from [1,s] = O (√ n log n log log n ) = o(n) bits. Example 8. 26 B = 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 M ′ = 6 10 14 M = 1 3 6 0 3 4 1 3 4 s = 3 s′ = 9 000 0 0 0 001 0 0 1 010 0 1 1 011 [...] 011 0 1 2 100 1 1 1 101 1 1 2 110 1 2 2 111 1 2 3 1 2 3V iP : A query rank1(B, i) is then decomposed into 3 sub-queries, as seen in the following picture: B = 1 2 3 i rank1(1, i) 1© superblock query: p [...] = I(Tu1) ◦ [r] ◦ · · · ◦ [r] ◦ I(Tuk). Hence, |I| = max(k − 1, 1) + ∑ 1≤i≤k |I(Tui)| ≤ max(k − 1, 1) + ∑ 1≤i≤k (2ni − 1) (by the induction hypothesis) = max(k − 1, 1)− k + 2 ∑ 1≤i≤k ni ≤ 1 + 2 ∑ 1≤i≤k ni …

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to crisp version (and more) for a,b  { 0, 1 } a b a ∧ b t(a,b) 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 a b a ∨ b s(a,b) 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 a a c(a) 0 1 1 1 0 0 Fuzzy Logic Lecture 07 G. Rudolph: C [...] T(p) ? x1 x2 x3 0.1 0.8 1.0 y1 y2 0.5 1.0 R x1 x2 x3 y1 1.0 0.7 0.5 y2 1.0 1.0 1.0 example: let Imp(a, b) = min{ 1, 1 – a + b } and consider fuzzy sets A: B:  z.B. R(x2 , y1 ) = Imp(A(x2 ), B(y1 )) = Imp(0 [...] IF X is A THEN Y is B x1 x2 x3 0.5 1.0 0.6 y1 y2 1.0 0.4 A: B: R x1 x2 x3 y1 1.0 1.0 1.0 y2 0.9 0.4 0.8 given fact y1 y2 0.9 0.7 B‘: with Imp(a,b) = min{1, 1-a+b } thus: B‘ ◦ R-1 = A‘ with max-min-composition …

to crisp version (and more) for a,b ∈ { 0, 1 } a b a ∧ b t(a,b) 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 a b a ∨ b s(a,b) 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 a a c(a) 0 1 1 1 0 0 Fuzzy Logic Lecture 07 G. Rudolph: C [...] T(p) ? x1 x2 x3 0.1 0.8 1.0 y1 y2 0.5 1.0 R x1 x2 x3 y1 1.0 0.7 0.5 y2 1.0 1.0 1.0 example: let Imp(a, b) = min{ 1, 1 – a + b } and consider fuzzy sets A: B: ⇒ z.B. R(x2, y1) = Imp(A(x2), B(y1)) = Imp(0 [...] IF X is A THEN Y is B x1 x2 x3 0.5 1.0 0.6 y1 y2 1.0 0.4 A: B: R x1 x2 x3 y1 1.0 1.0 1.0 y2 0.9 0.4 0.8 ⇒ given fact y1 y2 0.9 0.7 B‘: with Imp(a,b) = min{1, 1-a+b } thus: B‘ ◦ R-1 = A‘ with max-min-composition …

7 584 9,7 8,6 1,1 Bayern 26 96 27,1 325 5,4 11,4 -6,0 Berlin 1 1 100,0 1.093 18,2 9,5 8,7 Brandenburg 3 18 16,7 62 1,0 4,1 -3,1 Bremen 1 2 50,0 85 1,4 1,2 0,2 Hamburg 1 1 100,0 216 3,6 1,2 2,3 Hessen 11 [...] 825 10,2 8,6 1,6 Bayern 30 96 31,3 514 6,4 11,4 -5,1 Berlin 1 1 100,0 1.430 17,7 9,5 8,2 Brandenburg 4 18 22,2 137 1,7 4,1 -2,4 Bremen 1 2 50,0 139 1,7 1,2 0,5 Hamburg 1 1 100,0 325 4,0 1,2 2,8 Hessen [...] 7 640 9,6 8,6 1,0 Bayern 27 96 28,1 366 5,5 11,4 -5,9 Berlin 1 1 100,0 1.083 16,2 9,5 6,7 Brandenburg 4 18 22,2 132 2,0 4,1 -2,1 Bremen 1 2 50,0 83 1,2 1,2 0,0 Hamburg 1 1 100,0 327 4,9 1,2 0,7 Hessen 9 …

Praktische Optimierung

(x) = 1 2 x ′Ax + b′x + c = 1 2 (x1, x2) ( a11 a12 a12 a22 ) ( x1 x2 ) + (b1, b2) ( x1 x2 ) + c A symmetrisch = 1 2 (x1a11 + x2a12, x1a12 + x2a22) ( x1 x2 ) + b1x1 + b2x2 + c = 1 2 (a11x 2 1 + 2a12x1x2 + [...] (3) Berechne globales Optimum an Optimalstelle x∗ f (x∗) = f (−A−1b) = 1 2 (−A−1b)′AA−1b − b′A−1b + c = 1 2 b′A−1b − b′A−1b + c = −1 2 b′A−1b + c Nicola Beume (LS11) Optimalitätsbedingungen 17.10.2007 17 [...] + x2a22) + b1x1 + b2x2 + c Gradient ∇f (x) = ( fx1 fx2 ) = ( a11x1 + a12x2 + b1 a22x1 + a12x1 + b2 ) = Ax + b Nicola Beume (LS11) Optimalitätsbedingungen 17.10.2007 16 / 20 Optimum einer quadratischen …

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to crisp version (and more) for a,b ∈ { 0, 1 } a b a ∧ b t(a,b) 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 a b a ∨ b s(a,b) 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 a a c(a) 0 1 1 1 0 0 Fuzzy Logic Lecture 03 G. Rudolph: C [...] trueness T(p) ? x1 x2 x3 0.1 0.8 1.0 y1 y2 0.5 1.0 R x1 x2 x3 y1 1.0 0.7 0.5 y2 1.0 1.0 1.0 example: (discrete case) let Imp(a, b) = min{ 1, 1 – a + b } and consider fuzzy sets A: B: ⇒ z.B. R(x2, y1) = Imp(A(x2) [...] IF X is A THEN Y is B x1 x2 x3 0.5 1.0 0.6 y1 y2 1.0 0.4 A: B: R x1 x2 x3 y1 1.0 1.0 1.0 y2 0.9 0.4 0.8 ⇒given fact y1 y2 0.9 0.7 B‘: with Imp(a,b) = min{1, 1-a+b } thus: B‘ ◦ R-1 = A‘ with max-min-composition …

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to crisp version (and more) for a,b ∈ { 0, 1 } a b a ∧ b t(a,b) 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 a b a ∨ b s(a,b) 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 a a c(a) 0 1 1 1 0 0 Fuzzy Logic Lecture 03 G. Rudolph: C [...] trueness T(p) ? x1 x2 x3 0.1 0.8 1.0 y1 y2 0.5 1.0 R x1 x2 x3 y1 1.0 0.7 0.5 y2 1.0 1.0 1.0 example: (discrete case) let Imp(a, b) = min{ 1, 1 – a + b } and consider fuzzy sets A: B: ⇒ z.B. R(x2, y1) = Imp(A(x2) [...] IF X is A THEN Y is B x1 x2 x3 0.5 1.0 0.6 y1 y2 1.0 0.4 A: B: R x1 x2 x3 y1 1.0 1.0 1.0 y2 0.9 0.4 0.8 ⇒given fact y1 y2 0.9 0.7 B‘: with Imp(a,b) = min{1, 1-a+b } thus: B‘ ◦ R-1 = A‘ with max-min-composition …

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to crisp version (and more) for a,b ∈ { 0, 1 } a b a ∧ b t(a,b) 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 a b a ∨ b s(a,b) 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 a a c(a) 0 1 1 1 0 0 Fuzzy Logic Lecture 03 G. Rudolph: C [...] trueness T(p) ? x1 x2 x3 0.1 0.8 1.0 y1 y2 0.5 1.0 R x1 x2 x3 y1 1.0 0.7 0.5 y2 1.0 1.0 1.0 example: (discrete case) let Imp(a, b) = min{ 1, 1 – a + b } and consider fuzzy sets A: B: ⇒ z.B. R(x2, y1) = Imp(A(x2) [...] IF X is A THEN Y is B x1 x2 x3 0.5 1.0 0.6 y1 y2 1.0 0.4 A: B: R x1 x2 x3 y1 1.0 1.0 1.0 y2 0.9 0.4 0.8 ⇒given fact y1 y2 0.9 0.7 B‘: with Imp(a,b) = min{1, 1-a+b } thus: B‘ ◦ R-1 = A‘ with max-min-composition …

that c ∈DF (b1⟨1,−u1⟩ ⊥ . . . ⊥ br−1⟨1,−ur−1⟩) ∩ −DF (br⟨1,−ur⟩). By Lemma 7.1.15 we can thus assume br−1 = c and br = −c. This implies ψ ≅ b1⟨1,−u1⟩ ⊥ . . . ⊥ br−2⟨1,−ur−2⟩ ⊥ c⟨1,−ur−1,−1, ur⟩, the last [...] Q(x)((t1))⋯((t4)) we consider the forms ψ1 ∶= ⟨x,−(x + 4)⟩ ⊥ −t1⟨1,−(x + 4)⟩, ψ2 ∶= ⟨x,−(x + 1)⟩ ⊥ −2t1⟨1,−(x + 1)⟩ ρ1 ∶= ⟨1,−x,−t1t2(x + 4), t1t2x(x + 4)⟩ = ⟨⟨x, t1t2(x + 4)⟩⟩, ρ2 ∶= ⟨1,−x(x + 1)(x + 4) [...] Witt equivalent to ⟨−1, (x + 1)(x + 4),2t1(x + 1)(x + 2)(x + 4),−2t1(x + 2)⟩ ⊥ t3(⟨1,−(x + 4)⟩ ⊥ −t1⟨1,−(x + 4)⟩ ⊥ −t2(⟨1,−(x + 1)⟩ ⊥ −2t1⟨1,−(x + 1)⟩)) = − ⟨⟨(x + 1)(x + 4),−2t1(x + 2)⟩⟩ ´¹¹¹¹¹¹¹¹¹¹¹ …

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