Potts model on complex networks

We consider the general p-state Potts model on random networks with a given degree distribution (random Bethe lattices). We find the effect of the suppression of a first order phase transition in this model when the degree distribution of the network is fat-tailed, that is, in more precise terms, when the second moment of the distribution diverges. In this situation the transition is continuous and of infinite order, and size effect is anomalously strong. In particular, in the case of p = 1, we arrive at the exact solution, which coincides with the known solution of the percolation problem on these networks.Received: 3 December 2003, Published online: 17 February 2004PACS: 05.10.-a Computational methods in statistical physics and nonlinear dynamics - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.50. + q Lattice theory and statistics (Ising, Potts, etc.) - 87.18.Sn Neural networks     

Stationary and periodic solutions of the operator Riccati equation under a random perturbation

Sufficient conditions are presented for the existence of stationary and periodic solutions of the operator Riccati equation under a random perturbation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 609–615, May, 1993.     

On one condition of weak compactness of a family of measure-valued processes

We present a criterion for the weak compactness of continuous measure-valued processes in terms of the weak compactness of families of certain space integrals of these processes.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 885–891, July, 2004.     

Measurable Functionals and Finitely Absolutely Continuous Measures on Banach Spaces

We consider the structure of orthogonal polynomials in the space L ₂(B, ) for a probability measure on a Banach space B. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (B, ).     

Some remarks on a Wiener flow with coalescence

We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1327–1333, October, 2005.     

Consistency of the least squares estimator of an infinite-dimensional parameter

Kiev. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 65–69, July–August, 1992.