Some exact solutions and multi-mode invariant submanifolds were found for the Fermi-Pasta-Ulam (FPU) β-model by Poggi and Ruffo in Phys. D 103 (1997) 251. In the present paper we demonstrate how results of such a type can be obtained for an arbitrary Nparticle chain with periodic boundary conditions with the aid of our group-theoretical approach [Phys. D 117 (1998) 43] based on the concept of bushes of normal modes in mechanical systems with discrete symmetry. The integro-differential equation describing the FPU-α dynamics in the modal space is derived. The loss of stability of the bushes of modes for the FPU-α model, in particular, for the limiting case N → ∞ for the dynamical regime with displacement pattern having period twice the lattice spacing (π-mode) is studied. Our results for the FPU-α chain are compared with those by Poggi and Ruffo for the FPU-β chain.

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The paper describes an approach that allows solving the problem of extrinsic calibration of a multi-beam lidar and a stereo camera. The approach does not impose any restrictions on the place in which calibration should be performed. Calibration is performed using a calibration board, which is a flat rectangle with special markers. Three-dimensional correspondences are used for calibration. First, a search for the three-dimensional coordinates of the corner points of the calibration board in the coordinate systems of the stereo pair cameras as well as in the coordinate system of the lidar is made. Next, using the optimization methods, calibration parameters are calculated. The results of a series of virtual and real experiments show that the algorithm allows the calibration to be performed with an accuracy comparable to that of sensors. The proposed approach allows one to improve the calibration accuracy due to the simultaneous use of information from two cameras of the stereo pair and is suitable for lidars with both the low and high point density.