The theory of fractal homeomorphisms is applied to transform a Sierpinski triangle into what we call a Kigami
triangle. The latter is such that the corresponding harmonic functions and the corresponding Laplacian Δ take a
relatively simple form. This provides an alternative approach to recent results of Teplyaev. Using a second fractal
homeomorphism we prove that the outer boundary of the Kigami triangle possesses a continuous first derivative
at every point. This paper shows that IFS theory and the chaos game algorithm provide important tools for
analysis on fractals.
High quality video compression is necessary for reduction of transmission bandwidth and in archiving applications. We propose a compression scheme which, depending on the available bandwidth, can vary from lossless compression to lossy compression, but always with guaranteed quality. In the case of lossless compression, the customer receives the original content without any loss. Even the lower compression ratios obtained with lossless compression can represent significant savings in the communication bandwidth. In the case of lossy compression, the maximum error between recovered and the original video is mathematically bounded. The amount of compression achieved is a function of the error bounds. Furthermore, errors are statistically independent from the video content, and thus guaranteed not to create any type of artifacts. So the recovered video has the same quality, visually indistinguishable from the original, at all times and all motion conditions.