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The rapidly growing number of objects encircling our planet is an increasing concern. Collisions between these objects have already occurred and pose a potential threat in the future, resulting in the creation of countless debris fragments. In particular, the Low Earth Orbit (LEO) region is densely populated and highly contested. This underscores the critical importance of space surveillance in this area. Moreover, the utilization of Medium Earth Orbit (MEO) and Geosynchronous Earth Orbit (GEO) is also on the rise. To ensure the safety and functionality of operational satellites, it is paramount to accurately determine and continuously monitor the orbits of space objects, mitigating the risk of collisions. Precise and timely predictions of future trajectories are essential for this purpose. In response to these challenges, this survey paper provides a comprehensive review of various methods proposed in the literature for Orbit Determination (OD). It also identifies research gaps and suggests potential directions for future studies, emphasizing the pressing need for adequate Space Situational Awareness (SSA). [Display omitted] • Admissible Region is a robust, reliable cornerstone method for orbit determination. • Machine learning transforms orbit determination, boosting accuracy and efficiency. • Explainable AI is crucial for transparency and trust in orbit determination. • The recent focus on cislunar regime demands novel orbit determination methods. • Undervalued, space-based orbit determination is crucial for accurate cislunar missions. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of computing a preliminary orbit of a celestial body from one topocentric position vector P 1 and a very short arc (VSA) of optical observations A 2 . Using the conservation laws of the two-body dynamics, we write the problem as a system of 8 polynomial equations in 6 unknowns. We prove that this system is generically consistent, namely, for a generic choice of the data P 1 , A 2 , it always admits solutions in the complex field, even when P 1 , A 2 do not correspond to the same celestial body. The consistency of the system is shown by deriving a univariate polynomial v of degree 8 in the unknown topocentric distance at the mean epoch of the observations of the VSA. Through Gröbner bases theory, we also show that the degree of v is minimum among the degrees of all the univariate polynomials solving this problem. Even though we can find solutions to our problem for a generic choice of P 1 , A 2 , most of these solutions are meaningless. In fact, acceptable solutions must be real and have to fulfill other constraints, including compatibility with Keplerian dynamics. We also propose a way to select or discard solutions taking into account the uncertainty in the data, if present. The proposed orbit determination method is relevant for different purposes, e.g., the computation of a preliminary orbit of an Earth satellite with radar and optical observations, the detection of maneuvres of an Earth satellite, and the recovery of asteroids which are lost due to a planetary close encounter. We conclude by showing some numerical tests in the case of asteroids undergoing a close encounter with the Earth. [ABSTRACT FROM AUTHOR]

• Orbit maneuver-free approach to the angle-only relative orbit determination for is proposed. • The angles-only orbital observability problem is solved by introducing radial basis function neural network to capture the nonlinearity of the perturbed dynamics. • The RBFNN frame including training data acquisition, data pre-processing, and the network design for angles-only orbit determination problem is developed. • The validity of the proposed algorithm is evaluated by numerical simulations. • The anti-noise, generalization performance, and the sensitivity of the solution are checked. Angles-only orbit determination during space situational awareness missions for non-cooperative targets such as space debris is subject to the well-known observability problem, that is, the orbital state is weakly observable or even unobservable because of missing range measurements. The key point to solve this problem is how to quickly capture the nonlinearity of orbit dynamics if only one space-based passive sensor is available to observe the target and translational maneuvers are not executed. To solve the short-arc angles-only problem, this study develops a simple and fast orbit determination method by exploiting the advantages of radial basis function neural network (RBFNN). Emphasis was placed on the design of the RBFNN, which was trained to construct a mapping network model from line-of-sight (LOS) measurements to the relative orbit. First, the perturbed relative orbit dynamics considering the Earth's non-spherical J2 term and air drag was established with the LOS measurement model. Second, a RBFNN-based angles-only orbit determination frame was developed, in which the generator of the training dataset, data preprocessing algorithm, and RBF network structure were designed. Finally, the proposed method was verified and tested using a set of numerical simulations of low-Earth orbit cases. The anti-noise and generalization performance of the designed network and the sensitivity of the solution accuracy to the measurement frequency and span were presented and discussed. Additionally, the advantages of the RBFNN-based method were confirmed by comparison with the results obtained using the BPNN (Back Propagation Neural Network)-based method. The simulation results show that the average orbit prediction error for LEO scenarios is approximately 0.08% using the index of initial relative distance, where only three sets of LOS angles are measured within a span of 24 s. [ABSTRACT FROM AUTHOR]

We analysed the skulls and faces of Korean subjects using anthropometric methods to understand the anatomical characteristics of the eyeball and eye orbit region of Korean population and to determine the correlations between the hard and soft tissues around the eyeball and eye orbit region. In total, 82 sections in the region were measured to determine the correlations; among them, 34 showed significant differences by sex, and 6 showed significant differences by age. As the distance from the centre of the eye lens to the eye orbit is calculated as a ratio, we determined that the centre of the eye lens is located relatively on the lateral and superior position in each eye orbit in front view. Fourteen sections that could be used for craniofacial reconstruction/approximation in men and women were selected. Regression equations were derived according to the correlation of each section, and their reliabilities were verified by out of sample validation tests. Therefore, our results increase the accuracy of eyeball position determination, which would be useful for more efficient craniofacial reconstruction/approximation of the Korean population and should improve the efficiency of facial recognition. [ABSTRACT FROM AUTHOR]

The high-precision satellite velocities and accelerations calculated by the Global Navigation Satellite System (GNSS) are essential for tasks such as airborne gravity data processing. Users generally interpolate satellite positions in the precise ephemeris to calculate satellite velocity and acceleration. However, due to the edge effect, the accuracy of the interpolation is relatively low near day boundaries. In this study, a method for calculating GNSS satellite velocity and acceleration based on orbit fitting and orbit integration was proposed, and the high-precision transformation relationship between satellite velocity and acceleration in the Earth-Centered Inertial (ECI) coordinate system and the Earth-Centered, Earth-Fixed (ECEF) coordinate system was derived. The experimental results show that the satellite velocity accuracy is 1.5 × 10−6 m/s and the acceleration accuracy is 1.0 × 10−8 m/s2 according to the proposed method. Thus, the proposed method improves the accuracy of calculating satellite velocity and acceleration near day boundaries, and helps GNSS users to obtain satellite velocity and acceleration information with consistent precision throughout the day. [ABSTRACT FROM AUTHOR]

The BeiDou Global Navigation Satellite System (BDS-3) provides real-time precise point positioning (PPP) service via B2b signals, offering real-time decimeter-level positioning for users in China and surrounding areas. However, common interruptions and outliers in PPP-B2b services arise due to factors such as the Geostationary Orbit (GEO) satellite "south wall effect", Issue of Data (IOD) matching errors, and PPP-B2b signal broadcast priorities, posing challenges to continuous high-precision positioning. This study meticulously examines the completeness, continuity, and jumps in PPP-B2b orbit and clock correction using extensive observational data. Based on this analysis, a two-step method for detecting outliers in PPP-B2b orbit and clock corrections is devised, leveraging epoch differences and median absolute deviation. Subsequently, distinct prediction methods are developed for BDS-3 and GPS orbit and clock corrections. Results from simulated and real-time dynamic positioning experiments indicate that predicted corrections can maintain the same accuracy as normal correction values for up to 10 min and sustain decimeter-level positioning accuracy within 30 min. The adoption of predicted correction values significantly enhances the duration of sustaining real-time PPP during signal interruptions. [ABSTRACT FROM AUTHOR]

Classic initial orbit determination (IOD) methods are mainly based on the hypothesis of unperturbed Keplerian motion and are inapplicable for Libration point orbits (LPOs). To this end, this paper proposes a neural network–assisted method for accurate and efficient IOD of LPOs. The proposed method consists of a long short-term memory (LSTM) neural network–based initial guess generator (IGG) combined with a second-order optimal corrector. First, the LSTM neural network is developed to compensate for the residuals between the two-body IOD solution and the true solution to generate a more accurate initial guess. Two sample forms, one in the rotating frame and the other in the inertial frame, are investigated and compared to select the better form to train the LSTM neural network. Then, a second-order optimal corrector is proposed based on the second-order state transition tensor and a two-step procedure, which takes the initial guess from the LSTM neural network and iteratively obtains a more accurate IOD solution. Finally, the proposed method is applied to solve the IOD problem in two Earth–Moon LPO scenarios: a 4:1 synodic resonant Near-rectilinear Halo orbit (NRHO) and a transfer trajectory from a lunar orbit to the 4:1 NRHO. Numerical simulations show that the estimated errors are reduced by 99% using the proposed LSTM-based IGG. Moreover, for a comparable level of accuracy, the second-order optimal corrector converges one time faster than the first-order corrector. [ABSTRACT FROM AUTHOR]

Far-range rendezvous problem of geosynchronous orbit is studied based on the linear equation of near-circular deviation, and a two-layer nonlinear programming model is established based on the in-plane 3-pulse manuveur strategy. In the outer layer, the particle swarm optimization algorithm is used to optimize the arugment angle of the first and second pulses, which improves the traditional method of fixing at apogee or perigee. After determining the arugment angle of the first pulse and the second pulse, the inner layer uses the near-circular deviation linear equation to solve the arugment angle of the third pulse and the speed increment of the three pulses quickly. Simulation results show that the proposed planning method can reach the designated terminal in the specified time with less fuel cost, and has engineering application value. [ABSTRACT FROM AUTHOR]

Leveraging space-based optical platforms for space debris and defunct spacecraft detection presents several advantages, including a wide detection range, immunity to cloud cover, and the ability to maintain continuous surveillance on space targets. As a result, it has become an essential approach for accomplishing tasks related to space situational awareness. However, the prediction of the orbits of space objects is crucial for the success of such missions, and current technologies face challenges related to accuracy, reliability, and practical efficiency. These challenges limit the performance of space-based optical space situational awareness systems. To drive progress in this field and establish a more effective and reliable space situational awareness system based on space optical platforms, this paper conducts a retrospective overview of research advancements in this area. It explores the research landscape of orbit determination methods, encompassing orbit association methods, initial orbit determination methods, and precise orbit determination methods, providing insights from international perspectives. The article concludes by highlighting key research areas, challenges, and future trends in current space situational awareness systems and orbit determination methods. [ABSTRACT FROM AUTHOR]

We study a family of completely integrable systems attached to Takiff algebras g N , extending Toda systems for split simple Lie algebras g. With respect to Darboux coordinates on coadjoint orbits O , the potentials of the hamiltonians are products of polynomial and exponential functions. Explicit general solutions for their equations of motion are obtained using differential operators called jet transformations. The classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, quantizing the Poisson algebra of functions on O. These results are illustrated with concrete examples, including a 3-body problem based on sl (2) and an extension of soliton solutions for A ∞ to associated Takiff algebras. [ABSTRACT FROM AUTHOR]

AbstractThe chain group Ch(G) of a locally compact group G has one generator gρ for each irreducible unitary G-representation ρ, a relation gρ=gρ′gρ″ whenever ρ is weakly contained in ρ′⊗ρ″, and gρ*=gρ−1 for the representation ρ* contragredient to ρ. G satisfies chain-center duality if assigning to each gρ the central character of ρ is an isomorphism of Ch(G) onto the dual Z(G)̂ of the center of G. We prove that G satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. Müger’s result compact groups satisfy chain-center duality. [ABSTRACT FROM AUTHOR]

Let G be an abelian semigroup of matrices on K n ( K = C or R ). We show that if G is hypercyclic, then it has no non-zero finite orbit. This result fails if we drop the assumption that G is abelian. As a consequence, if G is abelian, it is not chaotic. On the other hand, we show that G is not minimal for n ≥ 3 , but it can be minimal for n = 1 ; for K = R , the critical number is n = 2 . [ABSTRACT FROM AUTHOR]

We introduce an initial orbit determination technique that uses only sequential range-rate measurements along fixed lines of sight and time between measurements. Compared to existing velocity-based initial orbit determination methods which impose additional requirements such as concurrent observations, our method may reduce the number of sensors required for a spacecraft to perform autonomous, or onboard, initial orbit determination. The measurements required may be obtained by a single range-rate sensor, such as an X-ray detector observing X-ray pulsars, and an onboard clock. The accuracy of our algorithm is compared with existing techniques for a variety of orbits. Orbit determination errors, in terms of estimated position, are characterized with Monte Carlo simulation techniques. Results indicate that our method achieves better performance compared to an initial orbit determination method utilizing velocity vectors and time of flight. The position error of our technique increases for lower velocities and shorter measurement times, which is consistent with existing velocity-based initial orbit determination methods. [ABSTRACT FROM AUTHOR]

For a smooth affine algebra R of dimension d ≥ 3 over a field k and an invertible alternating matrix χ of rank 2 n , the group S p (χ) of invertible matrices of rank 2 n over R which are symplectic with respect to χ acts on the right on the set U m 2 n (R) of unimodular rows of length 2 n over R. In this paper, we prove that S p (χ) acts transitively on U m 2 n (R) if k is algebraically closed, d ! ∈ k × and 2 n ≥ d. [ABSTRACT FROM AUTHOR]

Current orbit uncertainty propagation (OUP) and orbit determination (OD) methods suffer from drawbacks related to high computational burden, limiting their applications in deep space missions. To this end, this paper proposes a multivariate attention-based method for efficient OUP and OD of Earth–Jupiter transfer. First, a neural network-based OD framework is utilized, in which the orbit propagation process in a traditional unscented transform (UT) and unscented Kalman filter (UKF) is replaced by the neural network. Then, the sample structure of training the neural network for the Earth–Jupiter transfer is discussed and designed. In addition, a method for efficiently generating a large number of samples for the Earth–Jupiter transfer is presented. Next, a multivariate attention-based neural network (MANN) is designed for orbit propagation, which shows better capacity in terms of accuracy and generalization than the deep neural network. Finally, the proposed method is successfully applied to solve the OD problem in an Earth–Jupiter transfer. Simulations show that the proposed method can obtain a similar estimation to the UKF while saving more than 90% of the computational cost. [ABSTRACT FROM AUTHOR]

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C^\ast-algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C^\ast-algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space. [ABSTRACT FROM AUTHOR]

Satellite Laser Ranging (SLR) technology is one of the main technologies in the field of space geodesy, it has played an extremely important role in laser ranging data application and research. As a commonly used geodynamic satellites, the main contribution of LAGEOS and LAGEOS-2 is the solution of orbit determination and station coordinates. At present, there are 8 analysis centers of the International Laser Ranging Service (ILRS) to release the precise orbit products for the two LAGEOS satellites. However, there is currently no relevant research on the analysis of the consistency of these orbits. So, the authors would evaluate the accuracies of them, and the results show that the orbital accuracies of the two LAGEOS satellites are 3 ∼ 5 cm. SLR is regarded as one of the important input data of the International Terrestrial Reference Frame (ITRF), the analysis center of ILRS provides weekly solutions of station coordinates for ITRF. But, due to the limitation of SLR observation conditions, for example most observations of Chinese SLR stations can be made at night, and laser ranging test cannot be conducted in rainy days, the orbit solution of the seven-day arc may have the problem of less data. Therefore, this paper proposed a method for computing the station coordinates based on long arc orbit determination, and we developed the LAODGEO software (Long Arc Orbit Determination Software for Geodynamic Satellite), which is used to solve the orbits of two LAGEOS satellites and station coordinates of five SLR stations in China. The results show that the obtained orbital precision is basically superior to 3.70 cm, and the 3DRMS values between station coordinates solved by us and the coordinates published by SLRF2014 are mostly smaller than 0.013 m, which is consistent with research results domestic and overseas. [ABSTRACT FROM AUTHOR]

This paper introduces a novel dynamical model, building upon the existing dynamical model for Deimos in the current numerical ephemerides, which only encompasses the simple libration effects of Deimos. The study comprehensively incorporates the rotational dynamics of Deimos influenced by the torque exerted by the major celestial bodies (Mars, the Sun) in the solar system within the inertial space. Consequently, a full dynamical model is formulated to account for the complete coupling between the rotation and orbit of Deimos. Simultaneously, employing precision orbit determination methods used for artificial satellites, we develop an adjustment model for fitting data to the complete model. The 12-order Adams–Bashforth–Moulton (ABM) integration algorithm is employed to synchronously integrate the 12 state variables of the full model to obtain the orbit of Deimos.The difference in the orbits obtained by integrating the full model over a period of 10 years and those obtained by the simplified model is at the order of 10 km. After precise orbit determination, this difference decreases to below 100 m, so numerical simulation results indicate that the full dynamical model and adjustment model are stable and reliable. Simultaneously, the integration of the Deimos third-order gravity field in the full model over a 10-year period induces only meter-level positional changes. This suggests that when constructing the complete model, the utilization of a second-order gravity field alone is sufficient. Compared to the simple model, the polar axis of Deimos in the inertial space exhibits a more complex oscillation in the full model. Additionally, the full model calculates that the minimum moment of inertia principal axis of Phobos has an amplitude of approximately 0.5 degrees in the longitude direction and does not exceed 2 degrees in the latitude direction. This work further advances the current dynamical model for Deimos and establishes the foundational model for the generation of a new set of precise numerical ephemerides for Deimos. [ABSTRACT FROM AUTHOR]

Introduction/purpose: The paper presents a novel approach to certain well-established fixed point theorems for multivalued probabilistic contractions in b-Menger spaces, leveraging the boundedness of the orbits. The aim was to generalize and enhance the results previously derived by Fang and Hadžić. Methods: The boundedness of orbits in b-Menger spaces is used to establish their approach for multivalued probabilistic contractions. Results: The findings of the study not only generalized the existing fixed point theorems but also enhanced them significantly. The effectiveness of the approach in extending the results originally proposed by Fang and Hadžić was showcased. Moreover, the applicability of the coincidence fixed point theorem in fuzzy b-metric spaces was demonstrated. Conclusions: The study presented a novel perspective on fixed point theorems in multivalued probabilistic contractions within b-Menger spaces. By leveraging boundedness and introducing a coincidence fixed point theorem for fuzzy b-metric spaces, the work contributed to the advancement in this field. [ABSTRACT FROM AUTHOR]

Two types of recurrence sets are introduced for inverse semigroup partial actions in topological spaces. Some applications to the basic dynamical properties of the action are indicated, covering topics as topological transitivity, limit points and periodic points. We then explore the connections of these recurrence sets with similar notions for related types of imperfect symmetries (prefix inverse semigroup expansions, partial group and groupoid actions). [ABSTRACT FROM AUTHOR]

In a previous paper, the author and his collaborator studied the problem of lifting Hamiltonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups. Only even quantizations were considered there. In this paper, these results are generalized to the case of general quantizations with arbitrary periods. The key step is to introduce an enhanced version of the (truncated) period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth symplectic variety X, with values in the space of Picard Lie algebroid over X. As an application, we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition. [ABSTRACT FROM AUTHOR]

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work. [ABSTRACT FROM AUTHOR]

Space-based opportunistic positioning is a crucial component of resilient positioning, navigation, and timing (PNT) systems, and it requires the acquisition of orbit information for non-cooperative low Earth orbit (LEO) satellites. Traditional methods for orbit determination (OD) of non-cooperative LEO satellites have difficulty in achieving a balance between reliability, hardware costs, and availability duration. To address these challenges, this study proposes a framework for single-station orbit determination of non-cooperative LEO satellites. By utilizing signals of opportunity (SOPs) captured by a single ground station, the system performs initial orbit determination (IOD), precise orbit determination (POD), and orbit prediction (OP), enabling the long-term determination of satellite positions and velocities. Under the proposed framework, the reliability and real-time performance are dependent on the initial orbit determination and the orbit calculation based on the dynamical model. To achieve initial orbit determination, a three-step algorithm is designed. (1) An improved search method is employed to estimate a coarse orbit using single-pass Doppler measurements. (2) Data association is conducted to obtain multi-pass Doppler observations. (3) The least squares (LS) is implemented to determine the initial orbit using the associated multi-pass Doppler measurements and the coarse orbit. Additionally, to enhance computational efficiency, two fast orbit calculation algorithms are devised. These algorithms leverage the numerical stability of the Runge–Kutta integrator to reduce computations and exploit the strong correlation among nearby time intervals of orbits with small eccentricities to minimize redundant calculations, thereby achieving orbit calculation efficiently. Finally, through positioning experiments, the determined orbits are demonstrated to have accuracy comparable to that of two-line elements (TLE) updated by the North American Aerospace Defense Command (NORAD). [ABSTRACT FROM AUTHOR]

The strong isomorphism classes of extensions of finite groups are parametrized by orbits of a prescribed action on the second cohomology group. We study these orbits in the case of extensions of a finite abelian p-group by a cyclic factor of order p. As an application, we compute the number and sizes of these orbits when the initial p-group is generated by at most 3 elements. [ABSTRACT FROM AUTHOR]

This article uses appropriate transformations and the sub-ODE method to reveal multiwave, homoclinic breather, M-shaped rational, periodic cross rational, kink cross rational, periodic cross kink, and other soliton solutions for Einstein's vacuum field problem. Additionally, the interaction of rational solitons in the M shape with an exponential function and a double exponential form is disclosed. [ABSTRACT FROM AUTHOR]

The three-body tethered satellite system is a new potential technology for the purposes of orbital transportation. In contrast to conventional orbit transfer methods, this system is expected to transport space supplies to a predetermined orbital altitude without consuming fuel; however, the unwanted libration resulting from the Coriolis force acting on moving subsatellites may induce tumbling within the system. In order to analyze the strongly coupled characteristics of the libration motion and the variable-length tethers, a six-DOF dynamic model of the system based on Newton's law is established. By utilizing the dynamic equations and stability criterion for linear systems, three equilibrium configurations of the satellite system with two constant-length tethers are given. The coupling characteristics of the libration angles are analyzed based on mechanical features and simulations. The results demonstrate that part of the libration energy can transfer from out-of-plane motion to in-plane motion during out-of-plane libration, but not vice versa or when in-plane libration occurs alone. Furthermore, the dynamic characteristic of this system with a predesigned deployment strategy is surveyed. This investigation reveals that such a strategy can rapidly suppress the out-of-plane libration motion while maintaining purely sinusoidal oscillations in the in-plane motion. [ABSTRACT FROM AUTHOR]

The study is focused on examining 2,5‐Substituted 4‐Pyrone based compounds through quantum chemical and topological analysis techniques, evaluating the properties of these compounds, including their geometrical structure, intermolecular interactions and assess their possible applications. Additionally, the molecular stability, charge delocalization and UV‐Visible data was investigated and compared with the calculated energy and oscillator strength using the TD‐DFT approach. The researchers observed that charge transfer occurred within the molecule, indicated by the HOMO and LUMO energies. It was also found that the compound exhibited planarity and higher chemical reactivity. The calculated Mulliken charges and molecular electrostatic potential were used to interpret the Fukui index data that help predict reactive sites and understand the reactivity patterns of specific atoms in a compound. The study is aimed to understand the role of NCI in the molecule under investigation using electron localization functions and localized orbit locator methods. Molecular docking and ADMET studies were conducting involving a detailed MD simulation of a protein‐ligand complex using the OPLS3e force field and the SPC water model. These findings could prove to be beneficial in developing new therapeutic agents with various pharmacological effects and potential toxicities. [ABSTRACT FROM AUTHOR]

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSL n (q) (n ≥ 4) , PSL 3 (q) (q > 2) , or PSp 2 n (q) (n ≥ 3) , is isomorphic to a group algebra. [ABSTRACT FROM AUTHOR]

• A Hp-Adaptive Pseudospectral Hybrid Optimal Algorithm is proposed. • The optimal solution of the OSIP and OPEG problem can be robustly solved by this work. • Cases with different thrust configurations and initial conditions are reported. In this paper, a Cartesian model of the orbit pursuit-evasion game under J2 perturbations is constructed, and a step-by-step solution method applicable to this problem, namely the Hp-Adaptive Pseudospectral Hybrid Optimal Algorithm, is proposed by analyzing the principle of the pseudospectral method and the costate mapping principle. We first use the hp-adaptive Radau pseudospectral method to solve the one-sided interception problem to obtain a tentative guess of the costate and then get the saddle-point solution of the game problem by solving the two-point boundary value problem through the Newton method. The numerical results show that the hybrid method proposed in this work can efficiently find the optimal trajectory of interception and the optimal control laws of the game problem and have high accuracy. In addition, the effects of J2 perturbations and the initial orbital altitude of the evader on the game results and the algorithm itself are discussed in this paper. [ABSTRACT FROM AUTHOR]

• A relative orbit design method for CubeSats flying around space station is proposed. • The CubeSat's relative orbit is used for observing feature-points on surface of space station. • The design method considers the field-of-view constraints and occlusion effect. • The method could be used for other similar constrained close-range on-orbit service missions. This paper presents a novel relative orbit design method for CubeSats flying around the space station to perform visual inspection considering the Field-Of-View (FOV) constraints and occlusion effect. Firstly, a parametric model for Periodic Relative Orbits (PROs) is established by using the well-known Clohessy-Wiltshire equations. Based on this model, the relative orbit design problem for orbital target inspection is successfully integrated into a problem of two-parameters determination, one of which is the correlation coefficient and the other is the orbital amplitude. Secondly, by categorically evaluating the range of design variables, the design method for generating feasible solutions of PROs to satisfy the FOV constraints is proposed. Thirdly, considering the existence of the occlusion effect of large space structure, this paper presents an efficient calculating method to evaluate the degree of occlusion based on the analysis of line-of-sight blockage. Then, to avoid the occlusion effect, an improved relative orbit design method is developed. Finally, the designed algorithms are testified by carrying out some typical simulations with the China Space Station as the observed object and a small CubeSat assumed to perform the required inspection mission. The results illustrate well the validity and practicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we study the variety of one dimensional representations of a finite W -algebra attached to a classical Lie algebra, giving a precise description of the dimensions of the irreducible components. We apply this to prove a conjecture of Losev describing the image of his orbit method map. In order to do so we first establish new Yangian-type presentations of semiclassical limits of the W -algebras attached to distinguished nilpotent elements in classical Lie algebras, using Dirac reduction. [ABSTRACT FROM AUTHOR]

• A novel guidance algorithm for drag-modulated aerocapture is presented. • The algorithm is compared in detail to an efficient implementation of the state of the art solution, a numerical predictor-corrector, and is shown to achieve equivalent targeting performance for significantly lower computational expense. • This contribution could thus meaningfully advance the near-term technical feasibility of drag-modulated aerocapture. Aerocapture is a method of achieving orbit insertion via a single pass through the atmosphere of the central body. Single-event jettison drag modulation is a simple way of achieving control during atmospheric flight by effecting a discrete change in the aerodynamics of the vehicle. A novel guidance algorithm, energy reference guidance, is developed and implemented for a reference scenario of a small satellite aerocapture demonstration at Earth, and is compared to the baseline numerical predictor–corrector solution. The new approach is shown to achieve equivalent apoapsis targeting performance as the baseline algorithm with significantly lower CPU demand during atmospheric flight, although more onboard memory is required in exchange. The relationship between targeting performance and required memory is quantitatively explored for the new algorithm; the selected configuration generates approximately 3.3 MB of data, which is expected to be well within the capability of relevant avionics systems. [ABSTRACT FROM AUTHOR]

We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras, respectively. We then relate a construction of Efimov in the context of cohomological Hall algebras to the central zonotopal algebra of a graph |$G_{Q,\gamma }$| constructed from a symmetric quiver |$Q$| with enough loops and a dimension vector |$\gamma $|⁠. This provides a concrete combinatorial perspective on the former work, allowing us to identify the quantum Donaldson–Thomas (DT) invariants as the Hilbert series of the space of |$S_{\gamma }$| -invariants of the Postnikov–Shapiro slim subgraph space attached to |$G_{Q,\gamma }$|⁠. The connection with orbit harmonics in turn allows us to give a manifestly nonnegative combinatorial interpretation to numerical DT invariants as the number of |$S_{\gamma }$| -orbits under the permutation action on the set of break divisors on |$G$|⁠. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest. [ABSTRACT FROM AUTHOR]

In our recent work, homotopy types of nilpotent orbits are explicitly described in the case of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is not semisimple. Here we extend the above description of homotopy types of nilpotent orbits to the remaining cases of real simple classical Lie algebras forwhich anymaximal compact subgroup in the associated adjoint group is semisimple. [ABSTRACT FROM AUTHOR]

To realize rapid, safe and optimal avoidance for spacecraft in complex threat situations and incomplete-information conditions, and considering the problems in existing hierarchical decision-making modes, a learning-based spacecraft reactive anti-hostile-rendezvous (AHR) maneuver control framework is proposed in this paper. First, a fundamental AHR action planner based on the interfered fluid dynamical system (IFDS) is proposed, in which the avoidance strategies for hostile and non-adversarial threats are appropriately integrated. An avoidance strategy matched with the IFDS is further designed by specifically considering high-intensity confrontation situations where a threat maintains persistent pursuit. Then, the "state-action" solution methods based on deep reinforcement learning (DRL) and multi-agent-DRL (MADRL) methods for orbital avoidance maneuvers are proposed to control the avoidance time and directions by optimizing the IFDS parameters. In the offline training stage, the DRL method will first simulate the interactive processes between the spacecraft and non-adversarial threats so that the spacecraft can gradually adapt to complex space environments and generate the relative primary avoidance maneuver strategies. On this basis, for hostile threats, the MADRL-based solution method of orbit game strategies is further established, in which the spacecraft and hostile threats are constructed as two types of opposite agents to simulate the game processes so that the strategy levels can be bilaterally evolved and the AHR success rates can be improved. After training, the corresponding neural networks are extracted from the agents in the DRL-based and MADRL-based methods for reactive AHR maneuver control. Moreover, a progressive agent training mechanism matched with the above DRL-based and MADRL-based solution methods is proposed, which can reduce the invalid interactions in initial training episodes, improve the training efficiency, and enhance the generalization ability of orbital game strategies. Finally, the effectiveness of the proposed framework is demonstrated by simulations. [ABSTRACT FROM AUTHOR]

We describe the boundary of linear subvarieties in the moduli space of multi-scale differentials. Linear subvarieties are algebraic subvarieties of strata of (possibly) meromorphic differentials that in local period coordinates are given by linear equations. The main example of such are affine invariant submanifolds, that is, closures of SL(2, R)-orbits. We prove that the boundary of any linear subvariety is again given by linear equations in generalized period coordinates of the boundary. Our main technical tool is an asymptotic analysis of periods near the boundary of the moduli space of multi-scale differentials which yields further techniques and results of independent interest. [ABSTRACT FROM AUTHOR]

We study a class of Z -graded algebras introduced by Bell and Rogalski. Their construction generalizes in large part that of rank one generalized Weyl algebras (GWAs). We establish certain ring-theoretic properties of these algebras and study their connection to GWAs. We classify the simple weight modules in the infinite orbit case and provide a partial classification in the case of orbits of finite order. [ABSTRACT FROM AUTHOR]

Let G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$ -grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$ , let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$. We study the $G_0$ -equivariant projections $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to {\mathfrak g}_0$ and $\psi:\overline{{\mathcal O}_{\textsf{min}}}\to{\mathfrak g}_1$. It is shown that the properties of $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ essentially depend on whether the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is empty or not. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then both $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ contain a 1-parameter family of closed $G_0$ -orbits, while if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$ , then both are $G_0$ -prehomogeneous. We prove that $\overline{G{\cdot}\varphi({\mathcal O}_{\textsf{min}})}=\overline{G{\cdot}\psi({\mathcal O}_{\textsf{min}})}$. Moreover, if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then this common variety is the affine cone over the secant variety of ${\mathbb P}({\mathcal O}_{\textsf{min}})\subset{\mathbb P}({\mathfrak g})$. As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G -module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of ${\mathfrak g}$ in place of ${\mathfrak g}_0$ or spherical nilpotent G -orbits in place of ${\mathcal O}_{\textsf{min}}$. [ABSTRACT FROM AUTHOR]

We showed that isomorphism classes of idempotent evolution algebras are in bijection with the orbits of the semidirect product group of the symmetric group and the torus, considered the combinatoric problem of enumeration of isomorphism classes for these algebras over arbitrary finite fields, derived a general counting formula, and obtained explicit formulas for the numbers of isomorphism classes in dimensions 2, 3, and 4 over any finite field. [ABSTRACT FROM AUTHOR]

To assess the applicability of the tension estimation method using local vibration measurements in a thorough manner, this research work is devoted to systematically investigate the appropriate covering ranges of measurements for different cables. Four cables of three cable-stayed bridges are deliberately chosen to cover a wide range of the cable slenderness parameter. Numerical analyses with finite element models and field validations with real measurements are conducted for these stay cables to demonstrate that the covering range of local measurements can be undoubtedly reduced with the increase of cable slenderness. For a relatively short cable with the slenderness parameter at the order of 4000, the adoption of 1/3 coverage is sufficient to keep a high-level accuracy with at most 1% of error. Besides, 1/4, 1/6, and 1/7 coverages are found adequate to maintain the same level of accuracy for longer cables with the slenderness parameter at the orders of 30000, 55000, and 80000, respectively. With the solid validations presented in the current study, the covering range of the cable for this simplified method employing local vibration measurements can be confidently reduced to greatly alleviate the expense and hard work of sensor installation near the high end. [ABSTRACT FROM AUTHOR]

Classical methods for initial orbit determination (IOD) include Laplace method, Gauss method, and their variations. In addition to this, based on the characteristic of optical observation data nowadays, experts propose some other IOD methods, like Double- r method and admissible region method. One of the ways to determinate the orbit through double- r method is to guess distances of the target from the observer at two epochs—usually at the first and the last one. By doing so, we can solve the Lambert problem, and use its solution as the initial guess of the orbit. Furthermore, we can improve the initial guess by iterations to reduce the root mean square (RMS) of the observations. The admissible method is based on the concept of attributable (longitude, latitude, and their rates). With some conceptions, the admissible region described by the range and range rate from the observer is characterized. Using triangulation we can find the nodal point that makes the RMS minimal. In our work, we apply one intelligent optimization method—the particle swarm optimization method to the two methods, based on simulated and real data, and compare the results with that of modified Laplace method. At last, we briefly discuss the possibility of applying the double- r method to the orbit link problem. [ABSTRACT FROM AUTHOR]

Optical coherence tomography (OCT) has the advantage of high precision in measuring micrometre-scale precision structures, but its measurement range is limited by the correlation with Rayleigh length of Gaussian beam. Therefore, this study used the Airy beam to extend the focal depth of the OCT system. The system's coupled power was increased by 7.34 times when the sample position was at an optical defocus of 3 mm. Measurements of the resolution plate and silicon wafer sample showed that Airy beam OCT had a lateral resolution of 7.4 μm and the focal depth was >6 mm. While the focal depth of the Gaussian beam OCT with the equivalent lateral resolution was less than 1 mm. This improved OCT system demonstrates significant potential for sample measurements at large aspect ratios in the industrial surface detection field. [ABSTRACT FROM AUTHOR]

Very-low Earth orbit (VLEO) space below 200 km is essential for high-quality communications and near-Earth space environment detection. Due to the significant atmospheric drag, orbital maintenance is required for spacecraft staying here. Based on air-breathing electric propulsion (ABEP) technology, this paper analyzed the orbital boundary conditions of the spacecraft under the constraints of parameters including slenderness ratio, thrust-to-power ratio, drag coefficient, and effective specific impulse. The energy balance is the key constraint for low VLEO orbits, which is determined by the drag coefficient, slenderness ratio, and thrust-to-power ratio. Under the existing technical conditions, the lowest circular orbit (along the terminator) is about 170 km. An elliptical orbital flight scheme is also analyzed to reach a 150 km perigee. A half-period control method was proposed based on the on–off control method for the elliptical orbit, which could enable the spacecraft to maintain a stable 150–250 km elliptical orbit. [ABSTRACT FROM AUTHOR]

In this paper, we obtain explicit branching laws for all irreducible unitary representations of G=\operatorname {Spin}(N,1) when restricted to a parabolic subgroup P. The restriction turns out to be a finite direct sum of irreducible unitary representations of P. We also verify Duflo's conjecture for the branching laws of discrete series representations of G with respect to P. That is to show: in the framework of the orbit method, the branching law of a discrete series representation is determined by some geometric behavior of the moment map for the corresponding coadjoint orbit. [ABSTRACT FROM AUTHOR]

The "IADC space debris mitigation guidelines" requires that the post-mission orbit lifetime of low Earth orbiting (LEO) satellites should not be longer than 25 years, which is the key to debris mitigation. To analyze the degree of global compliance to this 25-year guideline, a fast and relatively accurate orbit life estimation algorithm is required. The traditional method uses numerical orbit propagation to accomplish the task, although its accuracy is high when a high-precision force model is used, the calculation is time-consuming. This paper proposes a new method for orbit lifetime estimation without relying on orbit propagation. A back propagation (BP) neural network is used to directly fit the relationship from the satellite parameters to the orbit lifetime. The space objects that have reentered the Earth's atmosphere are selected from the catalog database of the US Space Surveillance Network (SSN), and are used to generate samples for network training. The genetic algorithm is combined with the traditional back propagation algorithm for the training of the neural network. The results show that for the specific application background of identifying whether a satellite complies with the 25-year guideline, the orbit lifetime estimation accuracy of the BP network meets the demand to some extent. Moreover, because it does not involve orbit propagation, the BP network has a high computational efficiency in estimating orbit lifetime, showing a good application prospect. [ABSTRACT FROM AUTHOR]

Maneuvering flight can generate the asymmetry of the stiffness or damping to an engine rotor system, as so to influent the whirl direction and characteristics of rotor system. In this article, the identification method for orbit direction is provided for a rotor system under maneuvering flight condition. By a simulation example, the variable movement shapes about rotor under maneuvering flight are illustrated through cases as the mixed and the pending whirls. The results show that the original low-order critical speed in the backward or forward whirl mode changes to the mixed whirl while considering the impact of maneuvering flight, where some backward and forward critical speeds in the same order are almost overlapped as one critical speed which in the mixed whirl mode. Maneuvering flight results in dynamic eccentricity of the rotor shaft, and the orbits of nodes in the mixed whirl are a series of ellipses with variable inclination angles. The case with pending whirl direction is finally determined as a mixed whirl by the orbit direction graph. The whirl directions of orbits on bearings or the contact angle of bearings maybe different under maneuvering flight can be explained from the rotor dynamical view. [ABSTRACT FROM AUTHOR]

Let G = \operatorname {exp}(\mathfrak {g}) be a connected and simply connected real nilpotent Lie group with Lie algebra \mathfrak g, H = \operatorname {exp}(\mathfrak {h}) an analytic subgroup of G with Lie algebra \mathfrak h, \chi a unitary character of H and \tau = \text {ind}_H^G \chi the monomial representation of G induced by \chi. Let D_{\tau }(G/H) be the algebra of the G-invariant differential operators on the fiber bundle over the base space G/H associated to the data (H,\chi). We prove the polynomial conjecture due to Corwin-Greenleaf stating that if \tau is of finite multiplicities, the algebra D_{\tau }(G/H) is isomorphic to the algebra {{\mathbb C}[{\Gamma }_{\tau }]}^H of the H-invariant polynomial functions on the affine subspace {\Gamma }_{\tau } = \{\ell \in {\mathfrak g}^*; {\ell }|_{\mathfrak h} = -\sqrt {-1}d\chi \} of {\mathfrak g}^*. In this case, we show that any non-zero element of D_{\tau }(G/H) admits a fundamental tempered solution. [ABSTRACT FROM AUTHOR]

We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra L κ (sl n + 1) with n ∈ N of admissible level κ. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand–Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl 2 -induced modules in these orbits. In particular, simple admissible sl 2 -induced modules are fully classified for the principal nilpotent orbit. [ABSTRACT FROM AUTHOR]

We present precise spectrophotometric parameters of the two close visual binary systems, HD 200325 and HD 220077, which have not been spectrophotometrically studied before. Precise parameters are presented based on Al-Wardat's method for analyzing binary and multiple stellar systems. Which combines the spectroscopic analysis with the photometric analysis to construct the combined synthetic SED and its components of a binary with the aid of Kurucz's models. From the analysis of the synthetic models, we determine precise values for each of the spectrophotometric parameters, masses, and parallaxes of the systems. Precise spectrophotometric masses of both components were found to be M sph A = 1.30 ± 0.12 M ⊙ and M sph B = 0.85 ± 0.08 M ⊙ for HD 200325 and M sph A = 1.08 ± 0.10 M ⊙ and M sph B = 0.90 ± 0.10 M ⊙ for HD 220077 with an age of about 1.413 ± 0.10 Ga and 3.548 ± 0.35 Ga, respectively. Also, the orbital parameters are modified based on Tokovinin's method for solving orbits of binary stars. In our spectrophotometric and dynamical analysis, we arrive at new dynamical parallaxes as π dyn = 18.8096 ± 0.80 mas for HD 200325 and π dyn = 12.681 ± 0.48 mas for HD 220077. [ABSTRACT FROM AUTHOR]