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2.
Let K be ? or an imaginary quadratic number field, and q ∈ K an integer with ¦q¦ > 1. We give a quantitative version of Σn≥1 an/(qn ? 1) ? K for non-zero periodic sequences (an) 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 相似文献
3.
Rojo I Teixidor F Viñas C Kivekäs R Sillanpää R 《Chemistry (Weinheim an der Bergstrasse, Germany)》2004,10(21):5376-5385
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. 相似文献
4.
Rojo I Teixidor F Kivekäs R Sillanpää R Viñas C 《Journal of the American Chemical Society》2003,125(48):14720-14721
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. 相似文献
5.
Saukkoriipi J Sillanpää A Laasonen K 《Physical chemistry chemical physics : PCCP》2005,7(22):3785-3792
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. 相似文献
6.
Grüner B Stíbr B Kivekäs R Sillanpää R Stopka P Teixidor F Viñas C 《Chemistry (Weinheim an der Bergstrasse, Germany)》2003,9(24):6115-6121
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. 相似文献
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8.
Tutusaus O Viñas C Kivekäs R Sillanpää R Teixidor F 《Chemical communications (Cambridge, England)》2003,(19):2458-2459
Designed ligands have been synthesised to produce the first arene-like metallacarborane. For arene-like coordination the number of electronegative elements on the coordinating site must be kept to a minimum. Choosing ligands with bulky substituents on the heteroatom allows easy rearrangement and arene-like coordination. This is more hampered the higher the number of hetereoatoms to be re-located. 相似文献
9.
10.
Starting from ergosterol two novel 9,11-secosterols with modified side chains (1a) and (1c) were synthesized via eight main transformations. 相似文献