Let $X$ be a smooth complex projective variety. Using construction devised to Gathmann, we present recursive formula for some of the Gromov-Witten invariants $X$. We prove that, when is homogeneous, this gives number osculating rational curves at general point hypersurface This generalizes classical well known pairs inflexion (asymptotic) lines surfaces in $\mathbb{P}^{3}$ Salmon, as Darboux's ...

THEOREM. In order that a system S of o I curves represents the additivemultiplicative trajectories ofa given field, it is necessary (but not sufficient) that S possesses the dynamical Property I; this may be stated simply in either of these alternative geometric forms: Property Ia: For the o I curves of S passing through any lineal element E of the plane, construct the osculating parabolas at E...

In part II of this group of papers, the control of the gait of a biped robot during rectilinear walk was considered. The modeling approach and simulation, using Kane’s method with implementation leveraged by Autolev, a symbolic computational environment that is complementary, was discussed in part I. Performing turns during the walk is technically more complex than the rectilinear case and dese...

Etienne Ghys has recently discovered a beautiful theorem: given a diffeomorphism of the projective line, there exist at least four distinct points in which the diffeomorphism is unusually well approximated by projective transformations [8]. The points in question are the ones in which the 3-jet of the diffeomorphism is that of a projective transformation; in a generic point the order of approxi...

We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real. Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified pos...

In the method of variation of parameters, we express the Cartesian coordinates or the Euler angles via the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants.” Out of convenience, the Lagrange constr...

The Poynting-Robertson (P-R) effect has been applied to meteoroids for several decades. It is well known that the P-R effect produces only changes in the orbital plane of the particle. The differential equations governing the secular changes in both eccentricity and semi-major axis are known since the time of Robertson, provided the initial orbits are not near circular. A disadvantage of this t...