Explicit Free Parameterization of the Modified Tetrahedron Equation

The Modified Tetrahedron Equation (MTE) with affine Weyl quantum variables at N-th root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem. We show that the solutions can be parameterized in terms of eight free parameters and sixteen discrete phase choices, thus providing a broad starting point for the construction of 3-dimensional integrable lattice models. The Fermat curve points parameterizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters. An explicit formula for the density factor of the MTE is derived. For the example N=2 we write the MTE in full detail. We also discuss a solution of the MTE in terms of bosonic continuum functions.Comment: 28 pages, 3 figure

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

Generalized Calogero-Moser systems from rational Cherednik algebras

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify. In the case of classical Coxeter groups we also obtain generalized Calogero-Moser systems with added quadratic potential.Comment: 36 pages; the main change is an improvement of section 7 so that it now deals with an arbitrary complex reflection group; Selecta Math, 201

β-Polymorph of phenazepam: a powder study

The title compound [systematic name: 7-bromo-5-(2-chloro­phen­yl)-1H-1,4-benzodiazepin-2(3H)-one] (β-polymorph), C15H10BrClN2O, has been obtained via cryomodification of the known α-polymorph of phenazepam [Karapetyan et al. (1979 ▶). Bioorg. Khim. 5, 1684–1690]. In both polymorphs, the mol­ecules, which differ only in the dihedral angles between the aromatic rings [75.4 (2)° and 86.2 (3)° in the α- and β-polymorphs, respectively], are linked into centrosymmetric dimers via N—H⋯O hydrogen bonds. In the crystal structure of the β-polymorph, weak inter­molecular C—H⋯O hydrogen bonds further link these dimers into layers parallel to bc plane

Amplitude-Frequency Characteristic of a Neural Control Based DC Drive

The paper interprets characteristics of a neural-control-based DC servodrive in terms of the classical theory of automatic control. It also touches on the problem of choosing training patterns to synthesize a nonlinear PID-controller with a desired amplitude-frequency characteristic and analyses the efficiency of using for this purpose input signals in form of a step function and a harmonic one. Synthesis of the neurocontroller has been performed within the framework of a three-layer perceptron. To train it, a genetic algorithm has been developed