Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM ⁿ. The central objects in this theory are supplementary invariant Poisson structuresP _c which are incompatable with the original Poisson structureP ₁ for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP ₁ andP ₂ in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP ₁ nor with respect toP ₂. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337.     

The High-Resolution Fourier Transform Spectrum of Dideuterated Methane CH2D2in the Region between 1900 and 2400 cm

The infrared spectrum of doubly deuterated methane CH₂D₂has been recorded in the region from 1900 to 2400 cm⁻¹at almost Doppler-limited resolution by using two high-resolution Fourier transform spectrometers. The vibrational bands observed include 2ν₄, ν₄+ ν₇, 2ν₇, ν₂, ν₈, ν₄+ ν₉, and ν₇+ ν₉, which were analyzed by taking into account Coriolis and Fermi interactions among them and also those with ν₄+ ν₅, ν₃+ ν₇, and ν₅+ ν₇. Most of the centrifugal distortion constants were constrained to appropriate values, while the vibrational term value and three rotational constants in each of the seven excited states were adjusted along with Coriolis and Fermi interaction parameters by the least-squares analysis of the observed spectrum. The vibration–rotation interaction constants α_sthus determined for the ν₂and ν₈states were combined with those of other fundamental states already published to calculate the equilibrium C–H distance.     

Degrees of unsolvability of first order decision problems for finitely presented groups

We show that for any arithmetical -degree there is a first order decision problem such that has -degree for the free 2-step nilpotent group of rank 2. This implies a conjecture of Sacerdote.

     

Research on orotron oscillator with dispersive open resonant system

Theoretical and experimental results of an investigation into a new resonant system have been obtained. This system is named the sphere-corner-echelette open resonator (SCEOR) due to the employment of a mirror that was formed by two echelettes at the angles of 45° to the resonator axis. It turns ont that this resonator is excited on the specific modes not unique to others oscillating systems. There are presented the results of the experimental research of the orotron oscillator with the SCEOR. The spectrum of this device contains only the fundamental modes such as theT E M ₀₀₆,T E M ₀₀₇,T E M ₀₀₈. The efficiency of the orotron is improved, when all other factors are the same the orotron with a much used sphere-cylindrical open resonator.     

Planar Pyrochlore,Quantum Ice and Sliding Ice

We study quantum antiferromagnetism on the highly frustrated planar pyrochlore lattice, also known as the square lattice with crossings. The quantum Heisenberg antiferromagnet on this lattice is of interest as a two-dimensional analogue of the pyrochlore lattice magnet. By combining several approaches we conclude that this system is most likely ordered for all values of spin, S, with a two-fold degenerate valence-bond solid being the ground state for small S. We show next that the Ising antiferromagnet with a weak four-spin exchange, equivalent to square ice with the leading quantum dynamics, exhibits analogous plaquette order. As a byproduct of this analysis we obtain, in the system of weakly coupled ice planes, a sliding phase with XY symmetry; at intermediate couplings, long range “anti-ferroelectric” order is stabilized.     

Ordering in spatially anisotropic triangular antiferromagnets

We investigate the phase diagram of the anisotropic spin-1/2 triangular lattice antiferromagnet, with interchain diagonal exchange J' much weaker than the intrachain exchange J. We find that fluctuations lead to a competition between (commensurate) collinear antiferromagnetic and (zigzag) dimer orders. Both states differ in symmetry from the spiral order known to occur for larger J', and are therefore separated by quantum phase transitions from it. The zero-field collinear antiferromagnet is succeeded in a magnetic field by magnetically ordered spin-density-wave and cone phases, before reaching the fully polarized state. Implications for the anisotropic triangular magnet Cs2CuCl4 are discussed.     

A mini-landscape of exact MSSM spectra in heterotic orbifolds

We explore a “fertile patch” of the heterotic landscape based on a Z₆-II${\mathbb{Z}}_6\text{-II}$ orbifold with SO(10) and E₆ local GUT structures. We search for models allowing for the exact MSSM spectrum. Our result is that of order 100 out of a total 3×¹⁰⁴$3\times {10}^4$ inequivalent models satisfy this requirement.     

Tetra point wetting at the free surface of liquid Ga-Bi

A continuous surface wetting transition, pinned to a solid-liquid-liquid-vapor tetra coexistence point, is studied by x-ray reflectivity in liquid Ga-Bi binary alloys. The short-range surface potential is determined from the measured temperature evolution of the wetting film. The thermal fluctuations are shown to be insufficient to induce a noticeable breakdown of mean-field behavior, expected in short-range-interacting systems due to their d(u) = 3 upper critical dimensionality.