By Yury S. Popkov
ISBN-10: 3540199551
ISBN-13: 9783540199557
Macrosystems are platforms during which the stochastic behaviour of the weather is reworked into deterministic behaviour of the approach as an entire. This e-book discusses equilibrium in those platforms. Mathematical versions of desk bound states utilizing the primary of utmost entropy are offered. this can be constructed and generalized for macrosystems with restricted assets. Parametric houses which represent a model's reaction to facts diversifications are mentioned. the writer has built new computational tools for the computer-aided recognition of desk bound country versions. Algorithms and computing device experiments are evaluated. Mathematical modelling tools are utilized to difficulties of hierarchical constructions, interregional product alternate and snapshot reconstruction.
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Extra resources for Macrosystems Theory and its Applications: Equilibrium Models
Example text
This appears to be because the "sharpness" is a qualitative property of the maximum. We tend to consider the maximum as "sharp" if small deviations of the argument give rise to significant decreases of the function values. g. the ratio of dispersion and maximum value (Kittel, 1969), the width of rectangle having the length equal to the maximum value and the unit area (Landau, 1964), the value of the second derivative at the maximum point (in this case we assume that N is large so that it can be replaced by a real variable), etc.
Nm) < qk, for a nonlinear consumption (12) and ~= { N: y~ tknNn = qk; k E 1,p; tknNn < qk, k E 1 + p , r for a linear resource consumption. } (13) L e m m a 2. Let the conditions of Lemma 1 be salisfied. Then the solution of problem (10)-(12) is such that N* > 0 , n E 1, m. (14) The proof is similar to that of Lemma 1. The third type of MSS describes macrosystems with the Fermi-states but with small occupation numbers. Such macrosystems are characterized by the generalized Boltzmann entropy. 29) is called the B-model.
The law characterizes the way in which the system macrostates evolve in time. If a system is in a nonstationary state at a moment of time then the monotonic increase of entropy is the most probable during the subsequent movement of the system. e. the macrostate with the maximum entropy is the stationary state N* realized in the macrosystem. Although these considerations are quite logical, the physical nature of processes and mathematical problems arisen here are still far from being completed (Landau, Lifshitz, 1964, Prigogine, 1985).
Macrosystems Theory and its Applications: Equilibrium Models by Yury S. Popkov
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