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Mathematical Morphology with Noncommutative Symmetry Groups
Published in Edward R. Dougherty, Mathematical Morphology in Image Processing, 2018
Remark 2.12. In the references cited above, homogeneous spaces are defined in terms of group actions or, equivalently, permutation representations of an abstract group Γ on a set X. If the representation of this abstract group is faithful, then Γ is isomorphic to a group of permutations on X. In that case the above definition applies, which is more intuitive and sufficient for our purposes.
Algebraic Aspects
Published in Marlos A. G. Viana, Vasudevan Lakshminarayanan, Symmetry in Optics and Vision Studies, 2019
Marlos A. G. Viana, Vasudevan Lakshminarayanan
of an element s∈V generated by G coincides with V we say that the action φ is transitive, or that G acts transitively on V. The action described given by (1.20) is transitive and faithful. If s≠f then either Os∩Of=∅ or Os=Of. Moreover, because s∈Os for every s∈V, it follows that every group action on V decomposes it as the disjoint union of the resulting orbits, in each one of which the restricted action is transitive. Group orbits are the homogeneous spaces generated by the group action.
Consensus and coordination on groups SO(3) and S 3 over constant and state-dependent communication graphs
Published in Automatika, 2021
Aladin Crnkić, Milojica Jaćimović, Vladimir Jaćimović, Nevena Mijajlović
Mathematical formalization of some important engineering problems yields consensus problems on certain higher-dimensional Riemmanian manifolds. In order to develop a meaningful geometric consensus theory, it is necessary to impose some conditions on the class of Riemannian manifolds on which the problems are stated. One natural restriction is to work under the assumption that the underlying manifold is a homogeneous space. Problems of coordination on Lie groups, notably on and , are of a special interest, due to mathematical tractability and great number of applications. Among others, problems of attitude synchronization in space [13,14], formation flying [9,15,16], sensor networks [17–21] can be stated in a mathematically rigorous way as coordination problems on these Lie groups.
Measures and entropy on non compact dynamics
Published in Dynamical Systems, 2020
Finally we have the case of a nilmanifold, i.e. a differentiable manifold which has the action of a transitive nilpotent group of diffeomorphisms. A nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H of a nilpotent Lie group N by a closed subgroup H. This was introduced by A. Malčev in [30]. One way to construct a compact nilmanifold is to start with a simply connected nilpotent Lie group N and a discrete subgroup Γ. If Γ acts co-compactly (via right multiplication) on N, then the quotient manifold is a compact nilmanifold. As Malčev has shown, every compact nilmanifold is obtained this way [30]. W. Parry proved that an ergodic automorphism of a nilmanifold has completely positive entropy [34]. When the action is no cocompact, we could still get this result using the above theorem. Indeed, for any simply connected solvable analytic group N, there exists a sequence of closed simply connected analytic subgroups such that [25, Corollary 1.126], and so, by the above observation, we could get the result by induction on n.
Kinematical Lie Algebras and Invariant Functions of Algebras
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2019
J. M. Escobar, J. Núñez, P. Pérez-Fernández
All these groups have great importance in Physics, since through these symmetry groups of space-time the basic invariance of the laws of that discipline can be implemented, particularly in the special theory of relativity, with the ten-parameter Poincaré group containing (as transformation group of the four-dimensional Minkowski space-time) the time and space translations, space rotations, and boosts (inertial transformations). In 1986, Bacry and Nuyts classified in reference [3], under certain natural physical assumptions, all the abstract ten-dimensional real Lie algebras that contain as a subalgebra the algebra of the three-dimensional rotation group (generators J) and decompose under the rotation group into three-vector representation spaces (J itself, K, and P) and a scalar (generator H), showing the existence of a homogeneous space of dimension 4 in all cases.