Root stacks and periodic decompositions

For an effective Cartier divisor D on a scheme X we may form an nth root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is 2n-periodic. For n=2 this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of D. For n>2 we find a higher spherical functor in the sense of recent work of Dyckerhoff, Kapranov and Schechtman. We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.Comment: 21 pages. Revised introductio

Contraction algebra and invariants of singularities

In [7], Donovan and Wemyss introduced the contraction algebra of flop- ping curves in 3-folds. When the flopping curve is smooth and irreducible, we prove that the contraction algebra together with its A_\infty-structure recovers various invariants associated to the underlying singularity and its small resolution, including the derived category of singularities and the genus zero Gopakuma-Vafa invariants

Application of the vibration test in the evaluation of the technical condition of shock absorbers built into the vehicle

The paper presents an assessment of the technical condition of the vehicle shock absorbers built into the vehicle on the basis of the signals recorded during the vibration test. The test was performed on a test stand with kinematic harmonic forcing. The test stand can enforce vibrations in the range of 0-21 Hz and amplitude of 6 mm. The shock absorbers with indicated technical condition were built into the vehicle (the new shock absorber and the one with oil leak). The research was conducted on the twin-tube hydraulic shock with a modified structure that allows to change the amount of the fluid. The study was conducted for the front suspension shock absorber of passenger car with an independent McPherson front suspension. Acceleration was measured using capacitive sensors of accelerations type ADXL. Acceleration was recorded on test stand platform and near the axis of the wheel and on the upper point of attachment of the McPherson strut. Registered acceleration diagrams were subject to further treatment consisting of filtration and analysis. The results were compared for selected technical states