Linear Independence of q-Analogues of Certain Classical Constants

Let K be ? or an imaginary quadratic number field, and q ∈ K an integer with ¦q¦ > 1. We give a quantitative version of Σ_n≥1 a_n/(qⁿ ? 1) ? K for non-zero periodic sequences (a_n) in K of period length ≤ 2. As a corollary, we get a quantitative version of the linear independence over K of 1, the q-harmonic series, and a q-analogue of log 2. A similar result on 1, the q-harmonic series, and a q-analogue of ζ(2) is also proved. Mathematics Subject Classification (2000): 11J72, 11J82     

Whirling frequencies and amplitudes due to deviations of generator shape

Earlier measurements in large synchronous generators indicate the existence of backward whirling motion, and also relatively large deviations of shape in both the rotor and the stator. These non-symmetric geometries produce an attraction force between the rotor and the stator, called unbalanced magnetic pull (UMP). The target of this paper is to analyse the whirling frequencies and amplitude of the response for large synchronous generators with a high number of poles, due to deviations of shape in the rotor and stator. A mathematical model is developed to describe the shapes of the rotor and stator, and the corresponding UMP is obtained by using the law of energy conservation. The UMP is analysed due to different deviations of shape. The result gives the average angular frequency and the magnitude of the UMP for certain deviations of shape. From this result, the whirling frequency and the amplitude of the corresponding response can be approximated. Simulations of the response of a Jeffcott rotor model show good agreement with the theoretical results of the UMP for some generator geometries. The conclusion is that different whirling frequencies, both backward and forward whirling, can occur in these machines due to deviations in shape of the generator. Therefore, the shape of the generator can excite resonance vibrations on several other frequencies than the rotation frequency. During maintenance of hydropower generators the shapes of the rotor and stator are frequently measured. The results from this paper can be used to evaluate such measurements and to explain the existence of complicated whirling motion.     

The crystal structure of Hf3As

Hf₃As has a monoclinic unit cell of dimensions a = 15.3898(14) Å, b = 5.3795(5) Å, c = 15.330(14) Å, β = 90.291(6)°. A structure proposal based on space group $\text{C2}\text{c}$ (No. 15) has been refined by the least-squares method using a Rietveld-type fullprofile analysis of Guinier-Hägg X-ray powder film intensity data. The Hf₃As structure is an intermediate between the Fe₃P and the Ti₃P types. The atomic coordination follows rules formulated earlier for representatives of the Fe₃PTi₃PV₃S family of structures.     

Synthesis and coordinating ability of an anionic cobaltabisdicarbollide ligand geometrically analogous to BINAP

The anionic chelating ligand [1,1'-(PPh2)2-3,3'-Co(1,2-C2B9H10)2]- has been synthesized from [3,3'-Co(1,2-C2B9H11)2]- in very good yield in a one-pot process with an easy work-up procedure. The coordinating ability of this ligand has been studied with Group 11 metal ions (Ag, Au) and with transition-metal ions (Pd, Rh). The two dicarbollide halves of the [1,1'-(PPh2)2-3,3'-Co(1,2-C2B9H10)2]- ligand can swing about one axis in a manner analogous to the constituent parts of BINAP and ferrocenyl phosphine derivatives. All these ligands function as hinges, with the most important property in relation to the coordination requirements of the metal being the PP distance. [1,1'-(PPh2)2-3,3'-Co(1,2-C2B9H10)2]-, BINAP, ferrocenyl phosphine derivatives, and other hinge ligands present a range of different PP separations, and consequently different coordination spheres and dispositions around metal cations. To account for these differences, the equation Dphi2 = D02 + 4 R2cos2(90-phi/2) has been developed. It relates the PP distance (Dphi) in a complex with the minimum PP distance (D0) that is characteristic of the hinge-type ligand.     

Formation of bridging alkene and conjugated dialkenes exclusively generated from alkynes on the [3,3'-Co(1,2-C2B9H11)2]- platform. The unique hydroboration role of [3,3'-Co(1,2-C2B9H11)2]-

The unprecedented metal-mediated transformation of an alkyne into a B,B' bridging alkene is reported. Also, the unprecedented synthesis of a conjugated dialkene derivative of [3,3'-Co(1,2-C2B9H11)2]- generated only from an alkyne, contrary to the usual case where an alkyne and an alkene are needed, is described. This has been possible through the singular capacity of a B-H to produce hydroboration.     

Chemical fourling on the unit cell level as a structure-building operation in the solid state

Chemical fourling on the unit-cell level is introduced as a structure-building operation in the solid state. A chemical fourling structure is achieved when two twin planes are perpendicular to each other. Each fourling unit has infinite extension in only one direction and can be related to a well-known structure type or packing of atoms. The structures of the tetragonal tungsten bronzes M_xWO₃ and Pd(NH₃)₄Cl · H₂O are presented as examples of chemical fourlings of cubic close packing. The structures of SeO₂, Mn₂Hg₅, and Zr₂F₇O are shown to be a sequence of chemical fourlings of primitive cubic packing. Chemical fourling units of hexagonal close packing are found in the structures of tetragonal Ti₃Sb, α-V₃S and Sb₆O₇(SO₄)₂. The structures of CuAl₂, SeTl, NbTe₄, Ti₃Sb, and MnU₆ all have the same chemical fourling unit. Cr₂₃C₆ is described as a bounded chemical fourling of cubic close packing.     

Computational studies of the cationic aluminium(chloro) hydroxides by quantum chemical ab initio methods

Cationic aluminium(chloro) hydroxide complexes with two to four aluminium atoms were studied using quantum chemical methods. Complexes were studied in both gas and liquid phase. The liquid environment was modeled by using a conductor-like screening model (COSMO). COSMO calculations were carried out as a single point calculation at the optimized gas phase structures. Water (epsilon = 78.54) was used as the solvent. The minimum energy structures obtained from the gas phase studies were mostly compact cyclic structures. Aluminium preferred to be five-coordinated in oxygen rich clusters. Core oxygen preferred three-fold coordination but in the largest clusters the four-coordinated oxygen was observed. Water reacted dissociatively with hydrogen poor clusters. The COSMO calculations showed that the optimal structures of cationic aluminium(chloro) hydroxides tend to be more open in the liquid than in the gas phase.     

Diferratricarbaboranes of the subcloso-[(eta5-C5H5)2Fe2C3B8H11] type, the first representatives of the 13-vertex dimetallatricarbaborane series

Treatment of the [2-Cp-9-tBuNH-closo-2,1,7,9-FeC(3)B(8)H(10)] (1) ferratricarbollide (Cp = eta(5)-C(5)H(5) (-)) with Na(+) C(10)H(8) (-) in 1,2-dimethoxyethane (DME) at room temperature produced an air-sensitive transient anion with a tentatively identified nido-[tBuNH-CpFeC(3)B(8)H(10)](2-) constitution. In-situ reaction of this low-stability ion with [CpFe(CO)(2)I] or [CpFe(CO)(2)](2) generated three violet diferratricarbaboranes identified as paramagnetic subcloso complexes [4,5-Cp(2-)-4,5,1,6,7-Fe(2)C(3)B(8)H(11)] (2; yield 2 %), [4,5-Cp(2-)-4,5,1,7,12-Fe(2)C(3)B(8)H(11)] (3; yield 2 %), and [7-tBuNH-4,5-Cp(2-)-4,5,1,7,12-Fe(2)C(3)B(8)H(10)] (4; yield 14 %). These first representatives of the 13-vertex dimetallatricarbaborane family were characterized by EPR and IR spectroscopy, and mass spectrometry, and their structures were determined by X-ray diffraction analysis.