Macrosystems theory and its applications. Equilibrium models
The book is mainly concerned with systems in which stochastic behavior of the components is transformed into a determined behavior of the system as a whole. The equilibria of the macrosystems was investigated using mathematical modeling. The macrosystem structure is studied with the help of a stochastic model, thus the mapping \(\mathcal H\) which characterizes the ``input-output'' transformation is decomposed into \({\mathcal H}_1\) (``input-state'') -- which is decomposed further into \({\mathcal H}_{11}\) (``input-stochastic state'') and \({\mathcal H}_{12}\) (``stochastic state-state'') -- and \({\mathcal H}_2\) (``state-output''). The stationary states for homogeneous macrosystems are determined by means of the principle of entropy maximization. The models of the stationary states are based upon constrained extremum problems having entropy objective functions. For unconstrained macrosystems (i.e. with no influence from the outside) the study approaches three cases: (1) macrosystems having Fermi-states (i.e. each state in \(\sigma_n\) can be occupied by only one element with probability one); (2) macrosystems having Einstein-states, for which each state in \(\sigma_n\) can be occupied by an arbitrary number of elements; (3) macrosystems having Boltzmann-states, where the occupation numbers \(N_n\) are significantly less than the capacity \(G_n\) of the subset \(\sigma_n\). The morphological properties -- namely the existence of the maximum and its ``sharpness'' -- are interesting for determining the fluctuations from the most probable states (there is a study for each type of macrosystem). If the resources are unconstrained, the set \(\mathcal N\) contains all possible macrostates generated by the stochastic mechanism for distributing the elements among the states. If the resources are constrained, not all the macrostates from \(\mathcal N\) are feasible, but only a subset \(\mathcal D\) in \(\mathcal N\). The model of stationary states (MSS) is represented on the basis of a variational principle as follows: \(H(N)\Rightarrow \max\), \(N\in {\mathcal D}\subset {\mathcal N}\), where the feasible set \(\mathcal D\) is a non-empty subset of the possible macrostate set \(\mathcal N\) and \(H(N)\) is the generalized information entropy. The optimality conditions for the MSS and MSS of homogeneous macrosystems with linear resource consumption are also discussed in the second chapter. Chapter 3 deals with the parametric properties of MSS, i.e. the properties of macrostate under variations of the MSS parameters. These parameters are divided into three groups: (1) parameters characterizing the stochastic mechanism of distributing the elements among the subsets of states; (2) the characteristics of the resource consumption; (3) the parameters characterizing the resource stores. MSS with both complete and incomplete consumption of resources are discussed. Theorems on implicit functions are also given in the third chapter. There are demonstrations on the existence and continuity, homogeneous forms and posynoms, and differentiability of the local properties of these functions. As for the global properties of these functions, the existence and the differentiability are also proved. The MSS with incomplete consumption of resources are analyzed based on general results of the properties of perturbed mathematical programming problems. Chapter 4 describes the computational methods for determining the stationary states of homogeneous macrosystems. From the MSS -- as a mathematical object -- to its computer implementation, the author presents some steps in order to investigate: (a) the convergence of multiplicative algorithms for solving systems of nonlinear equations and convex programming problems; (b) the application of convergence theorems in the investigation of multiplicative algorithms for determining stationary states of macrosystems with complete and incomplete consumption of resources. There is also given a brief description of various computational methods. Chapter 5 consists of full description of probabilistic hierarchical structures. The need of a hierarchical order appears in studying and designing complex systems for structuring macrosystems and especially in decomposing the system, grouping the parts into homogeneous classes, and setting a sequence in their functioning (e.g. for large systems of distribution of material resources: power supply systems, water economy systems, systems of organization management, etc.). The probabilistic hierarchy is presented as a more viable alternative to the hierarchical order in which the distribution of elements among the subordination levels is fixed. Some models of ensembles are characterized: an urban system consisting of three subsystems (``the basic employment sector'', ``the service sector'', ``the population''), a production system which uses flexible technologies, etc. The formation of hierarchical chains consists in random and independent entering the subordination levels by their elements. The models that govern the mechanisms for forming the hierarchical structures are also presented. As an application of the theory, the modelling and analysis of interregional exchange of construction materials are discussed. Finally, there is a very interesting presentation of the image reconstruction. After a short introduction of the notions used (such as the image density function \(\varphi(x,y)\), the two-dimensional function \({\mathcal D}(x,y)\) and the link between these -- given by a linear integral functional) a variety of examples is given to the reader: (a) recovering ``blurred'' images; (b) computer tomography; (c) engineering geophysics. The author gives also a very useful multiplicative algorithm for image recovering. The book has a valuable Appendix comprising the MSS coded in Pascal language. The rich bibliography is very useful for the reader interested to enlarge his or her knowledge. With its detailed and various examples from different zones of human activity, with its accuracy of the demonstrations, this monograph is a valuable asset to any researcher interested especially in the investigation of the equilibrium of macrosystems.
stationary states for homogeneous macrosystems
scientific article; zbMATH DE number 766959