Fractal Geometry

In the present paper some of the mathematical methods for analysing unstable flows will be presented and described. Stability calculations for the Navier-Stokes equations have been carried out using the Orr-Sommerfeld equations. From this analysis conclusions can be drawn regarding the transformation conditions of a laminar in a turbulent flow. The Bénard-convection shows another specific example of instabilities in fluid dynamics, which will be discussed in the present paper. The Lorenz equations, as a further development, used to describe meteorological processes in the atmosphere, demonstrates that instabilities can lead to chaotic behaviour of the system. Such kind of unpredictable behaviour is called deterministic chaos. These processes are very important in other fields as well, for instance when describing air pollution phenomena. Following a discussion of the existing mathema- tical tools to describe these processes an overview of newer mathematical possibilities will be giv...

Arborization pattern was studied in pial vascular networks by treating them as fractals. Rather than applying elaborate taxonomy assembled from measures from individual vessel segments and bifurcations arranged in their branching order, the authors' approach captured the structural details at once in high-resolution digital images processed for the skeleton of the networks. The pial networks appear random and at the same time having structural elements similar to each other when viewed at different scales—a property known as self-similarity revealed by the geometry of fractals. Fractal (capacity) dimension, Dcap, was calculated to evaluate the network's spatial complexity by the box counting method (BCM) and its variant, the extended counting method (XCM). Box counting method and XCM were subject to numerical testing on ideal fractals of known D. The authors found that precision of these fractal methods depends on the fractal character (branching, nonbranching) of the struct...

BackgroundAttention-Deficit/Hyperactivity Disorder (ADHD) is a neurodevelopmental condition characterized by executive function (EF) dynamics disturbances. Notwithstanding, current advances in translational neuroscience, no ADHD objective, clinically useful, diagnostic marker is available to date.ObjectivesUsing a customized definition of EF and a new clinical paradigm, we performed a prospective diagnostic accuracy trial to assess the diagnostic value of several fractal measures from the thinking processes or inferences in a cohort of ADHD children and typically developing controls.MethodWe included children from age five to twelve diagnosed with a reference standard based on case history, physical and neurological examination, Conners 3rd Edition, and DSM-V™. The index test consisted of a computer-based inference task with a set of eight different instances of the “Battleships” game to be solved. A consecutive series of 18 cases and 18 controls (n = 36) recruited at the primary pa...