Solution of the equations resulting from a separation of the Helmholtz equations in special coordinate systems
Lie group use in solution of equations resulting from separation of Helmholtz equations in special coordinate systems
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Autonomous Control for Arbitrary Thruster Configurations and Mass Properties in Special Euclidean Group SE(3)
Most current methods for determining maneuvers and thrust firing sequences depend on explicit and predetermined commands generated by a combination of on-board systems and ground-based human-in-the-loop methods. For spacecraft and space structures with changing mass properties and thruster configurations, such as the Deep Space Gateway as it changes configurations throughout its lifetime, determining these commands can be time-consuming and computationally intensive. However, recent work within the Lie group SE(3) has offered ways of autonomously determining the location, power, precision, and capabilities of thrusters in any arbitrary position. Furthermore, a method for determining thruster firing sequences based on an arbitrary control input (both translational and rotational in a coupled, 6-element vector) and arbitrary thruster configurations has also recently been developed. When combining these methods, any spacecraft with any mass properties and thruster configurations can be understood in terms of controllability limits and thruster firing sequences can be generated quickly and with low computational load, thus extending the autonomous capabilities of deep space missions. In this work, this method is presented and explored in terms of convergence time to the desired pose. The capabilities of this method are also examined in the case of the Deep Space Gateway both in fully controllable configurations and uncontrollable configurations.
SE(3)↗
System theory as applied differential geometry
The invariants of input-output systems under the action of the feedback group was examined. The approach used the theory of Lie groups and concepts of modern differential geometry, and illustrated how the latter provides a basis for the discussion of the analytic structure of systems. Finite dimensional linear systems in a single independent variable are considered. Lessons of more general situations (e.g., distributed parameter and multidimensional systems) which are increasingly encountered as technology advances are presented.
Hermann, R.↗
Solid state and molecular theory group Semiannual progress report
Use of scattered wave method to compute molecular wave functions, augmented plane wave method for energy band calculations, and Casimir invariants as invariant operators in Lie groups
Source record↗
Bifurcations of relative equilibria
The characteristics of equivariant dynamical systems near relative equilibria (RE: group orbits which are invariant in the flow of an equivariant vector field) are investigated analytically, with a focus on the dynamics and bifurcation (B) behavior. The principles of Lie-group theory are reviewed; the decomposition of the vector field is explained; and particular attention is given to the Bs of RE occurring when an eigenvalue passes through zero, Hopf Bs of RE, the classification of generic secondary steady-state and Hopf Bs with symmetry group O(2), Bs of the zero solution of the Kuramoto-Shivashinsky equation, and possible generic steady-state Bs in the two-dimensional Benard problem. In the latter case, it is shown that the primary generic Bs are to two types of equilibria (hexagons and rolls), while the secondary Bs result in trajectories which are either equilibria or rotating waves.
Krupa, Martin↗
The stratigraphy of the Steep Rock Group, N.W. Ontario, with evidence of a major unconformity
The Steep Rock Group is exposed 6 km north of Atikokan, 200 km west of Thunder Bay. It is situated on the southern margin of the Wabigoon Belt of the Archaean Superior Province, N. W. Ontario. Reinvestigation of the geology of the Group has shown that the Group lies unconformably on the Tonalite Complex to the east. This unconformity has been previously suspected, from regional and ine mapping but no conclusive outcrop evidence for its existence has as yet been published. The strike of the group, comprised of Basal Conglomerate, Carbonate Member, Ore Zone and Ashrock is generally north-northwest dipping steeply to the southwest. Of the 7 contacts between the Steep Rock Group and the Tonalite Complex, 3 expose the unconformity (The Headland, S. Roberts Pit, Trueman Point), and 4 are faulted. These three outcrops demonstrate unequivocally that the Steep Rock group was laid down unconformably on the underlying Tonalite Complex, which is circa 3 Ga old.
Wilks, M. E.↗
Conservation laws of linear, homogeneous systems
Abstract operator represented for linear, homogeneous system as Lie group exhibiting metrics and conservation laws
Williams, J. R.↗
Estimation and detection of signals in multiplicative noise
A class of detection-estimation problems on matrix Lie groups is defined in which the observation noise is multiplicative in nature. By examining the differential versions of the hypotheses, which are bilinear in nature, it is possible to derive the relevant likelihood ratio formula and the associated optimal estimation equations for the signal given the observations and the assumption that the signal is present. These estimation equations are of interest in their own right, in that they represent a finite dimensional optimal solution to a nonlinear estimation problem and can be viewed as consisting of a Kalman-Bucy filter along with the on-line computation of the solution of the associated Riccati equation, which is driven by the observations. The usefulness of these results is illustrated via an example concerning the detection of an actuator failure in a rigid body rotational control system.
Willsky, A. S.↗
Estimation for bilinear stochastic systems
Three techniques for the solution of bilinear estimation problems are presented. First, finite dimensional optimal nonlinear estimators are presented for certain bilinear systems evolving on solvable and nilpotent lie groups. Then the use of harmonic analysis for estimation problems evolving on spheres and other compact manifolds is investigated. Finally, an approximate estimation technique utilizing cumulants is discussed.
Willsky, A. S.↗
A class of finite dimensional optimal nonlinear estimators
Finite dimensional optimal nonlinear state estimators are derived for bilinear systems evolving on nilpotent and solvable Lie groups. These results are extended to other classes of systems involving polynomial nonlinearities. The concepts of exact differentials and path-independent integrals are used to derive optimal finite dimensional estimators for a further class of nonlinear systems.
Marcus, S. I.↗
Application of system theory to power processing problems
The work in power processing is reported. Input-output models, and Lie groups in control theory are discussed along with the methods of analysis for time invariant electrical networks.
Brockett, R. W.↗
Estimation and detection of signals in multiplicative noise
We define a class of detection-estimation problems on matrix Lie groups in which the observation noise is multiplicative in nature. By examining the differential versions of the hypotheses, which are bilinear, we are able to derive the relevant likelihood ratio formula and the associated optimal estimation equations for the signal given the observations and the assumption that the signal is present. These estimation equations are of interest in their own right, in that they represent a finite-dimensional optimal solution to a nonlinear estimation problem and consist of a Kalman-Bucy filter along with the on-line computation of the solution of the associated Riccati equation, which is driven by the observations. The usefulness of these results is illustrated via an example concerning the detection of an actuator failure in a rigid-body rotational control system.
Willsky, A. S.↗
Analysis of bilinear noise models in circuits and devices
There are a number of applications in which linear noise models are inappropriate. In the paper, the use of bilinear noise models in circuits and devices is considered. Several physical problems are studied in this framework. These include circuits involving varying parameters (such as variable resistance circuits constructed using field-effect transistors), the effect of switching jitter on sampled data system performance and communication systems involving voltage-controlled oscillators and phase-lock loops. In addition, several types of analytical techniques for stochastic bilinear systems are considered. Specifically, the moment equations of Brockett for bilinear systems driven by white noise are discussed, and closed-form expressions for certain bilinear systems (those that evolve an Abelian or solvable Lie groups) driven by white or colored noise are derived. In addition, an approximate statistical technique involving the use of harmonic expansions is described.
Willsky, A. S.↗
Prolongation structures of nonlinear evolution equations. II
The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schroedinger equation. The prolongation structure in this case is explicitly given, and recurrence relations derived which support the conjecture that the structure is open - i.e., does not terminate as a set of structure relations of a finite-dimensional Lie group. We introduce the use of multiple pseudopotentials to generate multiple Baecklund transformation, and derive the double Baecklund transformation. This symmetric transformation concisely expresses the (usually conjectured) theorem of permutability, which must consequently apply to all solutions irrespective of asymptotic constraints.
Estabrook, F. B.↗
Nonlinear upper hybrid drift waves for a longitudinal electric field perpendicular to a uniform magnetic field in the Vlasov-Maxwell approximation
Upper hybrid drift waves are found as a special solution to a Vlasov-Maxwell plasma which has a longitudinal electric field and a perpendicular uniform magnetic field. A single-species plasma with a constant-density mobile neutralizing background supports spatially varying disturbances that oscillate at the upper hybrid frequency. The general functional dependences of the electric field, the plasma number density, and the one-particle distribution function for the special case are found from more general Vlasov-Maxwell equations invariant under a Lie group point transformation. The one-particle distribution function for the plasma is a function of the Liouville invariant, which is the energy in the generalized Bernstein-Greene-Kruskal (BGK) reference frame, and the momentum in the drift direction.
Abraham-Shrauner, B.↗
Particle paths and phase plane for time-dependent similarity solutions of the one-dimensional Vlasov-Maxwell equations
The phase trajectories of particles in a plasma described by the one-dimensional Vlasov-Maxwell equations are determined qualitatively, analyzing exact general similarity solutions for the cases of temporally damped and growing (sinusoidal or localized) electric fields. The results of numerical integration in both untransformed and Lie-group point-transformed coordinates are presented in extensive graphs and characterized in detail. The implications of the present analysis for the stability of BGK equilibria are explored, and the existence of nonlinear solutions arbitrarily close to and significantly different from the BGK solutions is demonstrated.
Roberts, Dana Aaron↗
Design of Multi-Parameter Steerable Functions Using Cascade Basis Reduction
A new cascade basis reduction method of computing the optimal least-squares set of basis functions steering a given function is presented. The method combines the Lie group-theoretic and the singular value decomposition approaches in such a way that their respective strengths complement each other. Since the Lie group-theoretic approach is used, the set of basis and steering functions computed can be expressed analytically. Because the singular value decomposition method is used, this set of basis and steering functions is optimal in the least-squares sense. Furthermore, the computational complexity in designing basis functions for transformation groups with large numbers of parameters is significantly reduced. The efficiency of the cascade basis reduction method is demonstrated by designing a set of basis functions that steers a Gabor function under the four-parameter linear transformation group.
Teo, P.↗
Photometry of resolved galaxies. IV - Holmberg I and Holmberg II
Colors and magnitudes are presented for 279 resolved stars in the Holmberg I dwarf galaxy and 468 resolved stars in Holmberg II. Both systems are Magellanic type dwarf members of the M81-NGC 2403 Group, which lies at approximately 3 Mpc from the Local Group. The photometry was done in the GRI passbands using CCD detectors. Color-magnitude diagrams and luminosity functions are constructed; these are compared with results for several Local Group galaxies and with theoretical work. Holmberg I is found to have a low present star formation rate, while Holmberg II is very active at present.
Hoessel, J. G.↗