ALBERT

Description:    We propose a construction of a Lagrangian torus fibration of the full flag variety in ℂ 3 . In contrast to the classical fibration obtained from the Gelfand-Zeitlin system, the proposed fibration is special Lagrangian. Content Type Journal Article Pages 567-576 DOI 10.1007/s11232-011-0042-x Authors N. A. Tyurin, Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We propose a method for describing quasiequilibrium processes in a quasistable anharmonic oscillator affected by heat noise. We introduce effective thermodynamic parameters analogous to those of an equilibrium system but differing from them in numerical value. We define the parameter characterizing the nonequilibrium of the system, prove the mathematical consistency of the proposed method, and obtain the quasiequilibrium analogue of the theorem on energy distribution over degrees of freedom. Content Type Journal Article Pages 636-644 DOI 10.1007/s11232-011-0048-4 Authors A. V. Kargovsky, Lomonosov Moscow State University, Moscow, Russia L. S. Bulushova, Lomonosov Moscow State University, Moscow, Russia O. A. Chichigina, Lomonosov Moscow State University, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We develop the recent proposal to use dimensional reduction from the four-dimensional space-time (D = 1 + 3 ) to the variant with a smaller number of space dimensions D = 1 + d, d 〈 3, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein-Gordon equation with a few toy examples (“educational toys”) of a space-time with a variable spatial geometry including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique for transforming the study of solutions of the Klein-Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schrödinger-type equation with potential generated by this variation is useful. We draw the following conclusions: ( 1 ) The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). ( 2 ) The specific spectrum of scalar excitations resembles the spectrum of real particles; it reflects the geometry of the transition region and represents its “fingerprints.” ( 3 ) The parity violation due to the asymmetric character of the construction of our models could be related to the CP symmetry violation. Content Type Journal Article Pages 680-691 DOI 10.1007/s11232-011-0052-8 Authors P. P. Fiziev, Sofia University St. Kliment Ohridski, Sofia, Bulgaria D. V. Shirkov, Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    In the mode-coupling approximation, we consider the transition to the glass state in a system of collapsing hard spheres (a system with the hard-core potential to which a repulsive step is added). We propose an approximation for the structure factor of the system, which we use to construct the phase diagram of the transition to the glass state. We show that there exists a maximum on the liquid-glass curve corresponding to the reentrant transition to the glass state in the system. In the framework of the proposed model, we consider bifurcations of solutions of the equations describing the transition to the glass state and show that there exist bifurcations of the “swallow-tail” type corresponding to the glass-glass transition. Content Type Journal Article Pages 645-653 DOI 10.1007/s11232-011-0049-3 Authors V. N. Ryzhov, Institute for High Pressure Physics, RAS, Troitsk, Moscow Oblast, Russia E. E. Tareyeva, Institute for High Pressure Physics, RAS, Troitsk, Moscow Oblast, Russia Yu. D. Fomin, Institute for High Pressure Physics, RAS, Troitsk, Moscow Oblast, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We consider two initial boundary value problems on an interval with homogeneous Dirichlet conditions. These problems were proposed by M. I. Rabinovich and D. I. Trubetskov and are given by nonlinear fourth-order Sobolev-type equations. We prove the local-in-time existence of a strong generalized solution of one problem, and for both problems, we obtain sufficient conditions for the destruction of their strong generalized solutions in a finite time. Content Type Journal Article Pages 577-583 DOI 10.1007/s11232-011-0043-9 Authors M. O. Korpusov, Lomonosov Moscow State University, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 2

Description:    We consider the evolution of the spatial distribution of fast atomic particles as they pass through a crystal. At small penetration depths where the particles move without significant loss of coherence, the joint probability density function becomes smoother at the expense of the gradient of the anharmonic potential of the planar channel. We show that as a result of this smoothing, there arises a quasiequilibrium (quasistationary) state of the subsystem of fast particles. The further evolution of the particle distribution is caused by nonelastic scattering and is described by the kinetic equation. At this stage, the anharmonic character of particle oscillations between the channel walls results in a renormalization of the total phonon scattering cross section of particles, and the renormalization is determined by the fourth-order anharmonic interaction. Content Type Journal Article Pages 506-516 DOI 10.1007/s11232-011-0038-6 Authors Yu. A. Kashlev, Baikov Institute for Metallurgy and Material Science, RAS, Moscow, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description:    We propose a method for calculating vacuum means of the scalar field energy-momentum tensor on Lie groups and homogeneous spaces. We use the generalized harmonic analysis based on the method of coadjoint representation orbits. Content Type Journal Article Pages 468-483 DOI 10.1007/s11232-011-0035-9 Authors A. I. Breev, Tomsk State University, Tomsk, Russia I. V. Shirokov, Omsk State Technical University, Omsk, Russia A. A. Magazev, Irtysh Branch of Novosibirsk State Academy of Water Transport, Novosibirsk, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description:    We construct the group bundle for the inverse Fibonacci semigroups in the axiomatic approach framework. The proper Fibonacci semigroup is the corresponding group bundle for the Penrose semigroups, which can be interpreted as the generating grammar of the morphogenetic synthesis of the pentasymmetric Penrose parquet in the numbers of tiling by golden rhombuses. This morphogenetic synthesis of the Penrose parquet satisfies the scaling principle. Parquet plates are not absolutely rigid, and the relations between their metric characteristics are governed by the golden section and other magic numbers. The characteristic form factors of three-level dual alphabets are the corresponding invariants. We realize the morphogenetic synthesis in the examples of square-octagonal and bihexagonal lattices. We consider cumulative properties of magic series and the evolutionary aspects of semigroup orbits in the entropy representation. Content Type Journal Article Pages 517-537 DOI 10.1007/s11232-011-0039-5 Authors V. V. Yudin, Far-East State University, Vladivostok, Russia E. S. Startzev, Far-East State University, Vladivostok, Russia I. G. Permyakova, Far-East State University, Vladivostok, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description:    We obtain an expression for the NMR line at low temperatures for a system of nuclear spins described by a Hamiltonian with equal spin-spin coupling constants. We show that in the case of “easy axis” anisotropy, the line has a logarithmic low-frequency singularity and an exponentially decreasing high-frequency asymptotic behavior at the temperature of an anomalous peak of heat capacity. In the case of “easy plane” anisotropy, the line has the traditional Gaussian form. We discuss the possibility of using NMR data to discover specific thermodynamic and magnetic properties of the considered model system. Content Type Journal Article Pages 496-505 DOI 10.1007/s11232-011-0037-7 Authors A. A. Khamzin, Kazan State Energy University, Kazan, Russia R. R. Nigmatullin, Kazan (Volga Region) Federal University, Kazan, Russia Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1

Description:    We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations. Content Type Journal Article Pages 421-443 DOI 10.1007/s11232-011-0033-y Authors A. V. Mikhailov, Applied Mathematics Department, University of Leeds, Leeds, UK Jing Ping Wang, School of Mathematics and Statistics, University of Kent, Kent, UK P. Xenitidis, Applied Mathematics Department, University of Leeds, Leeds, UK Journal Theoretical and Mathematical Physics Online ISSN 1573-9333 Print ISSN 0040-5779 Journal Volume Volume 167 Journal Issue Volume 167, Number 1