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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
A group isomorphism φ:G↦G is a one-one mapping from group G onto group G preserving the (abstract) group operation, thus making the two groups algebraically the same. By preserving the operation, we mean that φ(στ)=φ(σ)∗φ(τ) for all σ,τ∈G, so that the abstract operations στ can be carried out as φ(σ)∗φ(τ) in G and recovered back in G using φ−1. We write G≃G to indicate the isomorphic relation between the two structures. A group homomorphism is a one-one mapping from group Ginto group G preserving the (abstract) group operation.
Characteristic matrix functions for delay differential equations with symmetry
Published in Dynamical Systems, 2023
If are two spatio-temporal symmetries of , then and hence . Thus the map is a group homomorphism and is a normal subgroup of . Therefore and is a subgroup of . This implies that where denotes the cyclic group of order m. If , we say that the periodic solution is a rotating wave; if (i.e. the group of spatio-temporal symmetries modulo the group of purely spatial ones is a finite group), we say that the periodic solution is a discrete wave, cf. [5].
Topological speedups of ℤd-actions
Published in Dynamical Systems, 2022
Aimee S. A. Johnson, David M. McClendon
Let be the speedup of conjugate to . Let so that is the unital dimension group associated to . The conjugacy between and induces a unital dimension group isomorphism . Define by By Lemma 3.1, , so . Therefore is well-defined and surjective. The function gives the desired group homomorphism.
Vortex-type solutions for magnetic pseudo-relativistic Hartree equation
Published in Applicable Analysis, 2022
Let be a -vector potential and be a scalar potential which satisfies Assume and have finite symmetries given by the action of a closed subgroup Γ of the orthogonal group , i.e. and satisfy We look for solutions to problem (1) which satisfy where is a continuous group homomorphism.