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This paper presents Auto-RubikAI, a modular autonomous planning framework that integrates a symbolic Knowledge Base (KB), a vision-language model (VLM), and a large language model (LLM) to solve structured manipulation tasks exemplified by Rubik's Cube restoration. Unlike traditional robot systems based on predefined scripts, or modern approaches relying on pretrained networks and large-scale demonstration data, Auto-RubikAI enables interpretable, multi-step task execution with minimal data requirements and no prior demonstrations. The proposed system employs a KB module to solve group-theoretic restoration steps, overcoming LLMs' limitations in symbolic reasoning. A VLM parses RGB-D input to construct a semantic 3D scene representation, while the LLM generates structured robotic control code via prompt chaining. This tri-module architecture enables robust performance under spatial uncertainty. We deploy Auto-RubikAI in both simulation and real-world settings using a 7-DOF robotic arm, demonstrating effective Sim-to-Real adaptation without retraining. Experiments show a 79% end-to-end task success rate across randomized configurations. Compared to CFOP, DeepCubeA, and Two-Phase baselines, our KB-enhanced method reduces average solution steps while maintaining interpretability and safety. Auto-RubikAI provides a cost-efficient, modular foundation for embodied task planning in smart manufacturing, robotics education, and autonomous execution scenarios. Code, prompts, and hardware modules will be released upon publication.

Discretizing the Dirac equation on a uniform grid with the central difference formula often generates spurious states. We propose a staggered-grid scheme in the framework of the finite-difference method that suppresses these spurious states without introducing Wilson terms or ad-hoc filtering. In this approach, the large and small components of the Dirac equation are placed on interlaced nodes, and the first-order derivatives are evaluated between staggered points, yielding a Hamiltonian that breaks the unitary transformation between $H_\kappa$ and $H_{-\kappa}$. Benchmarks with the nuclear Woods-Saxon potentials demonstrate one-to-one agreement with the eigenvalues obtained from shooting method and asymmetric finite-difference method, rapid convergence for weakly bound states, and reduced box-size sensitivity. The method retains the simplicity of central differences and standard matrix diagonalization, while naturally extending to higher-order and multi-dimension systems. It provides a compact and efficient tool for relativistic bound-state and scattering calculations.

The radial oscillations of neutron stars are studied using equations of state derived from density-dependent relativistic mean-field (DDRMF) models, which effectively describe the ground-state properties of finite nuclei. A novel numerical approach, the finite volume method (FVM), is employed to solve the eigenvalue problem associated with oscillation frequencies. Compared to conventional methods such as the finite difference method and shooting method, the FVM avoids the numerical instability encountered at high frequencies with an equation of state that includes a discontinuous adiabatic index and offers greater computational efficiency. The oscillation frequencies of high-order modes exhibit a similar trend of change. The radial displacements and pressure perturbations are largely influenced by the EOSs of crust region. {The frequency of the first excited state shows a strong linear relationship with both the slope and skewness parameters of the symmetry energy.} These findings suggest that the density dependence of the symmetry energy can be constrained through observations of neutron star radial oscillation frequencies.

Polygonal collision avoidance (PCA) is short for the problem of collision avoidance between two polygons (i.e., polytopes in planar) that own their dynamic equations. This problem suffers the inherent difficulty in dealing with non-smooth boundaries and recently optimization-defined metrics, such as signed distance field (SDF) and its variants, have been proposed as control barrier functions (CBFs) to tackle PCA problems. In contrast, we propose an optimization-free smooth CBF method in this paper, which is computationally efficient and proved to be nonconservative. It is achieved by three main steps: a lower bound of SDF is expressed as a nested Boolean logic composition first, then its smooth approximation is established by applying the latest log-sum-exp method, after which a specified CBF-based safety filter is proposed to address this class of problems. To illustrate its wide applications, the optimization-free smooth CBF method is extended to solve distributed collision avoidance of two underactuated nonholonomic vehicles and drive an underactuated container crane to avoid a moving obstacle respectively, for which numerical simulations are also performed.

The nucleon-nucleon ($NN$) potential is the residual interaction of the strong interaction in the low-energy region and is also the fundamental input to the study of atomic nuclei. Based on the non-perturbative properties of the quantum chromodynamics (QCD), $NN$ potential is not yet directly accessible from QCD theory. Therefore, various models of $NN$ interactions have been constructed based on Yukawa's meson exchange pictures since the 1930s, including one-boson-exchange models, coordinate operator models and chiral effective field models. Analysis of extensive $NN$ scattering data has shown that the two-body nuclear force exhibits a short-range repulsion and intermediate-range attraction, and decays rapidly with increasing distance. A series of charge-dependent high-precision $NN$ interactions have been further developed in the past thirty years, such as the AV18 potential, CD-Bonn potential, pvCD-Bonn potentials, and the chiral effective nuclear potentials with momentum expansion up to the fifth order. In this work, the phase shifts at different channels, the cross sections, the entanglement entropy in spin space, and the equations of state of symmetric nuclear matter and pure neutron matter from these high-precision $NN$ interactions are calculated and systematically compared. It can be found that they have significant differences in the cases with high angular momentum, high laboratory energy, and high-density regions.

The parametrized equation of state (EOS) of neutron stars is investigated by Bayesian inference method with various constraints from both nuclear physics and modern astronomical observations. The expansion coefficients correspond to the properties of symmetric nuclear matter and the density dependence of the symmetry energy. The empirical values of the symmetry energy at subsaturation density and the density of crust-core phase transition are considered to limit the low-density behavior of EOS, i.e. $L_{\mathrm{sym}}$, while the speed of sound of neutron star matter and mass-radius observations of millisecond pulsars PSR J0030+0451 and PSR J0740+6620 are adopted to eliminate the high-order expansion coefficients, such as $Q_{\mathrm{sat}}$ and $Q_{\mathrm{sym}}$. Finally, our analysis reveals that the skewness coefficient $Q_{\mathrm{sat}}$ of the energy per nucleon in symmetric nuclear matter (SNM) exhibits the strongest correlation with the speed of sound, constrained to $Q_{\mathrm{sat}} = -69.50_{-31.93}^{+16.52} \, \mathrm{MeV}$, whose uncertainties are much smaller than those of the experiments of heavy-ion collisions. The symmetry energy parameters are determined as follows: slope $L_{\mathrm{sym}} = 34.32_{-11.85}^{+13.66} \, \mathrm{MeV}$, curvature $K_{\mathrm{sym}} = -58.45_{-89.46}^{+88.47} \, \mathrm{MeV}$, and skewness $Q_{\mathrm{sym}} = 302.28_{-231.89}^{+251.62} \, \mathrm{MeV}$. Additionally, the radii of canonical ($1.4 \, M_{\odot}$) and massive ($2.0 \, M_{\odot}$) neutron stars are predicted as $R_{1.4} = 11.85_{-0.15}^{+0.06} \, \text{km}$ and $R_{2.0} = 11.42_{-0.35}^{+0.23} \, \text{km}$, respectively, with a maximum mass of $M_{\mathrm{max}} = 2.12_{-0.05}^{+0.11} \, M_{\odot}$. The tidal deformability is $\Lambda_{1.4} = 303.57_{-45.22}^{+47.95}$ at $1.4 \, M_\odot$, which is consistent with the analysis of the GW170817 event.

As the residual interaction of quantum chromodynamics in low-energy region, the nucleon-nucleon (NN) potential can only be exactly described by the model picture now. In the Bonn potential, one of the most well-known NN interaction models, the nucleons interact with each other through exchanging the pion and several heavier mesons, where the pion plays an essential role. It provides a partial contribution of tensor force in the intermediate-range region and the main component in the long-range region in NN potential. However, it is very difficult to be treated in the nuclear many-body system due to its pseudovector or pseudoscalar property. Recently, three high-precision charge-dependent Bonn potentials were proposed with pseudovector coupling types and different pion-nucleon coupling strengths and applied them to study the properties of nuclear matter and neutron stars in the non-relativistic and relativistic frameworks. Furthermore, to properly deal with the strong short-range repulsion and tensor force of the NN potential, some new relativistic {\it ab initio} methods have also been developed in the past decade to discuss the role of pion and relativistic effects in nuclear matter.

Motion planning for aerial manipulators in constrained environments has typically been limited to known environments or simplified to that of multi-rotors, which leads to poor adaptability and overly conservative trajectories. This paper presents RINGO: Real-time Navigation with a Guiding Trajectory, a novel planning framework that enables aerial manipulators to navigate unknown environments in real time. The proposed method simultaneously considers the positions of both the multi-rotor and the end-effector. A pre-obtained multi-rotor trajectory serves as a guiding reference, allowing the end-effector to generate a smooth, collision-free, and workspace-compatible trajectory. Leveraging the convex hull property of B-spline curves, we theoretically guarantee that the trajectory remains within the reachable workspace. To the best of our knowledge, this is the first work that enables real-time navigation of aerial manipulators in unknown environments. The simulation and experimental results show the effectiveness of the proposed method. The proposed method generates less conservative trajectories than approaches that consider only the multi-rotor.

In this paper, the safety-critical control problem for uncertain systems under multiple control barrier function (CBF) constraints and input constraints is investigated. A novel framework is proposed to generate a safety filter that minimizes changes to reference inputs when safety risks arise, ensuring a balance between safety and performance. A nonlinear disturbance observer (DOB) based on the robust integral of the sign of the error (RISE) is used to estimate system uncertainties, ensuring that the estimation error converges to zero exponentially. This error bound is integrated into the safety-critical controller to reduce conservativeness while ensuring safety. To further address the challenges arising from multiple CBF and input constraints, a novel Volume CBF (VCBF) is proposed by analyzing the feasible space of the quadratic programming (QP) problem. % ensuring solution feasibility by keeping the volume as a positive value. To ensure that the feasible space does not vanish under disturbances, a DOB-VCBF-based method is introduced, ensuring system safety while maintaining the feasibility of the resulting QP. Subsequently, several groups of simulation and experimental results are provided to validate the effectiveness of the proposed controller.

It is of great interest to understand the equation of state (EOS) of the neutron star (NS), whose core includes highly dense matter. However, there are large uncertainties in the theoretical predictions for the EOS of NS. It is useful to develop a new framework, which is flexible enough to consider the systematic error in theoretical predictions and to use them as a best guess at the same time. We employ a deep neural network to perform a non-parametric fit of the EOS of NS using currently available data. In this framework, the Gaussian process is applied to represent the EOSs and the training set data required to close physical solutions. Our model is constructed under the assumption that the true EOS of NS is a perturbation of the relativistic mean-field model prediction. We fit the EOSs of NS using two different example datasets, which can satisfy the latest constraints from the massive neutron stars, NICER, and the gravitational wave of the binary neutron stars. Given our assumptions, we find that a maximum neutron star mass is $2.38^{+0.15}_{-0.13} M_\odot$ or $2.41^{+0.15}_{-0.14}$ at $95\%$ confidence level from two different example datasets. It implies that the $1.4 M_\odot$ radius is $12.31^{+0.29}_{-0.31}$ km or $12.30^{+0.35}_{-0.37}$ km. These results are consistent with results from previous studies using similar priors. It has demonstrated the recovery of the EOS of NS using a nonparametric model.

We develop a density-dependent quark mean-field (DDQMF) model to study the properties of nuclear matter and neutron stars, where the coupling strength between $\sigma$ meson and nucleon is generated by the degree of freedom of quarks, while other meson coupling constants are regarded as density-dependent ones. Two values for the nucleon effective mass, $M^*_{N0}/M_N=0.556,~0.70$ at the saturation density are chosen based on the consideration of the core-collapse supernova simulation and finite nuclei when the meson-nucleon coupling constants are fixed. We find that the equation of state (EOS) of nuclear matter, the symmetry energy, the mass-radius relations, and the tidal deformabilities of neutron stars with larger nucleon effective mass are more sensitive to the skewness coefficient $J_0$. The EOSs with $M^*_{N0}/M_N=0.70$ are softer when the skewness coefficient $J_0=-800$ MeV. However, the maximum masses of the neutron star can be around $2.32M_\odot$ with $J_0=400$ MeV regardless of the value of the nucleon effective mass. By manipulating the coupling strength of the isovector meson to generate different slopes of symmetry energy, we construct the neutron star EOSs that can satisfy the different variables from the simultaneous mass-radius measurements of PSR J0030+0451, PSR J0740+6620 by the NICER collaboration, the mass-radius relations of HESS J1731-347, and the radius constraints from the gravitational-wave signal GW170817 in the framework of DDQMF model. At the same time, most of these constructed EOSs can also satisfy the constraints of the tidal deformability from GW170817 event.

The compact object with a mass of $2.50-2.67~M_\odot$ observed by LIGO Scientific and Virgo collaborations in GW190814, as well as the recent report of a light compact object with a mass and radius of $M=0.77^{+0.20}_{-0.17}M_{\odot}$ and $R=10.4^{+0.86}_{-0.78}$ km within the supernova remnant HESS J1731-347, have posed a great challenge to the investigations into the supranuclear matter. In the inner core region of the neutron star, the strangeness degrees of freedom, such as the hyperons, can be present, which is also named as a hyperonic star. In this work, the neutron star consisting of nucleons and leptons, and the hyperonic star including the hyperons will be studied in the framework of the density-dependent relativistic mean-field (DDRMF) model. Some popular DDRMF parameterizations will be adopted to investigate the properties of nuclear matter and the mass, radius, tidal deformability, and other properties of neutron star and hyperonic stars. We find that the maximum masses of neutron star calculated by DD-MEX, DD-MEX1, DD-MEX2, DD-MEXY and DD-LZ1 sets can be around $2.5-2.6~M_\odot$ with quite stiff equations of state (EOSs) generated by their strong repulsive contributions from vector potentials at high densities. Moreover, by investigating the influence of the crust EOS and core EOS on the neutron stars, we find that the observational data from HESS J1731-347 suggest the requirement of a crust EOS with a higher $L$ parameter and a core EOS with a lower $L$ parameter, and the $M-R$ relations from the constructed EOSs can also be consistent with the observables of PSR J0740+6620, PSR J0030+0451 from NICER and the GW170817 event. With the inclusion of hyperons, the hyperonic star matter becomes softer compared to the neutron star matter. But the massive hyperonic star can also be obtained with DDRMF parameter sets if the vector coupling constants are strong.

The one-dimensional Kronig-Penney potential in the Schrödinger equation, a standard periodic potential in quantum mechanics textbooks known for generating band structures, is solved by using the finite difference method with periodic boundary conditions. This method significantly improves the eigenvalue accuracy compared to existing approaches such as the filter method. The effects of the width and height of the Kronig-Penney potential on the eigenvalues and wave functions are then analyzed. As the potential height increases, the variation of eigenvalues with the wave vector slows down. Additionally, for higher-order band structures, the magnitude of the eigenvalue significantly decreases with increasing potential width. Finally, the Dirac comb potential, a periodic $\delta$ potential, is examined using the present framework. This potential corresponds to the Kronig-Penney potential's width and height approaching zero and infinity, respectively. The numerical results obtained by the finite difference method for the Dirac comb potential are also perfectly consistent with the analytical solution.

Motivated by the latest research on feasible space monitoring of multiple control barrier functions (CBFs) as well as polytopic collision avoidance, this paper studies the Polytope Volume Monitoring (PVM) problem, whose goal is to design a control law for inputs of nonlinear systems to prevent the volume of some state-dependent polytope from decreasing to zero. Recent studies have explored the idea of applying Chebyshev ball method in optimization theory to solve the case study of PVM; however, the underlying difficulties caused by nonsmoothness have not been addressed. This paper continues the study on this topic, where our main contribution is to establish the relationship between nonsmooth CBF and parametric optimization theory through directional derivatives for the first time, so as to solve PVM problems more conveniently. In detail, inspired by Chebyshev ball approach, a parametric linear program (PLP) based nonsmooth barrier function candidate is established for PVM, and then, sufficient conditions for it to be a nonsmooth CBF are proposed, based on which a quadratic program (QP) based safety filter with guaranteed feasibility is proposed to address PVM problems. Finally, a numerical simulation example is given to show the efficiency of the proposed safety filter.