By P. Auger, George J. Klir
Probably the most basic and effective methods of conceptualizing complicated structures is to prepare them hierarchically. A hierarchically prepared method is represented by means of a community of interconnected subsystems, each one of which has its personal community of subsystems, and so forth, till a few straight forward subsystems are reached that aren't extra decomposed. This unique and significant publication proposes a basic mathematical conception of a hierarchical approach and indicates the way it might be utilized to very assorted themes akin to physics (Hamiltonian systems), biology (coupling the molecular and the mobile levels), ecology (coupling the person, inhabitants and ecosystem), and economics (coupling the sectoral, neighborhood and nationwide levels). the 1st try to boost mathematical frameworks for conceptualizing such structures in basic terms seemed within the early Nineteen Seventies, and it took one other decade prior to those mathematical frameworks have been utilized to varied particular contexts of the sciences of the normal. a lot of this paintings has been pioneered by means of the writer, and he provides a extra thorough paintings with the intention to have an immense impression on moving present pondering within the sciences of the ordinary to a extra holistic foundation during which numerous degrees of hierarchically prepared structures are seen in an built-in style.
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Extra resources for Dynamics and Thermodynamics in Hierarchically Organized Systems. Applications in Physics, Biology and Economics
In this model, matrices A only concern intra-group transitions, while matrices A«P only concern inter-group transitions. Under these conditions, if there is conservation of the elements along the transitions, one must consider matrices Aßß different from matrices Aß. dt leave the state k in group β. These last contributions correspond to inter-group exchanges and thus cannot be included into Aß which correspond to intra-group β exchanges. These terms define new matrices Aßß with diagonal elements ßß a kk w h i c h m u s t n o t b e counfounded with terms of matrices Aß.
Such a a system can have solutions qj* * 0 and > 0. Replacing qj* by v ? q a and after simplification by q , one sees that the equilibrium frequencies v? 41 ): Σ vi" - 1 . 41) for each possible value of q . In the case of a cyclic motion around the steady state point, (stable limit cycle for instance, see fig. 37)), we are going to consider that the average position in time inside the groups is counfounded with the a position q pf the attractive steady state point. Thus, nothing has to be q changed to calculate the frequencies.
Figure 5 showed hyperspheres in dotted lines centered on the steady state point in different situations. 8) is satisfied. 10) become more and more important. Thus, assuming the knowledge of the contour map of an E<* function, (resulting for instance of computer simulations), for each isolated group α and for each value of the group variables V as shown on fig. 4, we suppose that this critical radius can be estimated from this map. a a Thus, for each partition P, we obtain a critical radius associated to each group a , say now R ( P ) .