Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature

Let (M, g) be a compact, d -dimensional Riemannian manifold without boundary. Suppose further that (M, g) is either two dimensional and has no conjugate points or (M, g) has non-positive sectional curvature. The goal of this note is to show that the long time parametrix obtained for such manifolds by Bérard can be used to prove a logarithmic improvement for the remainder term of the Riesz means of the counting function of the Laplace operator

Repositioning forward-leaning passengers by seatbelt pre-pretensioning

Objective: The study determined the seatbelt pre-pretensioner force needed and the time required to reposition average male front-seat passengers from forward-leaning to upright using finite element simulations of the Active SAFER Human Body Model (Active SHBM). Methods: The Active SHBM was positioned in an initial forward-leaning position (29\ub0 forward from upright) on a deformable vehicle seat. A pre-pretensioner was modeled as a pre-loaded spring and its ability to reposition the forward-leaning Active SHBM to an upright position was simulated for twenty-four different pre-crash conditions. Four parameters were varied: (1) Automated Emergency Braking (AEB) active with 11 m/s2 or no AEB, (2) type of seatbelt system: Belt-In-Seat or B-pillar, (3) pre-pretensioner activation time (200 ms before, 100 ms before, or at the same time as AEB ramp-up), and (4) pre-pretensioner force (200 N, 300 N, 400 N, 600 N). The first thoracic vertebra fore-aft (T1 X) trajectories were compared against a reference upright position to determine the force and time needed to reposition and the effectiveness of repositioning in the different conditions. Results: The lowest force enabling repositioning in all simulations was 400 N (no AEB, Belt-In-Seat). It took about 350 ms. In the presence of AEB, activating the pre-pretensioner 200 ms before AEB and using 600 N pre-pretensioner force was needed for repositioning (taking 200 ms with Belt-In-Seat and 260 ms with B-pillar installations). Repositioning was faster and thus more effective with the Belt-In-Seat seatbelt in all simulations. Conclusions: All four parameters (presence of AEB, type of seatbelt system, pre-pretensioner activation time and force) affected the repositioning ability and time required. Far from all combinations repositioned a forward-leaning average male occupant model, but those found to be effective and fast appear as a feasible option for vehicle safety systems to reposition out-of-position occupants during pre-crash events