Randall A. LaViolette

Lacunarity Definition For Ramified Data Sets Based On Optimal Cover

Lacunarity is a measure of how data fills space. It complements fractal dimension, which measures how much space is filled. This paper discusses the limitations of the standard gliding box algorithm for calculating lacunarity, which leads to a re-examination of what lacunarity is meant to describe. Two new lacunarity measures for ramified data sets are then presented that more directly measure the gaps in a ramified data set. These measures are rigorously defined. An algorithm for estimating the new lacunarity measure, using Fuzzy-C means clustering algorithm, is developed. The lacunarity estimation algorithm is used to analyze two- and three-dimensional Cantor dusts. Applications for these measures include biological modeling and target detection within ramified data sets.

Do dynamical systems follow Benford's law?

Data compiled from a variety of sources follow Benfords law, which gives a monotonically decreasing distribution of the first digit (1 through 9). We examine the frequency of the first digit of the coordinates of the trajectories generated by some common dynamical systems. One-dimensional cellular automata fulfill the expectation that the frequency of the first digit is uniform. The molecular dynamics of fluids, on the other hand, provides trajectories that follow Benfords law. Finally, three chaotic systems are considered: Lorenz, Henon, and Rossler. The Lorenz system generates trajectories that follow Benfords law. The Henon system generates trajectories that resemble neither the uniform distribution nor Benfords law. Finally, the Rossler system generates trajectories that follow the uniform distribution for some parameters choices, and Benfords law for others. (c) 2000 American Institute of Physics.

Self organized spatial-temporal structure within the fractured Vadose Zone: Influence of fracture intersections

Under conditions of unsaturated flow, others have shown experimentally that fracture intersections can direct flow to a single exiting fracture. In addition, they have been found to gather water from above to release as a pulse below. We formulate a simple model where these two behaviors are embedded within a network. With slow steady inflow distributed randomly along the top of the network, the system self organizes to form avalanches of water that can penetrate to great depths. When all intersections split their outflow, flow diverges with depth and develops into a self-organized dynamical state where the distribution of avalanche sizes follows a power-law over many decades. As the fraction of intersections that direct outflow singly is increased, spatial structure passes from divergent through braided to a fully convergent, hierarchical flow regime where avalanche size is minimized along one-dimensional slender pathways.

Self organized spatio‐temporal structure within the fractured Vadose Zone: The influence of dynamic overloading at fracture intersections

Under low flow conditions (where gravity and capillary forces dominate) within an unsaturated fracture network, fracture intersections act as capillary barriers to integrate flow from above and then release it as a pulse below. Water exiting a fracture intersection is often thought to enter the single connected fracture with the lowest invasion pressure. When the accumulated volume varies between intersections, the smaller volume intersections can be overloaded to cause all of the available fractures exiting an intersection to flow. We included the dynamic overloading process at fracture intersections within our previously discussed model where intersections were modeled as tipping buckets connected within a two-dimensional diamond lattice. With dynamic overloading, the flow behavior transitioned smoothly from diverging to converging flow with increasing overload parameter, as a consequence of a heterogeneous field, and they impose a dynamic structure where additional pathways activate or deactivate in time.