Narrow Results

Spanning a planar graph the way D. Hilbert's curve does has various image processing and industrial applications. Spanning a planar graph by two disjoint curves with fractal properties has even more scientific and industrial uses. For example, given two liquids and an active osmosis through membrane between them, we would like to both cool the liquids and to find a cost-effective structure for the osmosis to occur. Another equivalent problem is to expose two liquids to light that passes through a transparent slab as the osmosis between them occurs. Two disjoint curves can be the answer for the required structure. Differences of lengths between the curves can also be useful. A fractal structure is obvious in the lungs, where osmosis of oxygen is vital. Fractal structures are often found in organic osmotic processes in Nature. In this article, a method for spanning a planar graph by two disjoint curves will be presented.

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F^α-integral, where α is the dimension of F. A derivative along the fractal curve called F^α-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect.

Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The F^α-integral and F^α-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and F^α-differentiability is generalized. Finally we touch upon an example of absorption along fractal paths, to illustrate the utility of the framework in model making.

We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin equation with a particular model of noise is proposed and solved using techniques of the F${}^{\alpha }$-Calculus.

The main objective of this research is to analyze some geometrical properties of a quaternary four-point interpolatory subdivision scheme ($Q_4$-scheme) with a shape parameter and then find the precise range of the shape parameter of the $Q_4$-scheme to generate fractal and convexity of limit curves. That allows the curve to change easily and be more flexible without altering its control points. Therefore, by taking the various values of tension parameters, the curve still preserves its characteristics and geometrical configuration. These geometric modeling examples show that our techniques can be easily performed, and they can also provide us with an alternative strong strategy for the modeling of complex figures.

In this article we examine fractal curves and synthesis algorithms in musical composition and research. First we trace the evolution of different approaches for the use of fractals in music since the 80's by a literature review. Furthermore, we review representative fractal algorithms and platforms that implement them. Properties such as self-similarity (pink noise), correlation, memory (related to the notion of Brownian motion) or non correlation at multiple levels (white noise), can be used to develop hierarchy of criteria for analyzing different layers of musical structure. L-systems can be applied in the modelling of melody in different musical cultures as well as in the investigation of musical perception principles. Finally, we propose a critical investigation approach for the use of artificial or natural fractal curves in systematic musicology.