Narrow Results

The study of the viscous fingering in Hele Shaw cell and the evolution of fingering pattern was presented and the growth velocity of pattern was determined. The fingers were recorded using a digital web camera. The movie frames were separated and the selected patterns were analyzed. The box counting technique was applied to the growth pattern and the Richardson plot technique was used for characterization of shapes in terms of structure and texture. The fingering patterns thus obtained in different frames were analyzed and the structural and textural analysis was presented. The scale invariance at different length scales was discussed.

We report on a fractal pattern formed in Rayleight-Taylor instability. The fractal dimension is 1.88 ± 0.06, which is constant in time t > t₁ = 70 s. The pattern area decreases according to exp(-t/τ), where t and τ are time and time constant 69.8 s, respectively. τ agrees with t₁. This result leads that it is important for the fractal formation that sufficient annihilation of the heavier solution at the surface.

Multiple sclerosis (MS) is a severe brain disease. Early detection can provide timely treatment. Fractal dimension can provide statistical index of pattern changes with scale at a given brain image. In this study, our team used susceptibility weighted imaging technique to obtain 676 MS slices and 880 healthy slices. We used synthetic minority oversampling technique to process the unbalanced dataset. Then, we used Canny edge detector to extract distinguishing edges. The Minkowski–Bouligand dimension was a fractal dimension estimation method and used to extract features from edges. Single hidden layer neural network was used as the classifier. Finally, we proposed a three-segment representation biogeography-based optimization to train the classifier. Our method achieved a sensitivity of 97.78±1.29%, a specificity of 97.82±1.60% and an accuracy of 97.80±1.40%. The proposed method is superior to seven state-of-the-art methods in terms of sensitivity and accuracy.

Determining the fractal dimension d_B of a complex network requires computing N(s), the minimal number of boxes of size s needed to cover the network. While effective approximation methods for this problem are known, the computation of a lower bound on N(s) has not been studied. We show that a lower bound can be obtained by formulating the covering problem as an uncapacitated facility location problem, and applying dual ascent to the dual of its linear programming relaxation. We illustrate the method on a small example, and provide numerical results on some larger problems. The upper and lower bounds on N(s) can be used to define a linear program which yields upper and lower bounds on d_B.