Instability of the Mandelbrot Set

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

A Generalized Mandelbrot Set Based On Distance Ratio

The iteration of complex function can generate beautiful fractal images. This paper presents a novel method based on the iteration of the distance ratio with two points, which generates a generalized Mandelbrot set according to distance ratio convergence times. This paper states the definition of distance ratio and its iteration. Then taking the complex function f(z)=z+c for example, it discuss...

Massive Media Event Data Analysis to Assess World-Wide Political Conflict and Instability

Mining massive daily news media data to infer patterns of cultural trends, including political conflicts and instabilities, is an important goal of computational social science and the new interdisciplinary field called “culturnomics.” While the sheer size of media data makes this task challenging, a greater hurdle is the nonstationarity of data, manifested in several ways, which invalidates su...

Dynamics of Mandelbrot Set with Transcendental Function

These days Mandelbrot set with transcendental function is an interesting area for mathematicians. New equations have been created for Mandelbrot set using trigonometric, logarithmic and exponential functions. Earlier, Ishikawa iteration has been applied to these equations and generate new fractals named as Relative Superior Mandelbrot Set with transcendental function. In this paper, the Mann it...

Local Connectivity of the Mandelbrot Set at Certain Infinitely Renormalizable Points

We construct a subset of the Mandelbrot set which is dense on the boundary of the Mandelbrot set and which consists of only infinitely renormalizable points such that the Mandelbrot set is locally connected at every point of this subset. We prove the local connectivity by finding bases of connected neighborhoods directly.