Orbital Sets Research Articles

Behavior Analysis of a Class of Discrete-Time Dynamical System with Capture Rate

In this paper, we study the stability and bifurcation analysis of a class of discrete-time dynamical system with capture rate. The local stability of the system at equilibrium points are discussed. By using the center manifold theorem and bifurcation theory, the conditions for the existence of flip bifurcation and Hopf bifurcation in the interior of R+2 are proved. The numerical simulations show that the capture rate not only affects the size of the equilibrium points, but also changes the bifurcation phenomenon. It was found that the discrete system not only has flip bifurcation and Hopf bifurcation, but also has chaotic orbital sets. The complexity of dynamic behavior is verified by numerical analysis of bifurcation, phase and maximum Lyapunov exponent diagram.

Multireference Approach to Normal and Resonant Auger Spectra Based on the One-Center Approximation.

A methodology to calculate the decay rates of normal and resonant Auger processes in atoms and molecules based on the One-Center Approximation (OCA), using atomic radial Auger integrals, is implemented within the restricted-active-space self-consistent-field (RASSCF) and the multistate restricted-active-space perturbation theory of second order (MS-RASPT2) frameworks, as part of the OpenMolcas project. To ensure an unbiased description of the correlation and relaxation effects on the initial core excited/ionized states and the final cationic states, their wave functions are optimized independently, whereas the Auger matrix elements are computed with a biorthonormalized set of molecular orbitals within the state-interaction (SI) approach. As a decay of an isolated resonance, the computation of Auger intensities involves matrix elements with one electron in the continuum. However, treating ionization and autoionization problems can be overwhelmingly complicated for nonexperts, because of many peculiarities, in comparison to bound-state electronic structure theory. One of the advantages of our approach is that by projecting the intensities on the atomic center bearing the core hole and using precalculated atomic radial two-electron integrals, the Auger decay rates can be easily obtained directly with OpenMolcas, avoiding the need to interface it with external programs to compute matrix elements with the photoelectron wave function. The implementation is tested on the Ne atom, for which numerous theoretical and experimental results are available for comparison, as well as on a set of prototype closed- and open-shell molecules, namely, CO, N2, HNCO, H2O, NO2, and C4N2H4 (pyrimidine).

Cislunar distributed architectures for communication and navigation services of lunar assets

The last decade saw a renewed interest on the Moon as a well suited training premise in preparation to manned mission to Mars, but also as an interesting target itself, for scientific investigations, technological developments and new markets opportunities. As a result, numerous and very different missions to the Moon are currently being studied and implemented, assuming to have our satellite quite crowded soon.Such a scenario motivates the settling of space infrastructures to offer recurrent services like data relays, communication links and navigation in the cislunar environment which would facilitate and enlighten the single mission’s implementation and operation.The paper presents the strategy adopted to address the design of the orbital configuration for a distributed architecture to answer the communication and navigation needs to serve at the best the diversified lunar missions scenario expected for the next decades. First, a set of parameters of merit are identified and explained in their mathematical expression and physical meaning. Then, different regions of interest for possible future missions are identified and mapped to the relevant performances wanted for that specific region. Last a Multi-Objective Optimisation framework is presented, both in the exploited genotype and the different objectives participating to the definition of the cost function, in order to provide a versatile tool.The paper critically discusses the effectiveness of the proposed approach in detecting the best suited distributed orbital architectures for the servicers according to the expected service performance in specific user regions, spread all over the Earth–Moon volume — from Earth vicinity to Lunar surface, considering also robustness aspects. The benefits in the exploitation of the multibody dynamical regime offered by the Earth–Moon system to set up the most promising orbital set with a minimum number of servicing spacecraft are underlined as well.

Large-scale Multiconfiguration Dirac–Hartree–Fock Calculations for Astrophysics: C-like Ions from O iii to Mg vii

Large-scale multiconfiguration Dirac–Hartree–Fock calculations are provided for the n ≤ 5 states in C-like ions from O iii to Mg vii. Electron correlation effects are accounted for by using large configuration state function expansions, built from sets of orbitals with principal quantum numbers n ≤ 10. An accurate and complete data set of excitation energies, wavelengths, radiative transition parameters, and lifetimes is offered for the 156 (196, 215, 272, 318) lowest states of the 2s 22p 2, 2s2p 3, 2p 4, 2s 22p3s, 2s 22p3p, 2s 22p3d, 2s2p 23s, 2s2p 23p, 2s2p 23d, 2p 33s, 2p 33p, 2p 33d, 2s 22p4s, 2s 22p4p, 2s 22p4d, 2s 22p4f, 2s2p 24s, 2s2p 24p, 2s2p 24d, 2s2p 24f, 2s 22p5s, 2s 22p5p, 2s 22p5d, 2s 22p5f, and 2s 22p5g configurations in O iii (F iv, Ne v, Na vi, Mg vii). By comparing available experimental wavelengths with the MCDHF results, the previous line identifications for the n = 5, 4, 3 → n = 2 transitions of Na vi in the X-ray and EUV wavelength range are revised. For several previous identifications discrepancies are found, and tentative new (or revised) identifications are proposed. A consistent atomic data set including both energy and transition data with spectroscopic accuracy is provided for the lowest hundreds of states for C-like ions from O iii to Mg vii.

Enhanced orbit determination for formation-flying satellites based on M-estimation

The classical least-squares estimation (LS) has been widely used for parameter estimation in satellite precise orbit determination (POD). However, the classical LS estimation with uniform weighting is very sensitive to outliers in the observations. To overcome this issue, we use the M-estimation on the basis of the LS estimation to improve the POD solutions. To demonstrate the added value of the M-estimation, we use one year of GPS data collected by the GRACE Follow-On mission, and two sets of orbits are produced using both the LS and M-estimation. The obtained orbits are assessed through orbit formal error analysis, orbit comparison, residual analysis of satellite laser ranging (SLR), and K-band ranging (KBR) measurements. Formal error analysis shows that the precision can be improved with the M-estimation notably by 25 and 40% for the dynamic and kinematic orbit, respectively. Orbit comparison reveals that the M-estimation can offer better solution consistency between the dynamic and kinematic orbit, with the differences being reduced by 23% when compared to the LS estimation. KBR validation reports that the relative precision can be improved with the M-estimation significantly by 23 and 40% for the dynamic and kinematic orbit, respectively. Specifically, the KBR residuals are reduced from 0.65 to 0.50 mm for dynamic orbit, and from 3.24 to 1.96 mm for kinematic orbit. These results demonstrate that the M-estimation can improve the orbit precision in both absolute and relative senses. Finally, we find that the current orbit accuracy is likely dominated by systematic errors, which can obscure the contribution of the M-estimation. The SLR measurements, with precision at 5–10 mm level, also seem to be not precise enough to validate the potential contribution. As a result, the improvements as indicated by the SLR validation are shown to be insignificant.

Symmetry reductions and exact solutions of two new generalized negative KdV type equations

We give a simple geometric interpretation of the mapping of the negative KdV equation (−ψxxψ)t=(ψ2)x as proposed by Qiao and Li (2011) [26] and the Fuchssteiner equation (utxxux)x+4(uutux)x+2ut=0 using geometry of projective connection on S1 or stabilizer set of the Virasoro orbit. We propose a similar connection between (−ψxxψ)t=(ψ3)x and (−ψxxψ)t=(ψ4)x with the higher-order negative KdV equations of Fuchssteiner type described as (utxxx3u2+uxx)x+10((uut)x3u2+uxx)x+3ut=0 and (utxxxxuxxx+16uux)x+20(uutxxuxxx+16uux)x+30(uxutxuxxx+16uux)x+18(uxxut+64u2utuxxx+16uux)x+4ut=0 respectively. Then we perform the symmetry analysis of these newly found equations and study one-dimensional optimal classifications of their inequivalent subalgebras. We then obtain several exact solutions for these two equations by using those subalgebras. Further we establish some traveling wave solutions for these two equations. Moreover we construct some new exact solutions of the negative KdV equation which have not been reported in earlier studies. We illustrate the physical significance of some of the obtained solutions by performing numerical simulations and they appeared to be several kinds of soliton solutions.

Periodic orbit evaluation of a spectral statistic of quantum graphs without the semiclassical limit

Energy level statistics of quantized chaotic systems have been evaluated in the semiclassical limit via their periodic orbits using the Gutzwiller and related trace formulae. Here we evaluate a spectral statistic of chaotic 4-regular quantum graphs from their periodic orbits without the semiclassical limit. The variance of the n-th coefficient of the characteristic polynomial is determined by the sizes of the sets of distinct primitive periodic orbits with n bonds which have no self-intersections, and the sizes of the sets with a given number of self-intersections which all consist of two sections of the pseudo orbit crossing at a single vertex. Using this result we observe the mechanism that connects semiclassical results to the total number of orbits regardless of their structure.

CRISM‐Based High Spatial Resolution Thermal Inertia Mapping Along Curiosity's Traverses in Gale Crater

AbstractThermal inertia is a key summary property describing the thermophysical characteristics of geologic materials. Orbital estimates of thermal inertia are especially useful in conjunction with rover‐based observations to provide additional constraints on material properties and to interpret the broader region surrounding the traverse. We have developed an approach to estimate apparent thermal inertias (ATI) from observations from the Mars Reconnaissance Orbiter's Compact Reconnaissance Imaging Spectrometer for Mars (CRISM) at a much higher spatial resolution than is available from any other orbital data set. We apply this method to along‐track oversampled scene FRT00021C92 over Gale Crater within Glen Torridon, where the Curiosity rover has traversed; the retrieved values are then analyzed along with remote sensing and in‐situ observations by Curiosity to characterize surface properties in and around Glen Torridon. CRISM‐based ATI estimates of basaltic sand fields indicate a correlation between higher ATI values and the abundance of large (up to 3 mm in size) grains on the crests and flanks of large ripples. On the Vera Rubin ridge, ATI estimates are used to identify a west‐to‐east increase in the abundance of rock fragments within unconsolidated surface cover. On the Greenheugh pediment, an unmixing approach is used to identify a thin layer of wind‐blown sand sourced from the nearby Sands of Forvie. Finally, within Glen Torridon, a north‐to‐south increase in the mean and width of the ATI distribution is identified as a consequence of increased bench‐like bedrock exposures.

Diffusion transitions in a 2D periodic lattice

Spatial diffusion of particles in periodic potential models has provided a good framework for studying the role of chaos in global properties of classical systems. Here a bidimensional “soft” billiard, classically modeled from an optical lattice Hamiltonian system, is used to study diffusion transitions under variation of the control parameters. Sudden transitions between normal and ballistic regimes are found and characterized by inspection of topological changes in phase-space. Transitions correlated with increases in global stability area are shown to occur for energy levels where local maxima points become accessible, deviating trajectories approaching them. These instabilities promote a slowing down of the dynamics and an island myriad bifurcation phenomenon, along with the suppression of long flights within the lattice. Other diffusion regime variations occurring within small intervals of control parameters are shown to be related to the emergence of a set of orbits with long flights, thus altering the total average displacement for long integration times but without global changes in phase-space.

Heavy-hole bilayer trions of transition metal dichalcogenides by analytical treatment to model He-isoelectronic ions upto dipole factor of Green’s function expansion of Coulomb interaction

Electronic structures of heavy-hole trions and He-isoelectronic ions show dependence on transitions among two-electron bound states constituted of hydrogenic orbitals for which Coulomb (exchange) interaction causes nontrivial secular divergence to Schrödinger equation (SE). Coulomb interaction between a pair of electrons triggers nonadiabatic transitions due conical intersections and larger vibrational amplitudes of nuclei. Therefore, existing state-of-art theories ubiquitously urge analytical integrals for multipoles of electrostatic Green’s function expansion. Born–Oppenheimer (BO) approximation separates the hamiltonian into electronic and nuclear coordinates at adiabatic limit. Employing associated Laguerre polynomial and Whittaker-M function basis sets of hydrogenic orbitals for the pair of electrons (electronic coordinate) furnishes analytical, terminable, simple and finitely summed integrals in terms of Lauricella functions. The exact integrals also remedy the difficulty arisen from different scaling factors of bound states for higher order perturbation calculations. Analytical perturbation calculations for monopole and dipole factors using singly and doubly excited states (|n2sn4s〉, |n2pn4p〉) with spherical and dumbbell symmetries show sufficient convergence of ground-state energy correction of He-isoelectronic series within 0.20–6.89% of reported results. Calculation of current-density with ground-state upto first order perturbation correction clearly shows that singly and doubly excited spherically symmetric states do not contribute to local current of He-isoelectronic series for monopole factor. Binding energies of heavy-hole trions of transition metal dichalcogenides (TMDCs) are found to be in good agreement with experimentally observed values.

Full configuration interaction quantum Monte Carlo treatment of fragments embedded in a periodic mean field.

We present an embedded fragment approach for high-level quantum chemical calculations on local features in periodic systems. The fragment is defined as a set of localized orbitals (occupied and virtual) corresponding to a converged periodic Hartree-Fock solution. These orbitals serve as the basis for the in-fragment post-Hartree-Fock treatment. The embedding field for the fragment, consisting of the Coulomb and exchange potential from the rest of the crystal, is included in the fragment's one-electron Hamiltonian. As an application of the embedded fragment approach, we investigate the performance of full configuration interaction quantum Monte Carlo (FCIQMC) with the adaptive shift. As the orbital choice, we use the natural orbitals from the distinguishable cluster method with singles and doubles. FCIQMC is a stochastic approximation to the full CI method and can be routinely applied to much larger active spaces than the latter. This makes this method especially attractive in the context of open shell defects in crystals, where fragments of adequate size can be rather large. As a test case, we consider dissociation of a fluorine atom from a fluorographane surface. This process poses a challenge for high-level electronic structure models as both the static and dynamic correlations are essential here. Furthermore, the active space for an adequate fragment (32 electrons in 173 orbitals) is already quite large even for FCIQMC. Despite this, FCIQMC delivers accurate dissociation and total energies.

Permanent variational wave functions for bosons

We study the performance of permanent states (the bosonic counterpart of the Slater determinant state) as approximating functions for bosons, with the intention to develop variational methods based upon them. For a system of N identical bosons, a permanent state is constructed by taking a set of N arbitrary (not necessarily orthonormal) single-particle orbitals, forming their product and then symmetrizing it. It is found that for the one-dimensional Bose–Hubbard model with the periodic boundary condition and at unit filling, the exact ground state can be very well approximated by a permanent state, in that the permanent state has high overlap (at least 0.96 even for 12 particles and 12 sites) with the exact ground state and can reproduce both the ground state energy and the single-particle correlators to high precision. For a generic model, we have devised a greedy algorithm to find the optimal set of single-particle orbitals to minimize the variational energy or maximize the overlap with a target state. It turns out that quite often the ground state of a bosonic system can be well approximated by a permanent state by all the criteria of energy, overlap, and correlation functions. And even if the error is apparent, it can often be remedied by including more configurations, i.e., by allowing the variational wave function to be a combination of multiple permanent states. The algorithm is used to study the stability of a two-particle system, with great success. All these suggest that permanent states are very effective as variational wave functions for bosonic systems, and hence deserve further studies.

Genome Sequence Resource of Sarocladium terricola TR, an Endophytic Fungus as a Potential Biocontrol Agent Against Meloidogyne incognita.

Genome Sequence Resource of Sarocladium terricola TR, an Endophytic Fungus as a Potential Biocontrol Agent Against Meloidogyne incognita.

Approximate functionals in hypercomplex Kohn–Sham theory

The recently developed hypercomplex Kohn–Sham (HCKS) theory shows great potential to overcome the static/strong correlation issue in density functional theory (DFT), which highlights the necessity of further exploration of the HCKS theory toward better handling many-electron problem. This work mainly focuses on approximate functionals in HCKS, seeking to gain more insights into functional development from the comparison between Kohn–Sham (KS) DFT and HCKS. Unlike KS-DFT, HCKS can handle different correlation effects by resorting to a set of auxiliary orbitals with dynamically varying fractional occupations. These orbitals of hierarchical correlation (HCOs) thus contain distinct electronic information for better considering the exchange–correlation effect in HCKS. The test on the triplet–singlet gaps shows that HCKS has much better performance as compared to KS-DFT in use of the same functionals, and the systematic errors of semi-local functionals can be effectively reduced by including appropriate amount of the HCO-dependent Hartree–Fock exchange. In contrast, KS-DFT shows large systematic errors, which are hardly reduced by the functionals tested in this work. Therefore, HCKS creates new channels to address to the strong correlation issue, and further development of functionals that depend on HCOs and their occupations is necessary for the treatment of strongly correlated systems.

Generalizations of the R-Matrix Method to the Treatment of the Interaction of Short-Pulse Electromagnetic Radiation with Atoms

Since its initial development in the 1970s by Phil Burke and his collaborators, the R-matrix theory and associated computer codes have become the method of choice for the calculation of accurate data for general electron–atom/ion/molecule collision and photoionization processes. The use of a non-orthogonal set of orbitals based on B-splines, now called the B-spline R-matrix (BSR) approach, was pioneered by Zatsarinny. It has considerably extended the flexibility of the approach and improved particularly the treatment of complex many-electron atomic and ionic targets, for which accurate data are needed in many modelling applications for processes involving low-temperature plasmas. Both the original R-matrix approach and the BSR method have been extended to the interaction of short, intense electromagnetic (EM) radiation with atoms and molecules. Here, we provide an overview of the theoretical tools that were required to facilitate the extension of the theory to the time domain. As an example of a practical application, we show results for two-photon ionization of argon by intense short-pulse extreme ultraviolet radiation.

Constructing periodic orbits of high-dimensional chaotic systems by an adjoint-based variational method.

Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatiotemporal systems including fluid turbulence is supported by nonchaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatiotemporally chaotic systems remains challenging as known methods either show poor convergence properties because they are based on time-marching of a chaotic system causing exponential error amplification, or they require constructing Jacobian matrices which is prohibitively expensive. We propose a new matrix-free method that is unaffected by exponential error amplification, is globally convergent, and can be applied to high-dimensional systems. The adjoint-based variational method constructs an initial value problem in the space of closed loops such that periodic orbits are attracting fixed points for the loop dynamics. We introduce the method for general autonomous systems. An implementation for the one-dimensional Kuramoto-Sivashinsky equation demonstrates the robust convergence of periodic orbits underlying spatiotemporal chaos. Convergence does not require accurate initial guesses and is independent of the period of the respective orbit.

Tomographic Inversion Methods for Retrieving the Tropospheric Water Vapor Content Based on the NDSA Measurement Approach

In this paper, we deal with the problem of retrieving maps of tropospheric Water Vapor (WV) concentration by means of a set of Low Earth Orbit (LEO) satellites orbiting in the same plane and along the same direction. It is assumed that a number of microwave links is deployed between a group of satellites with microwave transmitters onboard and another group with receivers. It is also assumed that the Normalized Differential Spectral Absorption (NDSA) approach is used to provide time series of Integrated Water Vapor (IWV) along each link. The set of links scans an annular region belonging to the orbital plane of the LEO satellites, so that the time series of the IWV measurements can be exploited to retrieve the WV concentration in such a region. This is a typical tomographic inversion problem. The geometry of the acquisition system and the path-integrated nature of measurements well fit the application of the Exterior Reconstruction Tomographic Algorithm (ERTA), which was proposed several decades ago in contexts different from remote sensing. In this paper, we investigate the potential of ERTA for the WV retrieval and compare its performance with that of a least square inversion approach already presented in the literature. The compared analysis has been carried out through simulations that have highlighted the peculiarities and retrieval capabilities of the two tomographic methods, as well as the impact of the richness of the satellite constellation on the overall performance.

Mapped adjoint control transformation method for low-thrust trajectory design

Variational approach to optimal control theory converts trajectory optimization problems into two- or multiple-point boundary-value problems, which consist of costates (i.e., Lagrange multipliers associated with the states). Estimating missing values of the non-intuitive costates is an important step in solving the resulting boundary-value problems. By leveraging costate vector mapping theorem, we extend the method of Adjoint Control Transformation (ACT), called Mapped ACT (MACT), to alternative sets of coordinates/elements for solving low-thrust trajectory optimization problems. In particular, extension of the ACT method to the set of modified equinoctial elements and an orbital element set based on the specific angular momentum and eccentricity vectors (h-e) is demonstrated. The computational and robustness efficiency of the MACT method is compared against the traditionally used random initialization of costates by solving 1) interplanetary rendezvous maneuvers and 2) an Earth-centered, orbit-raising problem with and without the inclusion of J2 perturbation. For the considered problems, numerical results indicate two to three times improvement in the percent of convergence of the resulting boundary-value problems when the MACT method is used compared to the random initialization method. Results also indicate that the h-e set is also a contender and suitable choice for solving low-thrust trajectory optimization problems.

Deformations of symplectic singularities and orbit method for semisimple Lie algebras

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of coadjoint orbits of a semisimple algebraic group to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras and conjecture that in that case the image consists of the primitive ideals corresponding to one-dimensional representations of W-algebras. Along the way, we get several new results on the Lusztig-Spaltenstein induction for coadjoint orbits.

Amenability, proximality and higher-order syndeticity

AbstractWe show that the universal minimal proximal flow and the universal minimal strongly proximal flow of a discrete group can be realized as the Stone spaces of translation-invariant Boolean algebras of subsets of the group satisfying a higher-order notion of syndeticity. We establish algebraic, combinatorial and topological dynamical characterizations of these subsets that we use to obtain new necessary and sufficient conditions for strong amenability and amenability. We also characterize dense orbit sets, answering a question of Glasner, Tsankov, Weiss and Zucker.