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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.
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.
Mathematicians’ beliefs, instruction, and students’ beliefs: how related are they?
Published in International Journal of Mathematical Education in Science and Technology, 2021
Classroom data were collected during three units in each class and were analyzed based on audio and video recordings. Alex’s class met twice each week in 75-minute class periods. Data were collected Week 3, in the middle of a unit on groups; Weeks 7–8, through the whole unit on group isomorphism; and Weeks 9–12, through the whole unit on quotient groups. Bailey’s class met three times each week in 50-minute class periods. Data were collected Weeks 3–4, in the middle of a unit on groups; Week 7, through the whole unit on group isomorphism; and Weeks 8–10, through the whole unit on quotient groups. Data were organized and analyzed with the Toolkit for Assessing Mathematics Instruction – Observation Protocol (TAMI-OP) (Hayward et al., 2017) and Inquiry-Oriented Instructional Measure (IOIM) (Kuster et al., 2019).
A survey of inconsistency indices for pairwise comparisons
Published in International Journal of General Systems, 2018
Barzilai (1998) used the logarithm as group isomorphism to pass from the multiplicative to the additive approach and proposed the following as a measure of inconsistency where w is estimated using the geometric mean method. is normalized since its values always range in the interval .