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I [ n+1] =Sum( ( - 1) ^k* x* l n( x) ^( n+1- k) * ( n+1) ! / ( n+1- k) ! , k=0. . n+1) ; InC1 = > k= 0 nC1 K1 k x ln x nC1Kk nC1 ! nC1Kk ! Die Rekursionsfomel gibt für n+1 InC1 = ln x nC1 xK nC1 In Setzen [...] x nC1 xK nC1 > k= 0 n x K1 k ln x nKk n! nKk ! expr essi on1: =Sum( ( - 1) ^k* x* l n( x) ^( n+1- k) * ( n+1) ! / ( n+1- k) ! , k=0. . n+1) ; expression1 := > k= 0 nC1 K1 k x ln x nC1Kk nC1 ! nC1Kk ! Jetzt [...] n+1) ; expression1 = ln x nC1 xC> k= 1 nC1 K1 k x ln x nC1Kk nC1 ! nC1Kk ! ' expr essi on2' =expr essi on2; expression2 = ln x nC1 xK nC1 > k= 0 n x K1 k ln x nKk n! nKk ! Nach einer kurzen Überlegung sieht …

Examples: a) standard fuzzy complement c(a) = 1 – a ad (A1): c(0) = 1 – 0 = 1 and c(1) = 1 – 1 = 0 ad (A2): c‘(a) = –1 < 0 (monotone decreasing) ad (A3):  ad (A4): 1 – (1 – a) = a Lecture 06 G. Rudolph: Computational [...] Examples ad (A1): c(0) = 1 and c(1) = 0 ad (A2):  ad (A3): is continuous as a composition of continuous functions ad (A4): e) c(a) = ( 1 – aw )1/w for w > 0 Yager class  (1 – aw)1/w ≥ (1 – bw)1/w ⇔ 1 – aw ≥ [...] generator g(-1)(x) pseudo-inverse Examples a) g(x) = x ⇒ g-1(x) = x ⇒ c(a) = 1 – a (Standard) b) g(x) = xw ⇒ g-1(x) = x1/w ⇒ c(a) = (1 – aw)1/w (Yager class, w > 0) c) g(x) = log(x+1)⇒ g-1(x) = ex – 1 ⇒ c(a) …

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Slide 1

0) f(-1)(x) = 1 – x/k c(a) = b) f(x) = 1 – xw f(-1)(x) = (1 – x)1/w c(a) = f-1(aw) = (1 – aw)1/w (Yager) Theorem c: [0,1] → [0,1] is involutive fuzzy complement iff continuous function f: [0,1] → with [...] Examples: standard fuzzy complement c(a) = 1 – a ad (A1): c(0) = 1 – 0 = 1 and c(1) = 1 – 1 = 0 ad (A2): c‘(a) = –1 < 0 (monotone decreasing) ad (A3):  ad (A4): 1 – (1 – a) = a Lecture 02 4 G. Rudolph: Com [...] Examples ad (A1): c(0) = 1 and c(1) = 0 ad (A2):  ad (A3): is continuous as a composition of continuous functions ad (A4): e) c(a) = ( 1 – aw )1/w for w > 0 Yager class  (1 – aw)1/w ≥ (1 – bw)1/w  1 – aw ≥ …

Slide 1

0) f(-1)(x) = 1 – x/k c(a) = b) f(x) = 1 – xw f(-1)(x) = (1 – x)1/w c(a) = f-1(aw) = (1 – aw)1/w (Yager) Theorem c: [0,1] → [0,1] is involutive fuzzy complement iff continuous function f: [0,1] → with [...] Examples: standard fuzzy complement c(a) = 1 – a ad (A1): c(0) = 1 – 0 = 1 and c(1) = 1 – 1 = 0 ad (A2): c‘(a) = –1 < 0 (monotone decreasing) ad (A3):  ad (A4): 1 – (1 – a) = a Lecture 06 4 G. Rudolph: Com [...] Examples ad (A1): c(0) = 1 and c(1) = 0 ad (A2):  ad (A3): is continuous as a composition of continuous functions ad (A4): e) c(a) = ( 1 – aw )1/w for w > 0 Yager class  (1 – aw)1/w ≥ (1 – bw)1/w  1 – aw ≥ …

Slide 1

0) f(-1)(x) = 1 – x/k c(a) = b) f(x) = 1 – xw f(-1)(x) = (1 – x)1/w c(a) = f-1(aw) = (1 – aw)1/w (Yager) Theorem c: [0,1] → [0,1] is involutive fuzzy complement iff continuous function f: [0,1] → with [...] Examples: standard fuzzy complement c(a) = 1 – a ad (A1): c(0) = 1 – 0 = 1 and c(1) = 1 – 1 = 0 ad (A2): c‘(a) = –1 < 0 (monotone decreasing) ad (A3):  ad (A4): 1 – (1 – a) = a Lecture 06 4 G. Rudolph: Com [...] Examples ad (A1): c(0) = 1 and c(1) = 0 ad (A2):  ad (A3): is continuous as a composition of continuous functions ad (A4): e) c(a) = ( 1 – aw )1/w for w > 0 Yager class  (1 – aw)1/w ≥ (1 – bw)1/w  1 – aw ≥ …

Slide 1

0) f(-1)(x) = 1 – x/k c(a) = b) f(x) = 1 – xw f(-1)(x) = (1 – x)1/w c(a) = f-1(aw) = (1 – aw)1/w (Yager) Theorem c: [0,1] → [0,1] is involutive fuzzy complement iff continuous function f: [0,1] → with [...] Examples: standard fuzzy complement c(a) = 1 – a ad (A1): c(0) = 1 – 0 = 1 and c(1) = 1 – 1 = 0 ad (A2): c‘(a) = –1 < 0 (monotone decreasing) ad (A3):  ad (A4): 1 – (1 – a) = a Lecture 06 4 G. Rudolph: Com [...] Examples ad (A1): c(0) = 1 and c(1) = 0 ad (A2):  ad (A3): is continuous as a composition of continuous functions ad (A4): e) c(a) = ( 1 – aw )1/w for w > 0 Yager class  (1 – aw)1/w ≥ (1 – bw)1/w  1 – aw ≥ …

Slide 1

0) f(-1)(x) = 1 – x/k c(a) = b) f(x) = 1 – xw f(-1)(x) = (1 – x)1/w c(a) = f-1(aw) = (1 – aw)1/w (Yager) Theorem c: [0,1] → [0,1] is involutive fuzzy complement iff continuous function f: [0,1] → with [...] Examples: standard fuzzy complement c(a) = 1 – a ad (A1): c(0) = 1 – 0 = 1 and c(1) = 1 – 1 = 0 ad (A2): c‘(a) = –1 < 0 (monotone decreasing) ad (A3):  ad (A4): 1 – (1 – a) = a Lecture 02 4 G. Rudolph: Com [...] Examples ad (A1): c(0) = 1 and c(1) = 0 ad (A2):  ad (A3): is continuous as a composition of continuous functions ad (A4): e) c(a) = ( 1 – aw )1/w for w > 0 Yager class  (1 – aw)1/w ≥ (1 – bw)1/w  1 – aw ≥ …

Bremen 1 1 3 1.9 0.10 1 12 1 1 2 1.8 0.12 1 40 Hamburg 3 1 6 5.2 0.26 1 50 2 1 4 3.4 0.20 1 40 Hessen 3 1 7 5.8 0.46 1 60 2 1 3 3.3 0.19 1 50 Mecklenburg-Vorpommern 2 1 5 4.3 0.24 1 50 2 1 3 3.1 0.21 1 45 [...] 14 1 25 Sachsen 2 1 5 3.9 0.22 1 60 1 1 4 3.1 0.15 1 30 Sachsen-Anhalt 1 1 3 3.1 0.18 1 27 1 1 3 2.7 0.21 1 102 Schleswig-Holstein 3 1 6 5.3 0.57 1 165 2 1 3 3.0 0.15 1 40 Thüringen 1 1 3 3.5 0.33 1 100 [...] Bremen 1 1 3 2.2 0.26 1 42 1 1 2 1.8 0.10 1 30 Hamburg 3 1 6 5.5 0.32 1 50 1 1 3 3.0 0.19 1 80 Hessen 4 1 8 6.6 0.54 1 83 2 1 3 3.2 0.20 1 53 Mecklenburg-Vorpommern 2 1 5 4.2 0.27 1 30 1 1 4 3.0 0.16 1 36 …

739 1,710 1,627 1,629 1,484 1,409 1,361 1,332 1,278 1,223 1,272 1,295 1,324 1,439 1,496 1,511 1,692 1,672 1,645 1,667 1,669 1,499 1,373 1,396 1,463 1,339 1,345 1,215 1,096 1,022 898 801 653 1,527 Anteil in [...] 539 1,352 Der Paritätische 53,169 6,503 1,736 1,754 1,716 1,577 1,431 1,313 1,279 1,208 1,200 1,236 1,174 1,131 1,091 1,042 1,036 1,053 1,150 1,069 1,104 1,230 1,232 1,257 1,378 1,326 1,403 1,363 1,370 [...] 2.0 1.9 1.9 1.9 1.8 1.9 1.8 1.7 -0.2 ● -0.2 ● Sachsen 1.5 1.6 1.6 1.6 1.7 1.6 1.6 1.5 1.5 1.4 1.4 -0.2 ● -0.2 ● Sachsen-Anhalt 2.2 1.8 1.7 1.7 1.9 1.9 1.9 2.0 1.9 1.9 1.9 -0.0 ● -0.3 ● Thüringen 5.3 5 …