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Differential Systems and Algebras
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
Michael K. Kinyon, Arthur A. Sagle
As examples, consider the notions of subalgebra and ideal, A subalgebra of an m-ary algebra (V,p) is a subspace W such that p(W,…, W) ⊆ W. An ideal of an m-ary algebra (V,p) is a subspace W such that p(V,…, V, W) ⊆ W. For p = p1+ … + pm, we generalize this as follows: a subobject of (V,Ẋ = p(X)) is a subspace W such that p1(W) ⊆ W and (W,pk) is a subalgebra of (V, pk) for each k > 1. An ideal of (V, Ẋ = p(X)) is a subspace W such that p1(W) ⊆ W and (W,pk) is an ideal of (V,pk) for each k > 1. Given an ideal W, the quotient V/W is well-defined, and the canonical mapping V → V/W is solution-preserving from Ẋ = p(X) into X¯˙=p¯(X¯) where p¯(X¯):=p(X)+W; see [35, Prop. 2.4].
One-sided topological conjugacy of normal subshifts and gauge actions on the associated C*-algebras
Published in Dynamical Systems, 2021
Let us define commutative -subalgebras of by where for a subset denotes the -subalgebra of generated by F. It is easy to see that the -subalgebras are naturally isomorphic to the commutative -algebras of continuous functions on , respectively. We note that the natural inclusion relations comes from the continuous surjection .
A screw theory approach to compute instantaneous rotation axes of indeterminate spherical linkages
Published in Mechanics Based Design of Structures and Machines, 2020
Juan Ignacio Valderrama-Rodríguez, José M. Rico, J. Jesús Cervantes-Sánchez
It follows that for spherical linkages, the corresponding Lie algebra is not se(3), but the subalgebra so(3), the Lie subalgebra associated with the special orthogonal subgroup, SO(3), also called the spherical subgroup. Furthermore, so(3) retains the properties of an orthogonal space under the restrictions of the Klein and Killing forms to so(3) itself. Moreover these restrictions have quite interesting properties different from the original forms in se(3).
A review on some classes of algebraic systems
Published in International Journal of Control, 2020
Víctor Ayala, Heriberto Román-Flores
At once you can see that the Lie algebra generated by the matrices is a Lie subalgebra of In fact, the Lie bracket for any In particular, there exists a connected Lie subgroup with Lie algebra ∇ (Warner, 1971).