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By Gianna Stefani, Ugo Boscain, Jean-Paul Gauthier, Andrey Sarychev, Mario Sigalotti

Honoring Andrei Agrachev's sixtieth birthday, this quantity offers contemporary advances within the interplay among Geometric keep an eye on thought and sub-Riemannian geometry. at the one hand, Geometric keep an eye on idea used the differential geometric and Lie algebraic language for learning controllability, movement making plans, stabilizability and optimality for keep watch over structures. The geometric process grew to become out to be fruitful in purposes to robotics, imaginative and prescient modeling, mathematical physics and so forth. however, Riemannian geometry and its generalizations, corresponding to sub-Riemannian, Finslerian geometry etc., were actively adopting equipment built within the scope of geometric regulate. software of those tools has resulted in very important effects relating to geometry of sub-Riemannian areas, regularity of sub-Riemannian distances, houses of the gang of diffeomorphisms of sub-Riemannian manifolds, neighborhood geometry and equivalence of distributions and sub-Riemannian buildings, regularity of the Hausdorff quantity, etc.

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Complexity of intersection of real quadrics and topology of symmetric determinantal varieties. 1444 17. : A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91, American Mathematical Society, Providence, RI (2002) 18. : Lectures on the h-cobordismtheorem. , Princeton Math. Notes (1965) How to Run a Centipede: a Topological Perspective Yuliy Baryshnikov and Boris Shapiro Abstract In this paper we study the topology of the configuration space of a device with d legs (“centipede”) under some constraints, such as the impossibility to have more than k legs off the ground.

The main idea of Agrachev’s approach is to study the Lebesgue sets of the positive inertia index function on W; i. e. q/ of a symmetric matrix representing q. W; W j / are just the canditates for the homology of XW . XW / are algebro-topological conditions. The way to make these statements precise is to use the language of spectral sequences (the above conditions on the canditates translate into them being in the kernels of the differentials of the spectral sequence). The reader is referred to [6] for a detailed treatment.

The i -th open pyramid Pyri consists then of exactly those leg positions, for which the i -th leg has the maximal height, i. e. 1 cos i > 1 cos j ; 8j ¤ i . i 1/-st leg and the i -th leg are at the maximal height among all legs. Additionally, their positions are not allowed to coincide ( i ¤ j ) and the corresponding angles are in the correct cyclic position (i. e. i > i C1 ). The Fig. 3 illustrates the positions in the cut and outside it. Proof (Proof of the Theorem 1). The torus T d with the cut Cod deleted can be constructed by identifying the d pyramids Pyri ; i D 1; : : : ; d along the pairs of exitentrance faces: the exit face of the pyramid Pyri is identified with the entrance face of Pyri C1 .

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