### A bornological approach to rotundity and smoothness applied to approximation.

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We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

Suppose that, for each point x in a given subset E ⊂ Rn, we are given an m-jet f(x) and a convex, symmetric set σ(x) of m-jets at x. We ask whether there exist a function F ∈ Cm,w(Rn) and a finite constant M, such that the m-jet of F at x belongs to f(x) + Mσ(x) for all x ∈ E. We give a necessary and sufficient condition for the existence of such F, M, provided each σ(x) satisfies a condition that we call "Whitnet w-convexity".

We prove that the expansion of the real field by a restricted C${}^{\infty}$-function is generically o-minimal. Such a result was announced by A. Grigoriev, and proved in a different way. Here, we deduce quasi-analyticity from a transcendence condition on Taylor expansions. This then implies o-minimality. The transcendance condition is shown to be generic. As a corollary, we recover in a simple way that there exist o-minimal structures that doesn’t admit analytic cell decomposition, and that there exist incompatible...

The global convergence of a direct method for determining turning (limit) points of a parameter-dependent mapping is analysed. It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is clasified as a $bifurcationsingularity$ (i.e., as a $degenerate$ turning point). Then, the Theorz for Imperfect Bifurcation offers a particular scenario for the split of the singular root into a finite number of regular roots (turning points) due to a given parameter imperfection. The relationship...