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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.
Universal approximation with neural networks on function spaces
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Wataru Kumagai, Akiyoshi Sannai, Makoto Kawano
where , , and . The second symmetry is represented by a group action on the data space and labelling function space . Suppose that group acts on input space . We denote the action of on as . Then, the action of on is defined as
On the equivariance properties of self-adjoint matrices
Published in Dynamical Systems, 2020
Michael Dellnitz, Bennet Gebken, Raphael Gerlach, Stefan Klus
Recall that the isotropy subgroup for a point x in some space X characterizes the symmetry of x with respect to a certain group action. More precisely consider a group action ϑ of a group G on a linear space X. Then the isotropy subgroup of is given by .If we let act on matrices by then in Proposition 4.1 is the isotropy subgroup of A with respect to this action.If we replace by (unitary matrices) or (invertible matrices) and the matrix by or , respectively, then we obtain an analogous result for the unitary and invertible isotropy subgroup. It is also possible to formulate Proposition 4.1 for general orthogonal, unitary or invertible operators.
Stabilisation of a relative equilibrium of an underactuated AUV on SE(3)
Published in International Journal of Control, 2019
Hamiltonian function H(z), z ∈ Q, is chosen as an equivalence relation to reduce entire phase space to its quotient space, that is, the reduced space. The symmetric mechanical system is H-invariant if the Hamiltonian function is invariant after group action, that is, H(z) = H(Φ(g, z)), where g ∈ K, K is a symmetry group and Φ(g, z) is a group action. Thus, the reduced phase space Q/K and momentum map are induced. Denoting as the μ-level set on which two arbitrary points that can be transformed one to the other by a group action are identified, where Kμ is an isotropy subgroup defined as {g ∈ K|Φ(g, μ) = μ}, and the momentum map value is invariant under a coadjoint action (Marsden, 2009). It is called a coadjoint orbit when a symmetry group acts on the momentum map by group action Φ. Qμ is a consequence of equivariance on the space since Kμ leaves invariant. Therefore, the reduced space can be decomposed as , where Qμ is also called symplectic leaf in Q/K (Marsden, 2009).