Using fractal self-similarity and functional-expectation relations, the classical theory of box integrals—being expectations on unit hypercubes—is extended to a class of fractal “string-generated Cantor sets” (SCSs) embedded in unit hypercubes of arbitrary dimension. Motivated by laboratory studies on the distribution of brain synapses, these SCSs were designed for dimensional freedom—a suitabl...

The diffraction of atoms from stepped surfaces is treated using the semiclassical S-matrix theory. Two different regimes of the behavior of the classical trajectories, regular and chaotic, are considered. For the latter case, the diffraction order function presents a fractal structure, and there is an infinite number of trajectories connecting the initial and final states. Using the scaling law...

Anomalous kinetics of infective (e.g., autocatalytic) reactions in open, nonhyperbolic chaotic flows are important for many applications in biological, chemical, and environmental sciences. We present a scaling theory for the singular enhancement of the production caused by the universal, underlying fractal patterns. The key dynamical invariant quantities are the effective fractal dimension and...

It is of great significance to understand the particle distribution characteristics at different heights effectively control pollution. Based on fractal theory, dimension outdoor particles in a high-rise building Xi’an and its relationship with concentration sizes are discussed analyzed this paper. The results indicate that atmosphere mainly composed fine average proportion ranging from 0 1.0 µ...

We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energ...

The forecasting slope deformation potential is required to evaluate slope safety during open-pit mining, allowing us to formulate and promote effective emergency strategies in advance to prevent slope failure disasters. Although fractal models have been used to predict slope deformation, such limitations as low prediction accuracy, poor stability and the requirement for large amounts of data mu...

The ideas of Benôıt Mandelbrot, about the geometrical properties of sets which he called fractal, was published as late as 1975. Since then, there have for fractals, and most commonly the estimation of the fractal dimension, been found uses in the most diverse applications. Fractal geometry has been used in information theory, economics, flow dynamics and image analysis, among many different ar...

Background:Leukemia is cancer of blood and bone marrow cells. In general, there are four types of leukemia: chronic myelogenous leukemia (CML), acute myeloid leukemia (AML), B-cell chronic lymphocytic leukemia (CLL) and acute lymphoblastic leukemia (ALL).  Fractal geometry can be introduced as one of the effective ways to detect this type of cancer. In this study, with introduc...

Fractal analysis of earthquakes, lineages and waterways are considered as one of the practical tools to evaluate tectonic activity. With the help of fractal analysis, structural maturity and tectonic dynamics can be evaluated. In this research, using fractal analysis and box counting method and fild studies, the measured fractal dimensions for the designed network of Alborz province were studie...

The size effect on structural strength is an important phenomenon with a very old history. Unfortunately, despite abundant experimental evidence, this phenomenon is still not taken into account in most specifications of the design codes for concrete structures, as well as the design practices for polymer composites, rock masses and timber. The main reason appears to be a controversy between two...