverse elements)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094962 |
1 | 230% | 10d | 5 |
nid:1766314094890
c1
DiskMat
right cancellation
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nid:1766314094890
Cloze c1
Cloze answer: right cancellation
Q: In a group, the {{c1::right cancellation}} law states: \(a = b\) {{c2::\(\Leftrightarrow\)}} {{c3::\(ac = bc\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314094992 |
1 | 260% | 18d | 6 |
nid:1766314094924
DiskMat
What is the group generated by a, denoted \(\langle a \rangl...
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nid:1766314094924
Q: What is the group generated by a, denoted \(\langle a \rangle\) defined as?
A: For a group \(G\) and \(a \in G\), the group generated by \(a\), denoted \(\langle a \rangle\), is defined as: \[\langle a \rangle \ \overset{\text{def}}{=} \ \{a^n \ | \ n \in \mathbb{Z}\}\]
This is a group, the smallest subgroup of \(G\) containing the element \(a\).
For finite groups: \(\langle a \rangle = \{e, a, a^2, \dots, a^{\text{ord}(a)-1}\}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095049 |
1 | 230% | 2d | 9 |
nid:1766314094925
c2
DiskMat
smallest
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nid:1766314094925
Cloze c2
Cloze answer: smallest
Q: The {{c2:: smallest}} subgroup of a group \(G\) containing \(a \in G\) is {{c1:: the group generated by \(a\), \(\langle a \rangle\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095051 |
1 | 230% | 17d | 9 |
nid:1766314094942
DiskMat
For which order is every group cyclic?
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nid:1766314094942
Q: For which order is every group cyclic?
A: If the order of the group is prime, it is cyclic!Every element has order 1 or \(|G|\) (Lagrange). Therefore, it is either the neutral element or a generator of the entire group.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095076 |
1 | 230% | 14d | 10 |
nid:1766314094972
c3
DiskMat
greatest \(i\) for which \(a_i \neq 0\)
1
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users
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nid:1766314094972
Cloze c3
Cloze answer: greatest \(i\) for which \(a_i \neq 0\)
Q: The {{c1::degree of \(a(x)\), denoted \(\deg(a(x))\)}}, is the {{c3::greatest \(i\) for which \(a_i \neq 0\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095116 |
1 | 230% | 14d | 7 |
nid:1766314094978
c2
DiskMat
also is an integral domain
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nid:1766314094978
Cloze c2
Cloze answer: also is an integral domain
Q: If \(D\) is an {{c1::integral domain}}, then \(D[x]\) {{c2::also is an integral domain}}.
A: Lemma 5.22(1)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095125 |
1 | 200% | 3d | 10 |
nid:1766314094988
DiskMat
How do you find the GCD of two polynomials?
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nid:1766314094988
Q: How do you find the GCD of two polynomials?
A: To find \(\gcd(a(x), b(x))\):
Find a common factor \((x - \alpha)\) using the roots method:
Try all possible elements of the field to find roots
If \(\alpha\) is a root of both, then \((x - \alpha)\) is a common factor
Use division with remainder to reduce to smaller polynomials
Repeat the process on the smaller polynomialsAfter they have no roots anymore, try all monic polynomials up to degree d/2 to
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095138 |
1 | 230% | 18d | 7 |
nid:1766314094992
c1
DiskMat
no roots
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nid:1766314094992
Cloze c1
Cloze answer: no roots
Q: An irreducible polynomial of degree \(\geq 2\) has {{c1:: no roots}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095142 |
1 | 230% | 7d | 10 |
nid:1766314094995
DiskMat
State Theorem 5.31 about the number of roots a polynomial ca...
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nid:1766314094995
Q: State Theorem 5.31 about the number of roots a polynomial can have.
A: Theorem 5.31: For a field \(F\), a nonzero polynomial \(a(x) \in F[x]\) of degree \(d\) has at most \(d\) roots.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095145 |
1 | 230% | 3d | 7 |
nid:1766314095059
c2
DiskMat
\(a*e = a\) (\(e*a = a\))
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nid:1766314095059
Cloze c2
Cloze answer: \(a*e = a\) (\(e*a = a\))
Q: A {{c1::right (left) neutral element}} is an element such that for all \(a \in G\), {{c2:: \(a*e = a\) (\(e*a = a\))}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314095227 |
1 | 215% | 3d | 7 |
nid:1766314111367
LinAlg
Is the empty set of vectors linearly dependent or independen...
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nid:1766314111367
Q: Is the empty set of vectors linearly dependent or independent?
A: It is linearly independent by definition, since there is no vector it could be a combination of.
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|---|---|---|---|---|---|
| jonas | cid:1766314111367 |
1 | 230% | 4d | 7 |
nid:1766314111406
LinAlg
What does \(N(A) = \mathbb{R}^n\) mean?
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nid:1766314111406
Q: What does \(N(A) = \mathbb{R}^n\) mean?
A: it means \(A = \boldsymbol{0}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766314111411 |
1 | 230% | 3d | 5 |
nid:1766940295633
c1
DiskMat
For any \(i\) and \(k\), if \(t_1, \dots, t_k\) are terms, t...
1
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1/4
users
200%
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nid:1766940295633
Cloze c1
Q: For any \(i\) and \(k\), if \(t_1, \dots, t_k\) are terms, then {{c1::\(P_i^{(k)}(t_1, \dots, t_k)\) is a formula}}, called an {{c2::atomic formula}}.
A: A formula in 1st order logic with no logical connectives (like \(\lnot, \land, \lor, \rightarrow \)) and no quantifiers (\(\forall, \exists\))
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295669 |
1 | 200% | 6d | 8 |
nid:1766940295636
c1
DiskMat
The {{c1::set of statements \(\mathcal{S}\)}} is {{c2:: a s...
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users
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nid:1766940295636
Cloze c1
Q: The {{c1::set of statements \(\mathcal{S}\)}} is {{c2:: a subset of the finite bit strings \(\Sigma^*\)}}.
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|---|---|---|---|---|---|
| jonas | cid:1766940295678 |
1 | 230% | 5d | 8 |
nid:1766940295674
c2
DiskMat
assigns to each formula \(F = (f_1, f_2, \dots, f_k) \in \La...
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users
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nid:1766940295674
Cloze c2
Cloze answer: assigns to each formula \(F = (f_1, f_2, \dots, f_k) \in \Lambda^*\) a subset \({{c1
Q: The {{c3::semantics}} of a logic defines a function {{c1::\(free\)}} which {{c2::assigns to each formula \(F = (f_1, f_2, \dots, f_k) \in \Lambda^*\) a subset \({{c1::free}}(F) \subseteq \{1, \dots, k\}\) of the indices}}.
A: If \(i \in free(F)\), then the symbol is said to occur free in \(F\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295752 |
1 | 200% | 10d | 8 |
nid:1766940295686
c1
DiskMat
closed
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nid:1766940295686
Cloze c1
Cloze answer: closed
Q: A formula is {{c1::closed}} if it {{c2::contains no free variables}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295776 |
1 | 230% | 10d | 9 |
nid:1766940295694
c1
DiskMat
formal language
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nid:1766940295694
Cloze c1
Cloze answer: formal language
Q: \( L = \{s \ | \ \tau(s) = 1\} \) is a set of strings called a {{c1:: formal language}}. It defines a {{c2:: predicate \(\tau\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295788 |
1 | 230% | 8d | 10 |
nid:1766940295695
c1
DiskMat
For a set \(Z\) of atomic formulas, a {{c1::truth assignment...
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users
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nid:1766940295695
Cloze c1
Q: For a set \(Z\) of atomic formulas, a {{c1::truth assignment \(\mathcal{A}\)}} is {{c2::a function \(\mathcal{A}: Z \rightarrow \{0, 1\}\)}}.
A: A truth assignment \(\mathcal{A}\) is suitable for a formula \(F\) if it contains all atomic formulas appearing in \(F\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295790 |
1 | 230% | 9d | 9 |
nid:1766940295714
c1
DiskMat
syntax
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nid:1766940295714
Cloze c1
Cloze answer: syntax
Q: The {{c1::syntax}} of a logic defines {{c2::an alphabet \(\Lambda\) (of allowed symbols)}} and specifies {{c2::which strings in \(\Lambda^*\) are formulas (i.e. syntactically correct)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295825 |
1 | 230% | 7d | 7 |
nid:1766940295715
DiskMat
For \(F \vdash_K G\), what is \(F\) called in a calculus?
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nid:1766940295715
Q: For \(F \vdash_K G\), what is \(F\) called in a calculus?
A: The premises or preconditions.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295826 |
1 | 215% | 6d | 9 |
nid:1766940295739
c2
DiskMat
A set \(M\) of formulas is {{c1::unsatisfiable}} if and only...
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users
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nid:1766940295739
Cloze c2
Q: A set \(M\) of formulas is {{c1::unsatisfiable}} if and only if {{c2::\(\mathcal{K}(M) \vdash_{Res} \emptyset\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295870 |
1 | 200% | 4d | 8 |
nid:1766940295754
DiskMat
Propositional logic is (in relation to predicate logic):
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users
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nid:1766940295754
Q: Propositional logic is (in relation to predicate logic):
A: embedded into predicate logic as a special case. We extend it by the concept of predicates.Predicates of the form \(P()\) act as propositional symbols.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295892 |
1 | 230% | 11d | 5 |
nid:1766940295762
c1
DiskMat
function symbol
1
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users
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nid:1766940295762
Cloze c1
Cloze answer: function symbol
Q: A {{c1::function symbol}} is of the form {{c2::\(f_i^{(k)}\) with \(i, k \in \mathbb{N}\)}}, where {{c2::\(k\) denotes the number of arguments (the arity) of the function}}.
A: Function symbols for \(k = 0\) are called constants.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295904 |
1 | 230% | 6d | 7 |
nid:1766940295780
c1
DiskMat
every true statement has a proof: \(\phi(s, p) = 1 \Longleft...
1
lapses
1/4
users
230%
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nid:1766940295780
Cloze c1
Cloze answer: every true statement has a proof: \(\phi(s, p) = 1 \Longleftarrow \tau(s) = 1\)
Q: A proof system is {{c2::complete}} if {{c1:: every true statement has a proof: \(\phi(s, p) = 1 \Longleftarrow \tau(s) = 1\)}}.
A: Note that the use of \(\Longleftarrow\) is not the correct formalism.For all \(s \in \mathcal{S}\) with \(\tau(s) = 1\) there exists a \(p \in \mathcal{P}\) such that \(\phi(s, p) = 1\), is the correct formal definition.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295938 |
1 | 230% | 7d | 6 |
nid:1766940295793
c1
DiskMat
they are of the same type
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users
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nid:1766940295793
Cloze c1
Cloze answer: they are of the same type
Q: We are allowed to swap quantifier order in a formula if:{{c1:: they are of the same type}}{{c2:: the variables never appear in the same predicate}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1766940295958 |
1 | 230% | 5d | 6 |
nid:1767734963666
DiskMat
What is really important for the prenex form due to the bind...
1
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users
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nid:1767734963666
Q: What is really important for the prenex form due to the binding of quantifiers?
A: We need to wrap the entire expression in parentheses \(\forall \exists (...)\) otherwise, it's not prenex!
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| jonas | cid:1767734963666 |
1 | 215% | 5d | 6 |
nid:1765372936327
A&D
Quicksort
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users
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nid:1765372936327
Q: Quicksort
A: Best Case: \(O(n \log n)\)Worst Case: \(O(n^2)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765383739474 |
1 | 230% | 82d | 8 |
nid:1766531635418
A&D
Worst case for search in a binary tree?
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users
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nid:1766531635418
Q: Worst case for search in a binary tree?
A: Binary trees are not necessarily balanced, hence it is possible that \(h >> \log_2 n\).Worst case example if inserted in ascending order:
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635418 |
1 | 230% | 83d | 8 |
nid:1766580161426
A&D
In every connected graph \(G\), when executing Kruskal using...
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users
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nid:1766580161426
Q: In every connected graph \(G\), when executing Kruskal using Union-Find, the representative repr[u] changes \(O(\dots)\) times:
A: \(O(\log_2 |V|)\), as we always at least double the size of the representative (we merge smaller into bigger, and repr[u] changes if it's the smaller one).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580161426 |
1 | 230% | 72d | 10 |
nid:1764867989717
c2
A&D
Hamiltonian cycle (Hamiltonkreis)
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users
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nid:1764867989717
Cloze c2
Cloze answer: Hamiltonian cycle (Hamiltonkreis)
Q: In graph theory, a {{c2::Hamiltonian cycle (Hamiltonkreis)}} is a {{c1::cycle that contains every vertex}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867989719 |
1 | 230% | 90d | 8 |
nid:1765655148922
A&D
Runtime Determine if Hamiltonian path exists?
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nid:1765655148922
Q: Runtime Determine if Hamiltonian path exists?
A: Hamiltonian walk - exponential, we have to brute-force
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765655148922 |
1 | 230% | 87d | 8 |
nid:1766271258597
c1
A&D
datastructure
1
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nid:1766271258597
Cloze c1
Cloze answer: datastructure
Q: A {{c1:: datastructure}} is the implementation of the wishlist of operations defined in our ADT.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766271258597 |
1 | 230% | 94d | 8 |
nid:1766531635615
c1
A&D
triangle inequality
1
lapses
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nid:1766531635615
Cloze c1
Cloze answer: triangle inequality
Q: The {{c1::triangle inequality}} in a weighted graph is {{c2::\(d(u, v) \leq d(u, w) + d(w, v)\)}}.
A: This holds, since if the path through \(w\) was actually cheaper, then \(d(u, v)\) would be wrong.Does not hold in graphs with negative cycles.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635615 |
1 | 230% | 84d | 10 |
nid:1764867989867
A&D
Pre- and Postordering in BFS:
1
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users
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ease
nid:1764867989867
Q: Pre- and Postordering in BFS:
A: Same as with pre-/postordering, we can use enter-/leave-ordering here: enter step at which vertex \(v\) is first encountered.leave step at which vertex \(v\) is dequeued
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867989867 |
1 | 230% | 89d | 8 |
nid:1765372936339
A&D
How does extract_max work for a maxHeap?
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users
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nid:1765372936339
Q: How does extract_max work for a maxHeap?
A: The extract max operation works by taking the root node, the biggest element in the heap by it’s definition and restoring the heap condition.We remove the root and replace it by the element that is most to the right (last element in the array storing the heap).Then we "versickern" this small element, until the heap condition is restored. We swap it with the larger of the child nodes, until it's bigger than both of it's children.&nb
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936339 |
1 | 230% | 93d | 11 |
nid:1766531635603
A&D
BFS (Breadth First Search)
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users
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nid:1766531635603
Q: BFS (Breadth First Search)
A: \(O(|V|+|E|)\) (Adjacency List)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635603 |
1 | 230% | 91d | 8 |
nid:1765372936200
c2
A&D
O(k^n)
1
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1/4
users
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nid:1765372936200
Cloze c2
Cloze answer: O(k^n)
Q: Choose a tight bound!\({{c1::O(n^k)}} \leq {{c2::O(k^n)}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936201 |
1 | 230% | 102d | 9 |
nid:1766531635503
A&D
How can we make Knapsack polynomial using approximation?
1
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nid:1766531635503
Q: How can we make Knapsack polynomial using approximation?
A: Round the profits and solve the Knapsack problem for those rounded profits:\(\overline{p_i} := K \cdot \lfloor \frac{p_i}{K} \rfloor\). We then only have to compute every K'th entry of the DP-table.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635503 |
1 | 230% | 94d | 10 |
nid:1766271258634
A&D
In what situation is the array the correct underlying datast...
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users
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nid:1766271258634
Q: In what situation is the array the correct underlying datastructure for a list?
A: When we have a fixed upper bound for the size of the list.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766271258635 |
1 | 230% | 99d | 8 |
nid:1765372936324
A&D
Merge Sort
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users
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nid:1765372936324
Q: Merge Sort
A: Best Case: \(O(n \log n)\)Worst Case: \(O(n \log n)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765383739470 |
1 | 230% | 107d | 10 |
nid:1765372936234
c2
A&D
Master Theorem: If {{c1:: \(b = \log_2(a)\)}} then {{c2:: \(...
1
lapses
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users
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nid:1765372936234
Cloze c2
Q: Master Theorem: If {{c1:: \(b = \log_2(a)\)}} then {{c2:: \(T(n) \leq O(n^{\log_2 a} \cdot \log n)\)}}.
A: The recursive and non-recursive work is balanced.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936234 |
1 | 230% | 96d | 12 |
nid:1765372936330
A&D
Heapsort
1
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users
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nid:1765372936330
Q: Heapsort
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765383739478 |
1 | 230% | 103d | 11 |
nid:1766580144028
c1
A&D
\(n-x\) components (different values)
1
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users
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nid:1766580144028
Cloze c1
Cloze answer: \(n-x\) components (different values)
Q: After adding \(x\) edges to the Union-Find datastructure, the repr array contains {{c1::\(n-x\) components (different values)}}.
A: Each added edge removes one unconnected component.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580144028 |
1 | 230% | 93d | 11 |
nid:1766531635474
A&D
Subsequence
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users
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nid:1766531635474
Q: Subsequence
A: Teilfolge
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635474 |
1 | 230% | 106d | 8 |
nid:1764867989708
c2
A&D
incident (inzident oder anliegend)
1
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users
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nid:1764867989708
Cloze c2
Cloze answer: incident (inzident oder anliegend)
Q: In an edge \(e = \{u, v\}\), we call \(u\) {{c1::adjacent (adjazent oder benachbart)}} to \(v\) (and the other way around) and \(e\) {{c2::incident (inzident oder anliegend)}} to \(u, v\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867989709 |
1 | 230% | 109d | 8 |
nid:1766531635499
A&D
What is pseudo-polynomial time?
1
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users
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nid:1766531635499
Q: What is pseudo-polynomial time?
A: Runtime dependent on a number \(W\) (like in knapsack) which is not correlated polynomially to input length but exponentially.The DP-table get's 10x for \(W = 10 \rightarrow 100\) but the input size (binary) only grows from \(\log_2(10) \approx 3 \rightarrow \approx 6\) so x2.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635500 |
1 | 230% | 98d | 11 |
nid:1765372936146
A&D
Simplify \(\frac{a^{kn}}{b^{k'n}} =\)
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users
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nid:1765372936146
Q: Simplify \(\frac{a^{kn}}{b^{k'n}} =\)
A: \(\frac{e^{\ln(a^{kn})}}{e^{\ln(b^{k'n})}} = e^{kn \cdot \ln(a) - k'n \cdot ln(b)}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936146 |
1 | 230% | 119d | 12 |
nid:1765198542351
A&D
What is the sum of all natural numbers between 1 and \(n\)?
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users
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nid:1765198542351
Q: What is the sum of all natural numbers between 1 and \(n\)?
A: \(= \frac{n(n+1)}{2}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765198542351 |
1 | 230% | 116d | 9 |
nid:1765372936167
c1
A&D
\(f, g\) are differentiable (for sufficiently large \(x\))
1
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users
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nid:1765372936167
Cloze c1
Cloze answer: \(f, g\) are differentiable (for sufficiently large \(x\))
Q: What are the prerequisites for \(f\) and \(g\) to apply l'Hôpital's?{{c1::\(f, g\) are differentiable (for sufficiently large \(x\))}}{{c2::\(\lim_{x \to \infty} f(x) = \lim_{x \to \infty} g(x) = \infty\) (or both \(= 0\))}}{{c3::\(g'(x
A: Then: \(\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{f'(x)}{g'(x)}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936167 |
1 | 230% | 101d | 12 |
nid:1766580157417
A&D
Can Kruskal's Algorithm be executed in \(O(|E| + |V|\log|V|)...
1
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users
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nid:1766580157417
Q: Can Kruskal's Algorithm be executed in \(O(|E| + |V|\log|V|)\) time?
A: No, we need to sort the edges which takes at least \(|E| \log |E|\) time.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580157417 |
1 | 230% | 108d | 11 |
nid:1766580143726
A&D
Floyd-Warshall
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nid:1766580143726
Q: Floyd-Warshall
A: \(O(|V|^3)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766580143728 |
1 | 230% | 104d | 11 |
nid:1764867989631
c2
A&D
cycle (Kreis)
1
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users
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nid:1764867989631
Cloze c2
Cloze answer: cycle (Kreis)
Q: In graph theory, a {{c2::cycle (Kreis)}} is a {{c1::closed walk without repeated vertices}} and {{c1::at least three vertices}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867989631 |
1 | 230% | 122d | 8 |
nid:1765372936324
A&D
Merge Sort
1
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1/4
users
230%
ease
nid:1765372936324
Q: Merge Sort
A: Best Case: \(O(n \log n)\)Worst Case: \(O(n \log n)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765383739472 |
1 | 230% | 123d | 11 |
nid:1765372936300
A&D
What do we have to pay attention to in the I.H. and the I.S....
1
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users
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nid:1765372936300
Q: What do we have to pay attention to in the I.H. and the I.S. in an induction proof?
A: We should change the variable name from \(n\) to \(k\) (for example) as not to confuse it.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936300 |
1 | 230% | 125d | 11 |
nid:1765198542383
A&D
Which functions \(f(n)\) have \(\lim_{n\rightarrow \infty} f...
1
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users
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nid:1765198542383
Q: Which functions \(f(n)\) have \(\lim_{n\rightarrow \infty} f(n)\) undefined?
A: Typically functions that oscilate as they approach infinity such as \(f(n) = \sin n\) or \(f(n) = (-1)^n\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765198542383 |
1 | 230% | 139d | 9 |
nid:1765372936291
c1
A&D
{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)::Sum}} \(\leq\) {{c2::...
1
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1/4
users
230%
ease
nid:1765372936291
Cloze c1
Q: {{c1:: \(\sum_{i = 1}^{n} i\log(i)\)::Sum}} \(\leq\) {{c2::\(O(n \log(n))\)::O-notation}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936292 |
1 | 230% | 163d | 9 |
nid:1766531635431
c2
A&D
The height of a 2-3 Tree for \(n\) keys is {{c1::\(\leq \log...
1
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1/4
users
230%
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nid:1766531635431
Cloze c2
Q: The height of a 2-3 Tree for \(n\) keys is {{c1::\(\leq \log_2(n)\)}} thus \(h={{c2::O(\log(n))::\textbf{O-notation} }}\).
A: Note that for the case \(n = 1\) the root has one leaf with the key.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635433 |
1 | 230% | 154d | 9 |
nid:1765372936244
A&D
If \(T(n) = aT(n/ 2) + Cn^b\), then we get which type of O-...
1
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users
230%
ease
nid:1765372936244
Q: If \(T(n) = aT(n/ 2) + Cn^b\), then we get which type of O-Notation?
A: \(T(n) = \Theta(...)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936244 |
1 | 230% | 173d | 11 |
nid:1766531635639
A&D
Bellman-Ford optimisation in a DAG?
1
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users
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nid:1766531635639
Q: Bellman-Ford optimisation in a DAG?
A: In an acyclic graph, topological sorting is already an algorithm that gives us the most-efficient order to calculate the cost in.Because we can be sure that any predecessors already have the correct \(l\)-good bound distance (guaranteed by topo-sort, no backedges), we can simply relax once.Thus we can compute the correct cheapest path in one "relaxation": \(O(|E|)\).Therefore with toposort: \(O(|V| + |E|)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766531635639 |
1 | 230% | 157d | 11 |
nid:1765372936222
A&D
What is the form of the recursive equations solved by the Ma...
1
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users
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nid:1765372936222
Q: What is the form of the recursive equations solved by the Master Theorem?
A: \(T(n) \leq aT(n/2) + Cn^b\)where \(a\), \(C > 0\) and \(b \geq 0\) are constants.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936222 |
1 | 230% | 180d | 11 |
nid:1765372936231
c2
A&D
\(T(n) \leq O(n^b)\)
1
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users
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ease
nid:1765372936231
Cloze c2
Cloze answer: \(T(n) \leq O(n^b)\)
Q: Master Theorem: If {{c1:: \(b > \log_2(a)\)}} then {{c2:: \(T(n) \leq O(n^b)\)}}.
A: This is the case for which the work outside the recursion dominates.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936232 |
1 | 230% | 181d | 11 |
nid:1765372936324
A&D
Merge Sort
1
lapses
1/4
users
230%
ease
nid:1765372936324
Q: Merge Sort
A: Best Case: \(O(n \log n)\)Worst Case: \(O(n \log n)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765383739471 |
1 | 230% | 180d | 9 |
nid:1765372936286
c2
A&D
\(\log(n!)\)
1
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1/4
users
230%
ease
nid:1765372936286
Cloze c2
Cloze answer: \(\log(n!)\)
Q: {{c1:: \(\sum_{i = 1}^{n} \log(i)\)::Sum}} \(=\) {{c2::\(\log(n!)\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936286 |
1 | 230% | 191d | 11 |
nid:1765372936324
A&D
Merge Sort
1
lapses
1/4
users
230%
ease
nid:1765372936324
Q: Merge Sort
A: Best Case: \(O(n \log n)\)Worst Case: \(O(n \log n)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765383739469 |
1 | 230% | 207d | 9 |
nid:1765372936182
c2
A&D
\(g \geq \Omega(f)\)
1
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1/4
users
230%
ease
nid:1765372936182
Cloze c2
Cloze answer: \(g \geq \Omega(f)\)
Q: {{c2::\(g \geq \Omega(f)\)}} \( \Leftrightarrow\) {{c1::\( f \leq O(g)\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765372936183 |
1 | 230% | 211d | 9 |
nid:1764867991099
c1
DiskMat
the group generated by \(a\), \(\langle a \rangle\)
1
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users
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nid:1764867991099
Cloze c1
Cloze answer: the group generated by \(a\), \(\langle a \rangle\)
Q: The {{c2:: smallest}} subgroup of a group \(G\) containing \(a \in G\) is {{c1:: the group generated by \(a\), \(\langle a \rangle\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991099 |
1 | 230% | 67d | 8 |
nid:1764867991253
c1
DiskMat
degree of \(a(x)\), denoted \(\deg(a(x))\)
1
lapses
1/4
users
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nid:1764867991253
Cloze c1
Cloze answer: degree of \(a(x)\), denoted \(\deg(a(x))\)
Q: The {{c1::degree of \(a(x)\), denoted \(\deg(a(x))\)}}, is the {{c3::greatest \(i\) for which \(a_i \neq 0\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991253 |
1 | 230% | 67d | 8 |
nid:1766448533167
c2
DiskMat
predicate \(\tau\)
1
lapses
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users
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ease
nid:1766448533167
Cloze c2
Cloze answer: predicate \(\tau\)
Q: \( L = \{s \ | \ \tau(s) = 1\} \) is a set of strings called a {{c1:: formal language}}. It defines a {{c2:: predicate \(\tau\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448533168 |
1 | 230% | 66d | 11 |
nid:1766448533306
c1
DiskMat
The notation {{c1::\(\mathcal{A} \models F\)}} means that {{...
1
lapses
1/4
users
230%
ease
nid:1766448533306
Cloze c1
Q: The notation {{c1::\(\mathcal{A} \models F\)}} means that {{c2::\(\mathcal{A}\) is a model for \(F\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448533306 |
1 | 230% | 66d | 8 |
nid:1766448533150
c1
DiskMat
efficiently computable
1
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users
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nid:1766448533150
Cloze c1
Cloze answer: efficiently computable
Q: We require that the proof verification function \(\phi\) is {{c1::efficiently computable}}, otherwise the proof system is not useful.
A: A proof system is useless if verification is infeasible.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448533150 |
1 | 230% | 63d | 8 |
nid:1764867991337
DiskMat
Why is a polynomial of degree \(d\) uniquely determined by \...
1
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1/4
users
230%
ease
nid:1764867991337
Q: Why is a polynomial of degree \(d\) uniquely determined by \(d + 1\) values of \(a(x)\)?
A: This \(a(x)\) is unique since if there was another \(a'(x)\) then \(a(x) - a'(x)\) would have at most degree \(d\) and thus at most \(d\) roots. But since \(a(x) - a'(x)\) has the same \(d + 1\) roots, it's \(0 \implies a(x) = a'(x)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991337 |
1 | 230% | 65d | 8 |
nid:1764867990892
DiskMat
What exponentiation operation is valid in modular arithmetic...
1
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users
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ease
nid:1764867990892
Q: What exponentiation operation is valid in modular arithmetic?
A: This is allowed:\(a \equiv_n b\) and then \(a^x \equiv_n b^x\)But this on the other hand is illegal:\(a \equiv_n b\) and \(c \equiv_n d\) and then doing \(a^c \equiv_n b^d\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990892 |
1 | 230% | 68d | 12 |
nid:1767648242888
DiskMat
How do we construct a field \(GF(p^q)\)?
1
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users
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ease
nid:1767648242888
Q: How do we construct a field \(GF(p^q)\)?
A: We take the field \(GF(p)[x]_{m(x)}\) where \(m(x)\) is an irreducible polynomial of degree \(q\).Then \(GF(p)[x]_{m(x)}\) has \({|F|}^q\) polynomials in it, as all of degree less than \(q\) are coprime to \(m(x)\), by definition of irreducible. And this field is isomorphic to \(GF(p^q)\).
Example: The field \(GF(2)[x]\) \({x^2 + x + 1}\) is isomorphic to \(GF(2^2 = 4)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1767648242888 |
1 | 230% | 67d | 11 |
nid:1764867990250
DiskMat
If two sets each dominate the other, what can we conclude?
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nid:1764867990250
Q: If two sets each dominate the other, what can we conclude?
A: For sets \(A\) and \(B\):
\[A \preceq B \land B \preceq A \quad \Rightarrow \quad A \sim B\]
If there's an injection \(f: A \to B\) and an injection \(g: B \to A\), then there's a bijection between \(A\) and \(B\).Bernstein-Schröder Theorem
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990250 |
1 | 230% | 74d | 9 |
nid:1766448533015
DiskMat
A ring has the following properties:
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Q: A ring has the following properties:
A: Additive Group:closureassociativityidentityinversecommutativeMultiplicative group:closureassociativityidentitydistributivity
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|---|---|---|---|---|---|
| lorenz | cid:1766448533015 |
1 | 230% | 66d | 10 |
nid:1764867991055
c1
DiskMat
In a group, for \(n \geq 1\), the positive power is defined ...
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nid:1764867991055
Cloze c1
Q: In a group, for \(n \geq 1\), the positive power is defined recursively: {{c1::\(a^n = a \cdot a^{n-1}\)}}.
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| lorenz | cid:1764867991055 |
1 | 230% | 74d | 8 |
nid:1764867991429
c2
DiskMat
number of positions at which the string is non-zero
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nid:1764867991429
Cloze c2
Cloze answer: number of positions at which the string is non-zero
Q: The {{c1::Hamming weight}} of a string in a finite alphabet \(\mathcal{A}\) is the {{c2::number of positions at which the string is non-zero}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991430 |
1 | 230% | 75d | 8 |
nid:1766448533885
c2
DiskMat
there is a literal \(L\) such that \(L \in K_1\), \(\lnot L ...
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nid:1766448533885
Cloze c2
Cloze answer: there is a literal \(L\) such that \(L \in K_1\), \(\lnot L \in K_2\)
Q: A clause \(K\) is {{c1::resolvent}} of clauses \(K_1\) and \(K_2\) if {{c2::there is a literal \(L\) such that \(L \in K_1\), \(\lnot L \in K_2\)}}.
A: \[K = (K_1 \setminus \{L\}) \cup (K_2 \setminus \{\lnot L\})\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448533886 |
1 | 230% | 67d | 11 |
nid:1765193120869
DiskMat
What is a zerodivisor and in which structure do they exist?
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nid:1765193120869
Q: What is a zerodivisor and in which structure do they exist?
A: A zerodivisor is an element \(a \neq 0\) in a commutative ring for which there exists a \(b \neq 0\) such that \(ab = 0\).This is commonly encountered for the polynomial rings formed over \(\text{GF}[x]_{m(x)}\) with \(m(x)\) not irreducible (i.e. it's not a field).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765193120869 |
1 | 230% | 75d | 8 |
nid:1766448533482
c1
DiskMat
axiom \(A\); these axioms
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nid:1766448533482
Cloze c1
Cloze answer: axiom \(A\); these axioms
Q: An {{c1::axiom \(A\)}} is a {{c2::statement taken as true in a theory}}. {{c3::Theorems}} are the statements which {{c4::follow from {{c1::these axioms}} (\(A \models T\))}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448533483 |
1 | 230% | 72d | 8 |
nid:1764867990867
c1
DiskMat
\(-\infty\)
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nid:1764867990867
Cloze c1
Cloze answer: \(-\infty\)
Q: The degree of the polynomial \(0\) is defined as {{c1::\(-\infty\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990867 |
1 | 230% | 78d | 8 |
nid:1764867990874
DiskMat
Reduce \(R_{11}(9^{2024})\)
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nid:1764867990874
Q: Reduce \(R_{11}(9^{2024})\)
A: As \(9^{10} \equiv_{11} 1\) (see Fermat little theorem and 11 prime), we can reduce the exponent modulo 10 (see Lagrange's theorem in chapter 5). Thus \(R_{11}(9^{2024}) = R_{11}(9^{4}) = R_{11}(-2^{4}) = 5\).For this to work however, we need the number and the order of the group (modulo remainder) to be coprime, i.e. \(\gcd(9, 11) = 1\).If the modulus itself is prime then it always works and the order of the element can be used to reduce the exponent
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990874 |
1 | 230% | 80d | 8 |
nid:1766448533987
c2
DiskMat
atomic formula
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Cloze c2
Cloze answer: atomic formula
Q: For any \(i\) and \(k\), if \(t_1, \dots, t_k\) are terms, then {{c1::\(P_i^{(k)}(t_1, \dots, t_k)\) is a formula}}, called an {{c2::atomic formula}}.
A: A formula in 1st order logic with no logical connectives (like \(\lnot, \land, \lor, \rightarrow \)) and no quantifiers (\(\forall, \exists\))
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| lorenz | cid:1766448533988 |
1 | 230% | 71d | 11 |
nid:1764867990075
DiskMat
How is composition of relations represented in matrix and gr...
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nid:1764867990075
Q: How is composition of relations represented in matrix and graph form?
A: Matrix: Matrix multiplication
Graph: Natural composition - there's a path from \(a\) to \(c\) if there's a path \(a \to b\) in graph 1 and \(b \to c\) in graph 2
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|---|---|---|---|---|---|
| lorenz | cid:1764867990075 |
1 | 230% | 71d | 12 |
nid:1764867991290
DiskMat
What is the GCD in a polynomial field?
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nid:1764867991290
Q: What is the GCD in a polynomial field?
A: The monic polynomial \(g(x)\) of largest degree such that \(g(x) \ | \ a(x)\) and \(g(x) \ | \ b(x)\) is called the greatest common divisor of \(a(x)\) and \(b(x)\), denoted \(\gcd(a(x), b(x))\).
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|---|---|---|---|---|---|
| lorenz | cid:1764867991290 |
1 | 230% | 79d | 8 |
nid:1764867991451
c2
DiskMat
at least \(n - k + 1\) positions
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nid:1764867991451
Cloze c2
Cloze answer: at least \(n - k + 1\) positions
Q: Two codewords in a polynomial code with degree \(k-1\) cannot agree at {{c1:: \(k\) positions (else they'd be equal)}}, so they disagree in {{c2:: at least \(n - k + 1\) positions}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991452 |
1 | 230% | 80d | 8 |
nid:1764867990269
c3
DiskMat
\(A^*\) (finite sequences) is countable
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nid:1764867990269
Cloze c3
Cloze answer: \(A^*\) (finite sequences) is countable
Q: Which operations preserve countability?Let \(A\) and \(A_i\) for \(i \in \mathbb{N}\) be countable sets. Then: {{c1::\(A^n\) (\(n\)-tuples) is countable }}{{c2::\(\bigcup_{i\in \mathbb{N} } A_i\) (countable union) is countabl
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|---|---|---|---|---|---|
| lorenz | cid:1766229398360 |
1 | 230% | 83d | 11 |
nid:1764867991079
c2
DiskMat
order of \(G\)
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nid:1764867991079
Cloze c2
Cloze answer: order of \(G\)
Q: For a finite group \(G\), {{c1::\(|G|\)}} is called the {{c2::order of \(G\)}}.
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| lorenz | cid:1764867991079 |
1 | 230% | 85d | 8 |
nid:1764867990795
c1
DiskMat
A function \(f:\mathbb{N}\to\{0,1\}\) is called computable i...
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nid:1764867990795
Cloze c1
Q: A function \(f:\mathbb{N}\to\{0,1\}\) is called computable if {{c1::there is a computer program that, for every \(n\in\mathbb{N}\), when given \(n\) as input, outputs \(f(n)\).}}
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|---|---|---|---|---|---|
| lorenz | cid:1764867990795 |
1 | 230% | 90d | 11 |
nid:1764867991005
DiskMat
Give an example of a group homomorphism involving the logari...
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nid:1764867991005
Q: Give an example of a group homomorphism involving the logarithm function.
A: The logarithm function is a group homomorphism from \(\langle \mathbb{R}^{>0}; \cdot \rangle\) to \(\langle \mathbb{R}; + \rangle\) because: \[\log(a \cdot b) = \log a + \log b\]
It's also an isomorphism because the logarithm is bijective on positive reals.
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|---|---|---|---|---|---|
| lorenz | cid:1764867991005 |
1 | 230% | 87d | 8 |
nid:1764867990443
c2
DiskMat
\(a \equiv_m b \Longleftrightarrow R_m(a) = R_m(b)\) (congr...
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nid:1764867990443
Cloze c2
Cloze answer: \(a \equiv_m b \Longleftrightarrow R_m(a) = R_m(b)\) (congruence iff same remainder)
Q: What are the two key properties of the remainder function \(R_m\)? (Lemma 4.16)(i) {{c1:: \(a \equiv_m R_m(a)\) (the remainder represents the equivalence class)}}(ii) {{c2:: \(a \equiv_m b \Longleftrightarrow R_m(a) = R_m(b)\) (congru
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|---|---|---|---|---|---|
| lorenz | cid:1766229398882 |
1 | 230% | 92d | 8 |
nid:1764867991186
DiskMat
Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather...
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nid:1764867991186
Q: Why do we need \(\mathbb{Z}_m^*\) for multiplication, rather than just using \(\mathbb{Z}_m\)?
A: \(\mathbb{Z}_m\) (with \(\oplus\)) is not a group with respect to multiplication modulo \(m\) because elements that are not coprime to \(m\) don't have a multiplicative inverse.
For example, in \(\mathbb{Z}_6\), the element \(2\) has no multiplicative inverse because \(\gcd(2, 6) = 2 \neq 1\).
Thus we need \(\mathbb{Z}_m^*\) (elements coprime to \(m\)) to form a group with \(\odot\) (multiplication mod \(m\)).
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|---|---|---|---|---|---|
| lorenz | cid:1764867991186 |
1 | 230% | 91d | 8 |
nid:1767535579762
c1
DiskMat
We can solve \(R_a(b^c)\) by using the fact that {{c1:: \(R_...
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nid:1767535579762
Cloze c1
Q: We can solve \(R_a(b^c)\) by using the fact that {{c1:: \(R_a(b^c) = R_a(b^{R_{\varphi(a)}(c)})\)}} if \(a, b\) coprime.
A: Note that we can't simply reduce by \(a\)!
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|---|---|---|---|---|---|
| lorenz | cid:1767535579762 |
1 | 230% | 90d | 8 |
nid:1764867991067
c2
DiskMat
If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is 2}},...
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nid:1764867991067
Cloze c2
Q: If {{c2:: the order \(\text{ord}(a)\) of \(a \in G\) is 2}}, {{c1:: a is it's own self-inverse}}.
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|---|---|---|---|---|---|
| lorenz | cid:1764867991068 |
1 | 230% | 92d | 8 |
nid:1765655179118
c2
DiskMat
cyclic for every \(n\)
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nid:1765655179118
Cloze c2
Cloze answer: cyclic for every \(n\)
Q: The group \(\langle \mathbb{Z}_n; \oplus \rangle\) is {{c2::cyclic for every \(n\)}}, where {{c3:: 1}} is always a generator.
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|---|---|---|---|---|---|
| lorenz | cid:1765655179118 |
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nid:1766448533677
c1
DiskMat
The semantics of propositional logic are defined as:{{c1::\(...
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nid:1766448533677
Cloze c1
Q: The semantics of propositional logic are defined as:{{c1::\(\mathcal{A}(F) = \mathcal{A}(A_i)\) for any atomic formula \(A_i\)}}for \(\land, \lor, \lnot\) the semantics are identical to before.
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|---|---|---|---|---|---|
| lorenz | cid:1766448533677 |
1 | 230% | 86d | 11 |
nid:1764867991454
DiskMat
What is the left cancellation law in a group?
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nid:1764867991454
Q: What is the left cancellation law in a group?
A: Left cancellation law: \(a * b = a * c \ \implies \ b = c\)
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|---|---|---|---|---|---|
| lorenz | cid:1764867991454 |
1 | 230% | 90d | 12 |
nid:1764867991020
c2
DiskMat
neutral element; nullspace
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nid:1764867991020
Cloze c2
Cloze answer: neutral element; nullspace
Q: For a homomorphism \(h: G \rightarrow H\), the {{c1::kernel \(\ker(h)\)}} is the set of all elements mapped to the {{c2::neutral element}} (essentially the {{c2::nullspace}}).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991020 |
1 | 230% | 94d | 11 |
nid:1766448533765
DiskMat
For CNF construction, how do you form literals from a row in...
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nid:1766448533765
Q: For CNF construction, how do you form literals from a row in the truth table?
A: - If \(A_i = 0\) in the row, take \(A_i\)- If \(A_i = 1\) in the row, take \(\lnot A_i\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448533765 |
1 | 230% | 87d | 11 |
nid:1764867990962
c1
DiskMat
left inverse
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nid:1764867990962
Cloze c1
Cloze answer: left inverse
Q: A function \(f: A \rightarrow B\) has a {{c1::left inverse}} if and only if \(f\) is {{c2::injective}} (not in script).
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|---|---|---|---|---|---|
| lorenz | cid:1764867990962 |
1 | 230% | 85d | 10 |
nid:1764867990755
c1
DiskMat
the truth value depends on the interpretation of the symbols
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nid:1764867990755
Cloze c1
Cloze answer: the truth value depends on the interpretation of the symbols
Q: A logical formula is generally not a mathematical statement, because {{c1::the truth value depends on the interpretation of the symbols}}.
A: (so we can't prove/disprove it)
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|---|---|---|---|---|---|
| lorenz | cid:1764867990755 |
1 | 230% | 113d | 9 |
nid:1764867990887
DiskMat
How many divisors does \(n\) expressed as a factor of prime ...
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nid:1764867990887
Q: How many divisors does \(n\) expressed as a factor of prime numbers \(n = \prod_{i = 1}^m p_i^{e_i}\) have?
A: \(n\) has \(\# _ {\text{div}(n)} = \prod_{i = 1}^m (e_i + 1)\) divisors.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990887 |
1 | 230% | 90d | 12 |
nid:1764867990296
DiskMat
Why is \((\mathbb{N}; |)\) NOT totally ordered?
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nid:1764867990296
Q: Why is \((\mathbb{N}; |)\) NOT totally ordered?
A: Because \(2 \nmid 3\) and \(3 \nmid 2\) (they are incomparable).
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|---|---|---|---|---|---|
| lorenz | cid:1764867990296 |
1 | 230% | 103d | 9 |
nid:1766448533830
c1
DiskMat
empty set \(\emptyset\)
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nid:1766448533830
Cloze c1
Cloze answer: empty set \(\emptyset\)
Q: The {{c1::empty set \(\emptyset\)}} is a {{c2::clause}}.
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|---|---|---|---|---|---|
| lorenz | cid:1766448533831 |
1 | 230% | 94d | 11 |
nid:1764867990475
DiskMat
What does "unique up to order" mean in the Fundamental Theor...
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nid:1764867990475
Q: What does "unique up to order" mean in the Fundamental Theorem of Arithmetic?
A: Every integer has exactly one prime factorization if we don't care about the order of factors. For example, \(12 = 2^2 \cdot 3 = 3 \cdot 2 \cdot 2 = 2 \cdot 3 \cdot 2\) are all the same factorization, just written differently.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990475 |
1 | 230% | 98d | 9 |
nid:1764867990348
DiskMat
Which of the following are countable: \(\mathbb{N}\), \(\mat...
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nid:1764867990348
Q: Which of the following are countable: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), \(\{0,1\}^*\), \(\{0,1\}^{\infty}\)?
A: Countable: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\{0,1\}^*\)
Uncountable: \(\mathbb{R}\), \(\{0,1\}^{\infty}\)
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|---|---|---|---|---|---|
| lorenz | cid:1764867990348 |
1 | 230% | 102d | 9 |
nid:1767291036660
c2
DiskMat
Associativity
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nid:1767291036660
Cloze c2
Cloze answer: Associativity
Q: An abelian group has the following properties:{{c1::Closure}}{{c2::Associativity}}{{c3::Identity}}{{c4::Inverse}}{{c5::Commutativity}}
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|---|---|---|---|---|---|
| lorenz | cid:1767291036662 |
1 | 230% | 99d | 12 |
nid:1764867990542
c2
DiskMat
Prove that \( T \) is false.
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nid:1764867990542
Cloze c2
Cloze answer: Prove that \( T \) is false.
Q: Proof method: Proof by Contradiction1. {{c1:: Find a suitable statement \( T\).}}2. {{c2:: Prove that \( T \) is false.}}3. {{c3:: Assume that \( S \) is false and prove that \( T \) is true (-> contradiction).}}
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|---|---|---|---|---|---|
| lorenz | cid:1766229399206 |
1 | 230% | 118d | 9 |
nid:1766920111886
c1
DiskMat
\(|a|\)
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nid:1766920111886
Cloze c1
Cloze answer: \(|a|\)
Q: \(\gcd(a, 0) = \) {{c1::\(|a|\)}}
A: This is why \(0\) isn't in \(Z_m^* \) and \(F[x]^*_{m(x)}\).
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|---|---|---|---|---|---|
| lorenz | cid:1766920111887 |
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nid:1764867989950
DiskMat
What is the logical principle behind case distinction?
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nid:1764867989950
Q: What is the logical principle behind case distinction?
A: For every \(k\) we have:
\[(A_1 \lor \dots \lor A_k) \land (A_1 \rightarrow B) \land \dots \land (A_k \rightarrow B) \models B\]
(If at least one case occurs, and all cases imply \(B\), then \(B\) holds)
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|---|---|---|---|---|---|
| lorenz | cid:1764867989950 |
1 | 230% | 114d | 9 |
nid:1764867990164
DiskMat
When is the lexicographic order on \(A \times B\) totally or...
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nid:1764867990164
Q: When is the lexicographic order on \(A \times B\) totally ordered?
A: When both \((A; \preceq)\) and \((B; \sqsubseteq)\) are totally ordered.
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|---|---|---|---|---|---|
| lorenz | cid:1764867990164 |
1 | 230% | 103d | 9 |
nid:1767534763076
DiskMat
Uncountability Proof by Complement (with example \([0,1] \se...
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nid:1767534763076
Q: Uncountability Proof by Complement (with example \([0,1] \setminus \mathbb{Q}\)):
A: Find \(B\) uncountable such that \(A \subseteq B\).
Show that \(B \backslash A\) countable which proves that \(A\) uncountable.
You have to prove this implication in the exam:
Assume \(A\) is countable towards contradiction.
We have shown that \(B \ \backslash \ A\) is countable.
Thus \(A \cup (B \ \backslash \ A)\) also countable (Theorem 3.22: Union of countable is countable).
But \(A \cup
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|---|---|---|---|---|---|
| lorenz | cid:1767534763076 |
1 | 230% | 106d | 12 |
nid:1764867990259
DiskMat
Is \(\mathbb{N} \times \mathbb{N}\) countable?
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nid:1764867990259
Q: Is \(\mathbb{N} \times \mathbb{N}\) countable?
A: Yes, the set \(\mathbb{N} \times \mathbb{N}\) (= \(\mathbb{N}^2\)) of ordered pairs of natural numbers is countable.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990259 |
1 | 230% | 108d | 9 |
nid:1764867989903
DiskMat
What is \(\lnot \forall x P(x)\) equivalent to?
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nid:1764867989903
Q: What is \(\lnot \forall x P(x)\) equivalent to?
A: \(\lnot \forall x P(x) \equiv \exists x \lnot P(x)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867989903 |
1 | 230% | 133d | 9 |
nid:1768521665087
c1
DiskMat
not
1
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1/4
users
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ease
nid:1768521665087
Cloze c1
Cloze answer: not
Q: \(0\) is {{c1::not}} in \(A^*\) where {{c2::\(A\) is a multiplicative algebra like \(\mathbb{Z}_{25}\)}}. Justification Included
A: \(\gcd(0, n) = n\) and not \(1\)!
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768521665088 |
1 | 230% | 117d | 12 |
nid:1764867991416
c1
DiskMat
The set \(\mathcal{C} = {{c1::\text{Im}(E)}}\) is called the...
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nid:1764867991416
Cloze c1
Q: The set \(\mathcal{C} = {{c1::\text{Im}(E)}}\) is called the {{c2::set of codewords}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991417 |
1 | 230% | 131d | 9 |
nid:1764867991229
DiskMat
What is the characteristic of \(\mathbb{Z}_m\)?
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nid:1764867991229
Q: What is the characteristic of \(\mathbb{Z}_m\)?
A: The characteristic of \(\mathbb{Z}_m\) is \(m\).
Explanation: The characteristic is the order of \(1\) in the additive group. In \(\mathbb{Z}_m\), adding \(1\) to itself \(m\) times gives: \[\underbrace{1 + 1 + \cdots + 1}_{m \text{ times}} = m \equiv_m 0\]
So \(\text{ord}(1) = m\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991229 |
1 | 230% | 136d | 9 |
nid:1764867991382
DiskMat
When does an element of \(F[x]_{m(x)}\) have an inverse?
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nid:1764867991382
Q: When does an element of \(F[x]_{m(x)}\) have an inverse?
A: Lemma 5.36: The congruence equation \[a(x)b(x) \equiv_{m(x)} 1\] for a given \(a(x)\) has a solution \(b(x) \in F[x]_{m(x)}\) if and only if \(\gcd(a(x), m(x)) = 1\). The solution is unique.
In other words: \[ F[x]_{m(x)}^* = \{a(x) \in F[x]_{m(x)} \ | \ \gcd(a(x), m(x)) = 1\} \]
This is analogous to \(\mathbb{Z}_m^*\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991382 |
1 | 230% | 134d | 9 |
nid:1764867990569
c2
DiskMat
an inverse relation
1
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users
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nid:1764867990569
Cloze c2
Cloze answer: an inverse relation
Q: The definition of {{c2::an inverse relation}} is \( a \ \rho \ b \iff{{c1:: b \ \hat{\rho} \ a}}\).
A: Example: Inverse of parent relation is childhood relation. Also written as \( \rho^{-1}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990569 |
1 | 230% | 157d | 9 |
nid:1764867991373
DiskMat
What are the equivalence classes modulo \(m(x)\) in a polyno...
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nid:1764867991373
Q: What are the equivalence classes modulo \(m(x)\) in a polynomial field?
A: Lemma 5.33: Congruence modulo \(m(x)\) is an equivalence relation on \(F[x]\), and each equivalence class has a unique representation of degree less than \(\deg(m(x))\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991373 |
1 | 230% | 139d | 9 |
nid:1764867991413
c1
DiskMat
codeword
1
lapses
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users
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nid:1764867991413
Cloze c1
Cloze answer: codeword
Q: The {{c2::output \((c_0, \dots, c_{n-1})\)}} of an encoding function is called a {{c1::codeword}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991414 |
1 | 230% | 141d | 9 |
nid:1764867990311
c1
DiskMat
For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), wh...
1
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users
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nid:1764867990311
Cloze c1
Q: For \(f: \mathbb{R} \to \mathbb{R}\) with \(f(x) = x^2\), what are:
1. Range: {{c1::\(\mathbb{R}^{\geq 0}\) (non-negative reals)}}2. Preimage of \([4, 9]\): {{c2::\([-3, -2] \cup [2, 3]\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990311 |
1 | 230% | 159d | 9 |
nid:1764867991315
DiskMat
\(\alpha \in F\) is a root of \(a(x)\) if and only if:
1
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nid:1764867991315
Q: \(\alpha \in F\) is a root of \(a(x)\) if and only if:
A: \((x - \alpha)\) divides \(a(x)\).
Corollary: An irreducible polynomial of degree \(\geq 2\) has no roots.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991315 |
1 | 230% | 149d | 9 |
nid:1766448533127
c1
DiskMat
A proof system \(\Pi\) is {{c1:: a quadruple \(\Pi = (\mathc...
1
lapses
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users
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nid:1766448533127
Cloze c1
Q: A proof system \(\Pi\) is {{c1:: a quadruple \(\Pi = (\mathcal{S, P}, \tau, \phi)\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448533127 |
1 | 230% | 146d | 9 |
nid:1764867990323
DiskMat
What is the principle behind the proof step of composing imp...
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lapses
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users
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nid:1764867990323
Q: What is the principle behind the proof step of composing implications?
A: If \(S \Rightarrow T\) and \(T \Rightarrow U\) are both true, then \(S \Rightarrow U\) is also true (transitivity of implication).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867990323 |
1 | 230% | 156d | 9 |
nid:1766448532960
c3
DiskMat
In a finite group the function \(x \rightarrow x^e\) is {{c1...
1
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users
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nid:1766448532960
Cloze c3
Q: In a finite group the function \(x \rightarrow x^e\) is {{c1:: a bijection}} if {{c2::\(e\) coprime to \(|G|\)}}.For \(x^e = y\), the inverse of \(y\) is {{c3:: the unique \(e\)-th root \(x = y^d\), with \(de \equiv_{|G|} 1\)}}.
A: Proof:We have \(ed = k \cdot |G| + 1\) for some \(k\). Thus, for any \(x \in G\) we have\[(x^e)^d = x^{ed} = x^{k \cdot |G| + 1} = \underbrace{(x^{|G|})^k}_{=1} \cdot x = x\]which means that the function \(y \mapsto y^d\) is the inverse function of the function \(x \mapsto x^e\) (which is hence a bijection). The under-braced term is equal to 1 because the order of \(x\) must divide the order of \(G\) (Lagrange).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1766448532961 |
1 | 230% | 149d | 9 |
nid:1768263609578
c1
EProg
Primitives - instanceof only works with reference types
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nid:1768263609578
Cloze c1
Cloze answer: Primitives - instanceof only works with reference types
Q: The cases where instanceof causes a compile error:{{c1::Primitives - instanceof only works with reference types}}{{c2::Generics - type erasure means List<String> becomes just List at runtime, so the check is impossible t instanceof List<Str
A: However:Animal a = getanimal() could get a Dog which might implement List thus a instanceof List is not a compile error.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768263609578 |
1 | 230% | 67d | 10 |
nid:1765655188119
c1
EProg
Terminal; Literal
1
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nid:1765655188119
Cloze c1
Cloze answer: Terminal; Literal
Q: Ein Symbol (auf der RHS) wie z.B. 1, a, A in EBNF wird {{c1::Terminal}} oder auch {{c1::Literal}} gennant.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765655188119 |
1 | 230% | 91d | 12 |
nid:1767918757948
EProg
What does 5 % 0 produce in Java?
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nid:1767918757948
Q: What does 5 % 0 produce in Java?
A: Runtime error, division by 0
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1767918757949 |
1 | 230% | 89d | 11 |
nid:1769307700300
c1
EProg
false
1
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nid:1769307700300
Cloze c1
Cloze answer: false
Q: The weakest precondition for an empty program with postcondition false is {{c1::false}}.
A: As only false implies false.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1769307700300 |
1 | 230% | 95d | 10 |
nid:1768263609443
EProg
instanceof can result in a Compile-/Runtime-/No error?
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nid:1768263609443
Q: instanceof can result in a Compile-/Runtime-/No error?
A: instanceof never throws an exception, just compile errors.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768263609443 |
1 | 230% | 98d | 11 |
nid:1765655188156
EProg
Which of the following is (or are) NOT a Java keyword? - vol...
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nid:1765655188156
Q: Which of the following is (or are) NOT a Java keyword? - volatile- mod- strictfp- loop- transient- do- use
A: loop, use and mod
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765655188156 |
1 | 230% | 126d | 12 |
nid:1765655188125
c1
EProg
last
1
lapses
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nid:1765655188125
Cloze c1
Cloze answer: last
Q: The convention for EBNF is that the rule being considered is written {{c1::last}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765655188125 |
1 | 230% | 145d | 11 |
nid:1765655188143
c2
EProg
repetition (Wiederholung)
1
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nid:1765655188143
Cloze c2
Cloze answer: repetition (Wiederholung)
Q: Not every EBNF language (Sprache) can be described just with{{c2:: repetition (Wiederholung)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765655188144 |
1 | 230% | 141d | 9 |
nid:1767918757856
EProg
5 == 5 || String.yourStupidAss() evaluates to ???
1
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nid:1767918757856
Q: 5 == 5 || String.yourStupidAss() evaluates to ???
A: Compile Error, even if it shortcircuits.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1767918757857 |
1 | 230% | 157d | 8 |
nid:1765655188137
c1
EProg
ihre Sprachen gleich sind.
1
lapses
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nid:1765655188137
Cloze c1
Cloze answer: ihre Sprachen gleich sind.
Q: Zwei EBNF-Beschreibungen sind äquivalent falls {{c1:: ihre Sprachen gleich sind.}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1765655188137 |
1 | 230% | 173d | 9 |
nid:1768182517703
LinAlg
How do we find the inverse of \(A\) using Gauss-Jordan?
1
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users
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nid:1768182517703
Q: How do we find the inverse of \(A\) using Gauss-Jordan?
A: We do \(\text{RREF}(A, I)\) which gives us \((R, j_1, \dots, j_r, M)\) where in the case that \(A\) is invertible:\(R\) is \(I\) and \(r = n\)\(M = A^{-1}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182517703 |
1 | 230% | 66d | 10 |
nid:1768608741184
c1
LinAlg
the vectors \(v \neq 0\), \(v \in N(A - \lambda_i I)\), in t...
1
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1/4
users
230%
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nid:1768608741184
Cloze c1
Cloze answer: the vectors \(v \neq 0\), \(v \in N(A - \lambda_i I)\), in the nullspace
Q: All the eigenvectors for \(\lambda_i\) are {{c1::the vectors \(v \neq 0\), \(v \in N(A - \lambda_i I)\), in the nullspace::subspace}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768608741184 |
1 | 230% | 65d | 10 |
nid:1767105283735
LinAlg
What can we use to speed up long matrix multiplications, for...
1
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1/4
users
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nid:1767105283735
Q: What can we use to speed up long matrix multiplications, for example \(w^\intercal (vw^\intercal) v\)?
A: We can use associativity: \(w^\intercal (vw^\intercal) v = (w^\intercal v)(w^\intercal v)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1767105283735 |
1 | 230% | 79d | 8 |
nid:1768425682248
c1
LinAlg
multiplicative
1
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nid:1768425682248
Cloze c1
Cloze answer: multiplicative
Q: The sign of a permutation is {{c1::multiplicative::property}}: \(\text{sgn}(\sigma \circ \lambda) = {{c1:: \text{sgn}(\sigma) \cdot \text{sgn}(\lambda)}}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768425682248 |
1 | 230% | 67d | 10 |
nid:1768608739489
c1
LinAlg
the same
1
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1/4
users
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nid:1768608739489
Cloze c1
Cloze answer: the same
Q: The eigenvectors of \(A^{-1}\) are {{c1::the same}} as those of \(A\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768608739489 |
1 | 230% | 67d | 10 |
nid:1768944603654
LinAlg
How to recover a matrix \(A\) from it's eigenvectors and eig...
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users
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nid:1768944603654
Q: How to recover a matrix \(A\) from it's eigenvectors and eigenvalues (complete set)?
A: \(V\) the matrix with the eigenvectors of \(A\), orthogonal. Then we know \(AV = VD\) (\(Av_i = \lambda_i v_i\) in matrix form), with \(D = \Lambda\) the matrix with the eigenvalues on the diagonal.Thus \(AVV^\top = VDV^\top \implies A = VDV^\top\) .
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768944603654 |
1 | 230% | 69d | 10 |
nid:1768263610143
LinAlg
How do we get the \(QR\) decomposition for \(A\) with linear...
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1/4
users
230%
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nid:1768263610143
Q: How do we get the \(QR\) decomposition for \(A\) with linearly independent columns?
A: \(Q\) is the result of Gram-Schmidt on \(A\)\(R = Q^\top A\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768263610143 |
1 | 230% | 74d | 11 |
nid:1768182517756
c1
LinAlg
For \(A\) a matrix and \(M\) an invertible matrix:\(C(A) = \...
1
lapses
1/4
users
230%
ease
nid:1768182517756
Cloze c1
Q: For \(A\) a matrix and \(M\) an invertible matrix:\(C(A) = \) {{c1::Not equal to \(\textbf{C}(MA)\), the column space changes!}}
A: \(\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}\) after RREF is \(\begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}\) which spans a completely different line.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182517756 |
1 | 230% | 80d | 10 |
nid:1768944603346
c2
LinAlg
\(v_1, \dots, v_n\) are an orthonormal basis of eigenvector...
1
lapses
1/4
users
230%
ease
nid:1768944603346
Cloze c2
Cloze answer: \(v_1, \dots, v_n\) are an orthonormal basis of eigenvectors (the \(V\) in diagonalisation) and \(\lambda_1, \dots, \lambda_n\) the associated eigenvectors
Q: We can write \(A\) as the sum of {{c1::rank \(1\) matrices}}: \[A = {{c2::\sum_{k = 1}^n \lambda_i v_i v_i^\top}}\]where {{c2:: \(v_1, \dots, v_n\) are an orthonormal basis of eigenvectors (the \(V\) in diagonalisation) and \(\lambda_1, \dots, \lambd
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768944603347 |
1 | 230% | 72d | 12 |
nid:1768608740128
c1
LinAlg
\(A,B\) share an EV
1
lapses
1/4
users
230%
ease
nid:1768608740128
Cloze c1
Cloze answer: \(A,B\) share an EV
Q: If \(AB = BA\), then {{c1::\(A,B\) share an EV::EVs of A, B}}.
A: Assume \(AB = BA\).If \(\lambda, v\) an EW-EV pair of \(A\) then \(A(Bv) = (AB)v = B(Av) = \lambda Bv\) thus \(Bv\) is an eigenvector of \(A\).Then \(Bv\) is a multiple of some \(v\) of that EW \(\lambda\) (easiest to see for \(A\) complete set of real EVs) \(\implies\) \(Bv = \lambda'v\) thus that \(v\) is also an EV of \(B\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768608740128 |
1 | 230% | 83d | 10 |
nid:1768944601791
c1
LinAlg
\(n - \dim(N(A))\) so it's \(n\) minus the geometric multip...
1
lapses
1/4
users
230%
ease
nid:1768944601791
Cloze c1
Cloze answer: \(n - \dim(N(A))\) so it's \(n\) minus the geometric multiplicity of \(\lambda = 0\)
Q: \(A \in \mathbb{R}^{n \times n}\) arbitrary non-symmetric has rank {{c1:: \(n - \dim(N(A))\) so it's \(n\) minus the geometric multiplicity of \(\lambda = 0\) ::in terms of multiplicities}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768944601791 |
1 | 230% | 80d | 10 |
nid:1768944602558
c1
LinAlg
all its eigenvalues are real and the geometric multiplicitie...
1
lapses
1/4
users
230%
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nid:1768944602558
Cloze c1
Cloze answer: all its eigenvalues are real and the geometric multiplicities are the same as the algebraic multiplicities of all it's eigenvalues
Q: A matrix has a complete set of real eigenvectors if {{c1::all its eigenvalues are real and the geometric multiplicities are the same as the algebraic multiplicities of all it's eigenvalues::in terms of multiplicities}}.
A: Example \(I\) has eigenvalue \(1\) with geometric multiplicity \(n\) (\(\dim(N(I - 1 \cdot I)) = n\)) and algebraic multiplicity \(n\) (As the characteristic polynomial of \(I\), \(P(z) = (z - 1)(z - 1) \dots (z - 1)\) with that repeated \(n\) times).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768944602558 |
1 | 230% | 85d | 10 |
nid:1764867991528
LinAlg
The Cauchy-Schwarz Inequality tells us that for \(\textbf{v}...
1
lapses
1/4
users
230%
ease
nid:1764867991528
Q: The Cauchy-Schwarz Inequality tells us that for \(\textbf{v}, \textbf{w} \in \mathbb{R}^m\)
A: \(|\textbf{v} \cdot \textbf{w}| \leq ||\textbf{v}|| \ ||\textbf{w}||\).This equality holds exactly if one vector is the scalar multiple of the other.This essentially means that: the length of the projecton of v onto w is smaller than the both of their lengths multiplied.This explains the equality part: if they are already aligned, their projection doesn't lose any length...
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1764867991528 |
1 | 230% | 105d | 11 |
nid:1768944603219
c1
LinAlg
a real eigenvalue \(\lambda\)
1
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users
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nid:1768944603219
Cloze c1
Cloze answer: a real eigenvalue \(\lambda\)
Q: Every symmetric matrix \(A \in \mathbb{R}^{n \times n}\) has {{c1::a real eigenvalue \(\lambda\)::existence}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768944603219 |
1 | 230% | 87d | 10 |
nid:1768182518317
c1
LinAlg
linearly independent columns; \(MA\) has linearly independen...
1
lapses
1/4
users
230%
ease
nid:1768182518317
Cloze c1
Cloze answer: linearly independent columns; \(MA\) has linearly independent colums
Q: For \(A\) a matrix and \(M\) an invertible matrix:\(A\) has {{c1::linearly independent columns}} if and only if {{c1::\(MA\) has linearly independent colums}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1768182518317 |
1 | 230% | 87d | 10 |
nid:1768263610700
c1
LinAlg
x_1 + N(A) ; \(x_1 \in R(A)\) is unique such that \(Ax_1 = ...
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Cloze c1
Cloze answer: x_1 + N(A) ; \(x_1 \in R(A)\) is unique such that \(Ax_1 = b\)
Q: Suppose that \(\{x \in \mathbb{R}^n \ | \ Ax = b \} \not = \emptyset\). Then \[ \{x \in \mathbb{R}^n \ | \ Ax = b \} = {{c1::x_1 + N(A) }}\] where {{c1:: \(x_1 \in R(A)\) is unique such that \(Ax_1 = b\)}}.
A: This means that if there's more than one solution to the system (i.e. the nullspace is not \(= \{0\}\)), then the set of all solutions is a specific solution + the entire nullspace.
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| lorenz | cid:1768263610700 |
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nid:1768182517631
c2
LinAlg
there is only one \(0\)
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Cloze c2
Cloze answer: there is only one \(0\)
Q: In a vector space \(V\) three important properties hold:{{c1::\(0v = 0\) for all \(v\)}}{{c2:: there is only one \(0\)}}{{c3:: one unique inverse \(-v\) for all \(v\)}}
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| lorenz | cid:1768182517633 |
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nid:1768608741058
c1
LinAlg
\(\frac{1}{z} = {{c1:: \frac{\overline{z} }{|z|^2} :: \text{...
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nid:1768608741058
Cloze c1
Q: \(\frac{1}{z} = {{c1:: \frac{\overline{z} }{|z|^2} :: \text{in terms of z} }}\)
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| lorenz | cid:1768608741058 |
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nid:1768944603363
c2
LinAlg
Given \(n\) vectors \(v_1, \dots, v_n \in \mathbb{R}^n\) we ...
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Cloze c2
Q: Given \(n\) vectors \(v_1, \dots, v_n \in \mathbb{R}^n\) we call their {{c1::Gram matrix}} the {{c2::\(n \times n\) matrix of inner products \(G_{ij} = v_i^\top v_j\)}}.
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|---|---|---|---|---|---|
| lorenz | cid:1768944603364 |
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nid:1768944601064
LinAlg
Proof that the Rayleigh Quotient has it's maximum and minimu...
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nid:1768944601064
Q: Proof that the Rayleigh Quotient has it's maximum and minimum at the largest/smallest EWs?
A: It is easy to see that \(R(v_{\max}) = \lambda_{\max}\) and \(R(v_{\min}) = \lambda_{\min}\). See: \(R(v_{\text{max}}) = \frac{v_{\text{max}}^\top A v_{\text{max}}}{v_{\text{max}}^\top v_{\text{max}}} = \frac{v_{\text{max}}^\top (\lambda_{\text{max}} v_{\text{max}})}{v_{\text{max}}^\top v_{\text{max}}} = \lambda_{\text{max}}\)
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|---|---|---|---|---|---|
| lorenz | cid:1768944601064 |
1 | 230% | 89d | 10 |
nid:1768608739773
c1
LinAlg
not correlated
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nid:1768608739773
Cloze c1
Cloze answer: not correlated
Q: The eigenvalues of \(AB\) and \(BA\) are {{c1::not correlated}}.
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|---|---|---|---|---|---|
| lorenz | cid:1768608739773 |
1 | 230% | 94d | 10 |
nid:1768344745450
LinAlg
Why is the pseudoinverse (for \(A\) with full row-rank) \(A^...
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nid:1768344745450
Q: Why is the pseudoinverse (for \(A\) with full row-rank) \(A^\top (AA^\top)^{-1}\)?
A: It uses the multiplication by \(A^\top\) to choose an \(\hat{x}\) that lies in the row-space, thus minimising the norm.
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|---|---|---|---|---|---|
| lorenz | cid:1768344745450 |
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nid:1768182517842
c1
LinAlg
\(R = I\)
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nid:1768182517842
Cloze c1
Cloze answer: \(R = I\)
Q: \(A\) is invertible if and only if for \(\text{RREF}(A,I) = (R, M)\) we have {{c1::\(R = I\)}}.
A: Since we have \(R = MA\), \(M\) is the inverse of \(A\).
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| lorenz | cid:1768182517842 |
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nid:1768263611504
c1
LinAlg
\(A\) has linearly independent columns
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nid:1768263611504
Cloze c1
Cloze answer: \(A\) has linearly independent columns
Q: \(A^\top A\) is invertible if and only if {{c1::\(A\) has linearly independent columns}}.
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|---|---|---|---|---|---|
| lorenz | cid:1768263611504 |
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nid:1768263611647
c1
LinAlg
\(A\) to have independent columns, i.e. they form a basis fo...
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nid:1768263611647
Cloze c1
Cloze answer: \(A\) to have independent columns, i.e. they form a basis for \(C(A)\)
Q: For a projection to exist using our formula \(P = A (A^\top A)^{-1} A^\top\) we need {{c1:: \(A\) to have independent columns, i.e. they form a basis for \(C(A)\)}}.
A: Otherwise the projection is not unique.
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| lorenz | cid:1768263611648 |
1 | 230% | 99d | 8 |
nid:1768263609972
c1
LinAlg
norm; inner product
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nid:1768263609972
Cloze c1
Cloze answer: norm; inner product
Q: Orthogonal matrices preserve the {{c1::norm}} and {{c1::inner product}} of vectors.
A: In other words, if \(Q \in \mathbb{R}^{n \times n}\) is orthogonal, then, for all \(x, y \in \mathbb{R}^n\):\[ ||Qx|| = ||x|| \text{ and } (Qx)^\top(Qy) = x^\top y \]
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| lorenz | cid:1768263609972 |
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nid:1768425680760
LinAlg
How can we use Gauss-Jordan to simplify the determinant calc...
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nid:1768425680760
Q: How can we use Gauss-Jordan to simplify the determinant calculations?
A: We can use Gauss-Jordan to make any matrix upper triangular (then the determinant is the product of the diagonals).We are allowed to use:Row addition / substractionExchanging rows (change sign)Multiply rows (multiply the determinant at the end)
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| lorenz | cid:1768425680760 |
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nid:1764867991521
LinAlg
The euclidian norm of \(\textbf{v}\) is defined as?
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nid:1764867991521
Q: The euclidian norm of \(\textbf{v}\) is defined as?
A: \(|| \textbf{v} || := \sqrt{\textbf{v} \cdot \textbf{v}}\)This is also called the 2-norm.
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| lorenz | cid:1764867991521 |
1 | 230% | 117d | 11 |
nid:1768608739855
c1
LinAlg
Every polynomial \(P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \d...
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nid:1768608739855
Cloze c1
Q: Every polynomial \(P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0\) with \(a_n \neq 0\) has {{c1:: a zero \(\lambda \in \mathbb{C} \)}}.
A: Fundamental theorem of algebra
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| lorenz | cid:1768608739855 |
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nid:1768425682505
c1
LinAlg
the parity of the number of row swaps necessary to get back ...
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nid:1768425682505
Cloze c1
Cloze answer: the parity of the number of row swaps necessary to get back to the identity
Q: The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is {{c1:: the parity of the number of row swaps necessary to get back to the identity ::swaps}}.
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| lorenz | cid:1768425682505 |
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nid:1768263611355
c2
LinAlg
\(z = 0\); \(z^\top b = 0 \neq 1\)
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nid:1768263611355
Cloze c2
Cloze answer: \(z = 0\); \(z^\top b = 0 \neq 1\)
Q: Applications of the certificate of no solutions:Assume \(A \in \mathbb{R}^{m \times n}\) has linearly independent rows.Since {{c1::the rows are linearly independent}}, the only solution to \(z^\top A = 0\) is {{c2::\(z = 0\)}}. Hence {{c2::\(z^\top b = 0 \neq 1\)}}.
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|---|---|---|---|---|---|
| lorenz | cid:1768527254333 |
1 | 230% | 112d | 8 |
nid:1768521670852
c1
LinAlg
The determinant expressed in terms of co-factors is: \[\det(...
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nid:1768521670852
Cloze c1
Q: The determinant expressed in terms of co-factors is: \[\det(A) = {{c1:: \sum_{j = 1}^n A_{ij}C_{ij} }}\]
A: in which we multiply the cofactor of every element by the element itself, as is clear in the example for a 3x3.
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|---|---|---|---|---|---|
| lorenz | cid:1768521670852 |
1 | 230% | 117d | 11 |
nid:1768263611327
c1
LinAlg
\(I - P\)
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nid:1768263611327
Cloze c1
Cloze answer: \(I - P\)
Q: Let \(S^\perp\) be the orthogonal complement of \(S\) and \(P\) the projection matrix onto \(S\).Then {{c1::\(I - P\)}} is the projection matrix that maps {{c2::\(b \in \mathbb{R}^m\) to \(\text{proj}_{S^\perp}(b)\)}}.Proof Included
A: Since \(b = e + \text{proj}_S(b) = e + Pb\) with \(e \in S^\perp\) Thus \[ (I - P)b = b - Pb = e = \text{proj}_{S^\perp}(b) \]This is true, since it holds that indeed \(I - P\) is also idempotent: \((I - P)^2 = I - 2P + P^2 = I -P - P + P= I - P\)
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| lorenz | cid:1768263611327 |
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nid:1768425681409
c1
LinAlg
Multilinearity of the determinant:\[ \begin{vmatrix} ta & tb...
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nid:1768425681409
Cloze c1
Q: Multilinearity of the determinant:\[ \begin{vmatrix} ta & tb \\ c & d \end{vmatrix} = {{c1:: t \cdot \begin{vmatrix} a & b \\ c & d \end{vmatrix} }}\]
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| lorenz | cid:1768425681409 |
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nid:1768263610888
c1
LinAlg
A; \(QQ^\top \) is the projection onto \(A\), and \(C(Q) = C...
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nid:1768263610888
Cloze c1
Cloze answer: A; \(QQ^\top \) is the projection onto \(A\), and \(C(Q) = C(A)\)
Q: \(QQ^\top A = {{c1::A}}\) because {{c1::\(QQ^\top \) is the projection onto \(A\), and \(C(Q) = C(A)\)}}.
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|---|---|---|---|---|---|
| lorenz | cid:1768263610888 |
1 | 230% | 121d | 8 |
nid:1768521672527
c1
LinAlg
Given a permutation matrix \(P \in \mathbb{R}^{n \times n}\)...
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nid:1768521672527
Cloze c1
Q: Given a permutation matrix \(P \in \mathbb{R}^{n \times n}\) corresponding to a permutation \(\sigma\), then \(\det(P) = {{c1::\text{sgn}(\sigma)}}\)
A: (this is as \(P\) is also an orthogonal matrix, see 3.). We sometimes write \(\text{sgn}(P)\).For the permutation matrix, each row contains only one entry: a \(1\). Thus the only permutation \(\sigma\) in the product that doesn't have a \(0\) factor is the permutation corresponding to the matrix \(P\) itself. The product is \(1 \cdot 1 \dots \cdot 1\) thus we get \(\text{sgn}(\sigma) = \text{sgn}(P)\).
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| lorenz | cid:1768521672527 |
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nid:1768608739736
LinAlg
Prove some \(x \in \mathbb{C}\) is actually in \(\mathbb{R}\...
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nid:1768608739736
Q: Prove some \(x \in \mathbb{C}\) is actually in \(\mathbb{R}\)?
A: Show that \(x = \overline{x} \implies x \in \mathbb{R}\)
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|---|---|---|---|---|---|
| lorenz | cid:1768608739736 |
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nid:1768944602222
c1
LinAlg
\(n\) real eigenvalues
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Cloze c1
Cloze answer: \(n\) real eigenvalues
Q: Spectral Theorem: Any symmetric matrix \(A \in \mathbb{R}^{n \times n}\) has {{c1::\(n\) real eigenvalues::EW}} and {{c1::an orthonormal basis of \(\mathbb{R}^{n \times n}\) consisting of it's eigenvectors::EV}}.
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|---|---|---|---|---|---|
| lorenz | cid:1768944602223 |
1 | 230% | 116d | 13 |
nid:1768263610707
LinAlg
Why does \(QR\) give \(A\)?
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nid:1768263610707
Q: Why does \(QR\) give \(A\)?
A: \(QQ^\top\) is the projection on the span of the \(q_i\)'s and thus also on the \(a_i\)'s (\(C(Q) = C(A)\)).Thus \(QQ^\top A = A\) and therefore \(QR = QQ^\top A = A\).
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|---|---|---|---|---|---|
| lorenz | cid:1768263610707 |
1 | 230% | 121d | 11 |
nid:1768263608594
c1
LinAlg
are orthogonal
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Cloze c1
Cloze answer: are orthogonal
Q: Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric matrix and \(\lambda_1 {{c2::\neq}} \lambda_2 \in \mathbb{R}\) two {{c2::distinct}} eigenvalues of \(A\) with corresponding eigenvectors \(v_1, v_2\).Then \(v_1\) and \(v_2\) {{c1::are orthogonal}}. Proof Included
A: \(\lambda_1 v_1 ^\top v_2 = (Av_1)^\top v_2\) \( = v_1^\top A ^\top v_2 = \) \(v_1^\top (Av_2)\) \( = \lambda_2 v_1^\top v_2\)
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| lorenz | cid:1768263608595 |
1 | 230% | 126d | 11 |
nid:1768608740846
LinAlg
What is the fundamental theorem of algebra?
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nid:1768608740846
Q: What is the fundamental theorem of algebra?
A: Any degree \(n\) polynomial \(P(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0\) (with \(n \geq 1\) and \(a_n \neq 0\)) has at least one zero \(\lambda \in \mathbb{C}\) such that \(P(\lambda) = 0\).
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|---|---|---|---|---|---|
| lorenz | cid:1768608740846 |
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nid:1764867991504
LinAlg
A linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2...
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nid:1764867991504
Q: A linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is affine if
A: \(\lambda_1 + \lambda_2 + \dots + \lambda_n = 1\)
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| lorenz | cid:1764867991504 |
1 | 230% | 124d | 12 |
nid:1768608742500
c1
LinAlg
For a complex vector \(v\) we have \(||v|| =\) {{c1:: \(v^*v...
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nid:1768608742500
Cloze c1
Q: For a complex vector \(v\) we have \(||v|| =\) {{c1:: \(v^*v = \overline{v}^\top v = \sum_{i = 1}^n \overline{v_i}v_i = \sum_{i = 1}^n |v_i|^2\)}}.
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|---|---|---|---|---|---|
| lorenz | cid:1768608742500 |
1 | 230% | 129d | 11 |
nid:1768944602019
c2
LinAlg
\(A^\top A\); \(AA^\top\)
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nid:1768944602019
Cloze c2
Cloze answer: \(A^\top A\); \(AA^\top\)
Q: Given a real matrix \(A \in \mathbb{R}^{n \times n}\), the {{c1::non-zero eigenvalues}} of {{c2::\(A^\top A\)}} are the same ones as of {{c2::\(AA^\top\)}}. Proof Included
A: Shared EWs: For \((A^\top A)v_k = \lambda_k v_k\) we get \(AA^\top A v_k = \lambda_k Av_k\) and thus \(Av_k\) EV and \(\lambda_k\) is an EW of \(AA^\top\).Orthogonality: For \(j \neq k\) we have \((Av_j)^\top (Av_k) = v_j^\top A^\top Av_k = v_j^\top \lambda_k v_k = \lambda_k v_j^\top v_k = 0\)
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| lorenz | cid:1768944602020 |
1 | 230% | 125d | 12 |
nid:1768263611621
LinAlg
In QR decomposition \(R\) is invertible because?
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nid:1768263611621
Q: In QR decomposition \(R\) is invertible because?
A: \(N(A) = \{0\}\) since \(A\) has independent columns and thus \(N(R) = \{0\}\):\(Rx = 0\) then \(Ax = QRx = 0\) thus \(Q\cdot 0 = 0\)Thus \(x \in N(A) \implies x = 0\)Thus \(R \in \mathbb{R}^{n \times n}\) (square) must be invertible.
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|---|---|---|---|---|---|
| lorenz | cid:1768263611621 |
1 | 230% | 127d | 14 |
nid:1768344745223
c1
LinAlg
\(C(A^\top)\)
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nid:1768344745223
Cloze c1
Cloze answer: \(C(A^\top)\)
Q: \(A^\dagger A\) is the projection matrix onto {{c1::\(C(A^\top)\)}}.
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|---|---|---|---|---|---|
| lorenz | cid:1768344745223 |
1 | 230% | 129d | 11 |
nid:1768608742013
c1
LinAlg
possibly with repetitions
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nid:1768608742013
Cloze c1
Cloze answer: possibly with repetitions
Q: Any degree \(n\) polynomial \(P(z)\) (with \(n \geq 1\)) has {{c1::\(n\) zeros \(\lambda_1, \dots, \lambda_n \in \mathbb{C}\)}}, {{c1::possibly with repetitions}}, such that \[P(z) = a_n (z-\lambda_1)(z - \lambda_2) \cdots (z - \lambda_n)\]
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|---|---|---|---|---|---|
| lorenz | cid:1768608742013 |
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nid:1768344745894
LinAlg
What is the pseudoinverse in the case where \(A \in \mathbb{...
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nid:1768344745894
Q: What is the pseudoinverse in the case where \(A \in \mathbb{R}^{n \times m}\) has independent columns?
A: Because \(rank(A) = r = n\) and thus \(m \geq n\)\(R(A)\) spans \(\mathbb{R}^n\)(rows span the space)\(C(A) \subseteq\) \(\mathbb{R}^m\) (as \(A\) is not necessarily square)We therefore first project \(b\) into \(C(A)\) and then invert, which is Least Squares.
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| lorenz | cid:1768344745895 |
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nid:1764867991560
c2
LinAlg
Name the three definitions for linear independence:{{c1::Non...
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nid:1764867991560
Cloze c2
Q: Name the three definitions for linear independence:{{c1::None of the vectors is a linear combination of the other ones.}}{{c2::There are no scalars \(\lambda_1, ..., \lambda_n\) besides 0, 0, ..., 0 such that \(\sum_{i = 1}^n \lambda_i v_i = \mathbf{0}\). (\
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| lorenz | cid:1766491319679 |
1 | 230% | 145d | 9 |
nid:1768608742035
LinAlg
Does \(Av = v\) mean \(1\) is an eigenvalue of \(A\)?
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nid:1768608742035
Q: Does \(Av = v\) mean \(1\) is an eigenvalue of \(A\)?
A: No, we need to have \(v \neq 0\) to have that relationship hold!
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| lorenz | cid:1768608742035 |
1 | 230% | 144d | 11 |
nid:1768263610822
c1
LinAlg
symmetric
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nid:1768263610822
Cloze c1
Cloze answer: symmetric
Q: A projection matrix is always {{c1:: symmetric ::property?}} (note that this needs to be reproven in the exam, proof included)
A: \(P^\top = (A(A^\top A)^{-1} A^\top)^\top =\) \((A^\top)^\top {(A^\top A)^{-1}}^\top A^\top = A(A^\top A)^{-1} A^\top = P\)We use the fact that for invertible matrices \({M^{-1}}^\top = {M^\top}^{-1}\).
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| lorenz | cid:1768263610823 |
1 | 230% | 146d | 11 |
nid:1768263610994
LinAlg
Certificate of no solutions:Given \(P = \{x \in \mathbb{R}^n...
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Q: Certificate of no solutions:Given \(P = \{x \in \mathbb{R}^n \mid Ax = b \}\) we have: \(P = \left\{ x \in \mathbb{R}^3 \;\middle|\; \begin{aligned} x_1 + 2x_2 - x_3 &= 1 \\ 2x_1 + 4x_2 - 2x_3 &= 0 \end{aligned} \right\}\)Provide the system
A: The system \(D = \{ z \in \mathbb{R}^m | A^\top z = 0, b^\top z = 1 \}\) then is: \[D = \left\{ z \in \mathbb{R}^2 \;\middle|\; \begin{aligned} z_1 + 2z_2 &= 0 \\ 2z_1 + 4z_2 &= 0 \\ -z_1 - 2z_2 &= 0 \\ z_1 &= 1 \end{aligned} \right\}\]One equation per each column of \(A\).\(P = \emptyset\) and \(D \neq \emptyset\) because \(z = (1, -\frac{1}{2})^\top \in D\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
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| lorenz | cid:1768263610994 |
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c1
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die Berechnung von \(low[]\)
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Cloze c1
Cloze answer: die Berechnung von \(low[]\)
Q: Die um {{c1::die Berechnung von \(low[]\)}} ergänzte {{c2::Tiefensuche}} berechnet in einem zusammenhängenden Graphen alle Artikulationsknoten und Brücken in Zeit \(O({{c3::m}})\).
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| lorenz | cid:1771527094550 |
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nid:1774487164608
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Die Anzahl der Möglichkeiten, \(k\) Objekte aus \(n\) Sorten...
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Cloze c1
Q: Die Anzahl der Möglichkeiten, \(k\) Objekte aus \(n\) Sorten mit Zurücklegen zu wählen (Reihenfolge egal, Multiset) ist:\[{{c2::\binom{n + k - 1}{k} }} = {{c1::\frac{(n+k-1)!}{k!\,(n-1)!} }} \]
A: Auch bekannt als „Sterne und Striche“ (Stars and Bars).Beispiel: Wie viele Möglichkeiten, 3 Kugeln aus {rot, blau, grün} mit Zurücklegen zu ziehen?\(\binom{3+3-1}{3} = \binom{5}{3} = 10\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
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| lorenz | cid:1774487164608 |
1 | 230% | 6d | 9 |
nid:1776174099848
c1
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Für zwei unabhängige Zufallsvariablen \(X\) und \(Y\) sei \(...
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Cloze c1
Q: Für zwei unabhängige Zufallsvariablen \(X\) und \(Y\) sei \(Z := X + Y\). Es gilt:\[f_Z(\alpha) = {{c1::\sum_{\beta \in W_X} f_X(\beta) \cdot f_Y(\alpha - \beta)}}.\]
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| lorenz | cid:1776174099848 |
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nid:1776175078408
c2
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Für jede {{c1::nicht-negative}} Zufallsvariable \(X\) und al...
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Cloze c2
Q: Für jede {{c1::nicht-negative}} Zufallsvariable \(X\) und alle \(t > 0\), gilt\[\Pr\left[X \geq t\right] \leq {{c2::\frac{\mathbb{E}[X]}{t} }}.\]
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| lorenz | cid:1776175078409 |
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nid:1773307783473
IO r1
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[Image Occlusion region 1]
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Cloze c1
Q: {{c1::image-occlusion:polygon:left=.011:top=.2474:points=.0836,.2506 .4728,.2474 .4728,.3534 .011,.3566 .011,.3052 .0836,.3052}}{{c2::image-occlusion:rect:left=.0572:top=.4433:width=.1363:height=.045}}{{c2::image-occlusion:rect:left=.0924:top=.5815:width=.1869:height=.0514}}{{c3::image-o
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| lorenz | cid:1773307783481 |
1 | 230% | 21d | 10 |
nid:1774487164522
c1
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Die Anzahl der Anordnungen von \(n\) Objekten, von denen\(n_...
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Cloze c1
Q: Die Anzahl der Anordnungen von \(n\) Objekten, von denen\(n_1\) vom Typ 1, …, \(n_r\) vom Typ \(r\) sind (\(n_1 + \cdots + n_r = n\)), ist:\[{{c1::\frac{n!}{n_1!\, n_2!\, \cdots\, n_r!} }} = \binom{n}{n_1, n_2, \ldots, n_r}\](Multinomialkoeffizient)
A: Speziell für \(r=2\): \(\frac{n!}{k!\,(n-k)!} = \binom{n}{k}\).Beispiel: Anordnungen von „MISSISSIPPI“: \(\frac{11!}{1!\cdot 4!\cdot 4!\cdot 2!} = 34{,}650\).
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|---|---|---|---|---|---|
| lorenz | cid:1774487164523 |
1 | 230% | 2d | 8 |
nid:1774631276995
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Sei \(A_1,\ldots,A_n\) eine Partition von \(\Omega\) mit \(\...
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Cloze c1
Q: Sei \(A_1,\ldots,A_n\) eine Partition von \(\Omega\) mit \(\Pr[A_i]>0\) für alle \(i\). Dann:\[ \mathbb{E}[X] = {{c1::\sum_{i=1}^{n}\mathbb{E}[X\mid A_i]\cdot\Pr[A_i]}}. \]Proof Included
A: (Gesetz der totalen Erwartung, nicht im Skript!) Proof:\[\begin{align} \mathbb{E}[X] &=\sum_{x}x\cdot\Pr[X=x] \\ &\overset{\text{totale W'keit}}{=}\sum_x x\sum_i\Pr[X=x|A_i]\Pr[A_i] \\ &=\sum_i\Pr[A_i]\underbrace{\sum_x x\Pr[X=x|A_i]}_{=\mathbb{E}[X|A_i]} \end{align}\](Verwendet das Gesetz der totalen Wahrscheinlichkeit um \(\Pr[X=x]\) zu expandieren, dann wird die Summationsreihenfolge vertauscht.)
| User | Card ID | Lapses | Ease | Interval | Reviews |
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| lorenz | cid:1774631276995 |
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c1
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Für \(n \geq 2\) heisst eine Zufallsvariable \(X\) mit Dicht...
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Cloze c1
Q: Für \(n \geq 2\) heisst eine Zufallsvariable \(X\) mit Dichte\[f_X(k) = \begin{cases} {{c1::\binom{k-1}{n-1} \cdot p^n \cdot (1 - p)^{k-n} }} & \text{für } k = 1, 2, \ldots \\ 0 & \text{sonst} \end{cases}\]{{c2::negativ binomialverteilt}} mit {{c3::Ordnung}} \(n\).
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| lorenz | cid:1776171659230 |
1 | 230% | 6d | 9 |
nid:1771364144766
c1
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Sei \(G = (V, E)\) ein zusammenhängender Graph. Der {{c4::Bl...
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Cloze c1
Q: Sei \(G = (V, E)\) ein zusammenhängender Graph. Der {{c4::Block-Graph}} von \(G\) ist der bipartite Graph \(T = (A \uplus B, E_T)\) mit\(A = {{c1::\{\text{Artikulationsknoten von } G\} }}\).
\(B = {{c2::\{\text{Blöcke von } G\} }}\).
\(\forall a \in A, b \i
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| lorenz | cid:1771364144769 |
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Für \(x, y \in \mathbb{R}\) und \(n \in \mathbb{N}_0\) gilt:...
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Cloze c1
Q: Für \(x, y \in \mathbb{R}\) und \(n \in \mathbb{N}_0\) gilt:\[(x + y)^n = {{c1::\sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k} }}\]
A: Speziell:\((1+x)^n = \sum_{k=0}^n \binom{n}{k} x^k\)\((1-1)^n = 0 = \sum_{k=0}^n (-1)^k \binom{n}{k}\) (genutzt im Siebformel-Beweis!)
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| lorenz | cid:1774487164478 |
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nid:1774631269283
c1
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vollkommen wurscht ob unabhängig, du dummbatzi
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Cloze c1
Cloze answer: vollkommen wurscht ob unabhängig, du dummbatzi
Q: Die Linearität der Erwartung hält, wenn \(X_1,\ldots,X_n\) {{c1::vollkommen wurscht ob unabhängig, du dummbatzi}} sind?
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| lorenz | cid:1774631269283 |
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c2
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2^n
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Cloze c2
Cloze answer: 2^n
Q: Es gilt:\[{{c1::\sum_{k=0}^{n} \binom{n}{k}::\text{Binomialsatz} }} = {{c2::2^n}}\]
A: Beweis: Setze \(x = y = 1\) im Binomialsatz: \((1+1)^n = \sum_{k=0}^n \binom{n}{k}\).Interpretation: Anzahl aller Teilmengen einer \(n\)-elementigen Menge ist \(2^n\).
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| lorenz | cid:1774693607608 |
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c2
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Seien \(\delta, \varepsilon > 0\). Falls \({{c1::N \geq 3\,\...
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Cloze c2
Q: Seien \(\delta, \varepsilon > 0\). Falls \({{c1::N \geq 3\,\frac{|U|}{|S|} \cdot \frac{1}{\varepsilon^2} \cdot \ln(\tfrac{2}{\delta})}}\), ist die Ausgabe \(Y\) von Target-Shooting mit Wahrscheinlichkeit mindestens \(1 - \delta\) im Intervall \[{
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| lorenz | cid:1776332605881 |
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nid:1774487164950
c1
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dreiecksfreien
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Cloze c1
Cloze answer: dreiecksfreien
Q: Für alle \(k \geq 2\) gibt es einen {{c1::dreiecksfreien}} Graphen \(G_k\) mit \(\chi(G_k) \geq k\).
A: (Mycielski-Konstruktion)Konstruktion: Aus \(G_k = (V_k, E_k)\) mit \(V_k = \{v_1,\ldots,v_n\}\) bilde \(G_{k+1}\):Füge Knoten \(w_1,\ldots,w_n, z\) hinzu. \(w_i\) ist mit allen Nachbarn von \(v_i\) verbunden (aber nicht mit \(v_i\) selbst). \(z\) ist mit allen \(w_i\) verbunden.Der neue Graph ist dreiecksfrei und braucht eine Farbe mehr als \(G_k\).
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| lorenz | cid:1774487164950 |
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nid:1774631277097
A&W
Wann ist der Erwartungswert \(\mathbb{E}[X] = \sum_{x\in W_X...
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Q: Wann ist der Erwartungswert \(\mathbb{E}[X] = \sum_{x\in W_X} x\cdot\Pr[X=x]\) undefiniert?
A: Falls die Summe nicht absolut konvergiert (z.B. positiver und negativer Anteil beide divergieren).Bemerkung:In der Vorlesung betrachten wir nur Zufallsvariablen mit definiertem Erwartungswert.Der Erwartungswert ist nur definiert, wenn die Summe absolut konvergiert, d.h. \(\sum_{x\in W_X}|x|\cdot\Pr[X=x]<\infty\).In endlichen Wahrscheinlichkeitsräumen ist dies immer erfüllt (endlich viele Terme). Bei unendlichen Räumen muss man aufpassen.
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| lorenz | cid:1774631277098 |
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A&W
Was besagt der Vierfarbensatz?
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Q: Was besagt der Vierfarbensatz?
A: Jeder planare Graph (jede Landkarte) lässt sich mit \(\leq 4\) Farben färben.Formal: Für jeden planaren Graphen \(G\) gilt \(\chi(G) \leq 4\).(Appel & Haken, 1976 - erster computergestützter Beweis)
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| lorenz | cid:1774487164532 |
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nid:1774631277414
c1
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Hamiltonkreis
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Cloze c1
Cloze answer: Hamiltonkreis
Q: Es existiert ein {{c1::Hamiltonkreis}} in einem Graphen \(G\) mit gerader Zahl Knoten \(\implies\)perfektes Matching existiert.
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| lorenz | cid:1774631277415 |
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nid:1774487164722
c1
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Eine Zufallsvariable auf \(\Omega\) ist {{c1::eine Funktion ...
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Cloze c1
Q: Eine Zufallsvariable auf \(\Omega\) ist {{c1::eine Funktion \(X\colon \Omega \to \mathbb{R}\)}}.\[\Pr[X = x] := {{c2::\Pr[\{\omega \in \Omega : X(\omega) = x\}]}}.\]
A: Zufallsvariablen abstrahieren Ergebnisse zu numerischen Werten.Beispiel: Bei 2 Würfelwürfen ist \(X =\) "Summe der Augenzahlen" eine Zufallsvariable.
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| lorenz | cid:1774487164722 |
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c1
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\Pr[A]
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Cloze c1
Cloze answer: \Pr[A]
Q: Für \(X_A\) eine Indikator-Zufallsvariable gilt \(\mathbb{E}[X_a] = {{c1:: \Pr[A] }}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631269215 |
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nid:1771363498414
c2
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Blöcke
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Cloze c2
Cloze answer: Blöcke
Q: Sei \(G = (V, E)\). Wir definieren eine {{c3::Äquivalenzrelation}} auf \(E\) durch
\[{{c1::e \sim f :\iff \begin{cases} e = f, & \text{oder} \\ \exists \text{ Kreis durch } e \text{ und } f \end{cases} }}\]
Die {{c3::Äquivalenzklassen}} nennen wir {{c2::Blöcke}}.
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| lorenz | cid:1771363498415 |
1 | 230% | 15d | 8 |
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c1
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Zu einer Zufallsvariablen \(X\) mit Wertebereich \(W_X\) def...
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Cloze c1
Q: Zu einer Zufallsvariablen \(X\) mit Wertebereich \(W_X\) definieren wir {{c2::den Erwartungswert \(\mathbb{E}[X]\)}} durch\[{{c2::\mathbb{E}[X]}} := {{c1::\sum_{\alpha \in W_X} \alpha \cdot \Pr[X = \alpha]}},\]sofern die Summe absolut konvergiert.
A: Ansonsten sagen wir, dass der Erwartungswert undefiniert ist.Intuition: Gewichteter Durchschnitt aller möglichen Werte.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165098 |
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nid:1774487165116
c1
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n^3
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Cloze c1
Cloze answer: n^3
Q: Für \(n\) gerade und \(\ell : \binom{[n]}{2} \to \mathbb{N}_0\) kann man in Zeit \(O({{c1::n^3}})\) ein {{c2::minimales (gewichtsminimales) perfektes Matching}} in \(K_n\) finden.
A: Das ist der Blossom-Algorithmus.Dies wird im Christofides-Algorithmus für das metrische TSP benötigt.
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|---|---|---|---|---|---|
| lorenz | cid:1774487165118 |
1 | 230% | 17d | 10 |
nid:1772549069397
c1
A&W
Big \(O\) von Matching-Algorithmen:Für bipartite Graphen
...
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Cloze c1
Q: Big \(O\) von Matching-Algorithmen:Für bipartite Graphen
\( O(|V|^{1/2} \cdot |E|) \) Hopcroft-Karp (ungewichtet)
\( O(|E|^{1+o(1)}) \) (mit polynominellen Gewichte)
Für allgemeine Graphen (mit polynominellen Gewichten)
\( O({{c1::|V|^{1/2
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|---|---|---|---|---|---|
| lorenz | cid:1772549069398 |
1 | 230% | 34d | 8 |
nid:1771360670876
c1
A&W
\(k\)-zusammenhängend
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Cloze c1
Cloze answer: \(k\)-zusammenhängend
Q: Ein Graph \(G = (V, E)\) heisst {{c1::\(k\)-zusammenhängend}}, falls {{c2::\(|V| \geq k + 1\) und für alle Teilmengen \(X \subseteq V\) mit \(|X| < k\) gilt: Der Graph \(G[V \setminus X]\) ist zusammenhängend}}.
A: Man muss mindestens \(k\)-Knoten (und die inzidenten Kanten) löschen, um den Zusammenhang zu zerstören.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771360670877 |
1 | 230% | 18d | 8 |
nid:1774917592720
c1
A&W
Die Grösse \(\sigma := {{c1::\sqrt{\operatorname{Var}[X]} }}...
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Cloze c1
Q: Die Grösse \(\sigma := {{c1::\sqrt{\operatorname{Var}[X]} }}\) heisst {{c2::Standardabweichung von \(X\)}}.
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|---|---|---|---|---|---|
| lorenz | cid:1774917592720 |
1 | 230% | 12d | 7 |
nid:1774917592774
c1
A&W
Für die Varianz gilt: \[\mathbb{E}[(X - \mu)^2] = {{c1::\sum...
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Cloze c1
Q: Für die Varianz gilt: \[\mathbb{E}[(X - \mu)^2] = {{c1::\sum_{x \in W_X} (x - \mu)^2 \cdot \Pr[X = x]::\text{Summe} }}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917592774 |
1 | 230% | 15d | 7 |
nid:1774631269382
c2
A&W
Hopcroft-Karp findet in einem {{c1::bipartiten}} Graphen in ...
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nid:1774631269382
Cloze c2
Q: Hopcroft-Karp findet in einem {{c1::bipartiten}} Graphen in \(O({{c2::\sqrt{|V|} \cdot |E|}})\) ein {{c3::maximales Matching}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774999768204 |
1 | 230% | 16d | 7 |
nid:1773913363614
c1
A&W
Elementarereignissen
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Cloze c1
Cloze answer: Elementarereignissen
Q: Ein diskreter Wahrscheinlichkeitsraum ist bestimmt durch eine Ergebnismenge \(\Omega = \{\omega_1, \omega_2, \ldots\}\) von {{c1::Elementarereignissen}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773913363617 |
1 | 230% | 20d | 11 |
nid:1776174922324
c1
A&W
Sei \(X\) eine Zufallsvariable, die nur nicht-negative Werte...
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nid:1776174922324
Cloze c1
Q: Sei \(X\) eine Zufallsvariable, die nur nicht-negative Werte annimmt. Dann gilt für alle \(t \in \mathbb{R}\) mit \(t > 0\), dass\[{{c1::\Pr\left[X \geq t\right] \leq \frac{\mathbb{E}[X]}{t}.}}\]Oder äquivalent dazu,\[{{c2::\Pr\left[X \geq t \cdot \mathbb{E}[X]\right] \leq \frac{1}{t}.}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776174922324 |
1 | 230% | 14d | 7 |
nid:1772046826522
c1
A&W
effizient entscheidbare Probleme; (einseitig) effizient veri...
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Cloze c1
Cloze answer: effizient entscheidbare Probleme; (einseitig) effizient verifizierbare Probleme
Q: \(P\) = {{c1::effizient entscheidbare Probleme}}
\(NP\) = {{c1::(einseitig) effizient verifizierbare Probleme}}
A: P = polynomiellNP = nichtdeterministisch polynomiell
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772046826522 |
1 | 230% | 36d | 8 |
nid:1776175111067
c2
A&W
Für eine {{c1::beliebige}} Zufallsvariable \(X\) und alle \(...
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nid:1776175111067
Cloze c2
Q: Für eine {{c1::beliebige}} Zufallsvariable \(X\) und alle \(t > 0\), gilt\[\Pr\left[|X - \mathbb{E}[X]| \geq t\right] \leq {{c2::\frac{\text{Var}[X]}{t^2} }}.\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776175111068 |
1 | 230% | 14d | 7 |
nid:1773753822869
c1
A&W
isomorph
1
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nid:1773753822869
Cloze c1
Cloze answer: isomorph
Q: Es ist kein polynomieller Algorithmus bekannt, um zu entscheiden, ob zwei Graphen {{c1::isomorph}} sind.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773753822869 |
1 | 230% | 28d | 8 |
nid:1774358417548
c1
A&W
Seien \(A\) und \(B\) Ereignisse mit \(\Pr[B] > 0\).
Die be...
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nid:1774358417548
Cloze c1
Q: Seien \(A\) und \(B\) Ereignisse mit \(\Pr[B] > 0\).
Die bedingte Wahrscheinlichkeit \(\Pr[A|B]\) von \(A\) gegeben \(B\) ist definiert durch: \[\Pr[A|B] := {{c1::\frac{\Pr[A \cap B]}{\Pr[B]} }}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774358417548 |
1 | 230% | 21d | 8 |
nid:1773330177039
A&W
Wahr oder falsch?Es gibt einen polynomiellen Algorithmus, de...
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nid:1773330177039
Q: Wahr oder falsch?Es gibt einen polynomiellen Algorithmus, der für jeden planaren Graphen eine geeignete Einfärbung mit 6 Farben findet.
A: Wahr
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773330177039 |
1 | 230% | 29d | 8 |
nid:1774917594111
c2
A&W
\(k\)-te zentrale Moment
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nid:1774917594111
Cloze c2
Cloze answer: \(k\)-te zentrale Moment
Q: Für eine Zufallsvariable \(X\) nennen wir \(\mathbb{E}[X^k]\) das {{c1::\(k\)-te Moment}} und \(\mathbb{E}[(X - \mathbb{E}[X])^k]\) das {{c2::\(k\)-te zentrale Moment}}.
A: Der Erwartungswert ist also das erste Moment.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917594112 |
1 | 230% | 16d | 7 |
nid:1774487164563
c1
A&W
Für den Binomialkoeffizienten gilt:\[\binom{n}{k} = {{c1::\b...
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nid:1774487164563
Cloze c1
Q: Für den Binomialkoeffizienten gilt:\[\binom{n}{k} = {{c1::\binom{n}{n-k} :: \text{Symmetrie} }}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487164564 |
1 | 230% | 22d | 11 |
nid:1773329930605
A&W
Wahr oder falsch?Wenn \(G\) ein zusammenhängender Graph mit ...
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nid:1773329930605
Q: Wahr oder falsch?Wenn \(G\) ein zusammenhängender Graph mit einem maximalen Grad von 100 ist, dann hat \(G\) eine korrekte Färbung mit 100 Farben, es sei denn, \(G\) ist ein vollständiger Graph.
A: Wahr
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773329930605 |
1 | 230% | 39d | 8 |
nid:1774631276980
c3
A&W
Welche drei Bestandteile bestimmen einen diskreten Wahrschei...
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nid:1774631276980
Cloze c3
Q: Welche drei Bestandteile bestimmen einen diskreten Wahrscheinlichkeitsraum?{{c1::Eine Ergebnismenge \(\Omega = \{\omega_1, \omega_2, \ldots\}\) von Elementarereignissen.}}{{c2::Eine Wahrscheinlichkeitszuweisung \(\Pr[\omega_i] \in [0,1]\) für je
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631276982 |
1 | 230% | 22d | 10 |
nid:1772545581602
c2
A&W
Mit einem Greedy-Algorithmus kann man in Zeit \( O({{c1::|E|...
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nid:1772545581602
Cloze c2
Q: Mit einem Greedy-Algorithmus kann man in Zeit \( O({{c1::|E|}}) \) ein {{c3::inklusionsmaximales}} Matching \( M_{\text{Greedy}} \) bestimmen mit\[{{c2:: |M_{\text{Greedy} }| \geq |M_{\text{max} }| / 2, }}\]wobei \( M_{\text{max}} \) ein kardinalitätsmaximales Matching ist.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772545581602 |
1 | 230% | 35d | 11 |
nid:1772547951495
c1
A&W
\forall X \subseteq A : |X| \leq |N(X)|
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nid:1772547951495
Cloze c1
Cloze answer: \forall X \subseteq A : |X| \leq |N(X)|
Q: Ein bipartiter Graph \( G = (A \uplus B, E) \) enthält ein Matching \( M \) der Kardinalität \({{c2:: |M| = |A|}} \iff {{c1::\forall X \subseteq A : |X| \leq |N(X)| }}\).
A: (Hall, Heiratssatz)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772547951497 |
1 | 230% | 34d | 11 |
nid:1774358482736
c1
A&W
Multiplikationssatz: Seien \(A_1, \ldots, A_n\) Ereignisse.
...
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nid:1774358482736
Cloze c1
Q: Multiplikationssatz: Seien \(A_1, \ldots, A_n\) Ereignisse.
Falls \(\Pr[A_1 \cap \cdots \cap A_n] > 0\), so gilt: \[\begin{align} \Pr[A_1 \cap \cdots \cap A_n] =& {{c1::\Pr[A_1] \cdot \Pr[A_2|A_1] \\ &\cdot \Pr[A_3|A_1 \cap A_2] \cdots \\ &\Pr[A_n|A_1 \cap \c
A: Proof: Expand each conditional probability by definition:\[ \Pr[A_1]\cdot\frac{\Pr[A_1\cap A_2]}{\Pr[A_1]}\cdot\frac{\Pr[A_1\cap A_2\cap A_3]}{\Pr[A_1\cap A_2]}\cdots\frac{\Pr[A_1\cap\cdots\cap A_n]}{\Pr[A_1\cap\cdots\cap A_{n-1}]}. \]All intermediate terms cancel (telescoping product), leaving \(\Pr[A_1\cap\cdots\cap A_n]\). \(\square\)Note: All conditional probabilities are well-defined because \(\Pr[A_1]\ge\Pr[A_1\cap A_2]\ge\cdots>0\).
Multiplikationssatz
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774358482737 |
1 | 230% | 36d | 8 |
nid:1774631276956
c3
A&W
n
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nid:1774631276956
Cloze c3
Cloze answer: n
Q: Für den Binomialkoeffizienten gelten:\(\binom{n}{0} = {{c1::1}}\)\(\binom{n}{n} = {{c2::1}}\)\(\binom{n}{1} = {{c3::n}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1775000929043 |
1 | 230% | 35d | 8 |
nid:1774487164608
c2
A&W
Die Anzahl der Möglichkeiten, \(k\) Objekte aus \(n\) Sorten...
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nid:1774487164608
Cloze c2
Q: Die Anzahl der Möglichkeiten, \(k\) Objekte aus \(n\) Sorten mit Zurücklegen zu wählen (Reihenfolge egal, Multiset) ist:\[{{c2::\binom{n + k - 1}{k} }} = {{c1::\frac{(n+k-1)!}{k!\,(n-1)!} }} \]
A: Auch bekannt als „Sterne und Striche“ (Stars and Bars).Beispiel: Wie viele Möglichkeiten, 3 Kugeln aus {rot, blau, grün} mit Zurücklegen zu ziehen?\(\binom{3+3-1}{3} = \binom{5}{3} = 10\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631269206 |
1 | 230% | 41d | 8 |
nid:1771526674685
c2
A&W
\(v = root\), und \(v\) hat mindestens zwei Kinder im DFS-Ba...
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nid:1771526674685
Cloze c2
Cloze answer: \(v = root\), und \(v\) hat mindestens zwei Kinder im DFS-Baum.
Q: \(v\) ist genau dann Artikulationsknoten, wenn:{{c1::\(v \neq root\), und \(v\) hat ein Kind \(u\) im DFS-Baum mit \(low[u] \geq dfs[v]\)}} oder {{c2::\(v = root\), und \(v\) hat mindestens zwei Kinder im DFS-Baum.}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771526674685 |
1 | 230% | 57d | 9 |
nid:1772496585226
IO r1
A&W
[Image Occlusion region 1]
1
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nid:1772496585226
Cloze c1
Q: {{c1::image-occlusion:rect:left=.186:top=.2984:width=.5344:height=.2754}}{{c2::image-occlusion:rect:left=.183:top=.5891:width=.8119:height=.3672}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772496585227 |
1 | 230% | 52d | 8 |
nid:1772626803535
c6
Analysis
1
1
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nid:1772626803535
Cloze c6
Cloze answer: 1
Q: \(\lim_{n\to\infty} x^{1/n} = {{c6::1}},\quad x > 0\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772626803536 |
1 | 230% | 25d | 8 |
nid:1774138447415
Analysis
Trick: Rationalisieren
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nid:1774138447415
Q: Trick: Rationalisieren
A: Binomische Formel \(a^2 - b^2 = (a - b)(a + b)\). Multipliziere die Gleichung mit \(\dots \times 1 = \dots \times \frac{\sqrt{n} + \sqrt{n + 1}}{\sqrt{n} - \sqrt{n + 1}}\). Beispiel: \({\sqrt{n^2 + 3} - n} \cdot 1 = \sqrt{n^2 + 3} - n \cdot \frac{\sqrt{n^2 + 3} + n}{\sqrt{n^2 + 3} + n}\) und dann mit \(a^2 - b^2\) vereinfachen.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774138447415 |
1 | 230% | 18d | 10 |
nid:1774138448149
c1
Analysis
1
1
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nid:1774138448149
Cloze c1
Cloze answer: 1
Q: Für alle Polynome \(P(n)\) mit \(P(n) > 0\), gilt für grosse \(n\): \[ \lim_{n \rightarrow \infty} \sqrt[n]{P(n)} = {{c1:: 1}} \]
A: (Die Wurzel dämpft diese vollständig ab.)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774138448149 |
1 | 230% | 4d | 9 |
nid:1774487165594
c1
Analysis
Seien \(a_n, b_n > 0\). Dann:{{c1::\(\lim \frac{a_n}{b_n} = ...
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users
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nid:1774487165594
Cloze c1
Q: Seien \(a_n, b_n > 0\). Dann:{{c1::\(\lim \frac{a_n}{b_n} = g\) mit \(0 < g < \infty\)}} \(\implies\) \(\sum a_n\) und \(\sum b_n\) haben dasselbe Konvergenzverhalten{{c2::\(\lim \frac{a_n}{b_n} = 0\) und&nbs
A: Grenzwertkriterium (Limitenvergleich)Beispiel:\[\sum \frac{1}{n^2+3n}\]Vergleich mit \(1/n^2\), Grenzwert \(= 1\) → konvergiert.Proof Sketch
Ist \(\lim_{n \to \infty} \frac{a_n}{b_n} = g\) mit \(0 < g < \infty\)
So gilt \(\frac{a_n}{b_n} \leq g + \varepsilon\) und daher \(a_n \leq (g + \varepsilon) \, b_n\) für ein geeignetes \(\varepsilon > 0\) und alle genügend großen \(n\).
Nach dem Majorantenkrit
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165594 |
1 | 230% | 10d | 10 |
nid:1774487165742
c1
Analysis
konvergiert, aber die Reihe der Beträge \(\sum |a_k|\) diver...
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nid:1774487165742
Cloze c1
Cloze answer: konvergiert, aber die Reihe der Beträge \(\sum |a_k|\) divergiert
Q: Eine Reihe heisst bedingt konvergent, wenn sie {{c1::konvergiert, aber die Reihe der Beträge \(\sum |a_k|\) divergiert}}.Counterexample included
A: (D.h. nicht absolut konvergiert..)Beispiel: \(\sum \frac{(-1)^n}{n}\) ist bedingt konvergent.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165742 |
1 | 230% | 19d | 7 |
nid:1774917594967
c2
Analysis
injektiv
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Cloze c2
Cloze answer: injektiv
Q: Jede {{c1::streng monotone::Adjektiv}} Funktion ist {{c2::injektiv::Funktionseigenschaft}}.Proof Included
A: Proof: Nehme an wir haben eine streng monotone Funktion \(f\) die nicht injektiv ist.Dann gilt \(\exists x_1, x_2 \in \mathbb{D}\) sodass \(f(x_1) = f(x_2)\) weil nicht injektiv.Aber oBdA \(x_1 < x_2 \implies f(x_1) < f(x_2)\) was ein Widerspruch ist.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917594967 |
1 | 230% | 10d | 7 |
nid:1774487165263
Analysis
Wie lautet die Cauchy-Schwarz Ungleichung im euklidischen Ra...
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nid:1774487165263
Q: Wie lautet die Cauchy-Schwarz Ungleichung im euklidischen Raum?
A: Für alle \(x, y \in \mathbb{R}^n\) gilt:\[|x \cdot y| \leq \|x\| \cdot \|y\|\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165263 |
1 | 230% | 19d | 7 |
nid:1774487165756
c1
Analysis
0; - also konvergiert die Reihe nur für \(z = 0\); \infty;...
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nid:1774487165756
Cloze c1
Cloze answer: 0; - also konvergiert die Reihe nur für \(z = 0\); \infty; - die Reihe konvergiert für alle \(z\)
Q: Wurzelkriterium:wenn \((c_k)^{1/k}\) nicht beschränkt ist, setzen wir \(\rho = {{c1::0}}\){{c1:: - also konvergiert die Reihe nur für \(z = 0\)}} wenn \((c_k)^{1/k}\) beschränkt ist und \(\limsup (c_k)^{1/k} = 0\), setzen wir \(\rho ={{c1:: \infty}}\){{c1::&nbs
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165757 |
1 | 230% | 4d | 9 |
nid:1774487165914
c1
Analysis
Reihen mit nicht-negativen Gliedern (ab einem Index \(N\)): ...
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nid:1774487165914
Cloze c1
Cloze answer: Reihen mit nicht-negativen Gliedern (ab einem Index \(N\)): \(0 \leq a_n \leq b_n\) für alle \(n \geq N\)
Q: Das Majoranten-/Minorantenkriterium gilt nur für {{c1::Reihen mit nicht-negativen Gliedern (ab einem Index \(N\)): \(0 \leq a_n \leq b_n\) für alle \(n \geq N\)}}.
A: Für alternierende Reihen ist es nicht direkt anwendbar, erst Absolutkonvergenz mit \(\sum |a_n|\) zeigen, dann folgt Konvergenz.Häufiger Fehler: Vergleich von \((-1)^n/n\) mit \(1/n\) über Majorante. Das scheitert, da \((-1)^n/n \not\geq 0\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165915 |
1 | 230% | 2d | 8 |
nid:1774487165318
c1
Analysis
Cauchy-Verdichtungssatz: Sei \((a_n)\) monoton fallend, \(a_...
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nid:1774487165318
Cloze c1
Q: Cauchy-Verdichtungssatz: Sei \((a_n)\) monoton fallend, \(a_n \geq 0\):\[\sum_{n=0}^\infty a_n \text{ conv.} \iff {{c1::\sum_{n=0}^\infty 2^n a_{2^n} \text{ conv.} }}\]Proof Included
A: Anwendung: \(\sum 1/n^s\) für \(s > 1\) konvergiert: \(\sum 2^n \cdot 2^{-ns} = \sum 2^{n(1-s)}\) geometrisch mit \(q = 2^{1-s} < 1\).Proof
Weil \(a_n\) monoton fällt gilt \(2^n a_{2^n} \ge a_{2^k + 1} + a_{2^k + 2} + \dots + a_{2^{k + 1} - 1}\).
Wir benutzen das Majorantenkriterium mit\[\begin{align} \sum^n_{k = 0} 2^k a_{2^k} &\ge \sum_{k = 0}^n (a_{2^k + 1} + a_{2^k + 2} + \dots + a_{2^{k + 1} - 1}) \\ &= \sum^
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165318 |
1 | 230% | 4d | 9 |
nid:1774487165294
c4
Analysis
Sei \(\rho = {{c4:: \limsup_{n\to\infty} |a_n|^{1/n} }}\). D...
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nid:1774487165294
Cloze c4
Q: Sei \(\rho = {{c4:: \limsup_{n\to\infty} |a_n|^{1/n} }}\). Dann:\(\rho < 1\) \(\implies\) {{c1::\(\sum a_n\) konvergiert absolut}}\(\rho > 1\) \(\implies\) {{c1::\(\sum a_n\) divergiert}}\(\rho = 1\) \(\implies\) {{c1::keine Aussage möglich}}
A: (Wurzelkriterium)Wenn Quotientenkriterium versagt (\(\rho=1\)), versagt auch das Wurzelkriterium — aber nicht umgekehrt.Proof:
Convergence \(L < 1\)
\(\sum a_n \geq 0\), \(\displaystyle L = \limsup_{n\to\infty} \left| {a_n}^{1/n} \right| < 1\).
Choose \(q\) with \(L < q < 1\). Since \(\limsup \left| {a_n}^{1/n} \right| = L\), there exists \(N\) such that for all
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|---|---|---|---|---|---|
| lorenz | cid:1774917594525 |
1 | 230% | 12d | 7 |
nid:1774917594762
c1
Analysis
Es sei \(f : \mathbb{D}(f) \to \mathbb{R}\). Dann gilt:\[ \l...
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users
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nid:1774917594762
Cloze c1
Q: Es sei \(f : \mathbb{D}(f) \to \mathbb{R}\). Dann gilt:\[ \lim_{x \to x_o} f(x) = L \] genau dann, wenn {{c1::für jede konvergente Folge \((x_n)_{n \in \mathbb{N}_0}\), welche gegen \(x_0\) konvergiert, gilt: \[ \lim_{n \to \infty} f(x_n) = L \]}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917594762 |
1 | 230% | 11d | 10 |
nid:1774917595832
c2
Analysis
R
1
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nid:1774917595832
Cloze c2
Cloze answer: R
Q: Eine Funktion \(f: {{c1::D}} \rightarrow {{c2::R}}\) hat {{c1::einen Definitionsbereich \(\text{domain}(f) = \mathbb{D}(f) = D\)}} und {{c2::einen Wertebereich \(\text{range/image}(f) = R\)}}.
A: Der Input heißt unabhängige Variable (Argument) und der Output abhängige Variable.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917595833 |
1 | 230% | 9d | 7 |
nid:1776774733437
c1
Analysis
\(x_0 \in \mathbb{R}\) ist ein Häufungspunkt eines Intervall...
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nid:1776774733437
Cloze c1
Q: \(x_0 \in \mathbb{R}\) ist ein Häufungspunkt eines Intervalls \(D\) falls gilt {{c1::\[ \forall \epsilon > 0 \quad ((x_0 - \epsilon, x_0 + \epsilon) \setminus \{x_0\}) \cap D \neq \emptyset \]}}
A: Jedes Intervall um \(x_0\) hat mindestens einen Punkt, der nicht \(x_0\) ist.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776774733438 |
1 | 230% | 6d | 6 |
nid:1774487165324
Analysis
Welches Konvergenzkriterium wähle ich wann?
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nid:1774487165324
Q: Welches Konvergenzkriterium wähle ich wann?
A: Notwendiges Kriterium zuerst: \(a_n \to 0\)? Falls nein → divergiert sofort.Geometrisch/direkter Vergleich: Vergleichbar mit \(q^n\) oder \(1/n^s\)?Quotientenkriterium: Terme mit \(n!\), \(a^n\) oder einfachen Quotienten?Wurzelkriterium: Terme der Form \((\cdot)^n\) — mindestens so gut wie Quotient.Leibniz: Alternierende Reihe mit monoton fallenden \(|a_n| \to 0\)?Grenzwertkriterium: Ähnelt asymptotisc
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165324 |
1 | 230% | 14d | 10 |
nid:1776290100388
c1
Analysis
\(\mathbb{Q}\) ist dicht in \(\mathbb{R}\) also {{c1::existi...
1
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users
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nid:1776290100388
Cloze c1
Q: \(\mathbb{Q}\) ist dicht in \(\mathbb{R}\) also {{c1::existiert eine Folge \((a_n) \to x\), \((a_n) \subset \mathbb{Q}\) für alle \(x \in \mathbb{R}\)::Folge}}.Proof included
A: Äquivalent: Für alle \(a, b \in \mathbb{R}\) mit \(a < b\) existiert ein \(q \in \mathbb{Q}\) mit \(a < q < b\).Beweis: Sei \(x \in \mathbb{R}\). Für jedes \(n \in \mathbb{N}\) wähle \(q_n \in \mathbb{Q}\) mit \[x < q_n < x + \frac{1}{n}\]was nach der archimedischen Eigenschaft und der Existenz rationaler Zahlen zwischen je zwei reellen Zahlen möglich ist. Dann gilt \(|q_n - x| < \frac{1}{n}\), also \(q_n \to x\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1776290100388 |
1 | 230% | 8d | 9 |
nid:1772928333495
c1
Analysis
\[ \tan\!\left(\frac{2\pi}{3}\right) = {{c1::-\sqrt{3} }} \]
1
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users
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ease
nid:1772928333495
Cloze c1
Q: \[ \tan\!\left(\frac{2\pi}{3}\right) = {{c1::-\sqrt{3} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333496 |
1 | 230% | 22d | 10 |
nid:1774917595110
c1
Analysis
Falls gilt \[{{c1:: \forall N > 0 \ \exists \delta > 0 \text...
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users
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nid:1774917595110
Cloze c1
Q: Falls gilt \[{{c1:: \forall N > 0 \ \exists \delta > 0 \text{ s.d. } \ \forall x \in C \ (0 < |x - c| < \delta \implies f(x) > N) }}\] hat \(f\) in \(c\) {{c2::den uneigentlichen Grenzwert \(\infty\) d.h. \(\lim_{x \to c} f(x) = \infty\)}}.
A: Das gleiche kann auch \(f(x) < -N\) für \(-\infty\) gelten.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917595110 |
1 | 230% | 19d | 7 |
nid:1774917595832
c1
Analysis
D
1
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users
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nid:1774917595832
Cloze c1
Cloze answer: D
Q: Eine Funktion \(f: {{c1::D}} \rightarrow {{c2::R}}\) hat {{c1::einen Definitionsbereich \(\text{domain}(f) = \mathbb{D}(f) = D\)}} und {{c2::einen Wertebereich \(\text{range/image}(f) = R\)}}.
A: Der Input heißt unabhängige Variable (Argument) und der Output abhängige Variable.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917595832 |
1 | 230% | 11d | 7 |
nid:1771973928582
Analysis
Archimedisches Prinzip (Epsilon Variante)
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users
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nid:1771973928582
Q: Archimedisches Prinzip (Epsilon Variante)
A: Für jedes \(\epsilon > 0\) existiert \(n \in \mathbb{N}\) mit \(\frac{1}{n} < \epsilon\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928582 |
1 | 230% | 30d | 11 |
nid:1772928333431
c1
Analysis
\[ \sin\!\left(\frac{3\pi}{4}\right) = {{c1::\frac{\sqrt{2} ...
1
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1/4
users
230%
ease
nid:1772928333431
Cloze c1
Q: \[ \sin\!\left(\frac{3\pi}{4}\right) = {{c1::\frac{\sqrt{2} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333431 |
1 | 230% | 33d | 8 |
nid:1774487165599
c1
Analysis
Sei \(\sum a_n\) {{c1::absolut konvergent und \(\phi: \mathb...
1
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users
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nid:1774487165599
Cloze c1
Q: Sei \(\sum a_n\) {{c1::absolut konvergent und \(\phi: \mathbb{N}_0 \to \mathbb{N}_0\) eine Bijektion}}.Dann {{c2::konvergiert \(\sum a_{\phi(n)}\) ebenfalls absolut und:\[\sum_{n=0}^\infty a_n = \sum_{n=0}^\infty a_{\phi(n)}\]}}
A: Umordnungssatz für absolut konvergente Reihen (Dirichlet)Merke: Bei absolut konvergenten Reihen darf man frei umordnen.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774918631827 |
1 | 230% | 22d | 7 |
nid:1772928333376
c1
Analysis
\[ \cos\!\left(\frac{7\pi}{6}\right) = {{c1::-\frac{\sqrt{3}...
1
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users
230%
ease
nid:1772928333376
Cloze c1
Q: \[ \cos\!\left(\frac{7\pi}{6}\right) = {{c1::-\frac{\sqrt{3} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333376 |
1 | 230% | 37d | 8 |
nid:1771973928588
c4
Analysis
Beweis: Für alle \(a < b\) in \(\mathbb{R}\) existiert ein \...
1
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users
230%
ease
nid:1771973928588
Cloze c4
Q: Beweis: Für alle \(a < b\) in \(\mathbb{R}\) existiert ein \(\mathbb{Q}\) mit \(a < q < b\){{c1:: Wähle nach Archimedischem Prinzip \(n \in \mathbb{N}\) so dass \(\frac{1}{n} < b - a\).}}{{c2:: \(\frac{m}{n} \mid m \in \mathbb{Z}\) diese
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771973928588 |
1 | 230% | 39d | 7 |
nid:1773149513656
c1
Analysis
Falls für eine Folge gilt:\[{{c1:: \forall M > 0 \ \exists N...
1
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users
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ease
nid:1773149513656
Cloze c1
Q: Falls für eine Folge gilt:\[{{c1:: \forall M > 0 \ \exists N > 0 \text{ sodass } \forall n > N \ : \ a_n > M }}\] sagen wir, dass die Folge gegen unendlich divergiert und schreiben \(\lim_{n \rightarrow \infty} a_n = \infty\).
A: Genauso kann die Folge auch gegen \(-\infty\) divergieren.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1773149513656 |
1 | 230% | 23d | 12 |
nid:1774487165343
c1
Analysis
Cauchy-Kriterium:\(\sum a_n\) konvergiert \(\iff\) für jedes...
1
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users
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nid:1774487165343
Cloze c1
Q: Cauchy-Kriterium:\(\sum a_n\) konvergiert \(\iff\) für jedes \(\varepsilon > 0\) existiert ein \(N\), so dass für alle \(n > m \geq N\) gilt:
\[{{c1::\left|\sum_{k=m+1}^n a_k\right| = |S_n - S_m| < \varepsilon}}\]
Proof Included
A: Direktes Cauchy-Kriterium auf die Partialsummenfolge.Man kann \(\sum_{k = m+1}^n a_k \) auch als \(S_n - S_{m} \) schreiben. Und für die Folge \(S_n\) gilt dann der Cauchy Satz. Falls also \(\exists N \in \mathbb{N}_0\) sodass \(\forall n > m > N gilt |S_n - S_m| < \epsilon\), konvergiert die Folge \(S_n\).
Die gilt per Annahme und deswegen konvergiert \(S_n\). Da die Folge der Partialsummen konvergiert, konvergiert die Reihe.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165343 |
1 | 230% | 20d | 10 |
nid:1774487165301
c2
Analysis
Sei \(\sum a_n\) {{c1::bedingt konvergent und \(L \in \mathb...
1
lapses
1/4
users
230%
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nid:1774487165301
Cloze c2
Q: Sei \(\sum a_n\) {{c1::bedingt konvergent und \(L \in \mathbb{R} \cup \{+\infty, -\infty\}\)}}.Dann {{c2::gibt es eine Bijektion \(\phi\), so dass:\[\sum_{n=0}^\infty a_{\phi(n)} = L\]}}
A: (Riemannscher Umordnungssatz)Merke: Bedingt konvergente Reihen können durch Umordnung jeden Grenzwert annehmen!
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631277518 |
1 | 230% | 18d | 10 |
nid:1774917594698
c1
Analysis
x \mapsto f(x) \quad \forall x \in D'
1
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nid:1774917594698
Cloze c1
Cloze answer: x \mapsto f(x) \quad \forall x \in D'
Q: Die Einschränkung (Restriktion) von \(f: \mathbb{D}(f) \to \mathbb{R}\) auf \(D' \subset \mathbb{D}(f)\) ist:\[ f\mid_{D'} : D' \to \mathbb{R}, \quad {{c1::x \mapsto f(x) \quad \forall x \in D'}}\]Gleiche Zuordnung, aber nur auf der Teilmenge \(D'\) definiert.
A: Man beachte, dass \(f\) und \(f\mid_{D'}\) a priori zwei verschiedene Funktionen sind.
Beispiel \(\overline{f} : \mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0\) \(f(x) = x^2\) ist bijektiv.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917594698 |
1 | 230% | 16d | 10 |
nid:1774138446824
Analysis
Wie kann man einen Ausdruck in die Form von \(e^x\) bringen?
1
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users
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nid:1774138446824
Q: Wie kann man einen Ausdruck in die Form von \(e^x\) bringen?
A: Via \(\lim_{n \rightarrow \infty} (1 + \frac{x}{n})^n = e^x\).
Beispiel: Zunächst formen wir um: \((\frac{n}{n + 1})^n = (\frac{n + 1}{n})^{-n}\).
Dann trennen wir \((1 + \frac{1}{n})^{-n}\) und extrahieren den Exponenten \(((1 + \frac{1}{n})^n)^{-1}\).
Schliesslich können wir den Limes berechnen und erhalten \(e^{-1}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774138446825 |
1 | 230% | 20d | 11 |
nid:1774487165212
c5
Analysis
Form
Strategie
1
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users
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nid:1774487165212
Cloze c5
Q: Form
Strategie
A: (\(0\) und \(\infty\) sind hier Kurzschreibweisen für das Verhalten im Grenzwert: \(0\) steht für „geht gegen \(0\)" und \(\infty\) für „geht gegen \(\infty\)".)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1775072804338 |
1 | 230% | 16d | 7 |
nid:1774487165276
Analysis
Welches Konvergenzkriterium ist stärker: das Wurzel- oder da...
1
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1/4
users
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ease
nid:1774487165276
Q: Welches Konvergenzkriterium ist stärker: das Wurzel- oder das Quotientenkriterium?
A: Das Wurzelkriterium. Liefert der Quotient ein Ergebnis, so auch die Wurzel - aber nicht umgekehrt.
In der Praxis ist das Quotientenkriterium oft bequemer, besonders bei \(n!\) oder Potenzen.
Beide versagen bei \(\rho = 1\), z.B. bei \(p\)-Reihen.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165276 |
1 | 230% | 22d | 11 |
nid:1772928333327
c1
Analysis
\[ {{c1::\sin^2\theta + \cos^2\theta :: \text{Identity} }} =...
1
lapses
1/4
users
230%
ease
nid:1772928333327
Cloze c1
Q: \[ {{c1::\sin^2\theta + \cos^2\theta :: \text{Identity} }} = {{c2::1}} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333327 |
1 | 230% | 41d | 8 |
nid:1772928333418
c1
Analysis
\[ \sin\!\left(\frac{\pi}{3}\right) = {{c1::\frac{\sqrt{3} }...
1
lapses
1/4
users
230%
ease
nid:1772928333418
Cloze c1
Q: \[ \sin\!\left(\frac{\pi}{3}\right) = {{c1::\frac{\sqrt{3} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333418 |
1 | 230% | 28d | 12 |
nid:1772928333410
c1
Analysis
\[ \sin\!\left(\frac{\pi}{6}\right) = {{c1::\frac{1}{2} }} \...
1
lapses
1/4
users
230%
ease
nid:1772928333410
Cloze c1
Q: \[ \sin\!\left(\frac{\pi}{6}\right) = {{c1::\frac{1}{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333410 |
1 | 230% | 42d | 8 |
nid:1774138446942
c1
Analysis
kleinste Häufungspunkt ; grösste Häufungspunkt
1
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1/4
users
230%
ease
nid:1774138446942
Cloze c1
Cloze answer: kleinste Häufungspunkt ; grösste Häufungspunkt
Q: \(\liminf_{n \rightarrow \infty} a_n\) ist der {{c1:: kleinste Häufungspunkt }} von \((a_n)\).\(\limsup_{n \rightarrow \infty} a_n\) ist der {{c1:: grösste Häufungspunkt }} von \((a_n)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774138446942 |
1 | 230% | 29d | 8 |
nid:1772928333491
c1
Analysis
\[ \tan\!\left(\frac{\pi}{2}\right) = {{c1::\text{undefined}...
1
lapses
1/4
users
230%
ease
nid:1772928333491
Cloze c1
Q: \[ \tan\!\left(\frac{\pi}{2}\right) = {{c1::\text{undefined} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333491 |
1 | 230% | 31d | 11 |
nid:1774487165212
c1
Analysis
L'Hôpital, kürzen, Taylor
1
lapses
1/4
users
230%
ease
nid:1774487165212
Cloze c1
Cloze answer: L'Hôpital, kürzen, Taylor
Q: Form
Strategie
A: (\(0\) und \(\infty\) sind hier Kurzschreibweisen für das Verhalten im Grenzwert: \(0\) steht für „geht gegen \(0\)" und \(\infty\) für „geht gegen \(\infty\)".)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165212 |
1 | 230% | 38d | 8 |
nid:1772928333518
c1
Analysis
\[ \tan\!\left(\frac{4\pi}{3}\right) = {{c1::\sqrt{3} }} \]
1
lapses
1/4
users
230%
ease
nid:1772928333518
Cloze c1
Q: \[ \tan\!\left(\frac{4\pi}{3}\right) = {{c1::\sqrt{3} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772928333518 |
1 | 230% | 48d | 8 |
nid:1774487165225
c1
Analysis
konvergente Teilfolgen
1
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1/4
users
230%
ease
nid:1774487165225
Cloze c1
Cloze answer: konvergente Teilfolgen
Q: Eine divergente Folge kann trotzdem {{c1::konvergente Teilfolgen}} besitzen.
A: Beispiel: \(a_n = (-1)^n\) divergiert, aber \(a_{2n} = 1\) und \(a_{2n+1} = -1\) konvergieren.Umkehrung: Eine Folge konvergiert gegen \(L\) genau dann, wenn jede Teilfolge gegen \(L\) konvergiert.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487165228 |
1 | 230% | 37d | 8 |
nid:1772116353056
DDCA
What is this?
1
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users
230%
ease
nid:1772116353056
Q: What is this?
A: Cross-Coupled Inverters.Has two stable states: \(Q=1\) or \(Q=0\).Has a third possible "metastable" state with both outputs oscillating between 0 and 1 (we will see this later).Not useful without a control mechanism for setting Q.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772116353056 |
1 | 230% | 39d | 8 |
nid:1772117522575
DDCA
Addressability?
1
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users
230%
ease
nid:1772117522575
Q: Addressability?
A: 3-bits
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772117522575 |
1 | 230% | 34d | 8 |
nid:1772114387917
DDCA
How do we model a BUS as a circuit?
1
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1/4
users
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ease
nid:1772114387917
Q: How do we model a BUS as a circuit?
A: You can have two tri-state buffers: one driven by CPU, the other memory; and ensure at most one is enabled at any time.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772114387917 |
1 | 230% | 36d | 8 |
nid:1772202497193
c2
DDCA
# flip-flops, but not necessarily output logic or next state...
1
lapses
1/4
users
230%
ease
nid:1772202497193
Cloze c2
Cloze answer: # flip-flops, but not necessarily output logic or next state logic
Q: {{c1::Binary Encoding (Full Encoding)}}:
Use {{c3::the minimum possible number of}} bits
{{c3::Use log₂(num_states) bits to represent the states}}
Minimizes {{c2::# flip-flops, but not necessarily output l
A: Example state encodings: 00, 01, 10, 11
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772202497193 |
1 | 230% | 35d | 8 |
nid:1772202848055
c1
DDCA
Output
1
lapses
1/4
users
230%
ease
nid:1772202848055
Cloze c1
Cloze answer: Output
Q: {{c1::Output}} Encoding:
Outputs are directly accessible in the state encodingFor the traffic light example, since we have 3 outputs (light color), encode state with 3 bits, wher
A: Example states: 001, 010, 100, 110Bit₀ encodes green light outputBit₁ encodes yellow light outputBit₂ encodes red light output
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772202848056 |
1 | 230% | 33d | 8 |
nid:1772201473957
c1
DDCA
edge-triggered state element
1
lapses
1/4
users
230%
ease
nid:1772201473957
Cloze c1
Cloze answer: edge-triggered state element
Q: A flip-flop is called an {{c1::edge-triggered state element}} because it captures data on the clock edge.
A: A latch is a level-triggered state element.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772201473957 |
1 | 230% | 32d | 12 |
nid:1772200781530
c1
DDCA
We need to store data at the beginning of every clock cycle
1
lapses
1/4
users
230%
ease
nid:1772200781530
Cloze c1
Cloze answer: We need to store data at the beginning of every clock cycle
Q: Which properties do we need to implement a state register?{{c1::We need to store data at the beginning of every clock cycle}}{{c2::The data must be available during the entire clock cycle}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772200781531 |
1 | 230% | 40d | 8 |
nid:1772199886495
c1
DDCA
length
1
lapses
1/4
users
230%
ease
nid:1772199886495
Cloze c1
Cloze answer: length
Q: Combinational logic evaluates for the {{c1::length}} of the clock cycle.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772199886495 |
1 | 230% | 40d | 8 |
nid:1772204350790
IO r2
DDCA
[Image Occlusion region 2]
1
lapses
1/4
users
230%
ease
nid:1772204350790
Cloze c2
Q: {{c1::image-occlusion:rect:left=.3893:top=.0178:width=.4877:height=.0674:oi=1}}{{c2::image-occlusion:rect:left=.5552:top=.3107:width=.4354:height=.1021:oi=1}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772204350791 |
1 | 230% | 40d | 8 |
nid:1772199856337
c1
DDCA
synchronizes state changes
1
lapses
1/4
users
230%
ease
nid:1772199856337
Cloze c1
Cloze answer: synchronizes state changes
Q: A clock {{c1::synchronizes state changes}} across many sequential circuit elements.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772199856337 |
1 | 230% | 51d | 9 |
nid:1772117467823
c1
DDCA
the address space
1
lapses
1/4
users
230%
ease
nid:1772117467823
Cloze c1
Cloze answer: the address space
Q: The entire set of unique locations in memory is referred to as {{c1::the address space}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772117467823 |
1 | 230% | 57d | 9 |
nid:1774631279995
c1
PProg
\(T_P \ge T_1 / p\); \(T_P \ge T_\infty\)
1
lapses
1/4
users
230%
ease
nid:1774631279995
Cloze c1
Cloze answer: \(T_P \ge T_1 / p\); \(T_P \ge T_\infty\)
Q: The work law is {{c1::\(T_P \ge T_1 / p\)}} and the span law is {{c1::\(T_P \ge T_\infty\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631279995 |
1 | 230% | 1d | 6 |
nid:1774310311659
PProg
What speed-up bound does Gustafson's Law specify?
1
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1/4
users
230%
ease
nid:1774310311659
Q: What speed-up bound does Gustafson's Law specify?
A: Consider an infinite number of processors. Additionally, we assume that \(f < 1\), which is the same as saying the program has a parallel part. It follows that \(1 - f > 0\). \[ \begin{aligned} \lim_{P \to \infty} S_P &= f + P \cdot (1 - f) \\ &= f + (1 - f) \cdot \lim_{P \to \infty} P \\ &= \infty \end{aligned} \] Since \(P\) grows infinitely large and \(1 - f > 0\), \(S_P\) does not converge, meaning the speedup is unlimited.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774310311659 |
1 | 230% | 16d | 7 |
nid:1774487167075
c1
PProg
Latency of the first element through a pipeline \(= {{c1::\s...
1
lapses
1/4
users
230%
ease
nid:1774487167075
Cloze c1
Q: Latency of the first element through a pipeline \(= {{c1::\sum_{i} \text{time}(\text{stage}_i)}}\)
A: "Latency" by default refers to the first element. For a balanced pipeline this equals num_stages × max(stage_time).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487167075 |
1 | 230% | 5d | 9 |
nid:1774487167626
PProg
What does an exclusive parallel prefix sum compute?
1
lapses
1/4
users
230%
ease
nid:1774487167626
Q: What does an exclusive parallel prefix sum compute?
A: For input array \(A[0..n-1]\), it produces output \(B\) where \(B[i] = \sum_{j=0}^{i-1} A[j]\) (sum of all elements before index \(i\)). So \(B[0] = 0\) always.Example: \(A = [3,1,4,1,5] → B = [0,3,4,8,9]\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487167626 |
1 | 230% | 9d | 7 |
nid:1774487168256
PProg
What are the four necessary conditions for deadlock (Coffman...
1
lapses
1/4
users
230%
ease
nid:1774487168256
Q: What are the four necessary conditions for deadlock (Coffman conditions)?
A: Mutual exclusion — at least one resource is held in non-shareable modeHold and wait — a thread holds at least one resource while waiting for anotherNo preemption — resources cannot be forcibly taken awayCircular wait — a cycle of threads each waiting for a resource held by the nextAll four must hold simultaneously. Breaking any one prevents deadlock.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487168256 |
1 | 230% | 2d | 8 |
nid:1771365476583
c2
PProg
a single answer from a collection via an associative operato...
1
lapses
1/4
users
230%
ease
nid:1771365476583
Cloze c2
Cloze answer: a single answer from a collection via an associative operator
Q: {{c1::Reductions}} produce {{c2::a single answer from a collection via an associative operator}}.
A: Examples: max, count, rightmost, sum.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771365476593 |
1 | 230% | 21d | 7 |
nid:1774487167070
c1
PProg
the serial fraction \(f\) in Amdahl's Law
1
lapses
1/4
users
230%
ease
nid:1774487167070
Cloze c1
Cloze answer: the serial fraction \(f\) in Amdahl's Law
Q: The span {{c2::\(T_\infty\)}} in a DAG corresponds to {{c1::the serial fraction \(f\) in Amdahl's Law}}.
A: (The longest chain of sequential dependencies that no amount of additional parallelism can overcome.)Designing parallel algorithms means decreasing span without increasing work too much - directly equivalent to reducing \(f\) in Amdahl's Law.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487167070 |
1 | 230% | 2d | 8 |
nid:1774487167926
PProg
Compare Big \(O\) of work, span and parallelism for these pa...
1
lapses
1/4
users
230%
ease
nid:1774487167926
Q: Compare Big \(O\) of work, span and parallelism for these parallel quicksort strategies:Parallelize only the recursive callsAlso parallelize the partition step (via pack)
A: VariantWorkSpanParallelismParallel recursive calls only\(O(n \log n)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487167926 |
1 | 230% | 9d | 10 |
nid:1774487167931
PProg
Why is \(T_p \geq T_\infty\) a strict lower bound?
1
lapses
1/4
users
230%
ease
nid:1774487167931
Q: Why is \(T_p \geq T_\infty\) a strict lower bound?
A: \(T_\infty\) is the length of the critical path - a chain of nodes where each depends on the previous. Even with infinite processors, these nodes must execute sequentially. No amount of parallelism can compress a dependency chain.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487167931 |
1 | 230% | 2d | 8 |
nid:1774487168266
c1
PProg
Bandwidth
1
lapses
1/4
users
230%
ease
nid:1774487168266
Cloze c1
Cloze answer: Bandwidth
Q: {{c1::Bandwidth}} of a pipeline is {{c2::the amount of work being processed in parallel at any given time}}.
A: Distinct from throughput (items/time) and latency (time/item). Bandwidth captures how many elements are simultaneously in-flight across all stages.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487168266 |
1 | 230% | 2d | 8 |
nid:1774487167488
c4
PProg
Supports delayed and periodic task execution.
1
lapses
1/4
users
230%
ease
nid:1774487167488
Cloze c4
Cloze answer: Supports delayed and periodic task execution.
Q: The four standard ExecutorService pool types:newFixedThreadPool(n) - {{c1::Fixed n threads; excess tasks are queued.}}newSingleThreadExecutor() - {{c2::Exactly 1 thread; tasks execute sequentially.}}new
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631279572 |
1 | 230% | 17d | 7 |
nid:1774487167528
c3
PProg
steals a task from the back of another thread's deque (the o...
1
lapses
1/4
users
230%
ease
nid:1774487167528
Cloze c3
Cloze answer: steals a task from the back of another thread's deque (the oldest = largest task)
Q: How does the Fork/Join work-stealing scheduler work?Each worker thread has its own {{c1::deque (double-ended queue) of tasks}}.It processes {{c2::its own tasks LIFO from the front}}.When a thread runs out of work, it {{c3::steals
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774631279612 |
1 | 230% | 11d | 7 |
nid:1774487167493
c2
PProg
atomicity of compound operations (e.g. i++)
1
lapses
1/4
users
230%
ease
nid:1774487167493
Cloze c2
Cloze answer: atomicity of compound operations (e.g. i++)
Q: The Java volatile keyword guarantees {{c1::visibility, every read of a volatile field sees the most recent write by any thread}}, but does not guarantee {{c2::atomicity of compound operations (e.g. i++)}}.
A: Use volatile for simple flags (e.g. volatile boolean running). For compound operations, use synchronized or AtomicInteger.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487167493 |
1 | 230% | 16d | 7 |
nid:1771365476576
c1
PProg
livelock
1
lapses
1/4
users
230%
ease
nid:1771365476576
Cloze c1
Cloze answer: livelock
Q: A {{c1::livelock}} is a situation in which {{c2::all threads starve by infinitely often trying to enter a critical section, but never succeeding}}.
A: Similar to a deadlock, the system makes no real progress, although the threads execute statements/use CPU time.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771365476585 |
1 | 230% | 44d | 8 |
nid:1774487168261
PProg
A pipeline has 4 stages with times [2, 4, 2, 2].Is it balanc...
1
lapses
1/4
users
230%
ease
nid:1774487168261
Q: A pipeline has 4 stages with times [2, 4, 2, 2].Is it balanced? What is the throughput? What is the latency of the 1st element? Of the 3rd?
A: Balanced? No — stage 2 takes 4 units; others take 2. Bottleneck is stage 2.Throughput \(= 1/\max(2,4,2,2) = 1/4\) items per time unit.Latency (1st element) \(= 2+4+2+2 = 10\)Latency (3rd element) \(= 10 + (4-2)\cdot(3-1) = 10+4 = 14\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487168261 |
1 | 230% | 10d | 10 |
nid:1772531107039
c1
PProg
notify()
1
lapses
1/4
users
230%
ease
nid:1772531107039
Cloze c1
Cloze answer: notify()
Q: {{c1::notify()}} wakes the highest-priority thread closest to front of object's internal queue.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1772531107039 |
1 | 230% | 33d | 8 |
nid:1774917598731
c1
PProg
NEW; RUNNABLE; start()
1
lapses
1/4
users
230%
ease
nid:1774917598731
Cloze c1
Cloze answer: NEW; RUNNABLE; start()
Q: A newly created Java thread starts in the {{c1::NEW}} state and transitions to {{c1::RUNNABLE}} when {{c1::start()}} is called.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774917598731 |
1 | 230% | 14d | 7 |
nid:1771365476510
c2
PProg
a property of a system: "nothing bad ever happens"
1
lapses
1/4
users
230%
ease
nid:1771365476510
Cloze c2
Cloze answer: a property of a system: "nothing bad ever happens"
Q: A {{c1::safety property}} is {{c2::a property of a system: "nothing bad ever happens"}}.
A: Can be violated in finite time.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771365476513 |
1 | 230% | 33d | 8 |
nid:1771365476415
c1
PProg
Mutual exclusion
1
lapses
1/4
users
230%
ease
nid:1771365476415
Cloze c1
Cloze answer: Mutual exclusion
Q: {{c1::Mutual exclusion}} means preventing {{c2::more than one thread from being in a critical section, i.e. to execute a piece of code, at a given moment in time}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771365476420 |
1 | 230% | 48d | 8 |
nid:1771365476472
c2
PProg
how a Java/JVM thread is related to an operating system thre...
1
lapses
1/4
users
230%
ease
nid:1771365476472
Cloze c2
Cloze answer: how a Java/JVM thread is related to an operating system thread
Q: {{c1::Thread mapping}} describes {{c2::how a Java/JVM thread is related to an operating system thread}}.
A: In native threading (most common), each JVM thread is mapped to a dedicated operating system thread. In green threading, the JVM maps several threads to a single operating system thread.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771365476473 |
1 | 230% | 47d | 8 |
nid:1771365476475
c2
PProg
Locally reason about one thread at a time
1
lapses
1/4
users
230%
ease
nid:1771365476475
Cloze c2
Cloze answer: Locally reason about one thread at a time
Q: Locality has several meanings in parallel programming:{{c2::Locally reason about one thread at a time}} (thread modularity) - simplifies correctness arguments.{{c3::Data locality}}: related memory locations are accessed shortly after each other - improves cache usage{{c
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1771365476481 |
1 | 230% | 29d | 12 |
nid:1774487167037
c1
PProg
Pipeline throughput bound \(= {{c1::\dfrac{1}{\max_i(\text{s...
1
lapses
1/4
users
230%
ease
nid:1774487167037
Cloze c1
Q: Pipeline throughput bound \(= {{c1::\dfrac{1}{\max_i(\text{stage_time}_i)} }}\)(infinite stream, one execution unit per stage)
A: Throughput is limited by the slowest (bottleneck) stage. For a balanced pipeline all stage times are equal, so throughput = 1/stage_time.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774487167037 |
1 | 230% | 27d | 8 |
nid:1774359475784
c1
PProg
T_1 / p + T_\infty
1
lapses
1/4
users
230%
ease
nid:1774359475784
Cloze c1
Cloze answer: T_1 / p + T_\infty
Q: FJ work stealing scheduler: \[T_p = O({{c1::T_1 / p + T_\infty}})\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774359475784 |
1 | 230% | 36d | 8 |
nid:1774362467471
IO r2
PProg
[Image Occlusion region 2]
1
lapses
1/4
users
230%
ease
nid:1774362467471
Cloze c2
Q: {{c1::image-occlusion:rect:left=.057:top=.0000:width=.9345:height=.9956}}{{c3::image-occlusion:rect:left=.057:top=.515:width=.9323:height=.4768}}{{c2::image-occlusion:rect:left=.057:top=.309:width=.9345:height=.6905}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| lorenz | cid:1774362467471 |
1 | 230% | 43d | 8 |
nid:1761029886806
LinAlg
If columns \(v_1, v_2, ..., v_n\) of \(A\) are linearly inde...
1
lapses
1/4
users
290%
ease
nid:1761029886806
Q: If columns \(v_1, v_2, ..., v_n\) of \(A\) are linearly independent and \(A\lambda = A\mu = x\) are two ways of writing vector x as a linear combination of the vectors v then:
A: \(\lambda \ \text{and} \ \mu\) are the exact same vector of coefficients.Linear combinations are unique if all vectors are independent.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761029886806 |
1 | 290% | 36d | 9 |
nid:1761491477291
DiskMat
If \(F \models G\) in predicate logic, what can we conclude ...
1
lapses
1/4
users
230%
ease
nid:1761491477291
Q: If \(F \models G\) in predicate logic, what can we conclude via validity?
A: If \(F\) is valid, then \(G\) is also valid. (Logical consequence preserves validity)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477292 |
1 | 230% | 3d | 5 |
nid:1761491477341
DiskMat
What is the cardinality of the power set of a finite set wit...
1
lapses
1/4
users
260%
ease
nid:1761491477341
Q: What is the cardinality of the power set of a finite set with cardinality \(k\)?
A: \(|\mathcal{P}(A)| = 2^k\) (hence the alternative notation \(2^A\))
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477342 |
1 | 260% | 77d | 6 |
nid:1761491477349
DiskMat
What are the idempotence laws for sets?
1
lapses
1/4
users
260%
ease
nid:1761491477349
Q: What are the idempotence laws for sets?
A: \(A \cap A = A\)
\(A \cup A = A\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477350 |
1 | 260% | 83d | 6 |
nid:1761491477351
DiskMat
What are the commutativity laws for sets?
1
lapses
1/4
users
260%
ease
nid:1761491477351
Q: What are the commutativity laws for sets?
A: \(A \cap B = B \cap A\)
\(A \cup B = B \cup A\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477352 |
1 | 260% | 29d | 5 |
nid:1761491477439
DiskMat
What is the greatest lower bound (glb) of a subset \(S\) in ...
1
lapses
1/4
users
230%
ease
nid:1761491477439
Q: What is the greatest lower bound (glb) of a subset \(S\) in a poset?
A: The greatest element (by the relation, not just integer ordering) of the set of all lower bounds of \(S\). Also called the infimum.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477440 |
1 | 230% | 43d | 9 |
nid:1761491477479
DiskMat
When does set \(B\) dominate set \(A\) (denoted \(A \preceq ...
1
lapses
1/4
users
245%
ease
nid:1761491477479
Q: When does set \(B\) dominate set \(A\) (denoted \(A \preceq B\))?
A: When \(A \sim C\) for some subset \(C \subseteq B\), or equivalently, when there exists an injection \(A \to B\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477480 |
1 | 245% | 70d | 11 |
nid:1761491477481
DiskMat
What does it mean for a set \(A\) to be countable?
1
lapses
1/4
users
260%
ease
nid:1761491477481
Q: What does it mean for a set \(A\) to be countable?
A: \(A \preceq \mathbb{N}\) (i.e., there exists an injection \(A \to \mathbb{N}\))
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477482 |
1 | 260% | 25d | 8 |
nid:1761491477489
DiskMat
What are the two types of countable sets?
1
lapses
1/4
users
260%
ease
nid:1761491477489
Q: What are the two types of countable sets?
A: \(A\) is countable if and only if \(A \sim \mathbb{N}\) or \(A \sim \mathbf{n}\) for some \(n \in \mathbb{N}\) (i.e., \(A\) is finite or equinumerous with \(\mathbb{N}\)).
Conclusion: No cardinality level exists between finite and countably infinite.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477490 |
1 | 260% | 49d | 11 |
nid:1761491477505
DiskMat
What is a computable function \(f: \mathbb{N} \to \{0, 1\}\)...
1
lapses
1/4
users
305%
ease
nid:1761491477505
Q: What is a computable function \(f: \mathbb{N} \to \{0, 1\}\)?
A: A function for which there exists a program that, for every \(n \in \mathbb{N}\), when given \(n\) as input, outputs \(f(n)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477506 |
1 | 305% | 32d | 10 |
nid:1761491477525
DiskMat
What fundamental property distinguishes finite from infinite...
1
lapses
1/4
users
275%
ease
nid:1761491477525
Q: What fundamental property distinguishes finite from infinite sets regarding proper subsets?
A: A finite set never has the same cardinality as one of its proper subsets. An infinite set can (e.g., \(\mathbb{N} \sim \mathbb{O}\) where \(\mathbb{O}\) is the set of odd numbers).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1761491477526 |
1 | 275% | 10d | 8 |
nid:1762106939300
DiskMat
What is \(\text{gcd}(a, b)\)?
1
lapses
1/4
users
245%
ease
nid:1762106939300
Q: What is \(\text{gcd}(a, b)\)?
A: The unique positive greatest common divisor of \(a\) and \(b\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762106939301 |
1 | 245% | 12d | 7 |
nid:1762106939342
c1
DiskMat
\(a \equiv_m R_m(a)\) (the remainder represents the equival...
1
lapses
1/4
users
245%
ease
nid:1762106939342
Cloze c1
Cloze answer: \(a \equiv_m R_m(a)\) (the remainder represents the equivalence class)
Q: What are the two key properties of the remainder function \(R_m\)? (Lemma 4.16)(i) {{c1:: \(a \equiv_m R_m(a)\) (the remainder represents the equivalence class)}}(ii) {{c2:: \(a \equiv_m b \Longleftrightarrow R_m(a) = R_m(b)\) (congru
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762106939343 |
1 | 245% | 33d | 9 |
nid:1762106939348
DiskMat
State the Chinese Remainder Theorem (Theorem 4.19).
1
lapses
1/4
users
260%
ease
nid:1762106939348
Q: State the Chinese Remainder Theorem (Theorem 4.19).
A: Let \(m_1, m_2, \dots, m_r\) be pairwise relatively prime integers and let \(M = \prod_{i=1}^{r} m_i\). For every list \(a_1, \dots, a_r\) with \(0 \leq a_i < m_i\), the system
\[\begin{align} x &\equiv_{m_1} a_1 \\ x &\equiv_{m_2} a_2 \\ &\vdots \\ x &\equiv_{m_r} a_r \end{align}\]
has a unique solution \(x\) satisfying \(0 \leq x < M\).Why unique: If there are two solutions, then, for all \(i\):\(x \equiv_{m_i} a_i\) and
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762106939349 |
1 | 260% | 88d | 6 |
nid:1762106939370
c1
DiskMat
\(a \equiv_m a\) since \(m \mid (a - a) = 0\) ✓
1
lapses
1/4
users
230%
ease
nid:1762106939370
Cloze c1
Cloze answer: \(a \equiv_m a\) since \(m \mid (a - a) = 0\) ✓
Q: Verify that \(\equiv_m\) is reflexive, symmetric, and transitive.Reflexive: {{c1:: \(a \equiv_m a\) since \(m \mid (a - a) = 0\) ✓}}Symmetric: {{c2:: \(a \equiv_m b \Rightarrow m \mid (a-b) \Rightarrow m \mid (b-a) \Righ
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762106939371 |
1 | 230% | 45d | 7 |
nid:1762856073563
LinAlg
Was ist eine konjugiert-transponierte (auch: Hermitesch-tran...
1
lapses
1/4
users
260%
ease
nid:1762856073563
Q: Was ist eine konjugiert-transponierte (auch: Hermitesch-transponierte) Matrix?
A: \( \mathbf{A}^* = (\overline{\mathbf{A}})^\top = \overline{\mathbf{A}^\top}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073563 |
1 | 260% | 25d | 10 |
nid:1762856073577
c1
DiskMat
symmetric
1
lapses
1/4
users
245%
ease
nid:1762856073577
Cloze c1
Cloze answer: symmetric
Q: A relation ρ on a set A is called {{c1::symmetric}} if {{c2::\( a \ \rho \ b \iff b \ \rho \ a\) is true, i.e. if \( \rho = \hat{\rho}\)}}
A: Examples: \( \equiv_m\), marriage
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073578 |
1 | 245% | 24d | 6 |
nid:1762856073621
c1
DiskMat
meet of \(a\) and \(b\) (also denoted \(a \land b\)).
1
lapses
1/4
users
260%
ease
nid:1762856073621
Cloze c1
Cloze answer: meet of \(a\) and \(b\) (also denoted \(a \land b\)).
Q: Consider the poset \((A;\preceq)\). If \(\{a,b\}\) have a {{c2::greatest lower bound}}, then it is called the {{c1::meet of \(a\) and \(b\) (also denoted \(a \land b\)).}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073629 |
1 | 260% | 71d | 6 |
nid:1762856073628
c1
DiskMat
dominates (denoted \(A \preceq B\))
1
lapses
1/4
users
245%
ease
nid:1762856073628
Cloze c1
Cloze answer: dominates (denoted \(A \preceq B\))
Q: The set \(B\) {{c1::dominates (denoted \(A \preceq B\))}} if {{c2::there exists an injective function \(A \rightarrow B\).}}
A: Example: \(f(x): \mathbb{N} \rightarrow \mathbb{R} = x\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073642 |
1 | 245% | 42d | 5 |
nid:1762856073660
c2
DiskMat
\(\langle R, +, -, 0 \rangle\) is a commutative group
1
lapses
1/4
users
215%
ease
nid:1762856073660
Cloze c2
Cloze answer: \(\langle R, +, -, 0 \rangle\) is a commutative group
Q: {{c1::A ring \(\langle R, +, -, 0, \cdot, 1 \rangle\)}} is an algebra with the properties that{{c2::\(\langle R, +, -, 0 \rangle\) is a commutative group}}{{c3::\(\langle R, \cdot, 1 \rangle\) is a monoid}}{{c4::\( a(b+c) = (ab) + (ac), (b+c)a = (ba)
A: Examples: \(\mathbb{Z}, \mathbb{R}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856073673 |
1 | 215% | 26d | 7 |
nid:1762856074477
c2
A&D
closed Eulerian walk (Eulerzyklus)
1
lapses
1/4
users
245%
ease
nid:1762856074477
Cloze c2
Cloze answer: closed Eulerian walk (Eulerzyklus)
Q: In graph theory, a {{c2::closed Eulerian walk (Eulerzyklus)}} is an {{c1::Eulerian walk (Eulerweg) that ends at the start vertex}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856074510 |
1 | 245% | 88d | 7 |
nid:1762856074631
c2
DiskMat
expression using the propositional symbols \(A, B, C, \dots\...
1
lapses
1/4
users
245%
ease
nid:1762856074631
Cloze c2
Cloze answer: expression using the propositional symbols \(A, B, C, \dots\) and logical operators \(\land, \lor, \lnot, \ldots\)
Q: An {{c2::expression using the propositional symbols \(A, B, C, \dots\) and logical operators \(\land, \lor, \lnot, \ldots\)}} is called a {{c1::formula (of propositional logic)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856074667 |
1 | 245% | 5d | 7 |
nid:1762856074659
c1
DiskMat
composite
1
lapses
1/4
users
245%
ease
nid:1762856074659
Cloze c1
Cloze answer: composite
Q: An integer greater than \(1\) that is not a prime is called {{c1::composite}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856074691 |
1 | 245% | 75d | 5 |
nid:1762856074680
c1
DiskMat
The Fermat-Euler theorem states that for all \(m\ge 2\) and ...
1
lapses
1/4
users
260%
ease
nid:1762856074680
Cloze c1
Q: The Fermat-Euler theorem states that for all \(m\ge 2\) and all \(a\) with \(\gcd(a,m) = 1\),{{c1:: \[a^{\varphi(m)} \equiv_m 1\]and so in particular, for every prime \(p\) and every \(a\) not divisible by \(p\): \(a^{p-1} \equiv_p 1\).}}
A: We know \(a^{\operatorname{order}(a)} \equiv_m 1\). Since \(\operatorname{order}(a)\) divides \(| \mathbb{Z}_m^* | = \varphi(m)\) (Lagrange's), \(a^{\varphi(m)} \equiv_m a^{k \cdot \operatorname{order}(a)} \equiv_m (a^{\operatorname{order}(a)})^k \equiv_m 1^k \equiv_m 1\)This theorem is used for RSA.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856074708 |
1 | 260% | 26d | 8 |
nid:1762856074690
c1
LinAlg
\(\det (A^{-1}) =\) {{c1::\((\det (A))^{-1}\)}}
1
lapses
1/4
users
275%
ease
nid:1762856074690
Cloze c1
Q: \(\det (A^{-1}) =\) {{c1::\((\det (A))^{-1}\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1762856074715 |
1 | 275% | 9d | 12 |
nid:1763362644469
c1
A&D
adjacent (adjazent oder benachbart)
1
lapses
1/4
users
260%
ease
nid:1763362644469
Cloze c1
Cloze answer: adjacent (adjazent oder benachbart)
Q: In an edge \(e = \{u, v\}\), we call \(u\) {{c1::adjacent (adjazent oder benachbart)}} to \(v\) (and the other way around) and \(e\) {{c2::incident (inzident oder anliegend)}} to \(u, v\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1763362644470 |
1 | 260% | 223d | 7 |
nid:1763363435750
c2
A&D
connected and has no cycles (Kreise)
1
lapses
1/4
users
200%
ease
nid:1763363435750
Cloze c2
Cloze answer: connected and has no cycles (Kreise)
Q: A graph \(G\) is a {{c1::tree}} if it is {{c2::connected and has no cycles (Kreise)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1763363435750 |
1 | 200% | 64d | 8 |
nid:1763364155947
c2
A&D
the subgraph obtained after removing it (keeping the vertice...
1
lapses
1/4
users
245%
ease
nid:1763364155947
Cloze c2
Cloze answer: the subgraph obtained after removing it (keeping the vertices) is disconnected
Q: An edge in a connected graph is a {{c1::cut edge}} if {{c2::the subgraph obtained after removing it (keeping the vertices) is disconnected}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1763364155947 |
1 | 245% | 36d | 8 |
nid:1763493474474
DiskMat
What do we need to state before using the decomposition of a...
1
lapses
1/4
users
260%
ease
nid:1763493474474
Q: What do we need to state before using the decomposition of an \(n \in \mathbb{Z}\) into prime factors?
A: That this is allowed by the fundamental theorem of arithmetic.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1763493474474 |
1 | 260% | 18d | 9 |
nid:1764746595604
c1
A&D
BFS
1
lapses
1/4
users
260%
ease
nid:1764746595604
Cloze c1
Cloze answer: BFS
Q: We find the shortest walk in a graph using {{c1:: BFS}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764746595604 |
1 | 260% | 140d | 5 |
nid:1764859231354
c1
DiskMat
1 by definition
1
lapses
1/4
users
275%
ease
nid:1764859231354
Cloze c1
Cloze answer: 1 by definition
Q: The order \(\text{ord}(e)\) of \(e \in G\) is {{c1:: 1 by definition}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231355 |
1 | 275% | 33d | 7 |
nid:1764859231539
DiskMat
When is a polynomial of degree \(2\) or \(3\) irreducible?
1
lapses
1/4
users
245%
ease
nid:1764859231539
Q: When is a polynomial of degree \(2\) or \(3\) irreducible?
A: Corollary 5.30: A polynomial \(a(x)\) of degree \(2\) or \(3\) over a field \(F\) is irreducible if and only if it has no root.
Important: This doesn't work for polynomials of higher degrees! A degree \(4\) polynomial might be the product of two irreducible degree \(2\) polynomials, each with no roots.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231540 |
1 | 245% | 8d | 4 |
nid:1764859231560
DiskMat
Which of the following are fields: \(\mathbb{Z}, \mathbb{Q},...
1
lapses
1/4
users
260%
ease
nid:1764859231560
Q: Which of the following are fields: \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_5, \mathbb{Z}_6, R[x]\)?
A: Fields: \(\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_5\) (where \(5\) is prime)
Not fields:
- \(\mathbb{Z}\) (not all nonzero elements have multiplicative inverse, e.g., \(2\))
- \(\mathbb{Z}_6\) (since \(6\) is not prime, e.g., \(2\) has no inverse)
- \(R[x]\) for any ring \(R\) (polynomials don't have multiplicative inverses)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231561 |
1 | 260% | 7d | 10 |
nid:1764859231579
DiskMat
Is \(F[x]_{m(x)}\) a monoid, group, ring, field?
1
lapses
1/4
users
260%
ease
nid:1764859231579
Q: Is \(F[x]_{m(x)}\) a monoid, group, ring, field?
A: Lemma 5.35: \(F[x]_{m(x)}\) is a commutative ring with respect to addition and multiplication modulo \(m(x)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231580 |
1 | 260% | 60d | 7 |
nid:1764859231602
c2
DiskMat
The {{c2::output \((c_0, \dots, c_{n-1})\)}} of an encoding ...
1
lapses
1/4
users
245%
ease
nid:1764859231602
Cloze c2
Q: The {{c2::output \((c_0, \dots, c_{n-1})\)}} of an encoding function is called a {{c1::codeword}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764859231604 |
1 | 245% | 17d | 5 |
nid:1764860289620
c1
DiskMat
\(0a = 0\)
1
lapses
1/4
users
245%
ease
nid:1764860289620
Cloze c1
Cloze answer: \(0a = 0\)
Q: In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \(a0 =\) {{c1::\(0a = 0\)}}.
A: The zero (neutral of additive group) pulls all other elements to 0 by multiplication.\(0a=(0-0)a=0a-0a=0\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764860289620 |
1 | 245% | 21d | 6 |
nid:1764860422155
c1
DiskMat
\(-(ab)\)
1
lapses
1/4
users
275%
ease
nid:1764860422155
Cloze c1
Cloze answer: \(-(ab)\)
Q: In any ring \(\langle R; +, -, 0, \cdot, 1 \rangle\), and for all \(a, b \in R\) \((-a)b =\) {{c1::\(-(ab)\)}}. (Proof included)
A: Proof: \(ab+(−a)b=(a+(−a))b=0⋅b=0\)Since \((−a)b\) satisfies \(ab+(−a)b=0\), we have \((−a)b=−(ab\)).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764860422155 |
1 | 275% | 29d | 8 |
nid:1764860775647
c1
DiskMat
\(a \ | \ c\), i.e. the relation | is transitive
1
lapses
1/4
users
245%
ease
nid:1764860775647
Cloze c1
Cloze answer: \(a \ | \ c\), i.e. the relation | is transitive
Q: In any commutative ring: If \(a \ | \ b\) and \(b \ | \ c\) then {{c1:: \(a \ | \ c\), i.e. the relation | is transitive}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1764860775647 |
1 | 245% | 8d | 4 |
nid:1765194177649
LinAlg
What is the rank of a matrix?
1
lapses
1/4
users
260%
ease
nid:1765194177649
Q: What is the rank of a matrix?
A: it is the number of independent columns, where independence is defined such that given a column vector \(v_j\) then \(v_j\) is not a linear combination of \(v_1, v_2 ... v_{j-1}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765194177649 |
1 | 260% | 75d | 7 |
nid:1765198200601
A&D
Runtime: Operations in an Adjacency List:
1
lapses
1/4
users
245%
ease
nid:1765198200601
Q: Runtime: Operations in an Adjacency List:
A: 1. Check if \(uv \in E \): \(O(1 + \min\{\text{deg}(u), \text{deg}(v) \})\) (we have to check the smaller of the two adjacency lists2. Vertex \(u\), find all adjacent vertices: \(O(1+\text{deg}(u) )\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765198200601 |
1 | 245% | 26d | 10 |
nid:1765294753798
c2
A&D
\(f \leq O(g)\) and \(f \neq \Theta(g)\)
1
lapses
1/4
users
230%
ease
nid:1765294753798
Cloze c2
Cloze answer: \(f \leq O(g)\) and \(f \neq \Theta(g)\)
Q: If \(\frac{f(n)}{g(n)}\) tends to {{c1:: 0}}, then {{c2::\(f \leq O(g)\) and \(f \neq \Theta(g)\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765294753799 |
1 | 230% | 5d | 8 |
nid:1765294947576
c1
A&D
\leq
1
lapses
1/4
users
260%
ease
nid:1765294947576
Cloze c1
Cloze answer: \leq
Q: If \(f \leq O(h)\) and \(g \leq O(h)\), then \(f + g {{c1::\leq}} O(h)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765294947576 |
1 | 260% | 19d | 12 |
nid:1765295484756
A&D
When \(f \geq \Omega(g)\), this means what exactly?
1
lapses
1/4
users
260%
ease
nid:1765295484756
Q: When \(f \geq \Omega(g)\), this means what exactly?
A: \(\exists C \ge 0 \quad \forall n \in \mathbb{N} \quad f(n) \ge C\cdot g(n)\)\(f\) grows asymptotically faster than \(g\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765295484757 |
1 | 260% | 2d | 14 |
nid:1765296240804
c2
A&D
O(n!)
1
lapses
1/4
users
230%
ease
nid:1765296240804
Cloze c2
Cloze answer: O(n!)
Q: Choose a tight bound!\({{c1::O(k^n)}} \leq {{c2::O(n!)}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765296240805 |
1 | 230% | 17d | 4 |
nid:1765296364773
c2
A&D
O(n)
1
lapses
1/4
users
230%
ease
nid:1765296364773
Cloze c2
Cloze answer: O(n)
Q: Choose a tight bound!\({{c1::O(\log(n))}}\leq {{c2::O(n)}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765296364774 |
1 | 230% | 17d | 5 |
nid:1765297403833
c1
A&D
\(b = \log_2(a)\)
1
lapses
1/4
users
260%
ease
nid:1765297403833
Cloze c1
Cloze answer: \(b = \log_2(a)\)
Q: Master Theorem: If {{c1:: \(b = \log_2(a)\)}} then {{c2:: \(T(n) \leq O(n^{\log_2 a} \cdot \log n)\)}}.
A: The recursive and non-recursive work is balanced.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765297403833 |
1 | 260% | 51d | 7 |
nid:1765297729656
A&D
For \(T(n) = 4T(n/2) + n\), which Master Theorem case applie...
1
lapses
1/4
users
230%
ease
nid:1765297729656
Q: For \(T(n) = 4T(n/2) + n\), which Master Theorem case applies?
A: Because \(b = 1\) and \(\log_2(a) = \log_2 4 = 2 > b\), therefore \(T(n) = \Theta(n^2)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765297729656 |
1 | 230% | 4d | 5 |
nid:1765298206873
c2
A&D
\(O(n \log(n))\)
1
lapses
1/4
users
260%
ease
nid:1765298206873
Cloze c2
Cloze answer: \(O(n \log(n))\)
Q: {{c1:: \(\sum_{i = 1}^{n} i\log(i)\)::Sum}} \(\leq\) {{c2::\(O(n \log(n))\)::O-notation}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765298206875 |
1 | 260% | 97d | 6 |
nid:1765298610771
A&D
Provide the outline of an induction proof.
1
lapses
1/4
users
245%
ease
nid:1765298610771
Q: Provide the outline of an induction proof.
A: We want to prove that ... for \(n \geq 5\)Base Case: Let \(n = 5\) .... So the property holds for \(n = 5\).Induction Hypothesis: We assume the property is true for some \(k \geq 5\)Induction Step: We must show that the property holds for \(k + 1\).By the principle of mathematical induction ... is true for all \(n \geq 5\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765298610771 |
1 | 245% | 57d | 6 |
nid:1765301887927
A&D
How do we create a maxHeap?
1
lapses
1/4
users
245%
ease
nid:1765301887927
Q: How do we create a maxHeap?
A: Insert the node \(v\) at the next free space in the tree, i.e. first to the left, then right (to conserve the tree structure). Then we restore the heap condition by reverse-“versickern” the element until it’s restored.Swap it with it’s parent nodes until the condition is restored.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765301887927 |
1 | 245% | 3d | 5 |
nid:1765300723241
A&D
Bubble Sort
1
lapses
1/4
users
275%
ease
nid:1765300723241
Q: Bubble Sort
A: Best Case: \(O(n^2)\) (\(O(n)\) if checking for swaps and aborting early)Worst Case: \(O(n^2)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765388610996 |
1 | 275% | 47d | 11 |
nid:1765300723241
A&D
Bubble Sort
1
lapses
1/4
users
245%
ease
nid:1765300723241
Q: Bubble Sort
A: Best Case: \(O(n^2)\) (\(O(n)\) if checking for swaps and aborting early)Worst Case: \(O(n^2)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765388610998 |
1 | 245% | 41d | 7 |
nid:1765300949586
A&D
Selection Sort
1
lapses
1/4
users
245%
ease
nid:1765300949586
Q: Selection Sort
A: Best Case: \(O(n^2)\)Worst Case: \(O(n^2)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765388611000 |
1 | 245% | 45d | 9 |
nid:1765653532362
c2
EProg
char
1
lapses
1/4
users
245%
ease
nid:1765653532362
Cloze c2
Cloze answer: char
Q: The 8 primitve types of Java are:{{c1:: byte}}{{c2:: char}}{{c3:: short}}{{c4:: int}}{{c5:: long}}{{c6:: float}}{{c7:: double}}{{c8:: boolean}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765653532368 |
1 | 245% | 30d | 7 |
nid:1765653532374
c1
EProg
copied
1
lapses
1/4
users
245%
ease
nid:1765653532374
Cloze c1
Cloze answer: copied
Q: Values given to a method in Java are always {{c1::copied}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1765653532379 |
1 | 245% | 44d | 9 |
nid:1766000828773
DiskMat
What is the number of generators of \(\mathbb{Z}_{25}^* \)?
1
lapses
1/4
users
245%
ease
nid:1766000828773
Q: What is the number of generators of \(\mathbb{Z}_{25}^* \)?
A: \(\varphi(\varphi(25)) = |\mathbb{Z}_{\varphi(25)}| = |\mathbb{Z}_{20}| = 8\) ( 1, 3, 7, 9, 11, 13, 17, 19 )
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766000828773 |
1 | 245% | 10d | 7 |
nid:1766245701439
c2
A&D
\(O(1)\) as we know the offset for each key
1
lapses
1/4
users
245%
ease
nid:1766245701439
Cloze c2
Cloze answer: \(O(1)\) as we know the offset for each key
Q: In an array we can:Insert in {{c1:: \(O(1)\) as we know the first empty cell in the array and can just write the key there}}Get in {{c2::\(O(1)\) as we know the offset for each key}}InsertAfter in {{c3::\(\Theta(l)\), since we ha
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766245701441 |
1 | 245% | 50d | 4 |
nid:1766246034328
c1
A&D
previous and next element
1
lapses
1/4
users
260%
ease
nid:1766246034328
Cloze c1
Cloze answer: previous and next element
Q: In a doubly linked list, we store a pointer to the {{c1:: previous and next element}} for each key.This increases {{c2::memory usage}} as a trade-off for {{c2:: speed}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766246034328 |
1 | 260% | 29d | 5 |
nid:1766246342851
c3
A&D
\(O(1)\) if we get the memory address of the element to ins...
1
lapses
1/4
users
275%
ease
nid:1766246342851
Cloze c3
Cloze answer: \(O(1)\) if we get the memory address of the element to insert after.
Q: In a singly and doubly linked list, the operation:Insert is {{c1::\(\Theta(1)\) as we know the memory address of the final element in the list and just have to set the null pointer to the new keys address. Without this pointer it's \(\Th
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766246342851 |
1 | 275% | 37d | 9 |
nid:1766248090341
c1
A&D
LIFO
1
lapses
1/4
users
245%
ease
nid:1766248090341
Cloze c1
Cloze answer: LIFO
Q: A stack is also called a {{c1:: LIFO}} queue.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766248090341 |
1 | 245% | 21d | 4 |
nid:1766319025292
c3
DiskMat
Describe the RSA protocol:{{c1:: Alice generates primes \(p\...
1
lapses
1/4
users
275%
ease
nid:1766319025292
Cloze c3
Q: Describe the RSA protocol:{{c1:: Alice generates primes \(p\) and \(q\)}}{{c2:: Set \(n = pq\) and \(f = \varphi(n) = (p - 1)(q - 1)\) }}{{c3:: Select \(e\): \(d \equiv_f e^{-1}\) the modular inverse (decryption)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766319025293 |
1 | 275% | 15d | 9 |
nid:1766319025292
c4
DiskMat
Send \(n\) and \(e\) to Bob
1
lapses
1/4
users
245%
ease
nid:1766319025292
Cloze c4
Cloze answer: Send \(n\) and \(e\) to Bob
Q: Describe the RSA protocol:{{c1:: Alice generates primes \(p\) and \(q\)}}{{c2:: Set \(n = pq\) and \(f = \varphi(n) = (p - 1)(q - 1)\) }}{{c3:: Select \(e\): \(d \equiv_f e^{-1}\) the modular inverse (decryption)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766319025297 |
1 | 245% | 9d | 4 |
nid:1766319174572
c1
DiskMat
Closure
1
lapses
1/4
users
245%
ease
nid:1766319174572
Cloze c1
Cloze answer: Closure
Q: A monoid has the following properties:{{c1::Closure}}{{c2::Associativity}}{{c3::Identity}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766319174572 |
1 | 245% | 3d | 5 |
nid:1766319253408
DiskMat
An abelian group has the following properties:
1
lapses
1/4
users
245%
ease
nid:1766319253408
Q: An abelian group has the following properties:
A: closureassociativityidentityinversecommutative
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766319253408 |
1 | 245% | 11d | 4 |
nid:1766319397636
DiskMat
A field has the following properties:
1
lapses
1/4
users
245%
ease
nid:1766319397636
Q: A field has the following properties:
A: Additive Group:closureassociativityidentityinversecommutativeMultiplicative group:closureassociativitydistributivityidentityno zero-divisorinverse
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766319397636 |
1 | 245% | 7d | 6 |
nid:1766408177022
c1
DiskMat
meaning or semantics
1
lapses
1/4
users
275%
ease
nid:1766408177022
Cloze c1
Cloze answer: meaning or semantics
Q: The truth function \(\tau : \mathcal{S} \rightarrow \{0,1\}\) defines the {{c1:: meaning or semantics}} in \(\mathcal{S}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766408177022 |
1 | 275% | 23d | 8 |
nid:1766418002697
c2
DiskMat
an alphabet \(\Lambda\) (of allowed symbols); which strings ...
1
lapses
1/4
users
275%
ease
nid:1766418002697
Cloze c2
Cloze answer: an alphabet \(\Lambda\) (of allowed symbols); which strings in \(\Lambda^*\) are formulas (i.e. syntactically correct)
Q: The {{c1::syntax}} of a logic defines {{c2::an alphabet \(\Lambda\) (of allowed symbols)}} and specifies {{c2::which strings in \(\Lambda^*\) are formulas (i.e. syntactically correct)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002700 |
1 | 275% | 19d | 7 |
nid:1766418002702
c2
DiskMat
An interpretation consists of {{c1::a set \(\mathcal{Z} \sub...
1
lapses
1/4
users
260%
ease
nid:1766418002702
Cloze c2
Q: An interpretation consists of {{c1::a set \(\mathcal{Z} \subseteq \Lambda\) of \(\Lambda\)}}, {{c2::a domain (a set of possible values) for each symbol in \(\mathcal{Z}\)}}, and {{c3::a function that assigns to each symbol in \(\mathcal{Z}\) a value in the a
A: A set of symbols \(\mathcal{Z} \subseteq \Lambda\)
\(\Lambda\) is the "alphabet" or collection of all available symbols
\(\mathcal{Z}\) is the subset of symbols we're actually interpreting
A domain for each symbol
For each symbol in \(\mathcal{Z}\), there's a set of possible values it could take
Often the domain is defined in terms of the universe \(U\) where a symbol can be a fu
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002710 |
1 | 260% | 16d | 7 |
nid:1766418002746
c2
DiskMat
\(F \lor G\); \(F \vdash G \lor F\)
1
lapses
1/4
users
215%
ease
nid:1766418002746
Cloze c2
Cloze answer: \(F \lor G\); \(F \vdash G \lor F\)
Q: {{c1::\(F\) }} \(\vdash\) {{c2::\(F \lor G\)}} and {{c2::\(F \vdash G \lor F\)}} are valid derivation rules.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002790 |
1 | 215% | 4d | 5 |
nid:1766418002749
c1
DiskMat
\(F \rightarrow G\) is a tautology and thus that \(F \models...
1
lapses
1/4
users
245%
ease
nid:1766418002749
Cloze c1
Cloze answer: \(F \rightarrow G\) is a tautology and thus that \(F \models G\)
Q: If in a sound calculus \(K\) one can derive \(G\) from the set of formulas \(F\) (\(F \vdash_K G\)), then one has proved that {{c1::\(F \rightarrow G\) is a tautology and thus that \(F \models G\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002795 |
1 | 245% | 13d | 7 |
nid:1766418002768
DiskMat
For DNF construction from truth table, which rows do you use...
1
lapses
1/4
users
245%
ease
nid:1766418002768
Q: For DNF construction from truth table, which rows do you use?
A: Rows evaluating to 1.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002825 |
1 | 245% | 11d | 4 |
nid:1766418002773
c2
DiskMat
clause
1
lapses
1/4
users
260%
ease
nid:1766418002773
Cloze c2
Cloze answer: clause
Q: The {{c1::empty set \(\emptyset\)}} is a {{c2::clause}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002831 |
1 | 260% | 13d | 5 |
nid:1766418002791
c2
DiskMat
\(k\) denotes the number of arguments of the predicate (the ...
1
lapses
1/4
users
260%
ease
nid:1766418002791
Cloze c2
Cloze answer: \(k\) denotes the number of arguments of the predicate (the arity)
Q: A {{c1::predicate symbol}} is of the form {{c2::\(P_i^{(k)}\) with \(i, k \in \mathbb{N}\)}}, where {{c2::\(k\) denotes the number of arguments of the predicate (the arity)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002863 |
1 | 260% | 9d | 7 |
nid:1766418002792
c1
DiskMat
A variable
1
lapses
1/4
users
230%
ease
nid:1766418002792
Cloze c1
Cloze answer: A variable
Q: A term is defined inductively: {{c1::A variable}} is a termif {{c2::\((t_1, \dots, t_k)\) are terms}}, then {{c3::\(f^{(k)}(t_1, \dots, t_k)\) is a term}}.
A: For \(k = 0\) one writes no parentheses (constants).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002866 |
1 | 230% | 4d | 8 |
nid:1766418002817
c1
DiskMat
no existence quantifiers
1
lapses
1/4
users
260%
ease
nid:1766418002817
Cloze c1
Cloze answer: no existence quantifiers
Q: Skolem normal form has {{c1::no existence quantifiers}}.It is {{c2::equisatisfiable (not equivalent!)}} to the original formula.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002920 |
1 | 260% | 13d | 7 |
nid:1766418002818
c1
DiskMat
replacing all variables bound to an \(\exists\) by a functio...
1
lapses
1/4
users
245%
ease
nid:1766418002818
Cloze c1
Cloze answer: replacing all variables bound to an \(\exists\) by a function
Q: The Skolem transformation works by {{c1::replacing all variables bound to an \(\exists\) by a function}} whose arguments are {{c2::the universally quantified variables that precede it}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002922 |
1 | 245% | 3d | 6 |
nid:1766418002830
c1
DiskMat
no variable occurs both as a bound and as a free variable
1
lapses
1/4
users
245%
ease
nid:1766418002830
Cloze c1
Cloze answer: no variable occurs both as a bound and as a free variable
Q: Rectified form:{{c1::no variable occurs both as a bound and as a free variable}}{{c2::all quantifiers use distinct variable names}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002938 |
1 | 245% | 6d | 5 |
nid:1766418002830
c2
DiskMat
all quantifiers use distinct variable names
1
lapses
1/4
users
245%
ease
nid:1766418002830
Cloze c2
Cloze answer: all quantifiers use distinct variable names
Q: Rectified form:{{c1::no variable occurs both as a bound and as a free variable}}{{c2::all quantifiers use distinct variable names}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418002939 |
1 | 245% | 4d | 8 |
nid:1766418355297
c2
DiskMat
the variables never appear in the same predicate
1
lapses
1/4
users
245%
ease
nid:1766418355297
Cloze c2
Cloze answer: the variables never appear in the same predicate
Q: We are allowed to swap quantifier order in a formula if:{{c1:: they are of the same type}}{{c2:: the variables never appear in the same predicate}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766418355297 |
1 | 245% | 5d | 5 |
nid:1766484505751
c2
A&D
insert(x, W) Insert the key x into W, as long as it’s not sa...
1
lapses
1/4
users
245%
ease
nid:1766484505751
Cloze c2
Cloze answer: insert(x, W) Insert the key x into W, as long as it’s not saved there yet
Q: The ADT Dictionary implements the following methods:{{c1::search(x, W) returns the position of the key x in memory}}{{c2::insert(x, W) Insert the key x into W, as long as it’s not saved there yet}}{{c3::delete(x, W) find and delete
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766484505753 |
1 | 245% | 33d | 4 |
nid:1766484756595
c4
A&D
\(O(\log n)\)
1
lapses
1/4
users
230%
ease
nid:1766484756595
Cloze c4
Cloze answer: \(O(\log n)\)
Q: Search
Insertion
Deletion
Non-sorted array
{{c1::\(O(n)\)}}
{{c2::\(O(1)\)}}
{
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766484756597 |
1 | 230% | 31d | 6 |
nid:1766484876704
c1
A&D
\(O(h)\), where \(h\) is the height
1
lapses
1/4
users
245%
ease
nid:1766484876704
Cloze c1
Cloze answer: \(O(h)\), where \(h\) is the height
Q: The runtime of search in a binary tree is {{c1::\(O(h)\), where \(h\) is the height}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766484876704 |
1 | 245% | 42d | 6 |
nid:1766495679168
A&D
Subset Sum (Teilsummenproblem)
1
lapses
1/4
users
260%
ease
nid:1766495679168
Q: Subset Sum (Teilsummenproblem)
A: \(\Theta(n \cdot b)\) (Pseudo-Polynomial)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766495679169 |
1 | 260% | 9d | 12 |
nid:1766496919198
A&D
Longest Ascending Subsequence (Längste Aufsteigende Teilfolg...
1
lapses
1/4
users
260%
ease
nid:1766496919198
Q: Longest Ascending Subsequence (Längste Aufsteigende Teilfolge)
A: \(\Theta(n \log n)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766496919198 |
1 | 260% | 20d | 7 |
nid:1766500164961
A&D
How can we find a cross edge via DFS?
1
lapses
1/4
users
245%
ease
nid:1766500164961
Q: How can we find a cross edge via DFS?
A: If we find vertex with both pre- and post-values set, there's a cross edge.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766500164961 |
1 | 245% | 24d | 6 |
nid:1766500713117
c1
A&D
an adjacency list is better; an adjacency matrix is better
1
lapses
1/4
users
245%
ease
nid:1766500713117
Cloze c1
Cloze answer: an adjacency list is better; an adjacency matrix is better
Q: Which datastructure is best for DFS?In a sparse graph {{c1:: an adjacency list is better}}, in a dense graph {{c1:: an adjacency matrix is better}}.
A: \(|E| \geq |V|^2 / 10\), then DFS has the same runtime in the worst-case using adjacency matrices or lists as \(|V| + |E| \leq |V| + |V|^2 \)which is \(O(n^2)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766500713117 |
1 | 245% | 34d | 8 |
nid:1766523328098
A&D
BFS (Breadth First Search)
1
lapses
1/4
users
260%
ease
nid:1766523328098
Q: BFS (Breadth First Search)
A: \(O(|V|+|E|)\) (Adjacency List)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766523328099 |
1 | 260% | 30d | 8 |
nid:1766524219271
A&D
Dijkstra's Algorithm
1
lapses
1/4
users
260%
ease
nid:1766524219271
Q: Dijkstra's Algorithm
A: \(O((|V| + |E|) \log |V|)\) (or \(O(|V|^2)\)The runtime is calculated from \(O(n + (\#\text{extract-min} + \#\text{decrease-key}) \cdot \log n)\) which gives \(O((n + m) \cdot \log n)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766524219271 |
1 | 260% | 19d | 8 |
nid:1766524219271
A&D
Dijkstra's Algorithm
1
lapses
1/4
users
230%
ease
nid:1766524219271
Q: Dijkstra's Algorithm
A: \(O((|V| + |E|) \log |V|)\) (or \(O(|V|^2)\)The runtime is calculated from \(O(n + (\#\text{extract-min} + \#\text{decrease-key}) \cdot \log n)\) which gives \(O((n + m) \cdot \log n)\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766524328968 |
1 | 230% | 12d | 7 |
nid:1766568238909
c1
A&D
never contains a vertex already in the MST
1
lapses
1/4
users
230%
ease
nid:1766568238909
Cloze c1
Cloze answer: never contains a vertex already in the MST
Q: Prim's Algorithm Invariants: The priority queue \(H = V \setminus S\) (\(V\) set of all vertices, \(S\) vertices currently in the MST) {{c1::never contains a vertex already in the MST}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766568238910 |
1 | 230% | 5d | 3 |
nid:1766568909602
A&D
Kruskal's Algorithm
1
lapses
1/4
users
230%
ease
nid:1766568909602
Q: Kruskal's Algorithm
A: \(O(|E| \log |E| + |V| \log |V|)\)Outer loop: Iterate \(|E|\) times at most:Inner loop: find and union take \(O(\log |V|)\) per call amortised, thus \(O(|V| \log |V|)\) total.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766568909604 |
1 | 230% | 5d | 3 |
nid:1766574057724
c1
A&D
always negative \(\leq 0\)
1
lapses
1/4
users
260%
ease
nid:1766574057724
Cloze c1
Cloze answer: always negative \(\leq 0\)
Q: The height \(h(v)\) in Johnson's Algorithm is {{c1::always negative \(\leq 0\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766574057725 |
1 | 260% | 36d | 6 |
nid:1766742464527
IO r1
A&D
[Image Occlusion region 1]
1
lapses
1/4
users
230%
ease
nid:1766742464527
Cloze c1
Q: {{c5::image-occlusion:rect:left=.592:top=.4403:width=.0786:height=.0963:oi=1}}{{c10::image-occlusion:rect:left=.5847:top=.571:width=.0859:height=.0963:oi=1}}{{c12::image-occlusion:rect:left=.444:top=.6983:width=.0786:height=.0963:oi=1}}{{c3::image-occlusion:rect:left=.7912:top=.313:width
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766742464536 |
1 | 230% | 3d | 2 |
nid:1764744892590
c2
A&D
spanning, it connects all vertices
1
lapses
1/4
users
230%
ease
nid:1764744892590
Cloze c2
Cloze answer: spanning, it connects all vertices
Q: A Minimum Spanning Tree is a subgraph of a {{c1:: connected, undirected, weighted}} graph that fullfills:{{c2:: spanning, it connects all vertices}}{{c3:: acylic, it's a tree}}{{c4:: minimal, the sum of all edge weights in the Tree is minimal}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1766992688141 |
1 | 230% | 3d | 2 |
nid:1767084587767
c2
LinAlg
\((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move...
1
lapses
1/4
users
230%
ease
nid:1767084587767
Cloze c2
Cloze answer: \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
Q: Scalar product properties: \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar.{{c1::\(v \cdot w = w \cdot v\) (symmetry / commutatitivity}}{{c2:: \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1767084587769 |
1 | 230% | 10d | 4 |
nid:1767087495269
c1
LinAlg
The {{c2::independent}} columns of \(A\), {{c1::span the col...
1
lapses
1/4
users
245%
ease
nid:1767087495269
Cloze c1
Q: The {{c2::independent}} columns of \(A\), {{c1::span the column space \(\textbf{C}(A)\) of \(A\)}}.
A: Proven by induction, adding elements that are a linear combination of other ones doesn't change span, thus we can iteratively remove the dependent columns.Lemma 2.11
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1767087495271 |
1 | 245% | 19d | 7 |
nid:1767439652577
LinAlg
What is the inverse of \(A = \begin{bmatrix} a & b \\ c & d ...
1
lapses
1/4
users
245%
ease
nid:1767439652577
Q: What is the inverse of \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)?
A: \[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1767439652577 |
1 | 245% | 5d | 5 |
nid:1767888505024
IO r3
EProg
[Image Occlusion region 3]
1
lapses
1/4
users
260%
ease
nid:1767888505024
Cloze c3
Q: {{c1::image-occlusion:rect:left=.2281:top=.3427:width=.0814:height=.2045:oi=1}}{{c2::image-occlusion:rect:left=.3053:top=.345:width=.1142:height=.2067:oi=1}}{{c3::image-occlusion:rect:left=.1625:top=.5221:width=.0693:height=.2181:oi=1}}{{c4::image-occlusion:rect:left=.1625:top=.713:width
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1767888505024 |
1 | 260% | 22d | 10 |
nid:1767888505024
IO r6
EProg
[Image Occlusion region 6]
1
lapses
1/4
users
260%
ease
nid:1767888505024
Cloze c6
Q: {{c1::image-occlusion:rect:left=.2281:top=.3427:width=.0814:height=.2045:oi=1}}{{c2::image-occlusion:rect:left=.3053:top=.345:width=.1142:height=.2067:oi=1}}{{c3::image-occlusion:rect:left=.1625:top=.5221:width=.0693:height=.2181:oi=1}}{{c4::image-occlusion:rect:left=.1625:top=.713:width
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1767888505025 |
1 | 260% | 14d | 11 |
nid:1767888762979
c1
EProg
&& is false
1
lapses
1/4
users
245%
ease
nid:1767888762979
Cloze c1
Cloze answer: && is false
Q: Java has short circuiting for the && and || operators.This means that if the left of {{c1:: && is false}} then the right isn't even executed{{c2:: || is true}} then the right i
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1767888762980 |
1 | 245% | 21d | 6 |
nid:1768138841525
c2
LinAlg
There is an \(m \times m\) matrix \(B\) such that \(BA = I\)...
1
lapses
1/4
users
230%
ease
nid:1768138841525
Cloze c2
Cloze answer: There is an \(m \times m\) matrix \(B\) such that \(BA = I\).
Q: Three equivalent statements:{{c1::\(T_A : \mathbb{R}^m \rightarrow \mathbb{R}^m\) is bijective.::Transformation}}{{c2::There is an \(m \times m\) matrix \(B\) such that \(BA = I\).}}{{c3::The columns of \(A\) are linearly independent.}}
A: The third one can be derived from the fact that if \(BA = I\), there is only a single \(x \in \mathbb{R}^m\) such that \(A \textbf{x} = 0\).It is also intuitively clear that if not all columns were linearly independent, we'd actually have a tall linear transformation and would be losing information.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768138841525 |
1 | 230% | 4d | 4 |
nid:1768138841525
c3
LinAlg
The columns of \(A\) are linearly independent.
1
lapses
1/4
users
230%
ease
nid:1768138841525
Cloze c3
Cloze answer: The columns of \(A\) are linearly independent.
Q: Three equivalent statements:{{c1::\(T_A : \mathbb{R}^m \rightarrow \mathbb{R}^m\) is bijective.::Transformation}}{{c2::There is an \(m \times m\) matrix \(B\) such that \(BA = I\).}}{{c3::The columns of \(A\) are linearly independent.}}
A: The third one can be derived from the fact that if \(BA = I\), there is only a single \(x \in \mathbb{R}^m\) such that \(A \textbf{x} = 0\).It is also intuitively clear that if not all columns were linearly independent, we'd actually have a tall linear transformation and would be losing information.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768138841526 |
1 | 230% | 6d | 7 |
nid:1768140101247
c1
LinAlg
\(R = MA\); \(M\) invertible
1
lapses
1/4
users
245%
ease
nid:1768140101247
Cloze c1
Cloze answer: \(R = MA\); \(M\) invertible
Q: For RREF on \(A, I\) we get \(R, M\) with the property that {{c1::\(R = MA\)::equation}} and {{c1::\(M\) invertible:: property of M}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768140101247 |
1 | 245% | 16d | 6 |
nid:1768146369419
c2
LinAlg
The {{c1::set of independent columns of \(A\)}} is {{c2::a b...
1
lapses
1/4
users
245%
ease
nid:1768146369419
Cloze c2
Q: The {{c1::set of independent columns of \(A\)}} is {{c2::a basis of the column space \(\textbf{C}(A)\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768146369420 |
1 | 245% | 22d | 6 |
nid:1768146519592
c1
LinAlg
has a basis \(B \subseteq G\)
1
lapses
1/4
users
230%
ease
nid:1768146519592
Cloze c1
Cloze answer: has a basis \(B \subseteq G\)
Q: Let \(V\) be a finitely generated vector space and let \(G \subseteq V\) be a finite subset with \(\textbf{Span}(G) = V\). Then \(V\) {{c1::has a basis \(B \subseteq G\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768146519592 |
1 | 230% | 6d | 8 |
nid:1768210767870
c1
LinAlg
that is closest to \(b\)
1
lapses
1/4
users
230%
ease
nid:1768210767870
Cloze c1
Cloze answer: that is closest to \(b\)
Q: The projection of a vector \(b \in \mathbb{R}^m\) onto a subspace \(S\) (of \(\mathbb{R}^m\)) is the point in \(S\) {{c1::that is closest to \(b\)}}. In other words \[ \text{proj}_S(b) = {{c1:: \text{argmin}_{p \in S} ||b - p|| }}\]
A: Where \(b = p + e \implies b - p = e\), with \(e\) the error.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768210767870 |
1 | 230% | 1d | 8 |
nid:1768240573172
IO r6
A&D
[Image Occlusion region 6]
1
lapses
1/4
users
230%
ease
nid:1768240573172
Cloze c6
Q: {{c1::image-occlusion:rect:left=.264:top=.1517:width=.4676:height=.1291:oi=1}}{{c2::image-occlusion:rect:left=.264:top=.3156:width=.4709:height=.1018:oi=1}}{{c3::image-occlusion:rect:left=.264:top=.4472:width=.472:height=.1043:oi=1}}{{c4::image-occlusion:rect:left=.2662:top=.5764:width=.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768240573177 |
1 | 230% | 3d | 2 |
nid:1768302182238
LinAlg
What is the pseudoinverse in the case where \(A \in \mathbb{...
1
lapses
1/4
users
245%
ease
nid:1768302182238
Q: What is the pseudoinverse in the case where \(A \in \mathbb{R}^{n \times m}\) has independent rows?
A: Because \(rank(A) = r = m\) and thus \(n \geq m\)\(C(A)\) spans \(\mathbb{R}^m\) (columns span the space)\(R(A) \subseteq\) \(\mathbb{R}^n\)There could be multiple \(x \in \mathbb{R}^n\) that map to \(T_A(x) = b\). We pick the one with the smallest norm \(||x||^2\).We know \(x = x_r + x_n\) for \(x_r \in R(A)\) and \(x_n \in N(A)\) thus we pick \(x = x_r + 0\) to get
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768302182238 |
1 | 245% | 7d | 6 |
nid:1768302385713
c1
LinAlg
For \(A \in \mathbb{R}^{m \times n}\) with \(\text{rank}(A) ...
1
lapses
1/4
users
230%
ease
nid:1768302385713
Cloze c1
Q: For \(A \in \mathbb{R}^{m \times n}\) with \(\text{rank}(A) = m\), we define the pseudo-inverse \(A^\dagger \in \mathbb{R}^{n \times m}\) as:\[ A^\dagger = {{c1::A^\top (A A^\top)^{-1} }}\]
A: For an \(A\) with full column-rank, we basically define, \(A^\dagger\) as the transpose of the pseudoinverse of the transpose:
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768302385713 |
1 | 230% | 3d | 5 |
nid:1768303179258
c1
LinAlg
any full rank (not just CR)
1
lapses
1/4
users
230%
ease
nid:1768303179258
Cloze c1
Cloze answer: any full rank (not just CR)
Q: We can compute the pseudoinverse from the {{c1:: any full rank (not just CR)}} factorisation of \(A\).
A: Note to Lorenz: Leave the "the" in, it's for maximum confusion .
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1768303179258 |
1 | 230% | 2d | 4 |
nid:1769360147747
A&D
Extra memory requirements of Heapsort?
1
lapses
1/4
users
230%
ease
nid:1769360147747
Q: Extra memory requirements of Heapsort?
A: \(O(1)\) as we simply arrange the array into a heap.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1769360147747 |
1 | 230% | 5d | 5 |
nid:1769376963519
c1
EProg
always a subtype of the static type
1
lapses
1/4
users
245%
ease
nid:1769376963519
Cloze c1
Cloze answer: always a subtype of the static type
Q: The dynamic type is {{c1::always a subtype of the static type}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1769376963519 |
1 | 245% | 6d | 6 |
nid:1769377883253
c1
EProg
casting to the static type of the parent
1
lapses
1/4
users
230%
ease
nid:1769377883253
Cloze c1
Cloze answer: casting to the static type of the parent
Q: We can access the parent's attribute of a subclass by {{c1:: casting to the static type of the parent}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1769377883253 |
1 | 230% | 2d | 5 |
nid:1769445714054
A&D
The depth \(h\) of a seach tree of any comparison-based algo...
1
lapses
1/4
users
230%
ease
nid:1769445714054
Q: The depth \(h\) of a seach tree of any comparison-based algorithm satisfies which bound?
A: \(h \geq \Omega(\log n)\) this is information theoretically the least amount of comparisons necessary.Note that \(h \not \leq O(n)\) necessarily as we could have a really stupid algorithm that compares thrice for example.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1769445714055 |
1 | 230% | 2d | 4 |
nid:1769445882673
A&D
Can (g, h) ever be in an MST? Prove it:
1
lapses
1/4
users
230%
ease
nid:1769445882673
Q: Can (g, h) ever be in an MST? Prove it:
A: No, because it's the heaviest edge in the cycle.If there was an MST containing it, we could remove it and replace it by another edge in the cycle.Then we preserve the tree property yet it's weight is strictly lower.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1769445882674 |
1 | 230% | 2d | 4 |
nid:1771363788400
c1
Analysis
eindeutig bestimmte Kenngrössen
1
lapses
1/4
users
260%
ease
nid:1771363788400
Cloze c1
Cloze answer: eindeutig bestimmte Kenngrössen
Q: Maximum und Minimum sind {{c1::eindeutig bestimmte Kenngrössen}} einer Menge, sofern {{c2::sie existieren}}.
A: (Es gibt nur ein Maximum und ein Minimum)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771363788400 |
1 | 260% | 155d | 9 |
nid:1771364277468
c2
PProg
some form of orchestration via threads
1
lapses
1/4
users
245%
ease
nid:1771364277468
Cloze c2
Cloze answer: some form of orchestration via threads
Q: {{c1::Synchronisation}} is {{c2::some form of orchestration via threads}}.
A: Typically used to prevent bad interleavings.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277507 |
1 | 245% | 30d | 4 |
nid:1771364277503
c3
PProg
increasing utilisation of a CPU's functional units
1
lapses
1/4
users
245%
ease
nid:1771364277503
Cloze c3
Cloze answer: increasing utilisation of a CPU's functional units
Q: {{c1::Instruction level parallelism (ILP)}} is {{c2::CPU-internal parallelisation}} of independent instructions, with the goal of improving performance by {{c3::increasing utilisation of a CPU's functional units}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277613 |
1 | 245% | 31d | 4 |
nid:1771364277511
c2
PProg
any resource (memory location, input source, output sink) sh...
1
lapses
1/4
users
230%
ease
nid:1771364277511
Cloze c2
Cloze answer: any resource (memory location, input source, output sink) shared by more than one thread
Q: A {{c1::shared resource}} is {{c2::any resource (memory location, input source, output sink) shared by more than one thread}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277641 |
1 | 230% | 26d | 7 |
nid:1771364277512
c3
PProg
additional management information
1
lapses
1/4
users
230%
ease
nid:1771364277512
Cloze c3
Cloze answer: additional management information
Q: Process context includes:{{c1::CPU state (registers, program counter)}}{{c2::program state (stack, heap, resource handles)}}{{c3::additional management information}}.
A: A thread also has a context, but it is typically much smaller.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277644 |
1 | 230% | 11d | 5 |
nid:1771364277518
c2
PProg
a management process, e.g. on the operating system level, th...
1
lapses
1/4
users
245%
ease
nid:1771364277518
Cloze c2
Cloze answer: a management process, e.g. on the operating system level, that performs context switches
Q: A {{c1::scheduler}} is {{c2::a management process, e.g. on the operating system level, that performs context switches}}.
A: I.e. it interrupts/pauses/sends to sleep the currently running process (or thread), performs a context switch, and selects the next process (or thread) to run.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771364277668 |
1 | 245% | 36d | 6 |
nid:1771366536186
c1
A&W
eine Brücke
1
lapses
1/4
users
230%
ease
nid:1771366536186
Cloze c1
Cloze answer: eine Brücke
Q: Sei \(G = (V, E)\) ein zusammenhängender Graph. Ist \(\{x, y\} \in E\) {{c1::eine Brücke::Eigenschaft?}}, so gilt:
\({{c2::\deg(x) = 1}}\) oder {{c3::\(x\) ist Artikulationsknoten}}.
A: (und analog für \(y\))Aber die Umkehrung gilt nicht!
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771366536187 |
1 | 230% | 8d | 5 |
nid:1771366536198
c1
A&W
\(k\)-kanten-zusammenhängend
1
lapses
1/4
users
260%
ease
nid:1771366536198
Cloze c1
Cloze answer: \(k\)-kanten-zusammenhängend
Q: Ein Graph \(G = (V, E)\) heisst {{c1::\(k\)-kanten-zusammenhängend}}, falls {{c2::für alle Teilmengen \(X \subseteq E\) mit \(|X| < k\) gilt: Der Graph \((V, E \setminus X)\) ist zusammenhängend}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771366536214 |
1 | 260% | 78d | 6 |
nid:1771535790926
c3
A&W
m
1
lapses
1/4
users
260%
ease
nid:1771535790926
Cloze c3
Cloze answer: m
Q: Die um {{c1::die Berechnung von \(low[]\)}} ergänzte {{c2::Tiefensuche}} berechnet in einem zusammenhängenden Graphen alle Artikulationsknoten und Brücken in Zeit \(O({{c3::m}})\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771535790928 |
1 | 260% | 55d | 5 |
nid:1771872607246
DDCA
What is this?
1
lapses
1/4
users
215%
ease
nid:1771872607246
Q: What is this?
A: A NOT gate/inverter.The bubble indicates inversion.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607246 |
1 | 215% | 5d | 4 |
nid:1771872607256
DDCA
How can we build NAND from OR and NOT?
1
lapses
1/4
users
260%
ease
nid:1771872607256
Q: How can we build NAND from OR and NOT?
A: NAND is equivalent to OR with inputs complemented.\(B=\overline{(XY)}=\overline X + \overline Y\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607256 |
1 | 260% | 27d | 7 |
nid:1771872607259
c1
DDCA
0V
1
lapses
1/4
users
290%
ease
nid:1771872607259
Cloze c1
Cloze answer: 0V
Q: On the p-type transistor, the circuit is closed when the gate is supplied with {{c1::0V}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607260 |
1 | 290% | 13d | 14 |
nid:1771872607271
c1
DDCA
"lookup tables" to perform logic functions
1
lapses
1/4
users
245%
ease
nid:1771872607271
Cloze c1
Cloze answer: "lookup tables" to perform logic functions
Q: Multiplexers can be used as {{c1::"lookup tables" to perform logic functions}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607271 |
1 | 245% | 85d | 5 |
nid:1771872607276
DDCA
How can we make an AND gate?
1
lapses
1/4
users
245%
ease
nid:1771872607276
Q: How can we make an AND gate?
A: We make an AND gate using one NAND gate and one NOT gate:
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607276 |
1 | 245% | 31d | 5 |
nid:1771872607296
c1
DDCA
X
1
lapses
1/4
users
230%
ease
nid:1771872607296
Cloze c1
Cloze answer: X
Q: \(X + X \bullet Y = {{c1::X}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607298 |
1 | 230% | 17d | 5 |
nid:1771872607312
c1
DDCA
X \bullet Y
1
lapses
1/4
users
260%
ease
nid:1771872607312
Cloze c1
Cloze answer: X \bullet Y
Q: \((X + \overline{Y}) \bullet Y ={{c1:: X \bullet Y}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607312 |
1 | 260% | 27d | 7 |
nid:1771872607320
c1
DDCA
I; 0; (holes carry charge); 0; I; (electrons carry charge)
1
lapses
1/4
users
260%
ease
nid:1771872607320
Cloze c1
Cloze answer: I; 0; (holes carry charge); 0; I; (electrons carry charge)
Q: MOS transistors are imperfect switches.pMOS transistors pass {{c1::I}}'s well but {{c1::0}}'s poorly {{c1::(holes carry charge)}}.nMOS transistors pass {{c1::0}}'s well but {{c1::I}}'s poorly {{c1::(electrons carry charge)}}.
A: This is why AND is built with NAND + NOT.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607320 |
1 | 260% | 40d | 7 |
nid:1771872607339
c2
PProg
Kernel-level thread: Managed by the OS
1
lapses
1/4
users
245%
ease
nid:1771872607339
Cloze c2
Cloze answer: Kernel-level thread: Managed by the OS
Q: The three levels of threads:{{c1::User-level thread: Managed by the application using a thread library}}{{c2::Kernel-level thread: Managed by the OS}}{{c3::CPU-level thread}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607340 |
1 | 245% | 34d | 6 |
nid:1771872607379
c1
PProg
an actual execution thread
1
lapses
1/4
users
260%
ease
nid:1771872607379
Cloze c1
Cloze answer: an actual execution thread
Q: Each call to start() method of a Thread object creates {{c1::an actual execution thread}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607379 |
1 | 260% | 104d | 5 |
nid:1771872607385
c2
PProg
instruction stream (independent execution units within a pro...
1
lapses
1/4
users
260%
ease
nid:1771872607385
Cloze c2
Cloze answer: instruction stream (independent execution units within a process)
Q: Each thread has its own {{c1::execution stack (method calls, local variables)}} and {{c2::instruction stream (independent execution units within a process)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607386 |
1 | 260% | 13d | 8 |
nid:1771872607479
IO r3
PProg
[Image Occlusion region 3]
1
lapses
1/4
users
230%
ease
nid:1771872607479
Cloze c3
Q: {{c1::image-occlusion:rect:left=.5516:top=.2782:width=.1174:height=.0851:oi=1}}{{c2::image-occlusion:rect:left=.3149:top=.504:width=.1095:height=.0818:oi=1}}{{c2::image-occlusion:rect:left=.2425:top=.7396:width=.2562:height=.0785:oi=1}}{{c3::image-occlusion:rect:left=.7726:top=.504:width
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607479 |
1 | 230% | 2d | 4 |
nid:1771872607479
IO r2
PProg
[Image Occlusion region 2]
1
lapses
1/4
users
230%
ease
nid:1771872607479
Cloze c2
Q: {{c1::image-occlusion:rect:left=.5516:top=.2782:width=.1174:height=.0851:oi=1}}{{c2::image-occlusion:rect:left=.3149:top=.504:width=.1095:height=.0818:oi=1}}{{c2::image-occlusion:rect:left=.2425:top=.7396:width=.2562:height=.0785:oi=1}}{{c3::image-occlusion:rect:left=.7726:top=.504:width
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771872607481 |
1 | 230% | 2d | 4 |
nid:1771969055150
c2
Analysis
\(A \neq \emptyset\), \(B \neq \emptyset\); \(\forall a \i...
1
lapses
1/4
users
260%
ease
nid:1771969055150
Cloze c2
Cloze answer: \(A \neq \emptyset\), \(B \neq \emptyset\); \(\forall a \in A \ \forall b \in B \ : \ a \leq b\)
Q: Ordnungsvollständigkeit:Seien \(A, B \subseteq \mathbb{R}\), sodass
{{c2:: \(A \neq \emptyset\), \(B \neq \emptyset\)}}
{{c2:: \(\forall a \in A \ \forall b \in B \ : \ a \leq b\)}}
Dann {{c1:: gibt es ein \(c \in \mathbb{R}\), sodass \[ \foral
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771969055150 |
1 | 260% | 153d | 7 |
nid:1771969257001
c1
Analysis
2|xy|
1
lapses
1/4
users
275%
ease
nid:1771969257001
Cloze c1
Cloze answer: 2|xy|
Q: Youngsche UngleichungFür jedes \(x, y \in \mathbb{R}\), \(\epsilon > 0\) gilt: \[ {{c1:: 2|xy| }} \leq {{c2:: \epsilon x^2 + \frac{1}{\epsilon} y^2 }}\]Proof Included
A: Proof: Setze \(\gamma = \sqrt{\epsilon} > 0\). OBDA gelte \(x \cdot y \geq 0\). \[ 0 \leq (\gamma x - \frac{y}{\gamma})^2 = \gamma^2 x^2 - 2x\cdot y + \frac{1}{\gamma^2}y^2 \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771969257001 |
1 | 275% | 140d | 9 |
nid:1771969381133
Analysis
Dreiecksungleichung (Vektoren)
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users
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nid:1771969381133
Q: Dreiecksungleichung (Vektoren)
A: Für alle \(x, y, z \in \mathbb{R}^n\) gilt: \[ ||x - z|| \leq ||x - y|| + ||y - z|| \]wo \(||x||\) die euklidische Norm von \(x\) ist.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771969381133 |
1 | 230% | 101d | 7 |
nid:1771969600985
c1
Analysis
|z|^2
1
lapses
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users
245%
ease
nid:1771969600985
Cloze c1
Cloze answer: |z|^2
Q: Für \(z \in \mathbb{C}\) gilt: \(z \cdot \bar{z} = {{c1:: |z|^2 }}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771969600985 |
1 | 245% | 109d | 5 |
nid:1771969965872
c1
Analysis
\(r = |z| \ge 0\) und \(\varphi \in (-\pi, \pi]\) der Polar...
1
lapses
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users
275%
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nid:1771969965872
Cloze c1
Cloze answer: \(r = |z| \ge 0\) und \(\varphi \in (-\pi, \pi]\) der Polarwinkel \(\arg(z)\) (Argument) ist
Q: In der Polarform wird \(z = a + ib\) als {{c1:: \(r \cdot e^{i \varphi}\)}} dargestellt wo {{c1:: \(r = |z| \ge 0\) und \(\varphi \in (-\pi, \pi]\) der Polarwinkel \(\arg(z)\) (Argument) ist::Def. r und Winkel}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1771969965872 |
1 | 275% | 45d | 8 |
nid:1772209100380
IO r1
A&W
[Image Occlusion region 1]
1
lapses
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users
260%
ease
nid:1772209100380
Cloze c1
Q: {{c3::image-occlusion:rect:left=.1591:top=.8923:width=.7185:height=.0742}}{{c2::image-occlusion:rect:left=.3252:top=.7428:width=.5272:height=.0923}}{{c1::image-occlusion:rect:left=.0549:top=.1782:width=.9041:height=.1203}}{{c4::image-occlusion:rect:left=.1645:top=.4824:width=.1234:height
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772209100382 |
1 | 260% | 41d | 7 |
nid:1772209100471
c1
DDCA
Cheap (one bit costs only one transistor plus one capacitor)
1
lapses
1/4
users
260%
ease
nid:1772209100471
Cloze c1
Cloze answer: Cheap (one bit costs only one transistor plus one capacitor)
Q: Pros and cons of Dynamic RAM (DRAM){{c1::Cheap (one bit costs only one transistor plus one capacitor)}}{{c2::Slower, reading destroys content (refresh), needs special process for manufacturing}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772209100472 |
1 | 260% | 29d | 7 |
nid:1772209100485
IO r2
DDCA
[Image Occlusion region 2]
1
lapses
1/4
users
245%
ease
nid:1772209100485
Cloze c2
Q: {{c1::image-occlusion:rect:left=.0089:top=.5477:width=.2313:height=.092}}{{c2::image-occlusion:rect:left=.0059:top=.8424:width=.7046:height=.1459}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772209100485 |
1 | 245% | 22d | 5 |
nid:1772209100529
DDCA
How do we determine the number of OR gates in a PLA?
1
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1/4
users
245%
ease
nid:1772209100529
Q: How do we determine the number of OR gates in a PLA?
A: The number of output columns in the truth table.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772209100529 |
1 | 245% | 6d | 5 |
nid:1772209100544
DDCA
What is the Uniting Theorem?
1
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users
230%
ease
nid:1772209100544
Q: What is the Uniting Theorem?
A: \(F=A\overline B+AB\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772209100545 |
1 | 230% | 2d | 4 |
nid:1772209100565
c1
DDCA
Q (inverse at Q')
1
lapses
1/4
users
230%
ease
nid:1772209100565
Cloze c1
Cloze answer: Q (inverse at Q')
Q: R-S LatchData is stored at {{c1::Q (inverse at Q')}}S and R are {{c2::control inputs}}
In quiescent (idle) state, {{c3::both S and R are held at 1}}S (set): {{c4::drive S to 0 (keeping R at 1) to change Q to 1}}R (reset): {{c4::drive R to 0 (k
A: S and R should not both be 0 at the same time.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772209100567 |
1 | 230% | 2d | 4 |
nid:1772569386183
c1
A&W
einen augmentierenden Pfad
1
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users
260%
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nid:1772569386183
Cloze c1
Cloze answer: einen augmentierenden Pfad
Q: Jedes Matching, das nicht {{c2::(kardinalitäts-)maximal}} ist, besitzt {{c1::einen augmentierenden Pfad}}.Theorem name included
A: (Berge, 1957)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386183 |
1 | 260% | 21d | 8 |
nid:1772569386183
c2
A&W
(kardinalitäts-)maximal
1
lapses
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users
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nid:1772569386183
Cloze c2
Cloze answer: (kardinalitäts-)maximal
Q: Jedes Matching, das nicht {{c2::(kardinalitäts-)maximal}} ist, besitzt {{c1::einen augmentierenden Pfad}}.Theorem name included
A: (Berge, 1957)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386184 |
1 | 260% | 21d | 6 |
nid:1772569386187
A&W
Wie funktioniert der Algorithmus um ein maximales Matching z...
1
lapses
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users
275%
ease
nid:1772569386187
Q: Wie funktioniert der Algorithmus um ein maximales Matching zu finden?
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386187 |
1 | 275% | 14d | 9 |
nid:1772569386190
c2
A&W
Ein Matching \( M \) heisst {{c1::perfektes Matching}}, wenn...
1
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users
245%
ease
nid:1772569386190
Cloze c2
Q: Ein Matching \( M \) heisst {{c1::perfektes Matching}}, wenn {{c2::jeder Knoten durch genau eine Kante aus \( M \) überdeckt wird, oder, anders ausgedrückt, wenn \( |M| = \frac{|V|}{2}\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386191 |
1 | 245% | 48d | 4 |
nid:1772569386198
c1
A&W
Jede Kante in \( M_{\text{Greedy}} \) kann höchstens {{c1::z...
1
lapses
1/4
users
245%
ease
nid:1772569386198
Cloze c1
Q: Jede Kante in \( M_{\text{Greedy}} \) kann höchstens {{c1::zwei Kanten aus \( M_{\text{max} } \)}} überdecken.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386198 |
1 | 245% | 13d | 8 |
nid:1772569386201
A&W
Inklusionsmaximal? Kardinalitätsmaximal?
1
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users
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ease
nid:1772569386201
Q: Inklusionsmaximal? Kardinalitätsmaximal?
A: Sowohl als auch.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386201 |
1 | 245% | 39d | 4 |
nid:1772569386222
c1
A&W
|V| \cdot |E|
1
lapses
1/4
users
230%
ease
nid:1772569386222
Cloze c1
Cloze answer: |V| \cdot |E|
Q: In bipartiten Graphen kann man in Zeit \( O({{c1::|V| \cdot |E|}}) \) ein perfektes Matching bestimmen. Ist dies optimal?
A: Note, es geht mit Hopcroft-Karp in \(O(\sqrt{|V|} \cdot |E|)\) schneller.Augmentierende-Pfade-AlgorithmusMan startet mit einem beliebigen Matching und sucht iterativ \(M\)-augmentierende PfadeDiese baut schichtweise einen Layer-Graphen auf: \(L_0\) sind die unüberdeckten Knoten in \(A\), ungerade Schichten erreicht man über Kanten in \(E \setminus M\), gerade über Kanten in \(M\). Findet man einen unüberdeckten Knoten in \(B\), liefert
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386222 |
1 | 230% | 5d | 11 |
nid:1772569386236
c1
A&W
|E|
1
lapses
1/4
users
230%
ease
nid:1772569386236
Cloze c1
Cloze answer: |E|
Q: In \( 2^k \)-regulären bipartiten Graphen kann man in Zeit \( O({{c1::|E|}}) \) ein perfektes Matching bestimmen.
A: Satz 1.54 - Eulertour-basierter Algorithmus\(2^k\)-regulärer bipartiter Graph ist eulersch (alle Knoten haben geraden Grad).In jeder Zusammenhangskomponente berechnet man eine Eulertour in \(O(|E|)\)Dann läuft man diese ab und entfernt jede zweite Kante. Der verbleibende Graph ist \(2^{k-1}\)-regulär. Nach \(k\) Iterationen ist der Graph \(2^0 = 1\)-regulär, also selbst ein perfektes Matching. Die Gesamtl
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772569386236 |
1 | 230% | 1d | 5 |
nid:1772569386178
c2
A&W
es kein Matching \( M' \) gibt mit \( M \subseteq M' \) und ...
1
lapses
1/4
users
245%
ease
nid:1772569386178
Cloze c2
Cloze answer: es kein Matching \( M' \) gibt mit \( M \subseteq M' \) und \( |M'| > |M| \)
Q: Ein Matching \( M \subseteq E \) ist ein {{c1::inklusionsmaximales Matching}}, wenn {{c2::es kein Matching \( M' \) gibt mit \( M \subseteq M' \) und \( |M'| > |M| \)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772570517431 |
1 | 245% | 43d | 5 |
nid:1772698768089
c1
Analysis
einen Häufungspunkt, der mit dem Grenzwert übereinstimmt
1
lapses
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users
290%
ease
nid:1772698768089
Cloze c1
Cloze answer: einen Häufungspunkt, der mit dem Grenzwert übereinstimmt
Q: Jede konvergente Folge hat genau {{c1:: einen Häufungspunkt, der mit dem Grenzwert übereinstimmt}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772698768090 |
1 | 290% | 129d | 12 |
nid:1772783275475
A&W
Wahr oder falsch?Für zwei Knoten \( a, b \) eines Graphen se...
1
lapses
1/4
users
245%
ease
nid:1772783275475
Q: Wahr oder falsch?Für zwei Knoten \( a, b \) eines Graphen sei \( a \sim b \) genau dann, wenn \( a = b \) gilt oder wenn \( a \) und \( b \) auf einem gemeinsamen Kreis liegen. Dann ist \( \sim \) eine Äquivalenzrelation.
A: Falsch.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772783275475 |
1 | 245% | 20d | 4 |
nid:1772783275526
c1
Analysis
0
1
lapses
1/4
users
260%
ease
nid:1772783275526
Cloze c1
Cloze answer: 0
Q: \(\forall x \in \mathbb{R}: \lim_{n\to\infty} \frac{x^n}{n!} ={{c1::0}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772783275527 |
1 | 260% | 128d | 9 |
nid:1772783275528
c1
Analysis
1
1
lapses
1/4
users
275%
ease
nid:1772783275528
Cloze c1
Cloze answer: 1
Q: \(\lim_{n\to\infty} n^{1/n} ={{c1::1}}\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772783275528 |
1 | 275% | 154d | 9 |
nid:1772788241820
c1
Analysis
\[ \sin\!\left(\frac{2\pi}{3}\right) = {{c1::\frac{\sqrt{3} ...
1
lapses
1/4
users
260%
ease
nid:1772788241820
Cloze c1
Q: \[ \sin\!\left(\frac{2\pi}{3}\right) = {{c1::\frac{\sqrt{3} }{2} }} \]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772788241821 |
1 | 260% | 47d | 6 |
nid:1772885493204
c1
Analysis
\sin x \cos y \pm \cos x \sin y
1
lapses
1/4
users
260%
ease
nid:1772885493204
Cloze c1
Cloze answer: \sin x \cos y \pm \cos x \sin y
Q: \[sin(x \pm y) = {{c1:: \sin x \cos y \pm \cos x \sin y }}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1772885493204 |
1 | 260% | 86d | 7 |
nid:1773134608434
c1
Analysis
eindeutigen
1
lapses
1/4
users
245%
ease
nid:1773134608434
Cloze c1
Cloze answer: eindeutigen
Q: Eine konvergente Folge besitzt genau einen {{c1::eindeutigen}} Grenzwert.Proof Included
A: Proof For contradiction, assume there are \(A, B\) limits.Then there exists \(N_A \in \mathbb{N}\) such that \(\forall n > N_A \ : \ |a_n - A| < \frac{\epsilon}{2}\) There must also be \(N_B\) such that \(\forall n > N_B \ : \ |a_n - B| < \frac{\epsilon}{2}\)But then for \(N := \max \{N_A, N_B\}\) it holds that \(n > N\) \(|A - B| \le |A - a_n| + |a_n -B| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon\). As this hold
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1773134608434 |
1 | 245% | 4d | 5 |
nid:1773420068085
c1
A&W
3/2-Approximation Metrisches TSP
Bestimme minimalen Spannb...
1
lapses
1/4
users
245%
ease
nid:1773420068085
Cloze c1
Q: 3/2-Approximation Metrisches TSP
Bestimme minimalen Spannbaum \(T\)es gilt: \( \ell(T) \leq \text{opt}(K_n, \ell) \)
' \(X:=\) Knoten mit ungeradem Grad in \(T\)Bestimme minimales Matching \(M\) für \(X\) es gilt: \
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1773420068085 |
1 | 245% | 10d | 8 |
nid:1773420068117
A&W
Wahr oder falsch?Jeder Graph ohne Dreieck hat eine chromatis...
1
lapses
1/4
users
245%
ease
nid:1773420068117
Q: Wahr oder falsch?Jeder Graph ohne Dreieck hat eine chromatische Zahl von höchstens 100.
A: FalschSiehe Mycielski-Konstruktion.Konstruktion:Aus \(G_k = (V_k, E_k)\) mit \(V_k = \{v_1,\ldots,v_n\}\) bilde \(G_{k+1}\):Füge Knoten \(w_1,\ldots,w_n, z\) hinzu. \(w_i\) ist mit allen Nachbarn von \(v_i\) verbunden (aber nicht mit \(v_i\) selbst). \(z\) ist mit allen \(w_i\) verbunden.Der neue Graph ist dreiecksfrei und braucht eine Farbe mehr als \(G_k\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1773420068117 |
1 | 245% | 3d | 8 |
nid:1773420068133
c1
A&W
k+1
1
lapses
1/4
users
230%
ease
nid:1773420068133
Cloze c1
Cloze answer: k+1
Q: Gilt für die (gewählte) Reihenfolge \(|N(v_i) \cap \{v_1, \ldots, v_{i-1}\}| \leq k\) \(\forall\, 2 \leq i \leq n\), dann benötigt der Greedy-Algorithmus höchstens \({{c1::k+1}}\) viele Farben.
A: Heuristik:\(v_n\) := Knoten vom kleinsten Grad. Lösche \(v_n\).\(v_{n-1}\) := Knoten vom kleinsten Grad im Restgraph. Lösche \(v_{n-1}\). Iteriere.Falls \(G=(V,E)\) erfüllt:In jedem Subgraphen gibt es einen Knoten mit Grad \(\leq k\)\(\Rightarrow\) Heuristik liefert Reihenfolge \(v_1,\ldots,v_n\) für die der Greedy-Algorithmus höchstens \(k+1\) Farben benötigt
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1773420068136 |
1 | 230% | 2d | 7 |
nid:1773420068135
c1
A&W
3/2-Approximation Metrisches TSP
Bestimme minimalen Spannb...
1
lapses
1/4
users
245%
ease
nid:1773420068135
Cloze c1
Q: 3/2-Approximation Metrisches TSP
Bestimme minimalen Spannbaum \(T\)es gilt: \( \ell(T) \leq \text{opt}(K_n, \ell) \)
' {{c1::\(X:=\) Knoten mit ungeradem Grad in \(T\)Bestimme minimales Matching \(M\) für \(X\) es gilt:&
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1773420068138 |
1 | 245% | 23d | 6 |
nid:1773773841684
c1
A&W
|V| + |E|
1
lapses
1/4
users
230%
ease
nid:1773773841684
Cloze c1
Cloze answer: |V| + |E|
Q: Einen 3-färbbaren Graphen kann man in Zeit \(O({{c1::|V| + |E|}})\) mit \(O({{c2::\sqrt{|V|} }})\) Farben färben.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1773773841685 |
1 | 230% | 1d | 5 |
nid:1774005500952
c1
Analysis
Für eine {{c1:: monotone Folge reeller Zahlen \((a_n)_{n \in...
1
lapses
1/4
users
230%
ease
nid:1774005500952
Cloze c1
Q: Für eine {{c1:: monotone Folge reeller Zahlen \((a_n)_{n \in \mathbb{N}_0}\)}} gilt: Sie konvergiert genau dann, wenn {{c2::sie beschränkt ist}}.
A: (Weierstrass)Falls die Folge monoton wachsend ist, gilt: \[ \lim_{n \rightarrow \infty} a_n = \sup \{a_n \mid n \in \mathbb{N}_0\} \]Falls die Folge monoton fallend ist, gilt:\[\lim_{n \rightarrow \infty} a_n = \inf \{ a_n \mid n \in \mathbb{N}_0\}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1774005500953 |
1 | 230% | 17d | 12 |
nid:1774005965819
c2
Analysis
\((a_n)_{n \in \mathbb{N}_0}\) {{c1::eine konvergente Folge:...
1
lapses
1/4
users
215%
ease
nid:1774005965819
Cloze c2
Q: \((a_n)_{n \in \mathbb{N}_0}\) {{c1::eine konvergente Folge::Property}} \(\Longleftrightarrow\) \[ \lim_{n \rightarrow \infty} a_n = {{c2:: \limsup_{n \rightarrow \infty} a_n = \liminf_{n \rightarrow \infty} a_n }}\]
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1774005965819 |
1 | 215% | 18d | 6 |
nid:1774006045853
c2
Analysis
für unendlich viele Elemente \(A - \epsilon < a_n < A + \eps...
1
lapses
1/4
users
215%
ease
nid:1774006045853
Cloze c2
Cloze answer: für unendlich viele Elemente \(A - \epsilon < a_n < A + \epsilon\) gilt.
Q: Sei \((a_n)_{n \in \mathbb{N}_0}\) eine beschränkte Folge mit \(A = \limsup_{n \rightarrow \infty} a_n\). Dann ist \(A\) ein Häufungspunkt und für alle \(\epsilon > 0\) gilt, dass:{{c1::es nur endlich viele Elemente \(a_n\) gibt, für welche \(a_n \ge A + \e
A: Eine analoge Aussage gilt auch für den Limes inferior.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1774006045855 |
1 | 215% | 19d | 6 |
nid:1774006491519
c1
Analysis
beschränkt
1
lapses
1/4
users
260%
ease
nid:1774006491519
Cloze c1
Cloze answer: beschränkt
Q: Jede Cauchy-Folge ist {{c1:: beschränkt}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1774006491519 |
1 | 260% | 48d | 7 |
nid:1774474839885
c1
Analysis
Sei \(\sum a_n\) {{c1::bedingt konvergent und \(L \in \mathb...
1
lapses
1/4
users
230%
ease
nid:1774474839885
Cloze c1
Q: Sei \(\sum a_n\) {{c1::bedingt konvergent und \(L \in \mathbb{R} \cup \{+\infty, -\infty\}\)}}.Dann {{c2::gibt es eine Bijektion \(\phi\), so dass:\[\sum_{n=0}^\infty a_{\phi(n)} = L\]}}
A: (Riemannscher Umordnungssatz)Merke: Bedingt konvergente Reihen können durch Umordnung jeden Grenzwert annehmen!
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1774474839895 |
1 | 230% | 1d | 5 |
nid:1774474839891
c2
Analysis
die Koeffizienten
1
lapses
1/4
users
230%
ease
nid:1774474839891
Cloze c2
Cloze answer: die Koeffizienten
Q: Eine Potenzreihe hat die Form \({{c5:: \displaystyle\sum_{k=0}^\infty c_k (x - a)^k }}\), wobei:\(a\) ist {{c1::der Entwicklungspunkt (Zentrum)}}\(c_0, c_1, \ldots\) sind {{c2::die Koeffizienten}}\(x\) ist {{c3::das Argument}}\((a - R,\, a + R)\) ist {{c
A: Spezialfall \(a = 0\): \(\sum c_k x^k\) - Entwicklungspunkt im Ursprung.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| niklas | cid:1774474839909 |
1 | 230% | 3d | 4 |
nid:1765551644290
c1
LinAlg
The span of m linearly independent vectors is {{c1::\(\mathb...
1
lapses
1/4
users
245%
ease
nid:1765551644290
Cloze c1
Q: The span of m linearly independent vectors is {{c1::\(\mathbb{R}^m\)}}.
A: This also means that a matrix in \(\mathbb{R}^{n \times n}\) with rank(A) = n spans the entire space.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551644290 |
1 | 245% | 14d | 4 |
nid:1765551666570
c2
A&D
incident
1
lapses
1/4
users
245%
ease
nid:1765551666570
Cloze c2
Cloze answer: incident
Q: The {{c1::degree (Knotengrad) \(\deg(v)\)}} of a vertex \(v\) is the number of edges that are {{c2::incident}} to \(v\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551666580 |
1 | 245% | 25d | 6 |
nid:1765551666576
A&D
Cycle
1
lapses
1/4
users
245%
ease
nid:1765551666576
Q: Cycle
A: Kreis
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551666591 |
1 | 245% | 65d | 6 |
nid:1765551666578
A&D
What is the length of a walk?
1
lapses
1/4
users
245%
ease
nid:1765551666578
Q: What is the length of a walk?
A: The length of a walk \((v_0, v_1, \dots, v_k)\) is \(k\), i.e. the number of vertices minus 1.A walk of length \(l\) connects \(l + 1\) vertices.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551666594 |
1 | 245% | 48d | 6 |
nid:1765551666580
c2
A&D
for every two vertices \(u, v \in V\) \(u\) reaches \(v\)
1
lapses
1/4
users
230%
ease
nid:1765551666580
Cloze c2
Cloze answer: for every two vertices \(u, v \in V\) \(u\) reaches \(v\)
Q: A graph \(G\) is {{c1::connected (Zusammenhängend)}} if {{c2::for every two vertices \(u, v \in V\) \(u\) reaches \(v\)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551666598 |
1 | 230% | 93d | 7 |
nid:1765551666585
c1
A&D
direct predecessor (Vorgänger); direct successor (Nachfolger
1
lapses
1/4
users
230%
ease
nid:1765551666585
Cloze c1
Cloze answer: direct predecessor (Vorgänger); direct successor (Nachfolger
Q: In a directed graph, for the edge \(e = (u, v)\), \(u\) is the {{c1::direct predecessor (Vorgänger)}} of \(v\) and \(v\) the {{c1::direct successor (Nachfolger}} of \(u\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551666606 |
1 | 230% | 17d | 8 |
nid:1765551666588
c1
A&D
The {{c1::out-degree \(\deg_{\text{out} }(v)\) (Ausgangsgrad...
1
lapses
1/4
users
230%
ease
nid:1765551666588
Cloze c1
Q: The {{c1::out-degree \(\deg_{\text{out} }(v)\) (Ausgangsgrad)}} of a vertex in a directed graph is the {{c2::number of edges that have \(v\) as the start-vertex}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551666610 |
1 | 230% | 27d | 5 |
nid:1765551666614
c1
A&D
shortest length of a walk from \(u\) to \(v\)
1
lapses
1/4
users
230%
ease
nid:1765551666614
Cloze c1
Cloze answer: shortest length of a walk from \(u\) to \(v\)
Q: The distance \(d(u, v)\) in a directed graph is defined as {{c1:: shortest length of a walk from \(u\) to \(v\)}}.
A: Keep in mind in a weighted graph, this might mean the cheapest, which refers to cost not length.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551666649 |
1 | 230% | 46d | 9 |
nid:1765551666619
c1
A&D
the enter order equals the leave order
1
lapses
1/4
users
230%
ease
nid:1765551666619
Cloze c1
Cloze answer: the enter order equals the leave order
Q: In BFS enter/leave ordering, the FIFO queue guarantees that {{c1:: the enter order equals the leave order}} within a given level.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1765551666654 |
1 | 230% | 57d | 9 |
nid:1765551656956
c2
DiskMat
Well-definedness: \(\forall a \in A \ \forall b, b' \in B :...
1
lapses
1/4
users
230%
ease
nid:1765551656956
Cloze c2
Cloze answer: Well-definedness: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)
Q: What two properties must a relation \(f: A \to B\) have to be a function?{{c1:: Total-definedness: \(\forall a \in A \ \exists b \in B : a \ f \ b\) }}{{c2:: Well-definedness: \(\forall a \in A \ \forall b, b' \in B : (a
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766410039197 |
1 | 230% | 1d | 4 |
nid:1766501315026
A&D
Find Closed Eulerian Path
1
lapses
1/4
users
230%
ease
nid:1766501315026
Q: Find Closed Eulerian Path
A: \(O(n+m)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766501315056 |
1 | 230% | 4d | 5 |
nid:1766501315033
c2
A&D
\(\lnot \exists\) directed closed walk
1
lapses
1/4
users
230%
ease
nid:1766501315033
Cloze c2
Cloze answer: \(\lnot \exists\) directed closed walk
Q: {{c1:: \(\exists\) toposort}} \(\Longleftrightarrow\) {{c2:: \(\lnot \exists\) directed closed walk}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766501315063 |
1 | 230% | 19d | 6 |
nid:1766501315038
c1
A&D
Cross edge, \(u, v\) in different subtrees
1
lapses
1/4
users
230%
ease
nid:1766501315038
Cloze c1
Cloze answer: Cross edge, \(u, v\) in different subtrees
Q: Pre-/Post-Ordering Classification for an edge \((u, v)\):\(\text{pre}(v) < \text{post}(v) < \text{pre}(u) < \text{post}(u)\): {{c1:: Cross edge, \(u, v\) in different subtrees}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766501315070 |
1 | 230% | 19d | 8 |
nid:1766576733264
c1
A&D
Prim's Algorithm Invariants:\(\forall v \not \in S, v \neq s...
1
lapses
1/4
users
230%
ease
nid:1766576733264
Cloze c1
Q: Prim's Algorithm Invariants:\(\forall v \not \in S, v \neq s\), \(d[v] = \) {{c1:: \(\min \{ w(u, v) \ | \ (u, v) \in E, u \in S \}\)(\(\infty\) if no such edge exists)}}.
A: The 3rd invariant \[d[v] = \begin{cases} 0, & \text{if } v = s \text{ (the starting vertex)} \\ \min_{(u,v) \in E : u \in S} {w(u,v)}, & \text{if } v \in V \setminus S \text{ and } \exists (u,v) \in E \text{ with } u \in S \\ \infty, & \text{if } v \in V \setminus S \text{ and } \nexists (u,v) \in E \text{ with } u \in S \end{cases}\]ensures that d[v] always reflects the minimum cost to reach vertex v from the current MST. We always want to add the
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766576733264 |
1 | 230% | 37d | 12 |
nid:1766576733286
c2
A&D
same(u,v) test if \(u, v\) in the same component
1
lapses
1/4
users
230%
ease
nid:1766576733286
Cloze c2
Cloze answer: same(u,v) test if \(u, v\) in the same component
Q: Union-Find datastructure methods:{{c1::make(u, v) creates the DS for \(F = \emptyset\)}}{{c2::same(u,v) test if \(u, v\) in the same component}}{{c3::union(u,v) merge ZHKs of \(u, v\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766576733293 |
1 | 230% | 6d | 7 |
nid:1766576733289
A&D
Floyd-Warshall
1
lapses
1/4
users
230%
ease
nid:1766576733289
Q: Floyd-Warshall
A: \(O(|V|^3)\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766576733296 |
1 | 230% | 8d | 7 |
nid:1766576739753
A&D
Floyd-Warshall, when is there a negative cycle?
1
lapses
1/4
users
230%
ease
nid:1766576739753
Q: Floyd-Warshall, when is there a negative cycle?
A: There exists a negative cycle \(\Leftrightarrow \exists v \in V \ : \ d^n_{v \rightarrow v} < 0\) In words: If there exists a path from a vertex to itself with negative weight (passing through any other vertex, i.e. \(n\)th iteration of the outer loop), then there exists a negative cycle that contains this vertex.We can perform a negative cycle check at the end, by going over all diagonals.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766576739753 |
1 | 230% | 16d | 8 |
nid:1766656891070
c1
A&D
a cycle; undirected
1
lapses
1/4
users
230%
ease
nid:1766656891070
Cloze c1
Cloze answer: a cycle; undirected
Q: A graph with more than \(n-1\) edges has {{c1::a cycle}} if it is {{c1::undirected}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1766656891070 |
1 | 230% | 17d | 8 |
nid:1767089638548
c4
LinAlg
\(v \cdot v \geq 0\) with equality if and only if \(v = 0\)...
1
lapses
1/4
users
230%
ease
nid:1767089638548
Cloze c4
Cloze answer: \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness
Q: Scalar product properties: \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar.{{c1::\(v \cdot w = w \cdot v\) (symmetry / commutatitivity}}{{c2:: \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1767089638552 |
1 | 230% | 1d | 3 |
nid:1771363954967
c1
PProg
Amdahl's Law
1
lapses
1/4
users
230%
ease
nid:1771363954967
Cloze c1
Cloze answer: Amdahl's Law
Q: {{c1::Amdahl's Law}} specifies {{c2::the maximum amount of speedup that can be achieved for a program with a given sequential part.}}
A: The pessimistic view on scalability.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363954970 |
1 | 230% | 11d | 9 |
nid:1771363954970
c1
PProg
Cache coherence protocols
1
lapses
1/4
users
230%
ease
nid:1771363954970
Cloze c1
Cloze answer: Cache coherence protocols
Q: {{c1::Cache coherence protocols}} are hardware protocols that {{c2::ensure consistency across caches}}, typically by {{c3::tracking which locations are cached, and synchronising them if necessary}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363954981 |
1 | 230% | 29d | 10 |
nid:1771363954971
c2
PProg
execute code and spawn new tasks if required
1
lapses
1/4
users
230%
ease
nid:1771363954971
Cloze c2
Cloze answer: execute code and spawn new tasks if required
Q: {{c1::Cilk-style programming}} is a parallel programming idiom: To compute a program, {{c2::execute code and spawn new tasks if required}}. Before returning, {{c3::wait for all spawned tasks to complete}}.
A: The system manages the eventual execution of the spawned tasks potentially in parallel.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363954985 |
1 | 230% | 17d | 7 |
nid:1771363954971
c1
PProg
Cilk-style programming
1
lapses
1/4
users
230%
ease
nid:1771363954971
Cloze c1
Cloze answer: Cilk-style programming
Q: {{c1::Cilk-style programming}} is a parallel programming idiom: To compute a program, {{c2::execute code and spawn new tasks if required}}. Before returning, {{c3::wait for all spawned tasks to complete}}.
A: The system manages the eventual execution of the spawned tasks potentially in parallel.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363954986 |
1 | 230% | 62d | 11 |
nid:1771363954976
c2
PProg
resources required to set up an operation
1
lapses
1/4
users
230%
ease
nid:1771363954976
Cloze c2
Cloze answer: resources required to set up an operation
Q: {{c1::Context switch overhead}} refers to {{c2::resources required to set up an operation}}.
A: In terms of context switch, CPU needs to store/save the local data, program pointer etc. of the current thread/process, and load the local data, program pointer etc. of the next thread/process to execute.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955002 |
1 | 230% | 10d | 11 |
nid:1771363954980
c2
PProg
recursively solving smaller sub-problems and combining their...
1
lapses
1/4
users
230%
ease
nid:1771363954980
Cloze c2
Cloze answer: recursively solving smaller sub-problems and combining their results
Q: {{c1::Divide and conquer style parallelism (also called recursive splitting)}} means: solve a problem by {{c2::recursively solving smaller sub-problems and combining their results}}.
A: Solve the sub-problems in separate threads to gain a speedup.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955014 |
1 | 230% | 21d | 8 |
nid:1771363954981
c1
PProg
Deadlock
1
lapses
1/4
users
230%
ease
nid:1771363954981
Cloze c1
Cloze answer: Deadlock
Q: {{c1::Deadlock}} is {{c2::circular waiting/blocking (no instructions are executed/CPU time is used) between threads, so that the system (union of all threads) cannot make any progress anymore}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955016 |
1 | 230% | 33d | 8 |
nid:1771363954983
c1
PProg
divide and conquer parallelism
1
lapses
1/4
users
230%
ease
nid:1771363954983
Cloze c1
Cloze answer: divide and conquer parallelism
Q: The ForkJoin framework embraces {{c1::divide and conquer parallelism}}.
A: Tasks can be spawned (forked) and joined by the framework.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955027 |
1 | 230% | 44d | 7 |
nid:1771363954984
c1
PProg
functional unit
1
lapses
1/4
users
230%
ease
nid:1771363954984
Cloze c1
Cloze answer: functional unit
Q: A {{c1::functional unit}} is a component of a CPU (or core) that {{c2::performs a certain task}}, an {{c3::execution unit}} is one such example.
A: performing a task - e.g. executing integer arithmetic operations
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955028 |
1 | 230% | 10d | 8 |
nid:1771363954984
c2
PProg
performs a certain task
1
lapses
1/4
users
230%
ease
nid:1771363954984
Cloze c2
Cloze answer: performs a certain task
Q: A {{c1::functional unit}} is a component of a CPU (or core) that {{c2::performs a certain task}}, an {{c3::execution unit}} is one such example.
A: performing a task - e.g. executing integer arithmetic operations
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955029 |
1 | 230% | 18d | 7 |
nid:1771363954987
c1
PProg
granularity
1
lapses
1/4
users
230%
ease
nid:1771363954987
Cloze c1
Cloze answer: granularity
Q: The trick with {{c1::granularity}} is to find a size that {{c2::minimizes overhead}} while {{c3::maximizing parallelism}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955040 |
1 | 230% | 10d | 8 |
nid:1771363954993
c2
PProg
a property of a system: "something good eventually happens"
1
lapses
1/4
users
230%
ease
nid:1771363954993
Cloze c2
Cloze answer: a property of a system: "something good eventually happens"
Q: A {{c1::liveness property}} is {{c2::a property of a system: "something good eventually happens"}}.
A: Can only be violated in infinite time. Infinite loops and starvation are typical liveness properties.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955058 |
1 | 230% | 32d | 6 |
nid:1771363955001
c2
PProg
The maximum possible speedup ({{c1::parallelism}}) is {{c2::...
1
lapses
1/4
users
230%
ease
nid:1771363955001
Cloze c2
Q: The maximum possible speedup ({{c1::parallelism}}) is {{c2::\(\frac{T_1}{T_\infty} \)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955092 |
1 | 230% | 5d | 5 |
nid:1771363955014
c2
PProg
extra time spent by the system or the algorithm
1
lapses
1/4
users
230%
ease
nid:1771363955014
Cloze c2
Cloze answer: extra time spent by the system or the algorithm
Q: {{c1::Scheduling overhead}} is the {{c2::extra time spent by the system or the algorithm}} to distribute work on {{c3::multiple threads/tasks}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955147 |
1 | 230% | 25d | 6 |
nid:1771363955022
c1
PProg
Span
1
lapses
1/4
users
230%
ease
nid:1771363955022
Cloze c1
Cloze answer: Span
Q: {{c1::Span}} is the {{c2::critical path (height)}} of the task graph. It corresponds to {{c3::T_∞}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955172 |
1 | 230% | 9d | 7 |
nid:1771363955022
c2
PProg
critical path (height)
1
lapses
1/4
users
230%
ease
nid:1771363955022
Cloze c2
Cloze answer: critical path (height)
Q: {{c1::Span}} is the {{c2::critical path (height)}} of the task graph. It corresponds to {{c3::T_∞}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771363955173 |
1 | 230% | 6d | 8 |
nid:1771364083961
c1
Analysis
halboffenes
1
lapses
1/4
users
230%
ease
nid:1771364083961
Cloze c1
Cloze answer: halboffenes
Q: Ein {{c1::halboffenes}} Intervall zwischen \(a\) und \(b\) wäre z.B.:\({{c2::[a, b)}}={{c3::\{x \in \mathbb{R} \mid a \leq x < b\} }}\).
A: Das Intervall kann selbstverständlich auch in die andere Richtung geöffnet sein:\((a, b]=\{x \in \mathbb{R} \mid a < x \leq b\}\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771364083966 |
1 | 230% | 2d | 5 |
nid:1771578182870
c2
PProg
load imbalance
1
lapses
1/4
users
230%
ease
nid:1771578182870
Cloze c2
Cloze answer: load imbalance
Q: Parallel execution can introduce inefficiencies such as {{c1::communication overhead}}, {{c2::load imbalance}}, and {{c3::idle time due to task dependencies or waiting for data exchange}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771578182870 |
1 | 230% | 3d | 5 |
nid:1771616145174
c1
Advanced Finance
reductions in a firm's value that arise from agency problems
1
lapses
1/4
users
230%
ease
nid:1771616145174
Cloze c1
Cloze answer: reductions in a firm's value that arise from agency problems
Q: Agency costs are {{c1::reductions in a firm's value that arise from agency problems}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771616145174 |
1 | 230% | 15d | 11 |
nid:1771616439344
c5
Advanced Finance
focusing on short-term results at the expense of long-term r...
1
lapses
1/4
users
230%
ease
nid:1771616439344
Cloze c5
Cloze answer: focusing on short-term results at the expense of long-term results
Q: Agency problems include a manager:{{c1:: not putting in sufficient effort}}{{c2:: wasting money on personal benefits}}{{c3:: overinvesting in search of power or prestige}}{{c4:: taking too many or too few risks}}{{c5:: focusing on short-term results at
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771616439347 |
1 | 230% | 17d | 9 |
nid:1771770315370
c1
Advanced Finance
incentive missalignment
1
lapses
1/4
users
230%
ease
nid:1771770315370
Cloze c1
Cloze answer: incentive missalignment
Q: Family controlled companies struggle less with {{c1::incentive missalignment}} because {{c2::the shareholders and management are one and the same}}, they may, however have problems with {{c3::exploitation of minority shareholders}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771770315370 |
1 | 230% | 19d | 7 |
nid:1771771254633
c2
Advanced Finance
smaller and more independent
1
lapses
1/4
users
230%
ease
nid:1771771254633
Cloze c2
Cloze answer: smaller and more independent
Q: Boards in {{c1::the U.S. and UK}} are typically {{c2:: smaller and more independent}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771771254634 |
1 | 230% | 14d | 7 |
nid:1771780392187
c1
DDCA
Unique mapping from input values to output values; The same ...
1
lapses
1/4
users
230%
ease
nid:1771780392187
Cloze c1
Cloze answer: Unique mapping from input values to output values; The same input values produce the same output value every time.; No memory (output does not depend on past input values)
Q: What does the "functional" in functional specification signify?{{c1::Unique mapping from input values to output values}}{{c1::The same input values produce the same output value every time.}}{{c1::No memory (output does not depend on past input values)}}
A: Example: Full 1-bit adder
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771780392187 |
1 | 230% | 3d | 12 |
nid:1771780392210
c1
DDCA
CNF
1
lapses
1/4
users
230%
ease
nid:1771780392210
Cloze c1
Cloze answer: CNF
Q: Product of Sums is equivalent to {{c1::CNF}}.
A: This is also the DeMorgan of SOP of \(\overline F\).
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771780392213 |
1 | 230% | 2d | 5 |
nid:1771780392218
c4
DDCA
Transistor (MOS)
1
lapses
1/4
users
230%
ease
nid:1771780392218
Cloze c4
Cloze answer: Transistor (MOS)
Q: By combining:
{{c1::Conductors (Metal)}}
{{c2::Insulators (Oxide)}}
{{c3::Semiconductors}}
We get a {{c4::Transistor (MOS)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771780392223 |
1 | 230% | 3d | 5 |
nid:1771780392220
c2
DDCA
broken (i.e., the circuit is open)
1
lapses
1/4
users
230%
ease
nid:1771780392220
Cloze c2
Cloze answer: broken (i.e., the circuit is open)
Q: If the gate of the n-type transistor is supplied with {{c1::zero}} voltage, the connection between the source and drain is {{c2::broken (i.e., the circuit is open)}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771780392227 |
1 | 230% | 3d | 5 |
nid:1771780392223
DDCA
How does a decoder work?
1
lapses
1/4
users
230%
ease
nid:1771780392223
Q: How does a decoder work?
A: \(n\) possible inputs and \(2^n\) outputsExactly one of the outputs is 1 and all the rest are 0sThe output that is logically 1 is the output corresponding to the input pattern that the logic circuit is expected to detectA decoder is an "input pattern detector".Example: 2-to-4 decoder
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771780392230 |
1 | 230% | 3d | 5 |
nid:1771780392226
DDCA
What's the formula for dynamic power consumption?
1
lapses
1/4
users
230%
ease
nid:1771780392226
Q: What's the formula for dynamic power consumption?
A: \(C\cdot V^2\cdot f\)\(C =\) capacitance of the circuit (wires and gates)\(V =\) supply voltage\(f =\) charging frequency of the capacitor
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771780392233 |
1 | 230% | 3d | 7 |
nid:1771794049785
c1
Advanced Finance
opinion that the statement is representative and in-line wit...
1
lapses
1/4
users
230%
ease
nid:1771794049785
Cloze c1
Cloze answer: opinion that the statement is representative and in-line with GAAP
Q: If an auditor finds no problems in a firm's financial statement, he issues an {{c1::opinion that the statement is representative and in-line with GAAP}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771794049785 |
1 | 230% | 11d | 11 |
nid:1771794112422
c2
Advanced Finance
the accounts of the firm have not been represented accuratel...
1
lapses
1/4
users
230%
ease
nid:1771794112422
Cloze c2
Cloze answer: the accounts of the firm have not been represented accurately
Q: If an auditor finds problems they can issue a {{c1::qualified opinion}} which states that {{c2::the accounts of the firm have not been represented accurately}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771794112423 |
1 | 230% | 14d | 11 |
nid:1771795613218
c2
Advanced Finance
syndicate ownership
1
lapses
1/4
users
230%
ease
nid:1771795613218
Cloze c2
Cloze answer: syndicate ownership
Q: {{c1::Keiretsu}} is a Japanese system of {{c2::syndicate ownership}} which centers around a main {{c3::bank}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771795613220 |
1 | 230% | 6d | 8 |
nid:1771795784415
c1
Advanced Finance
gives companies more space when they get in financial proble...
1
lapses
1/4
users
230%
ease
nid:1771795784415
Cloze c1
Cloze answer: gives companies more space when they get in financial problems
Q: The keiretsu system has positives in that it {{c1::gives companies more space when they get in financial problems}}.
A: This is because the company's lender is most likely the main group bank.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771795784415 |
1 | 230% | 14d | 6 |
nid:1771836465439
c1
PProg
the sequential part of a program
1
lapses
1/4
users
230%
ease
nid:1771836465439
Cloze c1
Cloze answer: the sequential part of a program
Q: Efficiency is heavily limited by {{c1::the sequential part of a program}}.
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771836465439 |
1 | 230% | 1d | 3 |
nid:1771836518739
c1
PProg
Efficiency
1
lapses
1/4
users
230%
ease
nid:1771836518739
Cloze c1
Cloze answer: Efficiency
Q: {{c1::Efficiency}} = {{c2::\(\frac{S_p}{p}\)}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771836518739 |
1 | 230% | 2d | 5 |
nid:1771836628438
c2
PProg
enforce mutual exclusion
1
lapses
1/4
users
230%
ease
nid:1771836628438
Cloze c2
Cloze answer: enforce mutual exclusion
Q: Locks are typically used to {{c2::enforce mutual exclusion}} by {{c1::guarding/protecting a critical section.}}
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771836628438 |
1 | 230% | 2d | 5 |
nid:1771914065795
Analysis
Ist die Menge \(A \neq \emptyset\) nach oben/unten unbeschrä...
1
lapses
1/4
users
230%
ease
nid:1771914065795
Q: Ist die Menge \(A \neq \emptyset\) nach oben/unten unbeschränkt, so definieren wir Supremum/Infinum:
A: \(\sup(A) = \infty\)/\(\inf(A) = -\infty\)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1771914065795 |
1 | 230% | 5d | 6 |
nid:1772090857637
c1
A&W
NP-vollständig
1
lapses
1/4
users
230%
ease
nid:1772090857637
Cloze c1
Cloze answer: NP-vollständig
Q: Das Problem „Gegeben ein Graph \(G = (V, E)\), enthält \(G\) einen Hamiltonkreis?" ist {{c1::NP-vollständig}}.
A: Karp (1972)
| User | Card ID | Lapses | Ease | Interval | Reviews |
|---|---|---|---|---|---|
| tomas | cid:1772090857637 |
1 | 230% | 3d | 5 |