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Many Hamiltonian problems in the Solar System are separable or separate into two analytically solvable parts, and thus give a great chance to the development and application of explicit symplectic integrators based on operator splitting and composing. However, such constructions cannot in general be available for curved spacetimes in general relativity and modified theories of gravity, because these curved spacetimes correspond to nonseparable Hamiltonians without the two part splits. Recently, several black hole spacetimes such as the Schwarzschild black hole were found to allow the construction of explicit symplectic integrators, since their corresponding Hamiltonians are separable into more than two explicitly integrable pieces. Although some other curved spacetimes including the Kerr black hole do not exist such multi part splits, their corresponding appropriate time transformation Hamiltonians do. In fact, the key problem for the obtainment of symplectic analytically integrable decomposition algorithms is how to split these Hamiltonians or time transformation Hamiltonians. Considering this idea, we develop explicit sympelcetic schemes in curved spacetimes. We introduce a class of spacetimes whose Hamiltonians are directly split into several explicitly integrable terms. For example, the Hamiltonian of rotating black ring has a 13 part split. We also present two sets of spacetimes whose appropriate time transformation Hamiltonians have the desirable splits. For instance, an 8 part split exists in a time-transformed Hamiltonian of Kerr-Newman solution with disformal parameter. In this way, the proposed symplectic splitting methods will be used widely for long-term integrations of orbits in most curved spacetimes we have known., Comment: 16pages,accepted by Astrophysical Journal

Water lettuce (Pistia stratiotes) is one of the most well-known aquatic weeds as it causes problems in watercourses all over the world. This invasive species is fast-growing and thus has the potential for its use in preparing growth medium of microorganisms. Toward it, the pretreatment and enzymatic saccharification are positively the decisive processes. But there are other crucial processes, which are usually ignored by researchers. To the best of my knowledge, this presented work is the first time discovering that the juice obtained from water lettuce is valuable. Except for sterilization, no treatment is needed for the water lettuce’s juice and it can be directly served for cell growth. For Escherichia coli, Saccharomyces cerevisiae, Pichia pastoris, Bacillus subtilis, Lactococcus Lactis, Lactobacillus rhamnosus, and Lactobacillus plantarum, the cell density in the broth of water lettuce’s juice as the only carbon source is 10–60% higher than that in LB, YPD, BHI, M17, MRS. We then examined the production of microbial lipid by YM prepared in the juice of water lettuce. In comparison to commercial medium YM broth, the YM with juice caused 84% increase in the production of microbial lipid. A simple process of collecting juice notably increased productivity.

Since the evolution of a mixed state in a unitary system is equivalent to the joint evolution of the eigenvectors contained in it, we could use the tool of instantaneous angular velocity for pure states to study the quantum speed limit (QSL) of a mixed state. We derive a lower bound for the evolution time of a mixed state to a target state in a unitary system, which automatically reduces to the quantum speed limit induced by the Fubini–Study metric for pure states. The computation of the QSL of a degenerate mixed state is more complicated than that of a non-degenerate mixed state, where we have to make a singular value decomposition (SVD) on the inner product between the two eigenvector matrices of the initial and target states. By combing these results, a lower bound for the evolution time of a general mixed state is presented. In order to compare the tightness among the lower bound proposed here and lower bounds reported in the references, two examples in a single-qubit system and in a single-qutrit system are studied analytically and numerically, respectively. All conclusions derived in this work are independent of the eigenvalues of the mixed state, which is in accord with the evolution properties of a quantum unitary system.

We introduce a metric in a geometric way for measuring the distance between two quantum states and derive the instantaneous angular velocity of a quantum state's dynamical evolution, which has two components: $\mathrm{\ensuremath{\Delta}}{E}_{t}/\ensuremath{\hbar}$ in the longitudinal direction and ${\overline{E}}_{t}/\ensuremath{\hbar}$ in the latitudinal direction. By applying the instantaneous angular velocities in the dynamical evolution governed by a time-independent Hamiltonian, the fastest way for a pure state to evolve to a target state is discussed. As a consequence, we generalize the Mandelstam-Tamm and Margolus-Levitin bounds without using the Heisenberg uncertainty relation and point out the conditions for saturating the two bounds. The fastest dynamical evolution of a quantum system with a particular time-dependent Hamiltonian is also briefly discussed. The quantum features in the dynamical evolution are revealed through phase change and lifetime of the initial state.

For each celestial body of multi-planet systems, there are two slowly varying quantities or quasi-integrals, Kepler energy and Laplace integral, which are closely associated with the orbital semimajor axis and eccentricity, respectively. To correct numerical errors of the quantities, we give an extension of Nacozy's approach and develop a new manifold correction method, where corresponding reference values of the quantities at every integration step are obtained from integral invariant relations, and only velocity corrections are used to approximately satisfy the two quasi-integrals. As a result, the scheme does enhance the quality of the integration by significantly raising the accuracy of the two elements. Especially, it is superior to the existing dual scaling method in the improvement of eccentricity in general when the adopted integrator provides a sufficient precision to the eccentricity.

Sufficient and necessary conditions for the stability of the libration points in the perturbed circular restricted three-body problem are given. The conditions are in terms of the coefficients of the characteristic equations of the variational equations of the libration points, and can be used very conveniently. The effect of the perturbation of an arbitrary external force on the stability of the triangular libration points in the classical circular restricted three-body problem and the effect of the perturbation of the inertial drag on the stability of the principal libration points in the Robe circular restricted three-body problem are discussed by taking advantage of the obtained conditions.