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13 39 2.66 56 2.70 309 40 2.66 44 2.68 17 2.01 310 28 2.42 45 2.69 28 2.43 310 57 2.69 22 2.23 21* 2.25 309 20 2.15 53 2.70 27 2.41 310 51 2.71 38 2.63 12* 1.71 309 62 2.61 33 2.53 5 1.13 308 81 2.14 16 [...] 43 2.73 50 2.76 535 52 2.76 35 2.63 13 1.83 532 9 1.49 35 2.64 56 2.74 534 53 2.78 22 2.36 25 2.52 533 25 2.43 54 2.76 21 2.19 534 58 2.74 34 2.61 9 1.56 533 72 2.49 26 2.44 2 0.74 532 58 2.74 23 2.28 [...] 2.91 30 2.61 275 61 2.81 25 2.51 14 1.91 278 19 2.38 47 2.90 34 2.74 278 47 2.91 31 2.64 22 2.44 278 57 2.89 39 2.84 4 1.12 276 85 2.05 13 1.90 2 0.87 274 78 2.36 18 2.18 4 1.12 276 13 2.04 73 2.65 14 …

+ d|2 ≥ Im2(cz + d) = c2y2 1 Grundlagen 7 und |z|2|cz+d|2 = |z(cz+d)|2 = | c|z|2+dz |2 = | c|z|2+dz |2 ≥ Im2(c|z|2+dz) = d2y2 sieht man (1+ |z|2)|cz + d|2 = |cz + d|2 + |z|2|cz + d|2 ≥ c2y2 + d2y2 = y2(c2 [...] |cz + d|2 ≤ (|c||z| + |d|)2 = c2|z|2 + 2|c||d||z| + d2 ≤ c2|z|2 + c2 + d2|z|2 + d2 = (|z|2 + 1)(c2 + d2). Für |z| ≥ 1 ist |z|2 + 1 ≤ 2|z|2 und für |z| < 1 haben wir |z|2 + 1 ≤ |z|2 +y−2 wegen y−2 > 1, womit [...] f(τ0 + p 2π t)e−2πin(τ0+ p 2π t)/pdt = 1 2π 2π∫ 0 f̂ (e2πi(τ0+ p 2π t)/p)e−2πin(τ0+ p 2π t)/pdt = 1 2πi 2π∫ 0 f̂ (e2πi(τ0+ p 2π t)/p) e−2πi(n+1)(τ0+ p 2π t)/p ie−2πi(τ0+ p 2π t)/pdt = 1 2πi ∫ |z|=e−2π Imτ0/p …

Derivation of Hierarchical Clustering Equations – Lecture Notes

Law of cosines:  \(d(A,B)^2=d(A,C)^2+d(B,C)^2-2d(A,C)d(B,C)\cos \gamma\) \(\Rightarrow 2d(A,C)d(B,C)\cos \gamma = d(A,C)^2+d(B,C)^2-d(A,B)^2\) Parallelogram law:  \(c^2=a^2+b^2+2ab\cos \gamma\) for \(c=d({A\cup [...] d(A\cup B,C)^2 & = \left(\tfrac {|A|}{|A|+|B|}d(A,C)\right)^2+ \left(\tfrac {|B|}{|A|+|B|}d(B,C)\right)^2 +\tfrac {|A||B|}{(|A|+|B|)^2}\cdot \underbrace{ 2d(A,C)d(B,C)\cos \gamma }_{=d(A,C)^2+d(B,C)^2-d(A,B)^2} [...] n_B(\mu _B{-}\mu _C)\right\rVert ^2 \\ & = \tfrac {n_C}{n_{AB}n_{ABC}}\Big( n_A^2\left\lVert \mu _A{-}\mu _C\right\rVert ^2 +n_B^2\left\lVert \mu _B{-}\mu _C\right\rVert ^2 +2n_An_B(\mu _A{-}\mu _C)^T(\mu _B{-}\mu …

Derivation of Hierarchical Clustering Equations – Lecture Notes

Law of cosines:  \(d(A,B)^2=d(A,C)^2+d(B,C)^2-2d(A,C)d(B,C)\cos \gamma\) \(\Rightarrow 2d(A,C)d(B,C)\cos \gamma = d(A,C)^2+d(B,C)^2-d(A,B)^2\) Parallelogram law:  \(c^2=a^2+b^2+2ab\cos \gamma\) for \(c=d({A\cup [...] d(A\cup B,C)^2 & = \left(\tfrac {|A|}{|A|+|B|}d(A,C)\right)^2+ \left(\tfrac {|B|}{|A|+|B|}d(B,C)\right)^2 +\tfrac {|A||B|}{(|A|+|B|)^2}\cdot \underbrace{ 2d(A,C)d(B,C)\cos \gamma }_{=d(A,C)^2+d(B,C)^2-d(A,B)^2} [...] n_B(\mu _B{-}\mu _C)\right\rVert ^2 \\ & = \tfrac {n_C}{n_{AB}n_{ABC}}\Big( n_A^2\left\lVert \mu _A{-}\mu _C\right\rVert ^2 +n_B^2\left\lVert \mu _B{-}\mu _C\right\rVert ^2 +2n_An_B(\mu _A{-}\mu _C)^T(\mu _B{-}\mu …

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+7,481 +2.5 +31,532 -2.5 +13,589 +2,464 +2.6 +11,125 -2.6 +5,277 +1,030 +2.5 +4,247 -2.5 +3,679 +830 +2.6 +2,849 -2.6 +2,167 +484 +3.3 +1,683 -3.3 +1,875 +279 +2.6 +1,596 -2.6 +12,426 +2,394 +2.0 +10,032 [...] 486 528 2,777 1,895 13,408 5.6 6.9 3.3 0.5 8.3 8.8 7.9 Sachsen-Anhalt 19,108 9,632 2,037 361 915 433 5,730 2.8 4.2 1.9 0.3 2.7 2.0 3.4 Thüringen 15,620 5,372 2,274 787 2,326 921 3,940 2.3 2.3 2.2 0.7 7.0 [...] +15,871 -2.3 +5,889 +1,329 +3.0 +4,560 -3.0 +2,877 +564 +2.0 +2,313 -2.0 +3,242 +499 +1.5 +2,743 -1.5 +888 +224 +2.8 +664 -2.8 +1,011 +154 +2.0 +857 -2.0 +5,978 +1,244 +2.1 +4,734 -2.1 25 - 34 +39,013 +7 …

U1−loop = 1 2 T ∑ n ∫ dp 2π { ln p2 + ω2 n +M2 σ p2 + ω2 n +m2 0 + (N − 1) ln p2 + ω2 n +M2 π p2 + ω2 n +m2 0 − M2 σ −m2 0 p2 + ω2 n +m2 0 − (N − 1) M2 π −m2 0 p2 + ω2 n +m2 0 } + 1 2 (M2 σ −m2 0)Bσ + N − [...] must be solved M2 σ = m2 − c+ 3 (λ1 + λ2)φ 2 0 + (λ1 + λ2) N2 f Tr ∆ (A.28a) M2 π = M2 σ − 2 (λ1 + λ2)φ 2 0 (A.28b) M2 η = M2 σ + 2 c− 2 (λ1 + λ2)φ 2 0 (A.28c) M2 a0 = M2 σ + 2 c− 2 λ1 φ 2 0 . (A.28d) We [...] M |M1|M2) = 1 (4π)4 [ 2 ǫ2 + 2 ǫ ( 1 2 − ln M2 µ̄2 R ) + 1 2 + π2 12 + ( − ln M2 µ̄2 R )2 − ln M2 µ̄2 R + f(a, b) ] + O(ǫ) (B.5) and 1 3 − n (M,M |M1|M2) = 1 (4π)4 [ − 2 ǫ2 − 2 ǫ (3 2 − ln M2 µ̄2 R ) − …

mw=1,73 md=2 s=0,88 48,5% 1 36,1% 2 11,3% 3 2,7% 4 0,8% 5 0,5% 6 Aus Ihrer eigenständigen Seminarleistung haben Sie ... 5.2) wenig gelerntviel gelernt n=840 mw=2,13 md=2 s=1,04 29,2% 1 41,8% 2 20,7% 3 4 [...] keine Angabe 1,5% Studienphase2.2) n=850Bachelor 42,6% Master 57,4% Sonstiges 0% Semesterzahl in dieser Studienphase2.3) n=8491-2 34,7% 3-4 22% 5-6 28,9% 7-8 11,1% 9-10 2,7% mehr als 10 0,6% Studienfach [...] E.=Enthaltung 25% 1 0% 2 50% 3 0% 4 25% 5 Relative Häufigkeiten der Antworten Std.-Abw. Mittelwert Median Skala Histogramm 2. Persönliche Angaben2. Persönliche Angaben Geschlecht2.1) n=842weiblich 79,6% …