Renyi limit

As has been noted above, the parameter which is equal to the difference of the Renyi limiting dimensions and (which are used in practice instead of and D, respectively), characterises the degree of chaos in a system, i.e., in the structure of the considered epoxy polymers [31]. Renyi limiting dimensions and for polymeric materials can be determined with the aid of the following equations [22] ... 

An interesting new model is provided by Eq. (97) with Kr= 1 for all r. In the limit off—> 00 it becomes the RAC model, fully equivalent to that described by Eq. (3) with Kij=ij. At bounded/it describes the evolution of/-trees, i.e., a process similar to that dealt with by Erdos and Renyi [20, 34], but with substitution degree restriction imposed on the vertices (units) and cycle formation disallowed. 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

Despite the fact that miscellaneous networks can be found practically everywhere, the first papers published by Erdos Renyi (1959,1960) appeared in the first half of the 20th century. The boom in networks dates back to the beginning of the 21st century and is connected with a paper by Barabasi Reka (1999). This fact has two causes. The first one involved a limitation due to the slow computational speed of computers. The second limitation was caused by numerous successes in continuum mechanics in physics which assumed that the real world can be viewed as continuous. It began to be clear at the beginning of the 21st century that this approach cannot be applied universally and is completely inappropriate in various areas of physics. 

The limiting Renyi dimensions and were calculated according to Equations 9.11 and 9.12, respectively, and dimension was determined as follows [22] ... 

This value of 0=oid, referred to as the jamming limit in one dimension, was obtained originally by Renyi and others [59, 60]. Interestingly, a very similar value of the jamming limit (i.e. 0.7506) was obtained in the diffusion RSA process solved analytically in Ref [61]. 

---