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.

Complex temperature zeros are calculated numerically from a polynomial form of the partition function of the two-dimensional Ising model in the absence of a magnetic field. In terms of a suitably chosen variable, z, the distribution of zeros is symmetrical under reflection in both the real and...

The distribution of complex temperature zeros of the partition function of the two-dimensional Ising model in the absence of a magnetic field is investigated in detail for a general anisotropic triangular lattice. When the three Ising interactions are all different the explicit expression for...

The distribution of complex temperature zeros of the partition function of the two-dimensional Ising model in the absence of a magnetic field is investigated. For anisotropic square and triangular lattices the distribution function is two-dimensional and satisfies a partial differential equation...

Some elegant exact general relations between energy and temperature fluctuations are obtained in the context of the classical mechanics of two (sub)systems in thermal contact, which together form an isolated combined system. The results apply especially to: (i) a thermometer used to measure the...

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.