PsiPsi - Vectors for Three Dimensional Models

In this paper we apply the method of psi-vectors to three dimensional statistical models. This method gives the correspondence between the Bazhanov -- Baxter model and its vertex formulation. Considering psi-vectors for the Planar model, we obtain its self-duality.Comment: 11 pages, LaTeX, no figure

Three-coloring statistical model with domain wall boundary conditions. I. Functional equations

In 1970 Baxter considered the statistical three-coloring lattice model for the case of toroidal boundary conditions. He used the Bethe ansatz and found the partition function of the model in the thermodynamic limit. We consider the same model but use other boundary conditions for which one can prove that the partition function satisfies some functional equations similar to the functional equations satisfied by the partition function of the six-vertex model for a special value of the crossing parameter.Comment: 16 pages, notations changed for consistency with the next part, appendix adde

Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites

We calculate the bulk contribution for the doubly degenerated largest eigenvalue of the transfer matrix of the eight vertex model with an odd number of lattice sites N in the disordered regime using the generic equation for roots proposed by Fabricius and McCoy. We show as expected that in the thermodynamic limit the result coincides with the one in the N even case.Comment: 11 pages LaTeX New introduction, Method change

Bethe roots and refined enumeration of alternating-sign matrices

The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.Comment: LaTeX 2e, 12 pages, minor corrections, references adde

Spin chains and combinatorics: twisted boundary conditions

The finite XXZ Heisenberg spin chain with twisted boundary conditions was considered. For the case of even number of sites $N$, anisotropy parameter -1/2 and twisting angle $2 \pi /3$ the Hamiltonian of the system possesses an eigenvalue $-3N/2$. The explicit form of the corresponding eigenvector was found for $N \le 12$. Conjecturing that this vector is the ground state of the system we made and verified several conjectures related to the norm of the ground state vector, its component with maximal absolute value and some correlation functions, which have combinatorial nature. In particular, the squared norm of the ground state vector is probably coincides with the number of half-turn symmetric alternating sign $N \times N$ matrices.Comment: LaTeX file, 7 page

Exact and simple results for the XYZ and strongly interacting fermion chains

We conjecture exact and simple formulas for physical quantities in two quantum chains. A classic result of this type is Onsager, Kaufman and Yang's formula for the spontaneous magnetization in the Ising model, subsequently generalized to the chiral Potts models. We conjecture that analogous results occur in the XYZ chain when the couplings obey J_xJ_y + J_yJ_z + J_x J_z=0, and in a related fermion chain with strong interactions and supersymmetry. We find exact formulas for the magnetization and gap in the former, and the staggered density in the latter, by exploiting the fact that certain quantities are independent of finite-size effects

Ground State of the Quantum Symmetric Finite Size XXZ Spin Chain with Anisotropy Parameter Δ=1/2\Delta = {1/2}

We find an analytic solution of the Bethe Ansatz equations (BAE) for the special case of a finite XXZ spin chain with free boundary conditions and with a complex surface field which provides for $U_q(sl(2))$ symmetry of the Hamiltonian. More precisely, we find one nontrivial solution, corresponding to the ground state of the system with anisotropy parameter $\Delta = {1/2}$ corresponding to $q^3 = -1$.Comment: 6 page

The development of blockchain technology in Russia : outlook and trends

Purpose: The article addresses the issue of new scientific decisions shaping with respect to the study of problems, current trends and perspectives of blockchain technology usage in the Russian Federation. Design/methodology/approach: To achieve the objectives of this study the increasing interest to blockchain technology in Russia was discussed. Findings: The article determined main problems in blockchain technology which includes gaps in legislative regulation; the existence of a considerable number of projects that are undergoing the development stage and that have not proved own economic feasibility yet; incomplete understanding of the blockchain spheres’ implementation by state officials, society and business representatives as well as expected outcomes according to the amount and time of their receiving; disputes on cryptocurrencies turnover in the country’s territory and their influence in the national economy. Practical implications: The study has demonstrated the interest growth mainly by businesses to the usage of blockchain technology in order to improve own competitiveness and to obtain additional benefits, including the form of their profits. Originality/value: The research has also determined the area of further key studies in blockchain technology usage in Russia and the world.peer-reviewe

A new representation for the partition function of the six vertex model with domain wall boundaries

We obtain a new representation for the partition function of the six vertex model with domain wall boundaries using a functional equation recently derived by the author. This new representation is given in terms of a sum over the permutation group where the partial homogeneous limit can be taken trivially. We also show by construction that this partition function satisfies a linear partial differential equation.Comment: 14 pages, v2: added references, accepted for publication in J. Stat. Mec

The role of orthogonal polynomials in the six-vertex model and its combinatorial applications

The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1-, 2- and 3-enumerations of Alternating Sign Matrices (ASMs), these polynomials specialize to classical ones (Continuous Hahn, Meixner-Pollaczek, and Continuous Dual Hahn, respectively). As a consequence, a unified and simplified treatment of ASMs enumerations turns out to be possible, leading also to some new results such as the refined 3-enumerations of ASMs. Furthermore, the use of orthogonal polynomials allows us to express, for generic values of the parameters of the model, the partition function of the (partially) inhomogeneous model in terms of the one-point boundary correlation functions of the homogeneous one.Comment: Talk presented by F.C. at the Short Program of the Centre de Recherches Mathematiques: Random Matrices, Random Processes and Integrable Systems, Montreal, June 20 - July 8, 200