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Flow, an optimal mental state merging action and awareness, significantly impacts our emotion, performance, and well-being. However, capturing its swift fluctuations on a fine timescale is challenging due to the sparsity of the existing flow detecting tools. Here we present a fine fingertip force control (F3C) task to induce flow, wherein the task challenge is set at a compatible level with personal skill, and to quantitatively track the flow state variations from synchronous motor control performance. We extract eight performance metrics from fingertip force sequence and reveal their significant differences under distinct flow states. Further, we built a learning-based flow decoder that aims to predict the continuous flow intensity during the user experiment through the selected performance metrics, taking the self-reported flow as the label. Cross-validation shows that the predicted flow intensity reaches significant correlation with the self-reported flow intensity (r=0.81). Based on the decoding results, we observe rapid oscillations in flow fluctuations during the intervals between sparse self-reporting probes. This study showcases the feasibility of tracking intrinsic flow variations with high temporal resolution using task performance measures and may serve as foundation for future work aiming to take advantage of flow' s dynamics to enhance performance and positive emotions.

Using the FermiDirac distribution function, Balart and Vagenas gave a charged spherically symmetric regular black hole, which is a solution of Einstein field equations coupled to a nonlinear electrodynamics. In fact, the regular black hole is a Reissner-Nordstrom (RN) black hole when a metric function is given a Taylor expansion to first order approximations. It does not have a curvature singularity at the origin,but the RN black hole does. Both black hole metrics have horizons and similar asymptotic behaviors, and satisfy the weak energy conditions everywhere. They are almost the same in photon effective potentials, photon circular orbits and photon spheres outside the event horizons. There are relatively minor differences between effective potentials, stable circular orbits and innermost stable circular orbits of charged particles outside the event horizons of the two black holes immersed in external magnetic fields. Although the twomagnetized black holes allow different construction methods of explicit symplectic integrators, they exhibit approximately consistent results in the regular and chaotic dynamics of charged particles outside the event horizons. Chaos gets strong as the magnetic field parameter or the magnitude of negative Coulomb parameter increases, but becomes weak when the black hole charge or the positive Coulomb parameter increases. A variation of dynamical properties is not sensitive dependence on an appropriate increase of the black hole charge. The basic equivalence between the two black hole gravitational systems in the dynamics of orbits outside the event horizons is due to the two metric functions having an extremely small difference. This implies that the RN black hole is reasonably replaced by the regular black hole without curvature singularity in many situations., Comment: 18 pages, 12 figures

Based on the scalar-tensor-vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is static spherical symmetric, it is integrable. However, the spherical symmetry and the integrability are destroyed when the black hole is immersed in an external asymptotic uniform magnetic field and the particle is charged. Although the magnetized modified Schwarzschild spacetime is nonintegrable and inseparable, it allows for the application of explicit symplectic integrators when its Hamiltonian is split into five explicitly integrable parts. Taking one of the proposed explicit symplectic integrators and the techniques of Poincare sections and fast Lyapunov indicators as numerical tools, we show that the charged particle can do chaotic motions under some circumstances. Chaos is strengthened with an increase of the modified gravity parameter from the global phase space structures. There are similar results when the magnetic field parameter and the particle energy increase. However, an increase of the particle angular momentum weakens the strength of chaos., Comment: 12 pages,5 figures

In recent publications, the construction of explicit symplectic integrators for Schwarzschild and Kerr type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various choices. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best one of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge-Kutta and Runge-Kutta-Nystr\"{o}m explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The former algorithms need small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge-Kutta and Runge-Kutta-Nystr\"{o}m explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracies during long-term integrations due to roundoff errors. The idea finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including the Kerr type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge-Kutta and Runge-Kutta-Nystr\"{o}m methods are worth recommending., Comment: 10 pages, 5 figures. accepted for publication in ApJ

A small deformation controlled by four free parameters to the Schwarzschild metric could be referred to a nonspinning black hole solution in alternative theories of gravity. Because such a non-Schwarzschild metric can be changed into a Kerr-like black hole metric via a complex coordinate transformation, the recently proposed time-transformed explicit symplectic integrators for the Kerr type spacetimes are suitable for a Hamiltonian system describing the motion of charged particles around the non-Schwarzschild black hole surrounded with an external magnetic field. The obtained explicit symplectic methods are based on a time-transformed Hamiltonian split into seven parts, whose analytical solutions are explicit functions of new coordinate time. Numerical tests show that such explicit symplectic integrators for intermediate time-steps perform good long-term performance in stabilizing Hamiltonian errors regardless of regular or chaotic orbits. One of the explicit symplectic integrators with the techniques of Poincar\'{e} sections and fast Lyapunov indicators is applied to investigate the effects of the parameters including the four free deformation parameters on the orbital dynamical behavior. From the global phase-space structure, chaotic properties are typically strengthened under some circumstances as any one of the energy and the magnitudes of magnetic parameter and negative deformation parameters increases. However, they are weakened when each of the angular momentum and positive deformation parameters increases., Comment: 27 pages, 14 figures