The newtonian and special-relativistic statistical predictions for the mean, standard deviation and probability density function of the position and momentum are compared for the periodically-delta-kicked particle at low speed. Contrary to expectation, we find that the statistical predictions, which are calculated from the same parameters and initial gaussian ensemble of trajectories, do not always agree if the initial ensemble is sufficiently well-localized in phase space. Moreover, the breakdown of agreement is very fast if the trajectories in the ensemble are chaotic, but very slow if the trajectories in the ensemble are non-chaotic. The breakdown of agreement implies that special-relativistic mechanics must be used, instead of the standard practice of using newtonian mechanics, to correctly calculate the statistical predictions for the dynamics of a low-speed system.
We show, contrary to expectation, that the trajectory predicted by general-relativistic mechanics for a low-speed weak-gravity system is not always well-approximated by the trajectories predicted by special-relativistic and newtonian mechanics for the same parameters and initial conditions. If the system is dissipative, the breakdown of agreement occurs for chaotic trajectories only. If the system is non-dissipative, the breakdown of agreement occurs for chaotic trajectories and non-chaotic trajectories. The agreement breaks down slowly for non-chaotic trajectories but rapidly for chaotic trajectories. When the predictions are different, general-relativistic mechanics must therefore be used, instead of special-relativistic mechanics (newtonian mechanics), to correctly study the dynamics of a weak-gravity system (a low-speed weak-gravity system).
The statistical predictions of Newtonian and special-relativistic mechanics, which are calculated from an initially Gaussian ensemble of trajectories, are compared for a low-speed scattering system. The comparisons are focused on the mean dwell time, transmission and reflection coefficients, and the position and momentum means and standard deviations. We find that the statistical predictions of the two theories do not always agree as conventionally expected. The predictions are close if the scattering is non-chaotic but they are radically different if the scattering is chaotic and the initial ensemble is well localized in phase space. Our result indicates that for low-speed chaotic scattering, special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to obtain empirically-correct statistical predictions from an initially well-localized Gaussian ensemble.
Structural magnetic resonance imaging (sMRI) is a popular technique that is widely applied in Alzheimer's disease (AD) diagnosis. However, only a few structural atrophy areas in sMRI scans are highly associated with AD. The degree of atrophy in patients' brain tissues and the distribution of lesion areas differ among patients. Therefore, a key challenge in sMRI-based AD diagnosis is identifying discriminating atrophy features. Hence, we propose a multiplane and multiscale feature-level fusion attention (MPS-FFA) model. The model has three components, (1) A feature encoder uses a multiscale feature extractor with hybrid attention layers to simultaneously capture and fuse multiple pathological features in the sagittal, coronal, and axial planes. (2) A global attention classifier combines clinical scores and two global attention layers to evaluate the feature impact scores and balance the relative contributions of different feature blocks. (3) A feature similarity discriminator minimizes the feature similarities among heterogeneous labels to enhance the ability of the network to discriminate atrophy features. The MPS-FFA model provides improved interpretability for identifying discriminating features using feature visualization. The experimental results on the baseline sMRI scans from two databases confirm the effectiveness (e.g., accuracy and generalizability) of our method in locating pathological locations. The source code is available at https://github.com/LiuFei-AHU/MPSFFA.