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Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
If H is a normal subgroup of G, then the quotient group (or factor group) of G modulo H is the group G/H = {aH|a ∊ G}, with binary operation aH ⋅ bH = (ab)H.
Topological machine learning for multivariate time series
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Chengyuan Wu, Carol Anne Hargreaves
The cycle group and boundary group are defined as and respectively. The th homology group is defined to be the quotient group . Informally, the rank of the th homology group (also called the th Betti number) counts the number of -dimensional holes in the simplicial complex . For instance, counts the number of connected components (0-dim holes), counts the number of ‘circular holes’ (1-dim holes), while counts the number of ‘voids’ or ‘cavities’ (2-dim holes). We show an example in Figure 3.
Degree of Dieudonné determinant defines the order of nonlinear system
Published in International Journal of Control, 2019
Ü. Kotta, J. Belikov, M. Halás, A. Leibak
Note that the addition and the multiplication of the left fractions b− 11a1 and b− 12a2 in are defined by where β2b1 = β1b2 and by where β2a1 = α1b2. The degree of Dieudonné determinant of the p × p matrix can be found by the algorithm described in the following. Note that in computations, the rules (6) and (7) have to be used. Since the degree of the Dieudonné determinant is unique for all elements in the equivalence class (quotient group ), we make computations not with the equivalence classes but with their representatives since this does not affect the result. The fact that the algorithm really computes the required degree is proved by Proposition 4, presented after the Algorithm.
The Garden of Eden theorem for cellular automata on group sets
Published in International Journal of Parallel, Emergent and Distributed Systems, 2019
Let be the origin (0, 0) and, for each point , let be the translation . The tuple is a coordinate system for . The stabiliser of under is the set , the quotient group is isomorphic to the group by , and, under this isomorphism, the right quotient group semi-action of on M is the map , .