Search Results (22)

To reduce the fuel consumption of ships’ oceanic voyages, this study incorporates the influence of ocean currents into the traditional level set algorithm and proposes a route planning algorithm capable of identifying energy-efficient routes in complex and variable sea conditions. The approach introduces the influence factor of ocean current to optimize routing in dynamically changing marie environments. First, models for the energy consumption of ships and flow fields are established. The level set curve is then evolved by integrating the flow environment and energy consumption gradient, solving the Hamilton–Jacobi equation with energy consumption parameters. The optimal path is subsequently determined through backtracking along the energy consumption gradient, enabling energy-efficient route planning from the starting point to the endpoint in complex ocean conditions. To verify the effectiveness of the proposed algorithm, its performance is evaluated through two case studies, comparing energy consumption under different environmental conditions. The experimental results demonstrate that, compared to the shortest path method based on the level set algorithm, the proposed approach achieves an energy saving rate of approximately 2.1% in obstacle-free environments and 1.4% in environments with obstacles. Full article

Using multifrequency observations, from radio to $\gamma$-rays of the blazar Mrk 501, we constructed their corresponding light curves and built periodograms using RobPer and Lomb–Scargle algorithms. Long-term variability was also studied using the power density spectrum and the detrended function analysis. Using the software VARTOOLS Version 1.40, we also computed the analysis of variance, box-least squares and discrete fourier transform. The result of these techniques showed an achromatic periodicity ≲$229\mathrm{d}$. This, combined with the result of pink-color noise in the spectra, led us to propose that the periodicity was produced via a secondary eclipsing supermassive binary black hole orbiting the primary one locked inside the central engine of Mrk 501. We built a relativistic eclipsing model of this phenomenon using Jacobi elliptical functions, finding a periodic relativistic eclipse occurring every ∼$224\mathrm{d}$ in all the studied wavebands. This implies that the frequency of the emitted gravitational waves falls slightly above $0.1$ mHz, well within the operational range of the upcoming LISA space-based interferometer, and as such, these gravitational waves must be considered as a prime science target for future LISA observations. Full article

An enhanced dung beetle optimization algorithm (EDBO) is proposed for nonlinear optimization problems with multiple constraints in manufacturing. Firstly, the dung beetle rolling phase is improved by removing the worst value interference and coupling the current solution with the optimal solution to each other, while retaining the advantages of the original formulation. Subsequently, to address the problem that the dung beetle dancing phase focuses only on the information of the current solution, which leads to the overly stochastic and inefficient exploration of the problem space, the globally optimal solution is introduced to steer the dung beetle, and a stochastic factor is added to the optimal solution. Finally, the dung beetle foraging phase introduces the Jacobi curve to further enhance the algorithm’s ability to jump out of the local optimum and avoid the phenomenon of premature convergence. The performance of EDBO in optimization is tested using the CEC2017 function set, and the significance of the algorithm is verified by the Wilcoxon rank-sum test and the Friedman test. The experimental results show that EDBO has strong optimization-seeking accuracy and optimization-seeking stability. By solving four engineering optimization problems of varying degrees, EDBO has proven to have good adaptability and robustness. Full article

In this paper, we propose an efficient numerical method to solve the problems of diffusive logistic models with free boundaries, which are often used to simulate the spreading of new or invasive species. The boundary movement is tracked by the level-set method, where the Hamilton–Jacobi weighted essentially nonoscillatory (HJ-WENO) scheme is utilized to capture the boundary curve embedded by the Cartesian grids via the embedded boundary method. Then the radial basis function–finite difference (RBF-FD) method is adopted for spatial discretization and the implicit–explicit (IMEX) scheme is considered for time integration. A variety of numerical examples are utilized to demonstrate the evolution of the diffusive logistic model with different initial boundaries. Full article

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials, respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes. Full article

In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for finding the right choice, it is better to introduce special variables first, and then investigate the problems that naturally lend themselves to these variables. This paper follows Jacobi’s prophetic observations by introducing certain “meta” variational problems on semi-simple reductive groups G having a compact subgroup K. We then use the Maximum Principle of optimal control to generate the Hamiltonians whose solutions project onto the extremal curves of these problems. We show that there is a particular sub-class of these Hamiltonians that admit a spectral representation on the Lie algebra of G. As a consequence, the spectral invariants associated with the spectral curve produce a large number of integrals of motion, all in involution with each other, that often meet the Liouville complete integrability criteria. We then show that the classical integrals of motion associated, with the Kowalewski top, the two-body problem of Kepler, and Jacobi’s geodesic problem on the ellipsoid can be all derived from the aforementioned Hamiltonian systems. We also introduce a rolling geodesic problem that admits a spectral representation on symmetric Riemannian spaces and we then show the relevance of the corresponding integrals on the nature of the curves whose elastic energy is minimal. Full article

This article presents some results of a geometric classification of Sasakian manifolds (SM) that admit an almost ∗-Ricci soliton (RS) structure $(g,\omega ,X)$. First, we show that a complete SM equipped with an almost ∗-RS with $\omega \ne$ const is a unit sphere. Then we prove that if an SM has an almost ∗-RS structure, whose potential vector is a Jacobi vector field on the integral curves of the characteristic vector field, then the manifold is a null or positive SM. Finally, we characterize those SM represented as almost ∗-RS, which are ∗-RS, ∗-Einstein or ∗-Ricci flat. Full article

We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results. Full article

This paper proposes an approximate optimal curve-path-tracking control algorithm for partially unknown nonlinear systems subject to asymmetric control input constraints. Firstly, the problem is simplified by introducing a feedforward control law, and a dedicated design for optimal control with asymmetric input constraints is provided by redesigning the control cost function in a non-quadratic form. Then, the optimality and stability of the derived optimal control policy is demonstrated. To solve the underlying tracking Hamilton–Jacobi–Bellman (HJB) equation in consideration of partially unknown systems, an integral reinforcement learning (IRL) algorithm is utilized using the neural network (NN)-based value function approximation. Finally, the effectiveness and generalization of the proposed method is verified by experiments carried out on a high-fidelity hardware-in-the-loop (HIL) simulation system for fixed-wing unmanned aerial vehicles (UAVs) in comparison with three other typical path-tracking control algorithms. Full article

We consider maximal gauged supergravity in 4D with the $\mathrm{ISO}\left(7\right)$ gauge group, which arises from a consistent truncation of massive IIA supergravity on a six-sphere. Within its G${}_2$-invariant sector, the theory is known to possess a supersymmetric AdS extremum, as well as two non-supersymmetric ones. In this context, we provide a first-order formulation of the theory by making use of the Hamilton–Jacobi (HJ) formalism. This allows us to derive a positive energy theorem for both non-supersymmetric extrema. Subsequently, we also find novel non-supersymmetric domain walls (DWs) interpolating between the supersymmetric extremum and each of the other two. Finally, we discuss a perturbative HJ technique that may be used in order to solve for curved DW geometries. Full article

We classify the magnetic Jacobi fields in cosymplectic manifolds of dimension 3, enriching the results in the study of magnetic Jacobi fields derived from uniform magnetic fields. In particular, we give examples of Jacobi magnetic fields in the Euclidean space ${\mathbb{E}}^3$ and we conclude with the description of magnetic Jacobi fields in the product spaces ${\mathbb{S}}^2\times \mathbb{R}$ and ${\mathbb{H}}^2\times \mathbb{R}$. Full article

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset $U\subseteq M$, such that $\pi \left(U\right)$ has a manifold structure and ${\pi }_{\left|U\right.}:U\to \pi \left(U\right)$, the restriction to U of the canonical projection $\pi :M\to M/G$, is a surjective submersion. If $X_{\left|U\right.}$ is not vertical with respect to ${\pi }_{\left|U\right.}$, we show that such complete solutions solve the reconstruction equations related to $X_{\left|U\right.}$ and G, i.e., the equations that enable us to write the integral curves of $X_{\left|U\right.}$ in terms of those of its projection on $\pi \left(U\right)$. On the other hand, if $X_{\left|U\right.}$ is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of $X_{\left|U\right.}$ up to quadratures. To do that, we give, for some elements $\xi$ of the Lie algebra $\mathfrak{g}$ of G, an explicit expression up to quadratures of the exponential curve $exp\left(\xi t\right)$, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of $exp\left(\xi t\right)$ is valid for all $\xi$ inside an open dense subset of $\mathfrak{g}$. Full article

We discuss a possibility that the entire universe on its most fundamental level is a neural network. We identify two different types of dynamical degrees of freedom: “trainable” variables (e.g., bias vector or weight matrix) and “hidden” variables (e.g., state vector of neurons). We first consider stochastic evolution of the trainable variables to argue that near equilibrium their dynamics is well approximated by Madelung equations (with free energy representing the phase) and further away from the equilibrium by Hamilton–Jacobi equations (with free energy representing the Hamilton’s principal function). This shows that the trainable variables can indeed exhibit classical and quantum behaviors with the state vector of neurons representing the hidden variables. We then study stochastic evolution of the hidden variables by considering D non-interacting subsystems with average state vectors, ${\overline{\mathbf{x}}}^1$, …, ${\overline{\mathbf{x}}}^D$ and an overall average state vector ${\overline{\mathbf{x}}}^0$. In the limit when the weight matrix is a permutation matrix, the dynamics of ${\overline{\mathbf{x}}}^{\mu }$ can be described in terms of relativistic strings in an emergent $D+1$ dimensional Minkowski space-time. If the subsystems are minimally interacting, with interactions that are described by a metric tensor, and then the emergent space-time becomes curved. We argue that the entropy production in such a system is a local function of the metric tensor which should be determined by the symmetries of the Onsager tensor. It turns out that a very simple and highly symmetric Onsager tensor leads to the entropy production described by the Einstein–Hilbert term. This shows that the learning dynamics of a neural network can indeed exhibit approximate behaviors that were described by both quantum mechanics and general relativity. We also discuss a possibility that the two descriptions are holographic duals of each other. Full article

In this study, singular integral solutions were studied to investigate scattering of Rayleigh waves by subsurface cracks. Defining a wave scattering model by objects, such as cracks, still can be quite a challenge. The model’s analytical solution uses five different numerical integration methods: (1) the Gauss–Legendre quadrature, (2) the Gauss–Chebyshev quadrature, (3) the Gauss–Jacobi quadrature, (4) the Gauss–Hermite quadrature and (5) the Gauss–Laguerre quadrature. The study also provides an efficient dynamic finite element analysis to demonstrate the viability of the wave scattering model with an optimized model configuration for wave separation. The obtained analytical solutions are verified with displacement variation curves from the computational simulation by defining the correlation of the results. A novel, verified model, is proposed to provide variations in the backward and forward scattered surface wave displacements calculated by different frequencies and geometrical crack parameters. The analytical model can be solved by the Gauss–Legendre quadrature method, which shows the significantly correlated displacement variation with the FE simulation result. Ultimately, the reliable analytic model can provide an efficient approach to solving the parametric relationship of wave scattering. Full article

A general formulation is considered for the free vibration of curved laminated composite beams (CLCBs) with alterable curvatures and diverse boundary restraints. In accordance with higher-order shear deformation theory (HSDT), an improved variational approach is introduced for the numerical modeling. Besides, the multi-segment partitioning strategy is exploited for the derivation of motion equations, where the CLCBs are separated into several segments. Penalty parameters are considered to handle the arbitrary boundary conditions. The admissible functions of each separated beam segment are expanded in terms of Jacobi polynomials. The solutions are achieved through the variational approach. The proposed methodology can deal with arbitrary boundary restraints in a unified way by conveniently changing correlated parameters without interfering with the solution procedure. Full article