# Global Perspectives on the Ecology of Human-Machine Systems - Vol. 1

By John Flach; Peter Hancock et al. | Go to book overview

Appendix A
Concepts and Theorems Required for Dimensionless Analysis and to Build a Fiber Bundle Geometry
Here we present the basic definitions and theorems used in this paper. We restrict ourselves to a pure mathematical language. The discussion follows the major steps of dimensional analysis as expressed by Kasprzak et al. ( 1990). Furthermore we also used Shaw et al. ( 1990) and Shaw et al. ( 1992).Definition: Group G= (G, *) is an algebraic structure on a set G with an operation * called multiplication if
 a. for any a, b∈G there is a unique c∈Gab = c b. for any a, b, and c∈Ga(bc) = (ab)c (associativity) c. there exists e∈G, for whichea = a for any a∈G d. for any a∈G exists a′∈G a′a = e

Furthermore if for any a, b∈G
ab = ba, then the group is commutative (Abelian).

Definition: We call G1 group an invariant group under transformation s∈G if for any a∈G1
GG1 and sa = as.

In other words, s is a commutator of G.

Definition: We call S subgroup an invariant subgroup of group G if for any a∈G and any s∈S
sa = as.

An invariant group is sometimes called a normal subgroup.

Definition: If N is a normal (invariant) subgroup of Gi , then aN a∈G classes are compatible classes. That is, if
ab a,b≠G, then aNbN = 0.

-346-

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