Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 603ing to cognitive appraisal about the self, cognitive appraisal about the environment,action tendency, action fantasy, synonym, antonym, and intensityrange of the emotion, respectively. These findings set the groundwork forthe construction of an instrument to assess emotions multicomponentially.4.9.8.1 Crowd HypothesisWe consider a human crowd C as a group of m autonomous agents A i(i = 1, ..., m), each of which carries its own nD motivational factor–structure. This nonlinear factor structure, which can be get using modernnonlinear factor analysis techniques (see [Yalcin and Amemiya (2001);Amemiya (1993); Wall and Amemiya (1998); Wall and Amemiya (2000)];also compare with subsection 3.11.4 below), is defined by n hypotheticalmotivational factor–coordinates q i = {q µ i }, (µ = 1, . . . , n), spanning thesmooth nD motivational factor manifold M i for each autonomous agentA i .We understand crowd representation as an environmental field–inducedcollective behavior of individual autonomous agents. To model it in a generalgeometrodynamical framework, we firstly define the behavior of eachagent A i as a motion π i along his motivational manifold M i , caused by hisown emotion field Φ i , which is an active (motor) subset of M i .Secondly, we formulate a collective geometrodynamical model for thecrowd, considered as a union C = ∪ i A i ,in the form of a divergence equationfor the total crowd’s SEM–tensor C. 144.9.8.2 Geometrodynamics of Individual AgentsTo formulate individual agents’ geometrodynamics, we firstly derive twohigher geometrical structures from a motivational factor manifold M i correspondingto an agent A i : (i) the agent’s velocity phase–space, defined as atangent bundle T M i , and (ii) the agent’s momentum phase–space, definedas a cotangent bundle T ∗ M i .Now, the sections of T M i we call the agent’s vector–fields v i , which canbe expanded in terms of the basis vector–fields {e i µ ≡ ∂ qµ }, as v i = v µ i i ei µ.Similarly, the sections of T ∗ M i we call the agent’s one–forms α i , which canbe expanded in terms of the basis one–forms {ω µ i ≡ dq µ i }, as α i = α i µ ω µ i .Here d denotes the exterior derivative (such that dd = 0). In particular,14 Throughout the text we use the following index convention: we label individualagents using Latin indices, and individual motivational factors using Greek indices; summationconvention is applied only to Greek factor indices.