In this paper we give, for the first time, a complete description of the latetime evolution of non-tilted spatially homogeneous cosmologies of Bianchi type VIII. The source is assumed to be a perfect fluid with equation of state p = (γ − 1)µ, where γ is a constant which satisfies 1 γ 2. Using the orthonormal frame formalism and Hubble-normalized variables, we rigorously establish the limiting behaviour of the models at late times, and give asymptotic expansions for the key physical variables.The main result is that asymptotic self-similarity breaking occurs, and is accompanied by the phenomenon of Weyl curvature dominance, characterized by the divergence of the Hubble-normalized Weyl curvature at late times.

Abstract. In this paper we give, for the first time, a qualitative description of the asymptotic dynamics of a class of non-tilted spatially homogeneous (SH) cosmologies, the so-called exceptional Bianchi cosmologies, which are of Bianchi type VI −1/9 . This class is of interest for two reasons. Firstly, it is generic within the class of non-tilted SH cosmologies, being of the same generality as the models of Bianchi types VIII and IX. Secondly, it is the SH limit of a generic class of spatially inhomogeneous G 2 cosmologies.Using the orthonormal frame formalism and Hubble-normalized variables, we show that the exceptional Bianchi cosmologies differ from the non-exceptional Bianchi cosmologies of type VI h in two significant ways. Firstly, the models exhibit an oscillatory approach to the initial singularity and hence are not asymptotically selfsimilar. Secondly, at late times, although the models are asymptotically self-similar, the future attractor for the vacuum-dominated models is the so-called Robinson-Trautman SH model instead of the vacuum SH plane wave models.

While standard Kalman-based filters, Gaussian assumptions, and covariance-weighted metrics are very effective in data-rich tracking environments, their use in the data-sparse environment of space surveillance is more limited. To properly characterize non-Gaussian density functions arising in the problem of long-term propagation of state uncertainties, a Gaussian sum filter adapted to the two-body problem in space surveillance is proposed and demonstrated to achieve uncertainty consistency. The proposed filter is made efficient by using only a onedimensional Gaussian sum in equinoctial orbital elements, thereby avoiding the expensive representation of a full six-dimensional mixture and hence the "curse of dimensionality." Additionally, an alternate set of equinoctial elements is proposed and is shown to provide enhanced uncertainty consistently over the traditional element set. Simulation studies illustrate the improvements in the Gaussian sum approach over the traditional unscented Kalman filter and the impact of correct uncertainty representation in the problems of data association (correlation) and anomaly (maneuver) detection. Nomenclature A= lower-triangular Cholesky factor of a covariance matrix a = semimajor axis a pert = perturbing acceleration a; h; k; p; q; ' = equinoctial orbital elements c = parameter controlling the accuracy of the Gaussian sum filter f = system dynamics vector G = process noise shape matrix h = measurement function vector k = (subscript) time index '= mean longitude N = number of mixture components N = Gaussian probability density function n = mean motion n; h; k; p; q; '= alternate equinoctial orbital elements P = covariance of a Gaussian distribution PE = prediction error p = probability density function Qt = process noise covariance matrix R k = measurement noise covariance matrix r = Cartesian Earth-Centered Inertial (ECI) position coordinates _ r = Cartesian ECI velocity coordinates r = Cartesian ECI acceleration coordinates t 0 , t = initial and current times u = orbital element coordinates (sixdimensional) fv 1 ; . . . ; v k g = Gaussian white noise sequence wt = Gaussian white noise process w ; w ; . . . = mixture weights x = dynamic state vector Z k fz 1 ; . . . ; z k g = measurement sequence ; ; . . . = (subscripts) indices of mixture components kj = Kronecker delta symbol = mean of a Gaussian distribution = Earth gravitational constant (398600:4418 km 3 =s 2 ) = standard deviation of a univariate Gaussian distribution = inverse solution flow r x = gradient operator with respect to x (column oriented)

Abstract. The problem of the invariant classification of the orthogonal coordinate webs defined in Euclidean space is solved within the framework of Felix Klein's Erlangen Program. The results are applied to the problem of integrability of the Calogero-Moser model.

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