We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of th...

The capacity of a wireless network with fractal and hierarchical social communications is studied in this paper. Specifically, we mathematically formulate the self-similarity of a fractal wireless network by a power-law degree distribution P (k), and we capture the direct social connection feature between two nodes with degree k1 and k2 by a joint probability distribution P (k1, k2). Firstly, f...

The capacity of a fractal wireless network with direct social interactions is studied in this paper. Specifically, we mathematically formulate the self-similarity of a fractal wireless network by a power-law degree distribution P(k), and we capture the connection feature between two nodes with degree k1 and k2 by a joint probability distribution P(k1, k2). It is proved that if the source node c...

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of th...

It is shown that the observed fractal distribution of galaxies is, in fact, consistent with homogeneity of the Universe and observational limits on ∆T /T , if the presence of dark matter and dark charge predicted by the Modified Field Theory is taken into account. It is discussed a new scenario of structure formation in which observed structures appear as a result of decay of the primordial fra...

It is shown that the observed fractal distribution of galaxies is, in fact, consistent with homogeneity of the Universe and observational limits on ∆T /T , if the presence of dark matter and dark charge predicted by the Modified Field Theory is taken into account. It is discussed a new scenario of structure formation in which observed structures appear as a result of decay of the primordial fra...

A parallel distributed processing approach to the computation of localized fractal dimension values in imagery is presented. This approach is a further development of the covering method which requires only nearest neighbor communication. A major benefit of our approach is the ability to readily incorporate any boundary information that may be available. Man y fractal textures or surfaces are f...

In the last three decades, interest in fractional calculus has experienced rapid growth and at present we can find many papers devoted its theoretical and application aspects (see the work of Machado et al. (2011) and the references therein). Fractional order models of real systems are often more adequate than the usually used integer order models in electrochemistry (Ichise et al., 1971), adve...

Research shows that multi-fractal can not only exhibit the singular probability distribution form of the fractal signal completely, but also increase the fine level of signal geometrical characteristics and local scaling behavior. Based on multi fractal dimension calculation of time-frequency matrix of vibration signal of rolling bearing in this paper, energy distribution characteristics of tim...

This paper seeks to check the validity of a hypothesis known as the “apparent fractal conjecture” (Ribeiro 1999: gr-qc/9909093), which states that the observed fractal structure of large-scale distribution of galaxies arises when some observational quantities, selected by their relevance in fractal characterization, are calculated along the past light cone. Since general relativity states that ...