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The eikonal approximation (EA) is widely used in various high-energy scattering problems. In this work we generalize this approximation from the scattering problems with time-independent Hamiltonian to the ones with periodical Hamiltonians, {\it i.e.}, the Floquet scattering problems. We further illustrate the applicability of our generalized EA via the scattering problem with respect to a shaking spherical square-well potential, by comparing the results given by this approximation and the exact ones. The generalized EA we developed is helpful for the research of manipulation of high-energy scattering processes with external field, {\it e.g.}, the manipulation of atom, molecule or nuclear collisions or reactions via strong laser fields.

This work delves into the exponential time differencing (ETD) schemes for the matrix-valued Allen-Cahn equation. In fact, the maximum bound principle (MBP) for the first- and second-order ETD schemes is presented in a prior publication [SIAM Review, 63(2), 2021], assuming a symmetric initial matrix field. Noteworthy is our novel contribution, demonstrating that the first- and second-order ETD schemes for the matrix-valued Allen-Cahn equation -- both being linear schemes -- unconditionally preserve the MBP, even in instances of nonsymmetric initial conditions. Additionally, we prove that these two ETD schemes preserve the energy dissipation law unconditionally for the matrix-valued Allen-Cahn equation. Some numerical examples are presented to verify our theoretical results and to simulate the evolution of corresponding matrix fields.

Mobility edge (ME), representing the critical energy that distinguishes between extended and localized states, is a key concept in understanding the transition between extended (metallic) and localized (insulating) states in disordered and quasiperiodic systems. Here we explore the impact of dissipation on a quasiperiodic system featuring MEs by calculating steady-state density matrix and analyzing quench dynamics with sudden introduction of dissipation, and demonstrate that dissipation can lead the system into specific states predominantly characterized by either extended or localized states, irrespective of the initial state. Our results establish the use of dissipation as a new avenue for inducing transitions between extended and localized states, and for manipulating dynamic behaviors of particles.

Extreme multistability (EM) is characterized by the emergence of infinitely many coexisting attractors or continuous families of stable states in dynamical systems. EM implies complex and hardly predictable asymptotic dynamical behavior. We analyse a model for pendulum clocks coupled by springs and suspended on an oscillating base, and show how EM can be induced in this system by a specifically designed coupling. First, we uncover that symmetric coupling can increase the dynamical complexity. In particular, the coexistence of multiple isolated attractors and continuous families of stable periodic states is generated in a symmetric cross-coupling scheme of four pendulums. These coexisting infinitely many states are characterized by different levels of phase synchronization between the pendulums, including anti-phase and in-phase states. Some of the states are characterized by splitting of the pendulums into groups with silent sub-threshold and oscillating behavior, respectively. The analysis of the basins of attraction further reveals the complex dependence of EM on initial conditions., Comment: 13 pages, 9 figures

Objective: Convolutional Neural Networks (CNNs) have shown great potential in the field of Brain-Computer Interfaces (BCIs). The raw Electroencephalogram (EEG) signal is usually represented as 2-Dimensional (2-D) matrix composed of channels and time points, which ignores the spatial topological information. Our goal is to make the CNN with the raw EEG signal as input have the ability to learn EEG spatial topological features, and improve its performance while essentially maintaining its original structure. Methods:We propose an EEG Topographic Representation Module (TRM). This module consists of (1) a mapping block from the raw EEG signal to a 3-D topographic map and (2) a convolution block from the topographic map to an output of the same size as input. According to the size of the kernel used in the convolution block, we design 2 types of TRMs, namely TRM-(5,5) and TRM-(3,3). We embed the TRM into 3 widely used CNNs, and tested them on 2 publicly available datasets (Emergency Braking During Simulated Driving Dataset (EBDSDD), and High Gamma Dataset (HGD)). Results: The results show that the classification accuracies of all 3 CNNs are improved on both datasets after using the TRM. With TRM-(5,5), the average accuracies of DeepConvNet, EEGNet and ShallowConvNet are improved by 6.54%, 1.72% and 2.07% on EBDSDD, and by 6.05%, 3.02% and 5.14% on HGD, respectively; with TRM-(3,3), they are improved by 7.76%, 1.71% and 2.17% on EBDSDD, and by 7.61%, 5.06% and 6.28% on HGD, respectively. Significance: We improve the classification performance of 3 CNNs on 2 datasets by the use of TRM, indicating that it has the capability to mine the EEG spatial topological information. In addition, since the output of TRM has the same size as the input, CNNs with the raw EEG signal as input can use this module without changing their original structures.

Accurately detecting and identifying drivers' braking intention is the basis of man-machine driving. In this paper, we proposed an electroencephalographic (EEG)-based braking intention measurement strategy. We used the Car Learning to Act (Carla) platform to build the simulated driving environment. 11 subjects participated in our study, and each subject drove a simulated vehicle to complete emergency braking and normal braking tasks. We compared the EEG topographic maps in different braking situations and used three different classifiers to predict the subjects' braking intention through EEG signals. The experimental results showed that the average response time of subjects in emergency braking was 762 ms; emergency braking and no braking can be well distinguished, while normal braking and no braking were not easy to be classified; for the two different types of braking, emergency braking and normal braking had obvious differences in EEG topographic maps, and the classification results also showed that the two were highly distinguishable. This study provides a user-centered driver-assistance system and a good framework to combine with advanced shared control algorithms, which has the potential to be applied to achieve a more friendly interaction between the driver and vehicle in real driving environment.