The analytical solutions to the parts of the boundary of the Mandelbrot set associated with cycles of order 7 and 8 are obtained by an arithmetic method in the form of polynomials. The “inverse orbit” method of calculating the values of cycle centres is extended to complex cases, and applied...

The boundaries of regions containing complex temperature zeros of the partition function of the two-dimensional Ising model on partially anisotropic triangular lattices are investigated in detail. A “diagonal” symmetry of one of the boundaries on lattices with one “odd” and two equal...

Special features of the complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular lattices with all “odd” interactions, such as one with interaction ratios 5:3:1, are investigated. A bifurcation of one of the boundary lines at...

The boundaries of regions containing complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular and quadratic lattices are investigated in detail. Numerical solutions of the boundary equations are presented for triangular lattices...

An explicit function is constructed to permit easy calculation of the asymptotic structure of the various spirals and branches around the main cardioid in the Mandelbrot set. Details of its application to “exterior” spirals are presented.

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.