The crossing number of Sierpiński graphs S(n, k) and their regularizations S(n, k) and S(n, k) is studied. Explicit drawings of these graphs are presented and proved to be optimal for S(n, k) and S(n, k) for every n ≥ 1 and k ≥ 1. These are the first nontrivial families of graphs of “fractal” type whose crossing number is known.

Model-Based Diagnosis (MBD) algorithms often yield a large number of diagnoses, severely limiting their practical utility. A practical way to deal with such diagnostic uncertainty is to use the active testing approach implemented by Fractal (FRamework for ACtive Testing ALgorithms). Fractal computes a sequence of control settings for reducing the number of diagnoses given a system description a...

The analytic, open-form solution for the steady-state, 2-dimensinal partial differential equation describing the concentration profile for the gas solute in a falling liquid film flowing under a steady-state laminar condition was shown. The overall mass transfer coefficient, c k , for a binary diffusion system, was evaluated in light of the analytic solution and was tested for oxygen absorption...

one of the important aspects of soil is, knowing the relationships between spatial features of soil and quantity in statistical model. the goal of this research is to estimate saturated hydraulic conductivity by regression and co-active neuro-fuzzy inference system (anfis) with using the parameters of bulk density, real density, porosity, fractal dimension of particle size, and clay percent, si...

BACKGROUND Right-sided heart failure has high morbidity and mortality, and may be caused by pulmonary arterial hypertension. Fractal dimension is a differentiated and innovative method used in histological evaluations that allows the characterization of irregular and complex structures and the quantification of structural tissue changes. OBJECTIVE To assess the use of fractal dimension in car...

We introduce the concept of the boundary 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 boundary nodes seen from a given node of complex networks. We find that for both Erdős-Rényi and scale-free model networks, as well as for several real networks, the boundaries have fractal proper...

This paper examines the fractal nature of the Narayana fractal, an object defined by N = {(i, j) ∈ N× N : N(i + j + 1, j + 1) = 1 (mod 2)}. where N(n, k) = 1 n ( n k )( n k − 1 ) are the Narayana numbers. This object closely resembles a fractal derived from Pascal’s triangle. This similarity is used to prove that the Hausdorff dimension of the Narayana fractal is log 3/ log 2, and the limit of ...