c1(x̃1 − x̃2) 2(x̃1 − x̃3) 2 . . . (x̃1 − x̃d) 2 [c1(x̃1 − x̃2)(x̃1 − x̃3) . . . (x̃1 − x̃d)]2 . But we have R′(x̃1) = 0, hence c1(x̃1 − x̃2) 2(x̃1 − x̃3) 2 . . . (x̃1 − x̃d) 2 [c1(x̃1 − x̃2)(x̃1 − x̃3) [...] Write (p′)2 − pp′′ = p2 (p′)2 p2 − p2pp′′ p2 = p2 (( p′ p )2 − p′′ p ) . We know that ( p′ p )2 = ( d∑ j=1 1 z − xj )2 , where xj are roots of p, 1 ≤ j ≤ d, hence p′′ p = ( d∑ j=1 1 z − xj )2 − d∑ j=1 1 [...] by ai = (i + 2)(i + 1)ai+2 i(i− 1)− d(d− 1) . (2.2.9) 2. If d is odd, then ad = 1, ad−(2b−1) = 0, b = 1, 2, ..., d+1 2 , and the other coeffi- cients are given by ai = (i + 2)(i + 1)ai+2 i(i− 1)− d(d− …

P:\TEX\SIOPT\19-4\70427\70427.dvi

> 1/(2c). Then, by (2.8), we have aj+1 ≤ (1 − 2cθ/j)aj + 1 2 θ2M2/j2. It follows by induction that (2.9) E [‖xj − x∗‖2 2 ] = 2aj ≤ Q(θ)/j, where (2.10) Q(θ) = max { θ2M2(2cθ − 1)−1, ‖x1 − x∗‖2 2 } . Suppose [...] y)‖ = √ ‖x‖2 x 2R2 x + ‖y‖2 1 2R2 y , and hence ‖(ζ, η)‖∗ = √ 2R2 x‖ζ‖2∗,x + 2R2 y‖η‖2∞. Let us assume uniform bounds: max 1≤i≤m E [‖Gi(x, ξ)‖2 ∗,x ] ≤ M2 ∗,x, E [ max 1≤i≤m |Fi(x, ξ)|2 ] ≤ M2 ∗,y, i = 1 [...] ex ≤ x + ex2 ) E[exp{γζt}|ξ[t−1]] ≤ E[exp { γ2ζ2 t } |ξ[t−1]] ≤ E [( exp { ζ2 t /σ2 t })γ2σ2 t |ξ[t−1] ] ≤ exp { γ2σ2 t } ; γσt > 1 ⇒ E[exp{γζt}|ξ[t−1]] ≤ E [ exp { [ 1 2 γ2σ2 t + 1 2 ζ2 t /σ2 t } |ξ[t−1] …

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+0 - # 2 )2 < 8 ! ! ! 0 ? 83 - ! " 6 < 6= ;6 < 5 6-2 0 / ! 8! L6 3! ;6 < , ? @D " / #5 76=F6 ?* -,.! ABBBH= 8 2 -!2 " =! +0 / -2 - 2 - " [...] 68 " , 6 < - ?+ 5!2 - 82 7! =2 =< ? + 2 !/ ! 65!2 ;6 < 0 - " ? 6 " * = # <0= $ " + +0 -- " ?" > 2 ,< = ?!. -?? -iA 0 = !6 ! 0 0 2 " ? " - !/ [...] ! 0 -< 6 0 =! 3 ! < 6 ? ( 2 8 = 0! / 2 +0!? 2 3 , = =! 2 3! - " ) - 3 ! < -=?3 ./ -8 0-= ? b ' 2 2 3 - " ?+5 3L ) " / - ! ) - +0! < …

Katharina 1 2 2 Ehrenberg 1 1 2 Flügel Thorsten 2 Peter 1 Funk Julia 1 2 Daniel 2 2 Stephan 1 Daniel 1 Lang Konstantin 2 Katrin 2 1 2 1 Löwen Alexander 1 1 2 1 Christoph 1 Kathrin 1 Rausch Tanja 2 Dirk 2 2 Stahl [...] Stahl 1 Raphael 1 2 Winter Christian 1 Böhmer Mirja Borg Chouikha Hamdi D'Addario Marianna Nikolas Faber Viktor Fitzner Marcel Fricke Gulinina Irina Haak Jablkowski Boguslaw Karlinski Kleefisch Linßen Lisakowski …

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+3 < * " =! ; - " * $*8&'(! RQ?;5 2 2 < - F3 !H-!/ # M 6 6B ! ;6 < ?8-2 6= "!/2 2 5 < 8= ?, = /=$" & 2 / 3 =2 ?>= 6= -PGX!5 " /! !! 5 = ? !8- 8= [...] 3+( # " ""////2 ) + 28= 7 =2 / 8- "! 2 ? " / =6 " 2?8 3 " 6 M 63 M 6 + ! 5 = 6 2 ? + / , " =!6" -2!- / = 6+!?5 /=" 2/,-. / [...] -= 0 ! . / 3 ( ! . ?9! ;6 < 2 6= 3 ! ?< 2 0 6 - / == =! / < '<- R-BAQ? +,2 2 - = =2 2+! ; 8 ! ?= 8 5 < ! ? + , !6 5 " ! $ &0 …

(µ1c 2 1 + µ2c 2 2) θ 2 c21c 2 2 + (c21 + c22) θ 2 + θ4 = aθ4 + 1 2 ab2θ2 + aθ4 + 1 2 b3θ2 + 2bθ4 θ4 + b2θ2 + 2θ4 + θ4 = 2(a+ b)θ2 + 1 2 (a+ b)b2 b2 + 4θ2 = (a+ b) ( 2θ2 + 1 2 b2 ) b2 + 4θ2 = 2λ1 ( 2θ2 + 1 [...] µ1c 2 2 + µ2c 2 1 + (µ1 + µ2)θ 2 c21c 2 2 + (c21 + c22) θ 2 + θ4 = 1 2 ab2 + aθ2 − 1 2 b3 − 2bθ2 + aθ2 θ4 + (b2 + 2θ2) θ2 + θ4 = 1 2 b2(a− b) + 2θ2(a− b) θ2 (b2 + 4θ2) = (a− b) ( 1 2 b2 + 2θ2 ) θ2 (b2 [...] b3 + bθ2 µ1c 2 1 + µ2c 2 2 = 1 2 ab2 + aθ2 + 1 2 b3 + 2bθ2 µ1c 2 2 = 1 2 (a+ s) ( 1 2 b2 + θ2 − 1 2 bs ) = 1 4 ab2 + 1 2 aθ2 − 1 4 abs+ 1 4 b2s+ 1 2 sθ2 − 1 4 b3 − bθ2 µ2c 2 1 = 1 2 (a− s) ( 1 2 b2 + θ2 …

Daten? 1. Φ1(x1, x2) = (x21, x2) 2. Φ2(x1, x2) = (x31 − 2x1, x2) 3. Φ3(x1, x2) = (x31, x2) x1 x2 y -1.5 -2.0 +1 -1.0 0.0 +1 -0.5 1.0 +1 0.0 2.0 +1 0.5 1.0 +1 1.0 2.0 +1 1.5 3.75 +1 -1.0 -2.0 -1 -0.5 -1.0 [...] uni-dortmund.de/LEHRE/VORLESUNGEN/KDD/SS13/DATA/svm-data.csv -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Abbildung 2: Datenpunkte im R2 1. Beschreiben Sie, wie Sie die gegebenen Daten trennen würden [...] kis, 1979, Akademie Vg. [UB-Dortmund: Signatur M20947] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 Abbildung 1: Datenpunkte im R2 Aufgabe 6.1 (3 Punkte) In dieser Aufgabe sollen Sie den naiven …

. , tn, (−1) n+2 2 t1 ⋅ . . . ⋅ tn⟩ ∈ I2F , we have GP2(ψ) = n 2 . Now, for the form ϕ ∶= ⟨⟨tn+1⟩⟩⊗ ψ ∈ I3F , which is of dimension 2(n + 2) ≤ 2( d 2 − 2 + 2) = d, we have GP3(ϕ) = n 2 = ⌊ d 4 ⌋ − 1. by [...] t1)) as the F2-linear map Φ ∶ F ∗/F ∗2 → E∗/E∗2 defined by aF ∗2 ↦ aE∗2 for all a ∈K∗; t1F ∗2 ↦ t1E ∗2; t2F ∗2 ↦ t2E ∗2 is a group isomorphism with Φ(−1) = −1 and Φ(DF (⟨x1, . . . , xn⟩)) =DE(⟨Φ(x1), [...] ⊥ tϕ2 and ϕ1 ≠ 0 ≠ ϕ2, where ϕ1, ϕ2 are the residue forms. Since we have ϕ1, ϕ2 ∈ In−1K by Lemma 3.4.1, the Holes Theorem implies dimϕ1,dimϕ2 ∈ {2n−1,2n−1 + 2n−2,2n}. If one of the dimensions is 2n−1, …