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In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus in terms of fractal limit, continuity, its derivative, and integral are formulated. The fractal fuzzy calculus is a new framework that includes fractal fuzzy…
Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal strings in the analysis of their geometry,…
We introduce the concept of boundaries of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundaries nodes of complex networks. We…
Exponential growth describes an extremely rapid process ubiquitous across mathematics and diverse physical, biological, and technological systems. Here, we introduce a class of fractal-inspired lattices that combine long-range periodic…
It is demonstrated that the dynamic structure is very important for the rate of drainage of a thin liquid film and it can be effectively taken into account by a dynamic fractal dimension. It is shown that the latter is a powerful tool for…
This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff's concepts of fractional dimension geometry. The…
The role played by non extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics has been discussed in details in many works and…
This paper introduces the concept of Fractal Frenet equations, a set of differential equations used to describe the behavior of vectors along fractal curves. The study explores the analogue of arc length for fractal curves, providing a…
We present an overview of a theory of complex dimensions of self-similar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several…
The Hausdorff fractal dimension has been a fast-to-calculate method to estimate complexity of fractal shapes. In this work, a modified version of this fractal dimension is presented in order to make it more robust when applied in estimating…
A macroscopic characterization of fractals showing up a structural transition from dense to multibranched growth is made using optical diffraction theory. Such fractals are generated via the numerical solution of the 2D Poisson and…
Recent developments in extracting and processing biological and clinical data are allowing quantitative approaches to studying living systems. High-throughput sequencing, expression profiles, proteomics, and electronic health records are…
We explored the fractal and multifractal characteristics of breast mammogram micrographs to identify quantitative biomarkers associated with breast cancer progression. In addition to conventional fractal and multifractal analyses, we…
Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like…
A homeomorphically irreducible spanning tree (HIST) is a spanning tree with no degree-2 vertices, serving as a structurally minimal backbone of a graph. While the existence of HISTs has been widely studied from a structural perspective, the…
The geometric characteristics of dust clouds provide important information on the physical processes that structure such clouds. One of such characteristics is the $2D$ fractal dimension $D$ of a cloud projected onto the sky plane. In…
Context. A porous and/or fractal description can generally be applied where particles have undergone coagulation into aggregates. Aims. To characterise finite-sized, porous and fractal particles and to understand the possible limitations of…
We consider the fragmentation process with mass loss and discuss self-similar properties of the arising structure both in time and space, focusing on dimensional analysis. This exhibits a spectrum of mass exponents $\theta$, whose exact…
In this article, we provide a simple and systematic way to represent general (inhomogeneous) fractals that may look different at different scales and places. By using set-valued compression maps, we express these general fractals as…
It is wellknown that the ordinary calculus is inadequate to handle fractal structures and processes and another suitable calculus needs to be developed for this purpose. Recently it was realized that fractional calculus with suitable…