On an Algebraic Property of the Disordered Phase of the Ising Model with Competing Interactions on a Cayley Tree

It is known that the disordered phase of the classical Ising model on the Caley tree is extreme in some region of the temperature. If one considers the Ising model with competing interactions on the same tree, then about the extremity of the disordered phase there is no any information. In the present paper, we first aiming to analyze the correspondence between Gibbs measures and QMC’s on trees. Namely, we establish that states associated with translation invariant Gibbs measures of the model can be seen as diagonal quantum Markov chains on some quasi local algebra. Then as an application of the established correspondence, we study some algebraic property of the disordered phase of the Ising model with competing interactions on the Cayley tree of order two. More exactly, we prove that a state corresponding to the disordered phase is not quasi-equivalent to other states associated with translation invariant Gibbs measures. This result shows how the translation invariant states relate to each other, which is even a new phenomena in the classical setting. To establish the main result we basically employ methods of quantum Markov chains.     

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

In the present paper, we define an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base , and study its properties. The defined coefficient is a generalization of the well-known the Dobrushin’s ergodicity coefficient. By means of the ergodicity coefficient we provide uniform asymptotical stability conditions for nonhomogeneous discrete Markov chains (NDMC). These results are even new in case of von Neumann algebras. Moreover, we find necessary and sufficient conditions for the weak ergodicity of NDMC. Certain relations between uniform asymptotical stability and weak ergodicity are considered.     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

In the present paper the Ising model with competing binary (J) and binary (J₁) interactions with spin values ±1, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model is studied. We completely describe the set of all periodic Gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type III then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too.     

Turbulence. 3. A new turbulent paradigm

An example of application of the mathematical apparatus developed in the first two parts of the present work [1, 2] to problems of mechanics of a continuous medium is given. The nature of the turbulent phenomenon is discussed from the viewpoint of the developed apparatus. A new program of turbulence research is suggested.     

Quinazolines. 2*. unsymmetric 1,3-dialkyl-6-chlorosulfonyl- quinazoline-2,4-diones in nucleo- philic substitution reactions

The corresponding 6-chlorosulfonylquinazoline-2,4-diones were synthesized by the reactions of 1-methylquinazoline-2,4-dione and its 3-alkyl-substituted derivatives with chlorosulfonic acid. Treatment of the products with nucleophilic agents (water or ammonia, aliphatic and cyclic amines) gave 2,4-dioxoquinazolin-6-sulfonic acids or their amides.     

The disordered phase of the inhomogeneous potts model is extremal on the Cayley tree

We find the sufficient condition on the parameters of the inhomogeneous Potts model on the Cayley tree under which the disordered phase becomes extremal. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 410–418, September, 2000.     

Algorithms of the auxiliary sources method for analysis of waveguide-dielectric periodic structures

Scientific-Research Radiophysics Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 34, No. 2, pp. 392–402, April, 1991.     

Some Properties of a Class of Diagonalizable States of von Neumann Algebras

In this paper, a class of representations of uniformly hyperfinite algebras is constructed and the corresponding von Neumann algebras are studied. It is proved that, under certain conditions, the Markov states generate factors of type IIIλ, where λ ∈ (0,1), in the GNS representation; this gives a negative answer to the conjecture that the factors corresponding to Hamiltonians with nontrivial interactions have type III. It is shown that, for a certain class of Hamiltonians, there exists a unique translation-invariant ground state.