Search

In the framework of the Multifractal Theory of Motion, which is expressed by means of the multifractal hydrodynamic model, complex system dynamics are explained through uniform and non-uniform flow regimes of multifractal fluids. Thus, in the case of the uniform flow regime of the multifractal fluid, the dynamics' description is "supported" only by the differentiable component of the velocity field, the non-differentiable component being null. In the case of the non-uniform flow regime of the multifractal fluid, the dynamics' description is "supported" by both components of the velocity field, their ratio specifying correlations through homographic transformations. Since these transformations imply metric geometries explained, for example, by means of Killing–Cartan metrics of the SL(2R)-type algebra, of the set of 2 × 2 matrices with real elements, and because these metrics can be "produced" as Cayleyan metrics of absolute geometries, the dynamics' description is reducible, based on a minimal principle, to harmonic mappings from the usual space to the hyperbolic space. Such a conjecture highlights not only various scenarios of dynamics' evolution but also the types of interactions "responsible" for these scenarios. Since these types of interactions become fundamental in the self-structuring processes of polymeric-type materials, finally, the theoretical model is calibrated based on the author's empirical data, which refer to controlled drug release applications. [ABSTRACT FROM AUTHOR]

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne's temporal measures in a more general manner. This work furthers the analysis published recently by our group in "Towards Interactions through Information in a Multifractal Paradigm". [ABSTRACT FROM AUTHOR]

In a multifractal paradigm of motion, Shannon's information functionality of a minimization principle induces multifractal–type Newtonian behaviors. The analysis of these behaviors through motion geodesics shows the fact that the center of the Newtonian-type multifractal force is different from the center of the multifractal trajectory. The measure of this difference is given by the eccentricity, which depends on the initial conditions. In such a context, the eccentricities' geometry becomes, through the Cayley–Klein metric principle, the Lobachevsky plane geometry. Then, harmonic mappings between the usual space and the Lobachevsky plane in a Poincaré metric can become operational, a situation in which the Ernst potential of general relativity acquires a classical nature. Moreover, the Newtonian-type multifractal dynamics, perceived and described in a multifractal paradigm of motion, becomes a local manifestation of the gravitational field of general relativity. [ABSTRACT FROM AUTHOR]