We investigate magnetic properties of a two-dimensional periodic structure with Ising spins and antiferromagnetic nearest neighbor interaction. The structure is topologically equivalent to the Archimedean (3, 122 ) lattice. The ground state energy is degenerate. In some ground states, the spin structure is translationally invariant, with the same configuration in each unit cell. Numerical results are reported on specific heat and static magnetic susceptibility against temperature. Both quantities show maxima at temperature T > 0. They reveal some sensitivity on the initial state in temperatures where the Edwards-Anderson order parameter is positive. For zero temperature and low frequency of the applied field, the magnetic losses are negligible. However, the magnetization curve displays some erratic behavior due to the metastable states.

Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barabási-Albert scale-free networks, where the probability of linking to a node is proportional to the number of its pre-existing links. In both cases, new nodes are linked to m = 1 nodes. Average node-node distance d is calculated numerically in evolving trees as dependent on the number of nodes N . The results for N not less than a thousand are averaged over a thousand of growing trees. The results on the mean node-node distance d for large N can be approximated by d = 2 ln(N ) + c1 for the exponential trees, and d = ln(N ) + c2 for the scale-free trees, where the ci are constant. We derive also iterative equations for d and its dispersion for the exponential trees. The simulation and the analytical approach give the same results.

A three-blocks Burridge-Knopoff model is investigated. The dimensionless velocity-dependent friction force F (v) ∝ (1 + av) −1 is linearized around a = 0. In this way, the model is transformed into a six-dimensional mapping x(t n ) → x(t n+1 ), where t n are time moments when a block starts to move or stops. Between these moments, the equations of motion are integrable. For a < 0.1, the motion is quasiperiodic or periodic, depending on the initial conditions. For the periodic solution, we observe a synchronization of the motion of the lateral blocks. For a > 0.1, the motion becomes chaotic. These results are true for the linearized mapping, linearized numerical and non-linearized numerical solutions.

A new and efficient algorithm is presented for the calculation of the partition function in the S = ±1 Ising model. As an example, we use the algorithm to obtain the thermaldependence of the magnetic spin susceptibility of an Ising antiferromagnet for a 8 × 8 square lattice with open boundary conditions. The results agree qualitatively with the prediction of the Monte Carlo simulations and the experimental data, and they are better than the mean field approach results. For the 8 × 8 lattice, the algorithm reduces the computation time by nine orders of magnitude.

Spin–spin correlation functions <S(0)S(R)> as dependent on interatomic distance R are studied in the random-site two-dimensional Ising S=1/2 ±J system. Oscillations of the correlation functions are found, which is not a case in the random-bond system.