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Statistics Indonesia (BPS) provides employment data through the National Labor Force Survey (SAKERNAS). SAKERNAS data collection is carried out quarterly under a rotating panel survey with a sample unit of the rotation group. Our study focuses on the Rao-Yu unit level model and its modifications. The modified Rao-Yu unit level model is required to handle pseudo panel units i.e., the units are replaced by the summary of units in the rotation group. In this context, these units are referred to as pseudo units. A modification was made by adding a measurement error pseudo unit into the Rao-Yu unit level model. The simulation study showed that the greater variance between areas, the better the capability of Rao-Yu pseudo unit level model compared with the Rao-Yu unit level model in estimating the parameters of interest. Our method proposal was applied to data on the quarterly unemployment rate at the district/city level in one of Indonesia's provinces according to the 2011-2014 SAKERNAS panel rotation data. [ABSTRACT FROM AUTHOR]

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a Z 3 -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 Z 3 -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. [ABSTRACT FROM AUTHOR]

We show that each irreducible tensor representation of weight 2 of the rotation group in the space of rank 3 tensors over three-dimensional space gives rise to an associative algebra with unity. We find the algebraic conditions to be satisfied by the generators of these algebras. Part of these relations is of binary, and another part is of ternary type. The structure of the latter one is based on the use of the cyclic group ℤ 3 generated by the primitive cubic root of unity given by q = exp (2 π i / 3). The subspace of each algebra spanned by triple products of generators is five-dimensional (5D) and is identical with the space of irreducible tensor representation of weight 2 of the rotation group SO (3). We define a Hermitian scalar product in this 5D subspace and construct its orthonormal basis in terms of triple products of generators. Then we find an explicit formula for the Lie algebra homomorphism so (3) → su (5). We suggest that the algebras constructed in this way, with binary and ternary constitutive relations, may find applications in the quark model and the Grand Unified Theories. [ABSTRACT FROM AUTHOR]

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.

Essentially, gauge fields are an important area of research in physics (quantum field theory, elementary particle theory) (six physicists won Nobel prizes in 1979, 1999 and 2004 for their work directly or indirectly related to gauge fields), and fiber bundles are an important area of research in mathematics (differential geometry, group theory, Lie algebras) (Donaldson won the Fields Medal in 1986 for his work on fiber bundle). In recent years, the research of fiber bundle and gauge field based on Yang-Mills equation is being carried out deeply. Therefore, this paper focuses on the formation and development of the related concepts of gauge field in quantum field theory and fiber bundle in differential geometry, and the relationship between gauge field and Yang-Mills equation, especially from the unification of electromagnetic force, weak force and strong force. It shows the importance and far-reaching significance of gauge field. In order to enable more relevant professional readers to quickly acquire necessary professional knowledge and generate enthusiasm for exploration and innovation in this important field, the basic concepts and processing methods from Abelian gauge field to non-Abelian gauge field and spontaneous symmetry breaking are discussed in detail in this review. In particular, the spatial properties of the Yang-Mills equation are discussed in order to better connect the gauge field and fiber bundle with the Yang-Mills equation, so as to deepen the understanding of the connection of the fiber bundles in a deeper level. Since this kind of problem is a meaningful research direction, it is worth to be explored by interested people. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory. Using the remarkable fact that the complex groups SO(3,C) and SL(2,C) are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory. We give a precise statement of the 3D geometric invariant moments computation, introducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint SL(2,C)-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group SO(3,C) to equivalent action of the complex Lie algebra sl2. The author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition. [ABSTRACT FROM AUTHOR]

We investigate the orientably regular non-abelian coverings of regular maps. A complete classification of dihedral coverings of the Platonic maps for branching over faces (or, dually, vertices) is given. As a result, we generalise the results of Jones and Surowski on regular cyclic coverings of the Platonic maps. [ABSTRACT FROM AUTHOR]

Creating a swarm of mobile computing entities, frequently called robots, agents or sensor nodes, with selforganization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three-dimensional Euclidean space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two-dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three-dimensional space. This article presents a necessary and sufficient condition for fully synchronous robots to solve the plane formation problem that does not depend on obliviousness, i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedron. [ABSTRACT FROM AUTHOR]

We derive an explicit form for the Cucker--Smale (CS) model on the special orthogonal group SOp3q by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and LaSalle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the \omega -limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results. [ABSTRACT FROM AUTHOR]

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.

Tracking on the rotation group is a key component of many modern systems for estimation and tracking of the orientation of rigid bodies. To address this problem, here, we describe a Bayesian algorithm that relies on directional measurements, acquired, for example, by an inertial measurements unit (IMU), for tracking on the special orthogonal (rotation) group. Its novelty lies in the use of maximum entropy distributions on these groups as models for the priors, and justifiable relaxation algorithms that permit the recursive implementation of such a model to provide a filter. We provide the solutions in a recursive closed form. In the two-dimensional case, the parameters of the prior and posterior distributions can be computed exactly and the solution has low complexity. Adoption of this approach eliminates the problem of angle wrapping. In higher dimensions, the exact solution cannot be computed, and it is necessary to make appropriate relaxations, which is done here. We demonstrate in simulations that, in contrast with some other approaches, our algorithm produces very accurate and statistically meaningful outputs. Pseudocode, specific to the IMU measurements is also provided. [ABSTRACT FROM AUTHOR]

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.

In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body attitude of an agent is modelled by a rotation matrix in R 3 as in Degond et al. (Math Models Methods Appl Sci 27(6):1005–1049, 2017). The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher-dimensional space from which we deduce the complete description of the possible equilibria. Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated with the stable equilibria in the spirit of Degond et al. (Arch Ration Mech Anal 216(1):63–115, 2015, Math Models Methods Appl Sci 27(6):1005–1049, 2017). [ABSTRACT FROM AUTHOR]

Single particle cryo-electron microscopy (EM) is a method for determining the 3-D structure of macromolecules from many noisy 2-D projection images of individual macromolecules whose orientations and positions are random and unknown. The problem of orientation assignment for the images motivated work on multireference alignment. The recent non-unique games framework provides a representation theoretic approach to alignment over compact groups, and offers a convex relaxation with certificates of global optimality in some cases. One of the great opportunities in cryo-EM is studying heterogeneous samples, containing two or more distinct conformations of molecules. Taking advantage of this opportunity presents an algorithmic challenge: determining both the class and orientation of each particle. We generalize multireference alignment to a problem of alignment and classification, and propose to extend non-unique games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying cryo-EM images and aligning them within their classes. [ABSTRACT FROM AUTHOR]

Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO (3) , with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on SO (3) . The variance of rotation matrices sampled by a certain determinantal point process is estimated, and formulas for the L 2 -norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interest. [ABSTRACT FROM AUTHOR]

The aim of this paper is to clear up the question of the connection between the geometric moment invariants and the invariant theory, considering a problem of describing the 2D geometric moment invariants as a problem of the classical invariant theory. We give a precise statement of the problem of computation of the 2D geometric invariant moments, introducing the notions of the algebras of simultaneous 2D geometric moment invariants, and prove that they are isomorphic to the algebras of joint SO (2) -invariants of several binary forms. Also, to simplify the calculating of the invariants, we proceed from an action of Lie group SO (2) to an action of its Lie algebra so 2 . Though the 2D geometric moments are not as effective as the orthogonal ones are, the author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition. [ABSTRACT FROM AUTHOR]

We investigate the structure of normal subgroups of the unit group of a quaternion algebra over a pythagorean field. [ABSTRACT FROM AUTHOR]

Denote by N1 = 2 ℓΣi=1 (mi + 1) + 2 the maximal possible number of non-zero coefficients of the Lyapunov differential system x˙ = y + ℓΣi=1 Pmi (x, y), y = -x + ℓ Σi=1 Qmi (x, y), where Pmi and Qmi are homogeneous polynomials of degree mi with respect to x and y, and 1 < m1 < m2 < ... < mℓ (ℓ < ∞). Then the upper bound of functionally independent focal quantities in the center and focus problem of considered system does not exceed N1 - 1. [ABSTRACT FROM AUTHOR]

Matrix-valued covariance functions are crucial to geostatistical modelling of multivariate spatial data. The classical assumption of symmetry of a multivariate covariance function is overly restrictive and has been considered as unrealistic for most of the real data applications. Despite of that, the literature on asymmetric covariance functions has been very sparse. In particular, there is some work related to asymmetric covariances on Euclidean spaces, depending on the Euclidean distance. However, for data collected over large portions of planet Earth, the most natural spatial domain is a sphere, with the corresponding geodesic distance being the natural metric. In this work, we propose a strategy based on spatial rotations to generate asymmetric covariances for multivariate random fields on the d-dimensional unit sphere. We illustrate through simulations as well as real data analysis that our proposal allows to achieve improvements in the predictive performance in comparison to the symmetric counterpart. [ABSTRACT FROM AUTHOR]

Motivated by the possibility to describe orientations of quantum triaxial rigid rotors, such as molecules, with respect to both internal (body-fixed) and external (laboratory) frames, we go through the theory of coherent states and design the appropriate family of coherent states on T ∗ SO ( 3 ) , the classical phase space of the freely rotating rigid body (the Euler top). We pay particular attention to the resolution of identity property in order to establish the explicit relation between the parameters of the coherent states and classical phase-space variables, actions and angles. [ABSTRACT FROM AUTHOR]

For every rotation $\unicode[STIX]{x1D70C}$ of the Euclidean space $\mathbb{R}^{n}$ ($n\geq 3$), we find an upper bound for the number $r$ such that $\unicode[STIX]{x1D70C}$ is a product of $r$ rotations by an angle $\unicode[STIX]{x1D6FC}$ ($0<\unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D70B}$). We also find an upper bound for the number $r$ such that $\unicode[STIX]{x1D70C}$ can be written as a product of $r$ full rotations by an angle $\unicode[STIX]{x1D6FC}$. [ABSTRACT FROM AUTHOR]