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Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a Z3-skew-symmetric covariant SO(3)-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group SO(3)↪SU(5). We find two SO(3)-invariants of Z3-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by K(z,z). We study the invariant properties of K(z,z) and find its stabilizer. Making use of these invariant properties, we define an SO(3)-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an SO(3)-irreducible geometric structure and find its curvature and torsion.

Recursive attitude estimation of a rigid body from inertial measurements is a crucial component of many modern systems and as such, has a rich historical background of proposed techniques. Recent work has been done on tracking rotations using maximal entropy distributions. However, there has been no evaluation done on the performance of this approach using real inertial data. In this work, we investigate the performance and limitations of classical and modern probabilistic Bayesian approaches and provide a rigorous comparison to attitude estimation on the special rotation group $\mathcal {SO}(3)$ using maximal entropy distributions. The extended Kalman Filter and the unscented Kalman filter are derived as benchmarks in attitude estimation from low-cost inertial measurement units, commonly found in smartphones. To evaluate robustness over multiple sampling intervals, we generated synthetic directional inertial measurements from a typical low-cost 3-axes inertial measurement unit and use the Frobenius Norm as our primary metric. To further our evaluation, we took advantage of a publicly available dataset where inertial measurements are recorded from a number of off-the-shelf smartphones and the ground truth is calculated using a Motion Capture system. Our experiments demonstrate that tracking rotations using maximum entropy distributions produce a more accurate and robust solution in contrast to alternate proven Kalman approaches.

The concept of the phase space plays a key role in the analysis of oscillating signals. For a 1-D signal, the coordinates of the 2-D phase space are the observation time and the instant frequency. For measurements of propagating wave fields, the time and instant frequency are linked to the spatial location and wave normal, defining a ray. In this case, the phase space is also termed the ray space. Distributions in the ray space find important applications in the analysis of radio occultation (RO) data because they allow the separation of interfering rays in multipath zones. Examples of such distributions are the spectrogram, Wigner distribution function (WDF), and Kirkwood distribution function (KDF). In this study, we analyze the application of the fractional Fourier transform (FrFT) to the construction of distributions in the ray space. The FrFT implements the phase space rotation. We consider the KDF averaged over the rotation group and demonstrate that it equals the WDF convolved with a smoothing kernel. We give examples of processing simple test signals, for which we evaluate the FrFT, KDF, WDF, and smoothed WDF (SWDF). We analyze the advantages of the SWDF and show examples of its application to the analysis of real RO observations.