Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle

After giving an introduction to the procedure dubbed slow polynomial mating and stating a conjecture relating this to other notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree 3 post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree 6 rational map. We give a conjectural interpretation of this in terms of pinched spheres and show further conformal representations.Comment: 35 page

Fixed point results for the complex fractal generation in the S -iteration orbit with s -convexity

Since the introduction of complex fractals by Mandelbrot they gained much attention by the researchers. One of the most studied complex fractals are Mandelbrot and Julia sets. In the literature one can find many generalizations of those sets. One of such generalizations is the use of the results from fixed point theory. In this paper we introduce in the generation process of Mandelbrot and Julia sets a combination of the S-iteration, known from the fixed point theory, and the s-convex combination. We derive the escape criteria needed in the generation process of those fractals and present some graphical examples

Applied Measure Theory for Probabilistic Modeling

Probabilistic programming and statistical computing are vibrant areas in the development of the Julia programming language, but the underlying infrastructure dramatically predates recent developments. The goal of MeasureTheory.jl is to provide Julia with the right vocabulary and tools for these tasks. In the package we introduce a well-chosen set of notions from the foundations of probability together with powerful combinators and transforms, giving a gentle introduction to the concepts in this article. The task is foremost achieved by recognizing measure as the central object. This enables us to develop a proper concept of densities as objects relating measures with each others. As densities provide local perspective on measures, they are the key to efficient implementations. The need to preserve this computationally so important locality leads to the new notion of locally-dominated measure solving the so-called base measure problem and making work with densities and distributions in Julia easier and more flexible