Analytical solutions to the parts of the boundary of the Mandelbrot set associated with cycles of orders 2, 3, 4, 5 are presented in the form of polynomial maps from the unit circle. Some applications of these polynomials are discussed.

Some scaling solutions of a class of radially symmetric non-linear diffusion equations in an arbitrary dimension d are obtained, (A) for an initial point source with a fixed total amount of material, and (B) for a radial flux of material through a hyper-spherical surface. In this macroscopic...

The analytical solution to the parts of the boundary of the Mandelbrot set associated with cycles of order six is obtained in the form of a polynomial. The associated algebraic elimination problem is considered for cycles of general order n.

A pair of alpha functions are used to quantify the (asymptotic) structure of the giant tentacles which occur on branches around the left-hand side of the main cardioid in the Mandelbrot set.

The density of the complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular lattices is investigated near real and complex critical points. The non-uniform behaviour of the density of zeros at interior and boundary critical...

The alpha function is used to quantify the (asymptotic) structure of the various branches and embedded spirals around the left-hand side of the main cardioid in the Mandelbrot set.