A dynamical model of remote-control model cars

Simple experiments for which differential equations cannot be solved analytically can be addressed using an effective model that satisfactorily reproduces the experimental data. In this work, the one-dimensional kinematics of a remote-control model (toy) car was studied experimentally and its dynamical equation modelled. In the experiment, maximum power was applied to the car, initially at rest, until it reached its terminal velocity. Digital video recording was used to obtain the relevant kinematic variables that enabled to plot trajectories in the phase space. A dynamical equation of motion was proposed in which the overall frictional force was modelled as an effective force proportional to the velocity raised to the power of a real number. Since such an equation could not be solved analytically, a dynamical model was developed and the system parameters were calculated by non-linear fitting. Finally, the resulting values were substituted in the motion equation and the numerical results thus obtained were compared with the experimental data, corroborating the accuracy of the model.Comment: 5 pages, 9 figure

Long-lived protoplanetary disks in multiple systems: the VLA view of HD 98800

The conditions and evolution of protoplanetary disks in multiple systems can be considerably different from those around single stars, which may have important consequences for planet formation. We present Very Large Array (VLA) 8.8 mm (34 GHz) and 5 cm (6 GHz) observations of the quadruple system HD 98800, which consists of two spectroscopic binary systems (Aa-Ab, Ba-Bb). The Ba-Bb pair is surrounded by a circumbinary disk, usually assumed to be a debris disk given its $\sim$10 Myr age and lack of near infrared excess. The VLA 8.8 mm observations resolve the disk size (5-5.5 au) and its inner cavity ($\approx$3 au) for the first time, making it one of the smallest disks known. Its small size, large fractional luminosity, and millimeter spectral index consistent with blackbody emission support the idea that HD 98800 B is a massive, optically thick ring which may still retain significant amounts of gas. The disk detection at 5 cm is compatible with free-free emission from photoionized material. The diskless HD 98800 A component is also detected, showing partial polarization at 5 cm compatible with non-thermal chromospheric activity. We propose that tidal torques from Ba-Bb and A-B have stopped the viscous evolution of the inner and outer disk radii, and the disk is evolving via mass loss through photoevaporative winds. This scenario can explain the properties and longevity of HD 98800 B as well as the lack of a disk around HD 98800 A, suggesting that planet formation could have more time to proceed in multiple systems than around single stars in certain system configurations.Comment: 14 pages, 4 figures, 3 tables; Submitted to ApJ May 14 2018; Accepted to ApJ August 3 2018. This version fixes a mistake in the reported position angle. The order of the figures has been changed to match that of the references in the tex

Three-structured smooth transition regression models based on CART algorithm

In the present work, a tree-based model that combines aspects of CART (Classification and Regression Trees) and STR (Smooth Transition Regression) is proposed. The main idea relies on specifying a parametric nonlinear model through a tree-growing procedure. The resulting model can be analysed either as a fuzzy regression or as a smooth transition regression with multiple regimes. Decisions about splits are entirely based on statistical tests of hypotheses and confidence intervals are constructed for the parameters within the terminal nodes as well as the final predictions. A Monte Carlo Experiment shows the estimators’ properties and the ability of the proposed algorithm to identify correctly several tree architectures. An application to the famous Boston Housing dataset shows that the proposed model provides better explanation with the same number of leaves as the one obtained with the CART algorithm.