By Gerald A. Edgar

Fractals are a tremendous subject in such diverse branches of technology as arithmetic, laptop technological know-how, and physics. Classics on Fractals collects for the 1st time the historical papers on fractal geometry, facing such subject matters as non-differentiable features, self-similarity, and fractional size. Of specific worth are the twelve papers that experience by no means sooner than been translated into English. Commentaries through Professor Edgar are incorporated to help the coed of arithmetic in examining the papers, and to put them of their historic standpoint. the amount includes papers from the next: Cantor, Weierstrass, von Koch, Hausdorff, Caratheodory, Menger, Bouligand, Pontrjagin and Schnirelmann, Besicovitch, Ursell, Levy, Moran, Marstrand, Taylor, de Rahm, Kolmogorov and Tihomirov, Kiesswetter, and naturally, Mandelbrot.

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Example text

Ym , y. , ym ). , ym )(y) ist also algebraisch mit Minimalpolynom g. , ym )(y) = F (y) = K(V ). Also sind V und W birational zueinander. ¨ Eine irreduzible Variet¨at V heißt rational, wenn sie birational Aquivalent zu AnK ist. Es gilt der folgende Satz. 22. Sei V eine irreduzible Variet¨at. Dann sind ¨aquivalent: (i) V ist birational. , Xn ). 36 (iii) Es gibt eine offene Teilmenge U0 ⊂ V , die isomorph zu einer offenen Teilmenge W ⊂ AnK ist. Beispiel. (1) Sei f : A1K → C 1 = {(x, y) ∈ A2K |y 2 − x3 = 0}, t → (t2 , t3 ) ist eine birationale Abbildung, aber kein Isomorphismus.

M,n ist irreduzibel. −1 n Beweis. Die Abbildungen π1 , π2 sind Morphismen. Sei P ∈ Pm K . Dann ist π1 ({P } × PK ) eine projektive Variet¨at. Wir k¨onnen diese mit PnK identifizieren. (PnK ∼ = = {P } × PnK ∼ n n sm,n ({P } × PK )) Insbesondere ist sm,n ({P } × PK ) irreduzibel. Analog sind die Fasern von π2 isomorph zu Pm K. Anngenommen: Σm,n = Y1 ∪ Y2 , wobei Y1 , Y2 abgeschlossene Teilmengen von Σm,n sind. Dann ist sm,n ({P } × PnK ) = sm,n ({P } × PnK ) ∩ Y1 ∪ sm,n ({P } × PnK ) ∩ Y2 . Da sm,n ({P } × PnK ) irreduzibel ist, ist sm,n ({P } × PnK ) ⊂ Y1 oder sm,n ({P } × PnK ) ⊂ Y2 .

Wir zeigen grad(f ) = 0: f : C− → K = K = P1K = {(: x : 1 :)|x ∈ K} ∪ {(0 : 1 :)} ∼ = A1K ∪ {∞}, sodass f : C− → P1K ⇒ f : C → P1K surjektiv. Es folgt (f ) = νP (f )P = νP (f )P + f (P )=0 νP (f )P f (P )=∞ νP (f )P − = νP 1 (P )=0 f f (P )=0 1 P f = grad(f ∗ (0)) − grad(f ∗ (∞)). Also (f ) = f ∗ (0) − f ∗ (∞). Also grad((f )) = grad(f ∗ (0)) − grad(f ∗ (∞)) = grad(f ) − grad(f ) = 0. Ist K = C, so kann man dieses Resultat auch mit Hilfe der Funktionentheorie beweisen. Denn ist K = C, so ist C eine kompakte Riemannsche Fl¨ache und f eine mereomorphe Funktion auf C.

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