By P. Auger, George J. Klir
The most primary and effective methods of conceptualizing complicated platforms is to prepare them hierarchically. A hierarchically prepared method is represented by way of a community of interconnected subsystems, each one of which has its personal community of subsystems, and so forth, until eventually a few undemanding subsystems are reached that aren't additional decomposed. This unique and significant e-book proposes a normal mathematical conception of a hierarchical process and exhibits the way it may be utilized to very varied subject matters equivalent 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 improve mathematical frameworks for conceptualizing such platforms merely seemed within the early Nineteen Seventies, and it took one other decade ahead of those mathematical frameworks have been utilized to varied particular contexts of the sciences of the normal. a lot of this paintings has been pioneered via the writer, and he provides a extra thorough paintings to be able to have a massive influence on transferring present considering within the sciences of the average to a extra holistic foundation within which a number of degrees of hierarchically prepared platforms are seen in an built-in type.
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Extra resources for Dynamics and Thermodynamics in Hierarchically Organized Systems. Applications in Physics, Biology and Economics
I is an index corresponding to different states s in the group of states a, i € [ 1 ,N |. 3): Ν- lΝ. (t) be the number of elements in the i t h state of the a l h group at time t or state variable. We use similar notations than in our papers, P. e. an upper index for the group number and a lower index for the state number. In this section, we consider an ordinary partition. Hierarchical partitions of the system are going to be defined further on. a n a e i " i a + Σ k a _ »ιΑ*ϊ a a _> , a b Σ n n ijk , k + a Σ Σ Σ c a a n U n i j k m η 1 + higher order terms .
T). (t). Consequently, we assume that for each Dt corresponding to nearly constant values of the group variables and for all possible values of n , there exists an intra-group equilibrium characterized by a equilibrium frequencies that we are going to calculate now. Firstly, we begin by studying the case either of a stable steady state point, or of a converging spiral or still of a converging unstable limit cycle, see fig. 5. Indeed, if one waits for equilibrium, the state point will converge to the steady state point and will stay there.
A? 22) aA). 24) Elements States Groups Fig. 7. Hierarchically organized transition graph. 25) AA+AAA J AA1 c) hierarchy assumption Placing the states into groups of states must be realized in order to reflect the hierarchy in the transitions of the elements. e. that for any (α,β) and (i,j,k,l) we have: lajjl» lajf I . 26) The components of the intra-group matrices A « are much larger than those of the inter-group matrices Α and A<*ß. e. to group together strongly coupled states inside each group of the hierarchical partitions.
Dynamics and Thermodynamics in Hierarchically Organized Systems. Applications in Physics, Biology and Economics by P. Auger, George J. Klir