Search Results (24)

The canonical quantization of gravity in general relativity is greatly simplified by the artificial decomposition of space time into a 3 + 1 formalism. Such a simplification appears to come at the cost of general covariance. This quantization procedure requires tangential and perpendicular infinitesimal diffeomorphisms generated by the symmetry group under the Legendre transformation of the given action. This gauge generator, along with the fact that Weyl curvature scalars may act as “intrinsic coordinates” (or a dynamical reference frame) that depend only on the spatial metric ($g_{ab}$) and the conjugate momenta ($p^{cd}$), allows for an alternative approach to canonical quantization of gravity. In this paper, we present the tensorial solution of the set of Weyl scalars in terms of canonical phase-space variables. Full article

The Red-Billed Blue Magpie Optimization (RBMO) algorithm is a metaheuristic method inspired by the foraging behavior of red-billed blue magpies. However, the conventional RBMO often suffers from premature convergence and performance degradation when solving high-dimensional constrained optimization problems due to its over-reliance on population mean vectors. To address these limitations, this study proposes an Improved Red-Billed Blue Magpie Optimization (IRBMO) algorithm through a multi-strategy fusion framework. IRBMO enhances population diversity through Logistic-Tent chaotic mapping, coordinates global and local search capabilities via a dynamic balance factor, and integrates a dual-mode perturbation mechanism that synergizes Jacobi curve strategies with Lévy flight strategies to balance exploration and exploitation. To validate IRBMO’s efficacy, comprehensive comparisons with 16 algorithms were conducted on the CEC-2017 (30D, 50D, 100D) and CEC-2022 (10D, 20D) benchmark suites. Subsequently, IRBMO was rigorously evaluated against ten additional competing algorithms across four constrained engineering design problems to validate its practical effectiveness and robustness in real-world optimization scenarios. Finally, IRBMO was applied to 3D UAV path planning, successfully avoiding hazardous zones while outperforming 15 alternative algorithms. Experimental results confirm that IRBMO exhibits statistically significant improvements in robustness, convergence accuracy, and speed compared to classical RBMO and other peers, offering an efficient solution for complex optimization challenges. Full article

The intelligent transformation of the wastewater treatment industry, as a core component of the modern environmental governance system, is of decisive significance for achieving sustainable development goals. This study focuses on the issue of multi-stakeholder collaborative governance in the intelligent transformation of the wastewater treatment industry, with differential game theory as the core framework. A tripartite game model involving the government, wastewater treatment enterprises, and digital twin platforms is developed to depict the dynamic interrelations and mutual influences of strategy choices, thereby capturing the coordination mechanisms among government regulation, enterprise technology adoption, and platform support in the transformation process. Based on the dynamic optimization properties of differential games, the Hamilton–Jacobi–Bellman (HJB) equation is employed to derive the long-term equilibrium strategies of the three parties, presenting the evolutionary paths under Nash non-cooperative games, Stackelberg games, and tripartite cooperative games. Furthermore, the Sobol global sensitivity analysis is applied to identify key parameters influencing system performance, while the response surface method (RSM) with central composite design (CCD) is used to quantify parameter interaction effects. The findings are as follows: (1) compared with Nash non-cooperative and Stackelberg games, the tripartite cooperative strategy based on the differential game model achieves global optimization of system performance, demonstrating the efficiency-enhancing effect of dynamic collaboration; (2) the most sensitive parameters are β, α, μ₃, and η₃, with β having the highest sensitivity index (ST_i = 0.459), indicating its dominant role in system performance; (3) significant synergistic enhancement effects are observed among α–β, α–μ₃, and β–μ₃, corresponding, respectively, to the “technology stability–benefit conversion” gain effect, the “technology decay–platform compensation” dynamic balance mechanism, and the “benefit conversion–platform empowerment” performance threshold rule. Full article

In this paper, we provide a complete analysis of the most general spherical solution of the Lyra scalar-tensor (LyST) gravitational theory based on the proper definition of a Lyra manifold. Lyra’s geometry features the metric tensor and a scale function as fundamental fields, resulting in generalizations of geometrical quantities such as the affine connection, curvature, torsion, and non-metricity. A proper action is defined considering the correct invariant volume element and the scalar curvature, obeying the symmetry of Lyra’s reference frame transformations and resulting in a generalization of the Einstein–Hilbert action. The LyST gravity assumes zero torsion in a four-dimensional metric-compatible spacetime. In this work, geometrical quantities are presented and solved via Cartan’s technique for a spherically symmetric line element. Birkhoff’s theorem is demonstrated so that the solution is proven to be static, resulting in the Lyra–Schwarzschild metric, which depends on both the geometrical mass (through a modified version of the Schwarzschild radius $r_S$) and an integration constant dubbed the Lyra radius $r_L$. We study particle and light motion in Lyra–Schwarzschild spacetime using the Hamilton–Jacobi method. The motion of massive particles includes the determination of the $r_{\mathrm{ISCO}}$ and the periastron shift. The study of massless particle motion shows the last photon’s unstable orbit. Gravitational redshift in Lyra–Schwarzschild spacetime is also reviewed. We find a coordinate transformation that casts Lyra–Schwarzschild spacetime in the form of the standard Schwarzschild metric; the physical consequences of this fact are discussed. Full article

With the increasing deployment of Unmanned Aerial Vehicle (UAV) swarms in uncertain and dynamically changing environments, optimal cooperative control has become essential for ensuring robust and efficient system coordination. To this end, this paper designs a data-driven optimal bipartite containment tracking control scheme for multi-UAV systems under compound uncertainties. A novel Dynamic Iteration Regulation Strategy (DIRS) is proposed, which enables real-time adjustment of the learning iteration step according to the task-specific demands. Compared with conventional fixed-step data-driven algorithms, the DIRS provides greater flexibility and computational efficiency, allowing for better trade-offs between the performance and cost. First, the optimal bipartite containment tracking control problem is formulated, and the associated coupled Hamilton–Jacobi–Bellman (HJB) equations are established. Then, a data-driven iterative policy learning algorithm equipped with the DIRS is developed to solve the optimal control law online. The stability and convergence of the proposed control scheme are rigorously analyzed. Furthermore, the control law is approximated via the neural network framework without requiring full knowledge of the model. Finally, numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed DIRS-based optimal containment tracking scheme for multi-UAV systems, which can reduce the number of iterations by 88.27% compared to that for the conventional algorithm. Full article

This review discusses two triatomic Hamiltonians and their applications to some non-adiabatic spectroscopic and collision problems. Carter and Handy in 1984 presented the first Hamiltonian in bond lengths–bond angle coordinates, that is here applied for studying the NO₂ spectroscopy: vibronic states, internal dynamics, and interaction with the radiation due to the $\tilde{\mathrm{X}}$²A′(A₁)−$\tilde{\mathrm{A}}$²A′(B₂) conical intersection. The second Hamiltonian was reported by Tennyson and Sutcliffe in 1983 in Jacobi coordinates and is here employed in the study of the Kr + OH(A²Σ⁺) electronic quenching due to conical intersection and Renner–Teller interactions among the 1²A′, 2²A′, and 1²A″ electronic species. Within the non-relativistic approximation and the expansion method in diabatic electronic representations, the formalism is exact and allows a unified study of various non-adiabatic interactions between electronic states. The rotation, inversion, and nuclear permutation symmetries are considered for defining rovibronic representations, which are symmetry adapted for ABC and AB₂ molecules, and the matrix elements of the Hamiltonians are then computed. Full article

We present a finite difference method working on sparse grids to solve higher dimensional heterogeneous agent models. If one wants to solve the arising Hamilton–Jacobi–Bellman equation on a standard full grid, one faces the problem that the number of grid points grows exponentially with the number of dimensions. Discretizations on sparse grids only involve $\mathcal{O}\left(N{(logN)}^{d-1}\right)$ degrees of freedom in comparison to the $\mathcal{O}\left(N^d\right)$ degrees of freedom of conventional methods, where N denotes the number of grid points in one coordinate direction and d is the dimension of the problem. While one can show convergence for the used finite difference method on full grids by using the theory introduced by Barles and Souganidis, we explain why one cannot simply use their results for sparse grids. Our numerical studies show that our method converges to the full grid solution for a two-dimensional model. We analyze the convergence behavior for higher dimensional models and experiment with different sparse grid adaptivity types. Full article

A new (2 + 1)-dimensional breaking soliton equation with the help of the nonisospectral Lax pair is presented. It is shown that the compatible solutions of the first two nontrivial equations in the (1 + 1)-dimensional Kaup–Newell soliton hierarchy provide solutions of the new breaking soliton equation. Then, the new breaking soliton equation is decomposed into the systems of solvable ordinary differential equations. Finally, a hyperelliptic Riemann surface and Abel–Jacobi coordinates are introduced to straighten the associated flow, from which the algebro-geometric solutions of the new (2 + 1)-dimensional integrable equation are constructed by means of the Riemann $\theta$ functions. Full article

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux $Y(s_1,\dots ,s_k)$ with $k\ge 2$ rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a $k=2$ Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic $sp\left(2k\right)$ algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the $su(1,1)$ algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for $Y(s_1,s_2)$ is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin $(1,1)$ is obtained. Full article

In this paper, the Kadometsev–Petviashvili equation and the Bargmann system are obtained from a second-order operator spectral problem $L\phi =({\partial }^2-v\partial -\lambda u)\phi =\lambda {\phi }_x$. By means of the Euler–Lagrange equations, a suitable Jacobi–Ostrogradsky coordinate system is established. Using Cao’s method and the associated Bargmann constraint, the Lax pairs of the differential equations are nonlinearized. Then, a new kind of finite-dimensional Hamilton system is generated. Moreover, involutive representations of the solutions of the Kadometsev–Petviashvili equation are derived. Full article

We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in ${SE}_2$ and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part. Full article

This article presents alternative Hamiltonian and Lagrangian formalisms for relativistic mechanics using proper time and proper Lagrangian coordinates in 1 + 1 dimensions as parameters of evolution. The Lagrangian and Hamiltonian formalisms for a hypothetical particle with and without charge are considered based on the relativistic equation for the dynamics and integrals of particle motion. A relativistic invariant law for the conservation of energy and momentum in the Lorentz representation is given. To select various generalized coordinates and momenta, it is possible to modify the Lagrange equations of the second kind due to the relativistic laws of conservation of energy and momentum. An action function is obtained with an explicit dependence on the velocity of the relativistic particles. The angular integral of the particle motion is derived from Hamiltonian mechanics, and the displacement Hamiltonian is obtained from the Hamilton–Jacobi equation. The angular integral of the particle motion $\theta$ is an invariant form of the conservation law. It appears only at relativistic intensities and is constant only in a specific case. The Hamilton–Jacobi–Lagrange equation is derived from the Hamilton–Jacobi equation and the Lagrange equation of the second kind. Using relativistic Hamiltonian mechanics, the Euler–Hamilton equation is obtained by expressing the energy balance through the angular integral of the particle motion $\theta$. The given conservation laws show that the angular integral of the particle motion reflects the relativistic Doppler effect for particles in 1 + 1 dimensions. The connection between the integrals of the particle motion and the doubly special theory of relativity is shown. As an example of the applicability of the proposed invariant method, analyses of the motion of relativistic particles in circularly polarized, monochromatic, spatially modulated electromagnetic plane waves and plane laser pulses are given, and comparisons are made with calculations based on the Landau and Lifshitz method. To allow for the analysis of the oscillation of a particle in various fields, a phase-plane method is presented. Full article

We report a computational study of the potential energy surface (PES) and vibrational bound states for the ground electronic state of ${Li}_2^{+}Kr$. The PES was calculated in Jacobi coordinates at the Restricted Coupled Cluster method RCCSD(T) level of calculation and using aug-cc-pVnZ (n = 4 and 5) basis sets. Afterward, this PES is extrapolated to the complete basis set (CBS) limit for correction. The obtained interaction energies were, then, interpolated numerically using the reproducing kernel Hilbert space polynomial (RKHS) approach to produce analytic expressions for the 2D-PES. The analytical PES is used to solve the nuclear Schrodinger equation to determine the bound states’ eigenvalues of ${Li}_2^{+}Kr$ for a $J$= 0 total angular momentum configuration and to understand the effects of orientational anisotropy of the forces and the interplay between the repulsive and attractive interaction within the potential surface. In addition, the radial and angular distributions of some selected bound state levels, which lie below, around, and above the T-shaped 90° barrier well, are calculated and discussed. We note that the radial distributions clearly acquire a more complicated nodal structure and correspond to bending and stretching vibrational motions “mode” of the $Kr$ atom along the radial coordinate, and the situation becomes very different at the highest bound states levels with energies higher than the T-shaped 90° barrier well. The shape of the distributions becomes even more complicated, with extended angular distributions and prominent differences between even and odd states. Full article

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions of the index k. The consideration of the case of the derivative of the Weierstrass rho-function, ${\wp }_z(z{,}_{}\tau )$, leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points $m+n\tau$ and “demi-lattice” points $m+1/2+n\tau$, $m+(n+1/2)\tau$, $m+1/2+(n+1/2)\tau$. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative ${\theta }_1\prime (z|\tau )$ of the elliptic theta-function and the Weierstrass $\zeta$-function, and that far from the origin of coordinates such zeroes of the $\zeta$-function tend to the positions of the simple poles of this function. Full article

The on-orbit operation of insertion and extraction of space robots is a technology essential to the assembly and maintenance in orbit, satellite fuel filling, failed satellite recovery, especially modular in-orbit assembly of micro-spacecraft. Therefore, the force/posture impedance control for the on-orbit operation of insertion and extraction is studied. Firstly, the dynamic model of space robots’ system in the form of uncontrolled carrier position and controlled attitude is derived by using the momentum conservation principle. Through the kinematic constraints of the replacement component plug, the Jacobi relationship of the plug motion in the base coordinate system is established. Secondly, to achieve the output force control of the plug during the on-orbit operation of insertion and extraction, a second-order linear impedance model is established based on the dynamic relationship between the plug posture and its output force and the impedance control principle. Then, in order to improve the stability, robustness, and adaptability of the controller, an adaptive Radial Basis Function Neural Network (RBFNN) is used to approximate the uncertainties in the dynamic model for the force/posture control of the plug. Finally, the stability of the system is verified by the Lyapunov principle. The simulation results show that the designed neural network impedance control strategy can achieve a control accuracy of less than ${10}^{-3}$ rad for the plug’s attitude tracking error, less than ${10}^{-3}$ m for its position tracking error, and less than 0.5 N for its output force tracking error. Full article