By Goryunov V. I., Lyashko O. V.
The speculation of singularities is a vital a part of a number of branches of arithmetic: algebraic geometry, differential topology, geometric optics, and so on. right here the point of interest is at the singularities of gentle maps and functions to dynamical platforms - specifically, bifurcations. This comprises the learn of bifurcations of intersections of sturdy and volatile cycles. in addition to the formal algebraic and analytic facets of the idea, the authors examine worldwide topological difficulties on the topic of invariants. The authors keep in mind a pupil reader, mathematician, or physicist, who needs to benefit the fashionable thoughts of neighborhood mathematical research as an device for utilized reports or a expert in a single of the utilized parts who's searching for the mandatory mathematical instruments.
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Extra resources for Dynamical Systems VI: Singularity Theory I
Sample text
Vari´et´es presque rationnelles, leurs points rationnels et leurs d´eg´en´erescences 33 On dispose alors d’un morphisme d’´evaluation M 0,2 (X, e) → X × X. La fibre g´en´erale de ce morphisme est un analogue de l’espace des chemins a` points base en topologie. La vari´et´e (projective et lisse) X est dite rationnellement simplement connexe si pour e ≥ 1 suffisamment grand il existe une composante M de M 0,2 (X, e) dominant X × X telle que la fibre g´en´erique de M → X × X soit une vari´et´e rationnellement connexe.
6 (d) de [17] et les r´esultats de [20]. 11 pour les surfaces (projectives et lisses) g´eom´etriquement rationnelles d´efinies sur C(t) impliquerait l’unirationalit´e des vari´et´es de dimension 3 sur C qui admettent une fibration en coniques sur le plan projectif. Il s’agit l`a d’une question largement ouverte. 12 Soient K un corps de nombres et X une K-vari´et´e rationnellement connexe. Le quotient X (K)/R est-il fini ? C’est connu dans les cas suivants : (i) La vari´et´e X est une compactification lisse d’un groupe lin´eaire connexe G.
Supposons p ≡ 1 mod 3, et soit a ∈ Z× p non cube. Qu’en est-il pour l’hypersurface x3 + y3 + z3 + p(u31 + au32) + p2 (v31 + av32) = 0 dans P6Q p ? Vari´et´es presque rationnelles, leurs points rationnels et leurs d´eg´en´erescences 37 Sur le corps K = C((a))((b)), en utilisant la th´eorie de l’intersection sur un mod`ele au-dessus de C((a))[[b]], Madore [58] a montr´e que pour l’hypersurface cubique lisse X ⊂ P4K d’´equation x3 + y3 + az3 + bu3 + abv3 = 0, on a A0 (X) = 0. 3 Intersections lisses de deux quadriques Soit K un corps p-adique, et soit X ⊂ PnK , avec n ≥ 4, une intersection compl`ete lisse de deux quadriques poss´edant un K-point.
Dynamical Systems VI: Singularity Theory I by Goryunov V. I., Lyashko O. V.
by Ronald
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