Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations

In this paper we consider three-dimensional quantum q-oscillator field theory without spectral parameters. We construct an essentially big set of eigenstates of evolution with unity eigenvalue of discrete time evolution operator. All these eigenstates belong to a subspace of total Hilbert space where an action of evolution operator can be identified with quantized discrete BKP equations (synonym Miwa equations). The key ingredients of our construction are specific eigenstates of a single three-dimensional R-matrix. These eigenstates are boundary states for hidden three-dimensional structures of U_q(B_n^1) and U_q(D_n^1)$.Comment: 13 page

Quantum 2+1 evolution model

A quantum evolution model in 2+1 discrete space - time, connected with 3D fundamental map R, is investigated. Map R is derived as a map providing a zero curvature of a two dimensional lattice system called "the current system". In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical one and it corresponds to known operator-valued R-matrix. The current system is a kind of the linear problem for 2+1 evolution model. A generating function for the integrals of motion for the evolution is derived with a help of the current system. The subject of the paper is rather new, and so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page

The modified tetrahedron equation and its solutions

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator in the space of a triple Weyl algebra. This operator is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for this operator follows without further calculations. If the Weyl parameter is taken to be a root of unity, the mapping operator decomposes into a matrix conjugation and a C-number functional mapping. The operator of the matrix conjugation satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equation. The matrix elements of this operator can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.Comment: 24 pages, 6 figures using epic/eepic package, Contribution to the proceedings of the 6th International Conference on CFTs and Integrable Models, Chernogolovka, Spetember 2002, reference adde

BLR kinematics and Black Hole Mass in Markarian 6

We present results of the optical spectral and photometric observations of the nucleus of Markarian 6 made with the 2.6-m Shajn telescope at the Crimean Astrophysical Observatory. The continuum and emission Balmer line intensities varied more than by a factor of two during 1992-2008. The lag between the continuum and Hbeta emission line flux variations is 21.1+-1.9 days. For the Halpha line the lag is about 27 days but its uncertainty is much larger. We use Monte-Carlo simulation of the random time series to check the effect of our data sampling on the lag uncertainties and we compare our simulation results with those obtained by random subset selection (RSS) method of Peterson et al. (1998). The lag in the high-velocity wings are shorter than in the line core in accordance with the virial motions. However, the lag is slightly larger in the blue wing than in the red wing. This is a signature of the infall gas motion. Probably the BLR kinematic in the Mrk 6 nucleus is a combination of the Keplerian and infall motions. The velocity-delay dependence is similar for individual observational seasons. The measurements of the Hbeta line width in combination with the reverberation lag permits us to determine the black hole mass, M_BH=(1.8+-0.2)x10^8 M_sun. This result is consistent with the AGN scaling relationships between the BLR radius and the optical continuum luminosity (R_BLR is proportional to L^0.5) as well as with the black-hole mass-luminosity relationship (M_BH-L) under the Eddington luminosity ratio for Mrk 6 to be L_bol/L_Edd ~ 0.01.Comment: 17 pages, 10 figures, accepted for publication in MNRA

Superanalogs of the Calogero operators and Jack polynomials

A depending on a complex parameter $k$ superanalog ${\mathcal S}{\mathcal L}$ of Calogero operator is constructed; it is related with the root system of the Lie superalgebra ${\mathfrak{gl}}(n|m)$. For $m=0$ we obtain the usual Calogero operator; for $m=1$ we obtain, up to a change of indeterminates and parameter $k$ the operator constructed by Veselov, Chalykh and Feigin [2,3]. For $k=1, \frac12$ the operator ${\mathcal S}{\mathcal L}$ is the radial part of the 2nd order Laplace operator for the symmetric superspaces corresponding to pairs $(GL(V)\times GL(V), GL(V))$ and $(GL(V), OSp(V))$, respectively. We will show that for the generic $m$ and $n$ the superanalogs of the Jack polynomials constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of ${\mathcal S}{\mathcal L}$; for $k=1, \frac12$ they coinside with the spherical functions corresponding to the above mentioned symmetric superspaces. We also study the inner product induced by Berezin's integral on these superspaces

Elucidating the structural composition of a Fe-N-C catalyst by nuclear and electron resonance techniques

Fe–N–C catalysts are very promising materials for fuel cells and metal–air batteries. This work gives fundamental insights into the structural composition of an Fe–N–C catalyst and highlights the importance of an in‐depth characterization. By nuclear‐ and electron‐resonance techniques, we are able to show that even after mild pyrolysis and acid leaching, the catalyst contains considerable fractions of α‐iron and, surprisingly, iron oxide. Our work makes it questionable to what extent FeN4 sites can be present in Fe–N–C catalysts prepared by pyrolysis at 900 °C and above. The simulation of the iron partial density of phonon states enables the identification of three FeN4 species in our catalyst, one of them comprising a sixfold coordination with end‐on bonded oxygen as one of the axial ligands

Functional Tetrahedron Equation

We describe a scheme of constructing classical integrable models in 2+1-dimensional discrete space-time, based on the functional tetrahedron equation - equation that makes manifest the symmetries of a model in local form. We construct a very general "block-matrix model" together with its algebro-geometric solutions, study its various particular cases, and also present a remarkably simple scheme of quantization for one of those cases.Comment: LaTeX, 16 page