International Journal of Modern Physics B

The natural variability in physiological structure is herein related to the geometric concept of a fractal. The average dimensions of the branches in the tracheobronchial tree, long thought to be exponential, are shown to be an inverse power law of the generation number modulated by a harmonic variation. A similar functional form is found for the power spectrum of the QRS-complex of the healthy human heart. These results follow from the assumption that the bronchial tree and the cardiac conduction system are fractal forms. The fractal concept provides a mechanism for the morphogenesis of complex structures which are more stable than those generated by classical scaling (i.e., they are more error tolerant).

First order phase transitions are rounded in solids of finite size. It is shown here that the above rounding is monitored by the correlation length ξ_L of the finite system, or equivalently by the so-called mass gap for the quantum Hamiltonian version of the model. Scaling with size is studied as a function of variable boundary conditions for the cylinder geometry (infinite strips with finite width), and a striking crossover is found in the mass gap behavior when the coupling g along the boundary becomes anti-periodic. For g>0, the rounding is exponential with size and an accurate determination of the spontaneous magnetization (order parameter) of the infinite system is obtained from numerical extrapolations.

The critical behaviour of the two-dimensional Ising model in annealed random fields is studied. Two different kinds of random field are considered and exact results obtained for the transition temperature when the field distribution is symmetric. In the first case the random field is unrestricted and the explicit results obtained for discrete and gaussian field distributions show the annealed field favours the ordering of the system. In the second case the random field satisfies a constraint and explicit results are also obtained for discrete and gaussian field distributions. In the discrete distribution the field assumes the values ±h₀,0 and it is shown that there is a critical field h_c such that for h₀>h_c the system presents a reentrant phase transition. In this case the ferromagnetic ground state becomes unstable and there is the appearance of a superantiferromagnetic phase. For h₀<h_c the ferromagnetic state is stable and the transition temperature decreases due to the presence of the random field.

X-ray absorption near edge structure (XANES) measurements on L_III absorption edge of Sm are employed to study the intermediate valence in Sm_1−xLa_xSe(x=0.08, 0.10, 0.15 and 0.25) alloys wherein the substitution of the trivalent cation, La^3a+, for Sm²⁺ drives the latter to a higher valence state from divalent towards trivalent in the SmSe lattice. Two absorption peaks corresponding to two different valent states of Sm provide direct evidence for intermediate valence of Sm in these alloys. The split white lines at the L_III edge of Sm are used to determine the Sm mean valence in the alloys. Average valence data deduced from Sm L_III edge XANES shows a continuous valence transition with increasing La concentration. Lattice parameters for these alloys measured by XRD also show a regular decrease with La concentration showing an agreement with the XANES results.

A general method to derive the generating functions for the area-weighted moments of the number of convex polygons on the square lattice is described. We apply this method to calculate the generating functions for the jth moments up to j=10. The empirical formula of Enting and Guttmann for the second moment is verified.

We describe the symmetries of the chiral checkerboard Potts model (duality, inversion relation, …) and write down the algebraic variety corresponding to the integrable case advocated by Baxter, Perk, Au-Yang. We examine some of its subvarieties, in different limits and for various lattices, with a special emphasis on q=3. This yields for q=3, a new algebraic variety where the standard scalar checkerboard Potts model is solvable. By a comparative analysis of the parametrization of the integrable four-state chiral Potts model and the one of the symmetric Ashkin-Teller model, we bring to light algebraic subvarieties for the q-state chiral Potts model which are invariant under the symmetries of the model. We recover in this manner the Fateev-Zamolodchikov points.