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, Illustration de la discontinuité induite par la formule d'intégration convexe : en bleu le cercle f 0 : t ? exp(i2?t), en orange, à gauche l'application construite par intégration convexe et à droite celle construite par le procédé de corrugation pour N =, vol.5

. .. , x), où ?(x) = cos(12?x, vol.2, p.2

, Une nouvelle immersion de RP 2 : obtenue par le procédé de corrugation, voir la section 3.3

, 9 4 Illustration of discontinuity introduced by the convex integration formula: in blue a graph of a circle given by f 0 : t ? exp(i2?t), in orange, at left a map built by convex integration and at right by corrugation process for N = 5 and ?(x, t) = (cos(?(x) cos(2?t))f 0 (x)+sin(?(x) cos(2?t))(?if 0 (x)))/J 0 (?(x), where ?(x) = cos(12?x)

, A new immersion of RP 2 : obtained by the Corrugation Process, see Section 3.3

, Plucker's Conoid (before a sphere inversion) which allows to obtained a new immersion of RP 2 by a sphere inversion (see Section 3.3)

. .. , 32 1.2 Approach the relation of isometries. The relation of isometries for g and two open relations of the sequence of k -isometries for g k . For m = 1 and n = 2, the relation Is is a circle, for a general geometric description

. .. =-1, Left: The slice of Is and its Convex hull, Right: the loop ?, p.63

, Values of ? in IntConv(I, d? x , u x ): ? = ? 0 over Z 0 and ? = 0 over Z ? . In between, ? is a smooth interpolation

, 36 (see point (P 1 ) of Proposition 28), The C 0 -density phenomenon: Corrugation Process with N = 6, vol.12

, Construction of v 1 : for a given x 1 , the image of the map x 2 ? ? 1 f 0 (x 1 , x 2 ) is a right line crossing zero and the image of x 2 ? v 1 (x 1 , x 2 ) (in blue) coincides with the first map for x 2 ? ?1 or x 2 ? 1

, Corrugation Process with ?

, the pictures

;. .. Cross-cap, Plücker's conoid, p.73

.. .. Plücker's-conoid,

, Desingularization of the Plücker's Conoid obtained as an image of M 2 by f 1 with N = 5.5 and ?(x 1 , x 2 ) = 0.5 (sin(0.5?x 1 ) + 1)? max (x 2 ), ?(x) = ?0 2 (cos(?x 1 + ?) + 1)

]. .. , 77 3.11 Immersions of RP 2 via an inversion. Left: the center of a Plücker's Conoid and its inversion for O I = (0, 0, 4) and k = ?12. Right: the center of a desinfularized Plücker's conoid and its inversion with the same parameters

, Immersions of RP 2 via an inversion. A desingularization of the Plücker's conoid and its inversion