Some remarks on finite-gap solutions of the Ernst equation

The relationship between the class of algebro-geometrical (finite-gap) solutions of the Ernst equation constructed by Korotkin and Matveev [Theor. Math. Phys. 77 (1989) 1018; St. Petersburg Math. J. 1 (1990) 379] and the solutions recently constructed by Meinel and Neugebauer [Phys. Lett. A 210 (1996) 160] is discussed. A new formula for the general algebro-geometrical solution is obtained.     

Star-triangle relation for a three-dimensional model

The solvablesl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising-type model on the body-centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to the spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly.On leave of absence from the Institute for High Energy Physics, Protvino, Moscow Region, 142284, Russia.     

New rational solutions of Yang-Baxter equation and deformed Yangians

In this paper a new class of quantum groups, deformed Yangians, is used to obtain new matrix rational solutions of the Yang-Baxter equation (YBE). The deformed Yangians arise from rational solutions of the classical Yang-Baxter equation of the form c ₂/u + const. The image of the universal quantum R-matrix for the deformed Yangian in finite-dimensional representations gives these new matrix rational solutions of YBE.     

Volume effects in the theory of equilibrium and quasiequilibrium states of multicomponent solid solutions

A phenomenological theory of equilibrium and quasiequilibrium states of multicomponent solid solutions is constructed taking account of volume effects. Quasiequilibrium states are characterized by the fact that only some of the conditions for thermal dynamic equilibrium of the system are satisfied. The short-range parts of the interatomic interactions are taken into account by introducing the proper volumes of the atoms based on a generalized lattice model. The long-range parts of the potentials are taken into account in the effective-field approximation. The equations for the quasiequilibrium components in the solutions are introduced taking account of the nonuniformity in the distributions of the less mobile nonequilibrium components. The conditions for spinodal decomposition of a solid solution with an arbitrary number of components in the equilibrium and quasiequilibrium cases are obtained. An equation for equilibrium spinodal decomposition of a three-component microheterogeneous solid solution is found. Fiz. Tverd. Tela (St. Petersburg) 41, 1609–1613 (September 1999)     

Integrable boundary Boltzmann weights and surface free energy inversion relations of the chiral Potts model

The integrability of the chiral Potts model with boundaries is considered in this paper. The boundary star-triangle relation determining the boundary Boltzmann weights for the chiral Potts model is presented. By solving the boundary star-triangle relation the boundary Boltzmann weights are obtained. The fusion procedure is then applied to derive the functional relations of the transfer matrices of the model with boundaries. From these functional relations the inversion relations of the surface free energies are extracted when the system size is big enough. Surprisingly, the inversion relation of the local surface free energy is as simple as those of other non-chiral models, but it has still to be solved.     

Yang-Baxterization of braid group representations

For a given braid group representation (BGR), a process of the Yang-Baxterization is formulated to generate solutions of the Yang-Baxter equation (YBE). When a BGR admits the Birman-Wenzl (BW) algebraic structure, this process can be explicitly passed through and two types of trigonometric solutions of YBE are generated from such a BGR. These two solutions have, the essential difference to each other and both of them, preserve the crossing symmetry property if the given BGR has. By taking certain, reduction on the BW algebra, the rational solution is also generated. A practical condition to judge whether a BGR satisfies the BW algebra is given, from which one finds that not only the familiar BGRs of [5,7,9], but also some new, ones obtained recently in [12] have the BW structure. Thus they can be explicitly Yang-Baxterized to solutions of the YBE.     

三维精确可解统计模型──Baxter－Bazhanov模型的可积性条件

从手征Ｐｏｔｔｓ模型推导出三维精确可解Ｂａｘｔｅｒ－Ｂａｚｈａｎｏｖ模型的“可逆性”及“星一方”关系,从而说明其可积性条件──四面体方程是手征Ｐｏｔｔｓ模型星──三角关系的一个结论．若把玻尔兹曼权参变数表示为Ｚａｍｏｌｏｄｃｈｉｋｏｖ角变量形式,其附加条件自然成立．值得指出的是,由本文处理方法可以得出三维可解统计模型的星－三角关系,它包含了Ｂａｚｈａｎｏｖ和Ｂａｘｔｅｒ的结论．     

A class of elliptic solutions to the star-triangle relation

The star-triangle relation for the N-state IRF model is investigated. A class of elliptic solutions is found. The corresponding lattice model splits into mutually non-interacting diagonal lattices with the interactions between nearest-neighbour spins.     

The median-based resolution of the St. Petersburg paradox

The St. Petersburg Paradox contributed to establishing expected utility theory by demonstrating that decision making based on the expectation (expected value, or mean, average) leads to an unreasonable behavior. Although the expected value is commonly used as an optimization criterion in various fields of mathematical sciences, such paradoxical problems as arbitrariness and intransitivity have led many researchers to forsake expected utility theory in search of more useful alternatives. Here we show an analytical solution of the St. Petersburg Paradox based on the median of the probability distribution. The present method provides a reasonable solution to any related problem. The median payout of repeated games suggests a new scaling relation in the limit of a large number of repetitions, while the game has no characteristic scale in terms of expected value.     

Degeneracy of the vacuum state according to chirality in the sigma-model with massless fermions

The degeneracy of the vacuum according to chirality is proved. It is shown that the local source of the fermionic field leads to a change of the chiral charge density. This signifies screening of the chiral charge and, possibly, confinement. Both effects are due to topological fluctuations of the sigma-field.St. Petersburg Higher Military Engineering College of Communications. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 77–80, November, 1993.     

Quasiequilibrium states of solid solutions

A phenomenological theory of quasiequilibrium states of solid solutions with at least three components is constructed under the assumption that the atomic mobilities of the components differ in order of magnitude. The short-range parts of the interatomic interactions are taken into account by introducing atomic self-volumes. The long-range parts of the potentials are taken into account in the effective-field approximation. The equations for the quasiequilibrium components in the solutions are derived neglecting the mobility of the least mobile component. It is shown that, for a nonuniform distribution of this component, the nonequilibrium critical temperature ceases being unique within the sample and acquires a local character. In consequence, opposing processes occur in the system in the temperature interval between T _c ^min and T _c ^max — homogenization in some sections and formation of interphase boundaries in others. Fiz. Tverd. Tela (St. Petersburg) 41, 60–63 (January 1999)     

Estimation of field in plane impedance waveguide with local geometrical irregularity. II

The author investigates effect on the field of a point source in a plane impedance waveguide of an irregularity in the form of spherical surface that projects from (or is embedded into) the plane of the, ionospheric wall of the waveguide. In a scalar approximation, the problem is reduced to a two-dimensional integral equation over the surface of the irregularity. A solution is constructed by successive approximations, for which the solution of the problem for a regular impedance waveguide is used as the initial approximation. Numerical results are given for estimation of the effect of a local ionospheric irregularity on the field of an electric dipole in the earth-ionosphere waveguide.St. Petersburg State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 35, No. 6/7, pp. 569–578, June–July, 1992.     

X-ray microanalysis of quaternary semiconductor solid solutions and its application to the (SnTe-SnSe):In system

A reliable technique of local chemical characterization of multicomponent semiconductor solid solutions has been developed, and the possibility of its application to the SnTe-SnSe quaternary solid solutions doped with 16 at.% In verified. The behavior of the electrical resistivity of samples of these solid solutions at low temperatures, 0.4–4.2 K, has been studied. The critical temperature T _c and the second critical magnetic field H _c2 of the superconducting transition and their dependences on the solid-solution composition have been determined. The superconducting transition at T _c≈2–3 K is due to hole filling of the In-impurity resonance states, and the observed variation of the superconducting transition parameters with increasing Se content in the solid solution is related to the extrema in the valence band and the In band of resonance states shifting with respect to one another. Fiz. Tverd. Tela (St. Petersburg) 41, 612–617 (April 1999)     

The influence of quantum-mechanical athermal phenomena on the thermal instability of plastic deformation in crystals at low temperatures

The way in which athermal (quantum) mechanisms that allow dislocations to overcome local barriers affect the thermal instability of plastic deformation of crystals is discussed theoretically for the case of low and ultralow (<1 K) temperatures. Calculations show that increasing the athermal component of the dislocation activation leads to a considerable narrowing of the temperature/strain-rate region where discontinuous strains appear, and that further increases in the quantum component of the process by which dislocations surmount local barriers leads to the complete disappearance of these discontinuous strains. Experimental situations in which the effects of thermal instability and athermal effects are observed at the same time in a number of crystals under conditions of low-temperature strain are discussed in light of these results. Fiz. Tverd. Tela (St. Petersburg) 39, 1392–1398 (August 1996)     

Misfit disclinations at crystal/crystal and crystal/glass interfaces

Theoretical concepts have been developed for a new type of misfit defects, misfit disclinations, at crystal/crystal and crystal/glass interfaces. It is shown, in particular, that the formation of misfit disclinations is an efficient physical micromechanism of misfit stress relaxation at crystal/crystal interfaces. A model describing misfit disclinations at crystal/glass interfaces has been constructed. The energy characteristics of phase boundaries with misfit disclination ensembles are estimated. Fiz. Tverd. Tela (St. Petersburg) 41, 1637–1643 (September 1999)     

Some remarks on the differential approach to the star-triangle relation

Some working hypotheses are proposed to solve the star-triangle relation by the differential method. They are applied to and checked by three-and four-state IRF model with a symmetry condition. Several solutions involving elliptic or trigonometric functions are presented.     

Neutron spectroscopy study of the local modes of interstitials in transition metals

Astract Local modes of carbon, nitrogen, oxygen, and hydrogen have been measured in a number of transition-metal solid solutions by inelastic neutron scattering. The metal-impurity force constants have been calculated from experimental data and constructed as functions of the metal-impurity distance. The correlation of these functions with the data obtained in effective-medium theory, as well as the effect of interstitial atoms on the metal-metal bonding are discussed. Fiz. Tverd. Tela (St. Petersburg) 39, 23–27 (January 1997)     

Use of the method of integral equations for estimating electromagnetic absorption in biological objects

We propose a method for calculating the specific absorbed power in biological objects using the Fredholm integral equation of the second kind with respect to the surface density of the electric charge. the regularization of the equation allows us to perform numerical calculation in the case of singularity of the kernel of the integral equation and obtain stable solutions. We calculate the specific absorbed power in a biological object represented by the operator of an industrial high-frequency welding plant. The dependence of the specific absorption on the type of floor coating in a production room and the air gap between the floor and the operator’s feet is estimated. Baltiyskii State Technical University “Voenmekh,” St. Petersburg, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 2, pp. 176–182, February 1999.     

Cole-Davidson dielectric relaxation as a self-similar relaxation process

An effort has been undertaken to understand the nature of one type of non-exponential relaxation, namely Cole-Davidson relaxation. Toward this end, a model of relaxation as a self-similar process is proposed. An equation containing operators of fractional integration and differentiation is obtained and solved, which the relaxation function obeys in this case. Fiz. Tverd. Tela (St. Petersburg) 39, 101–105 (January 1997)