3 research outputs found
Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds I
No full textAbstractIn this paper and its sequel we give results and methods for evaluating the Nielsen type numbers NΦn(f) and NPn(f) for periodic points of self maps f of nilmanifolds and solvmanifolds (which includes the Klein bottle). Using fibre space techniques we relate these numbers to the various ordinary Nielsen numbers N(fm) of the mth iterate (for m¦n), and thus capitalize on known techniques for our calculations. In this first paper we deal with the specific situation where for each m¦n either N(fm) = R(fm) (the Reidemeister number of fm) or N(fm) = 0. In fact this is the vast majority of cases. In the case where N(fn) = R(fn) we derive the simple formula NΦm(f) = N(fm) for all m¦n
Addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps
No full textAbstractIn this paper we generalize well known product formulae for the Nielsen number of a fibre preserving map, to give addition formulae for such maps. We give necessary and sufficient conditions for when a naïve addition formula expressing the Nielsen number of the fibre map as a simple sum of Nielsen numbers on the fibres is valid. In the second part of the paper we extend to the nonorientable situation the definition and properties of a Nielsen type number of a fibre preserving map introduced by the first author
Synthesis and Characterization of a Series of Ag(I) and Cu(I) NHC Bridged Triangular
No full textAbstractA series of picolyl-substituted NHC-bridged triangular complexes of Ag(I) and Cu(I)were prepared upon reaction of the corresponding ligand precursors, [HSim(CH2py)2]BF4 (1),[HSim(CH2pyCH3)2]BF4 (2), [HSim(CH2pyCl)2]PF6 (3), [HSim(CH2pyBr)2]PF6 (4),[HSim(CH2isoquin)2]PF6 (5), [Him(CH2py-3,4-(OMe)2)2]BF4 (6), [Him(CH2py-3,5-Me2-4-OMe)2]BF4 (7), [HCl2benzim(CH2py)2]BF4 (8), [Hcyclohexim(CH2py)2]PF6 (9),[Hbenzim(CH2py)2]BF4 (10), [H(CH3)2benzim(CH2py)2]BF4 (11), [Hbenzim(CH2py-3,4-(OMe)2)2]BF4 (12), [Hbenzim(CH2py-3,5-Me2-4-OMe)2]BF4 (13), [H(CH3)2benzim(CH2py-3,4-(OMe)2)2]BF4 (14), [H(CH3)2benzim(CH2py-3,5-Me2-4-OMe)2]BF4 (15),[H(OCH3)benzim(CH2py)2]PF6 (16), with Ag2O and Cu2O in acetonitrile, respectively.Complexes [Ag3(Sim(CH2py)2)3](BF4)3 (26), [Ag3(Sim(CH2pyCl)2)3](PF6)3 (27),[Ag3(Sim(CH2pyBr)2)3](PF6)3 (28), [Ag3(Sim(CH2isoquin)2)3](PF6)3 (29), [Ag3((R,R)-cyclohexylim(CH2py)2)3](PF6)3 (30), [Ag3(im(CH2py-3,4-(OMe)2)2)3](BF4)3 (31),[Ag3(im(CH2py-3,5-Me2-4-OMe)2)3](BF4)3 (32), [Ag3(5,6-Cl2-benzim(CH2py)2)3](BF4)3 (33),[Ag3(5,6-Me2-benzim(CH2py)2)3](BF4)3 (34), [Ag3(benzim(CH2py-3,4-(OMe)2)2)3](BF4)3 (35),[Ag3(benzim(CH2py-3,5-Me2-4-OMe)2)3](PF6)3 (36), [Ag3(5,6-Me2-benzim(CH2py-3,4-(OMe)2)2)3](BF4)3 (37), [Ag3(5,6-Me2-benzim(CH2py-3,5-Me2-4-OMe)2)3](BF4)3 (38), [Ag3(5-OMe-benzim(CH2py)2)3](PF6)3 (39), [Cu3(im(CH2py-3,4-(OMe)2)2)3](BF4)3 (44),[Cu3(im(CH2py-3,5-Me2-4-OMe)2)3](BF4)3 (45), [Cu3(5,6-Cl2-benzim(CH2py)2)3](BF4)3 (46),[Cu3(benzim(CH2py)2)3](BF4)3 (48), [Cu3(5,6-Me2-benzim(CH2py)2)3](BF4)3 (49),[Cu3(benzim(CH2py-3,4-(OMe)2)2)3](BF4)3 (50), [Cu3(benzim(CH2py-3,5-Me2-4-OMe)2)3](PF6)3(51), [Cu3(5,6-Me2-benzim(CH2py-3,4-(OMe)2)2)3](BF4)3 (52), [Cu3(5,6-Me2-benzim(CH2py-3,5-Me2-4-OMe)2)3](BF4)3 (53), and [Cu3(5-OMe-benzim(CH2py)2)3](PF6)3 (54) were easily preparedby this method. Complexes [Cu3(Sim(CH2py)2)3](BF4)3 (40), [Cu3(Sim(CH2pyCl)2)3](PF6)3 (41),[Cu3(Sim(CH2pyBr)2)3](PF6)3 (42), [Cu3(Sim(CH2isoquin)2)3](PF6)3 (43), and [Cu3((R,R)-iicyclohexylim(CH2py)2)3](PF6)3 (47), containing a saturated imidazolidine backbone, wereprepared by a carbene transfer reaction with CuI and the corresponding silver triangle. Ligandprecursor [HSim(CH2pyCH3)2]BF4 (2), did not react with Ag2O.A series of picolyl-substituted NHC-bridged rhomboidal complexes of Ag(I) wereprepared by treatment of the corresponding ligand precursors [py(CH2HimCH2py-3,4-(OMe)2](PF6)2 (18), ([py(CH2HimCH2py-3,5-Me2-4-OMe)2](PF6)2, (19),[py(CH2H(CH3)2benzimCH2py)2](PF6)2 (20), and ([py(CH2HbenzimCH2py)2](PF6)2 (21) withAg2O in acetonitrile to yield Ag4 complexes [Ag4(py(CH2HimCH2py-3,4-(OMe)2)2](PF6)4 (56),[Ag4(py(CH2HimCH2py-3,5-Me2-4-OMe)2)2](PF6)4 (57),[Ag4(py(CH2H(CH3)2benzimCH2py)2)2](PF6)4 (58), and [Ag4(py(CH2HbenzimCH2py)2)2](PF6)4(59). The mixed-metal Cu2Ag2 rhomboidal analogues [Cu2Ag2(py(CH2HimCH2py-3,4-(OMe)2)2](PF6)4 (61), [Cu2Ag2(py(CH2HimCH2py-3,5-Me2-4-OMe)2)2](PF6)4 (62),[Cu2Ag2(py(CH2HbenzimCH2py)2)2](PF6)4 (63), and [Cu2Ag2(py(CH2HbenzimCH2py)2)2](PF6)4(64) were prepared via treatment of the corresponding Ag4 complex with two equivalents of CuIin acetonitrile. The Cu2Ag2 complex [Cu2Ag2(py(CH2HimCH2py)2)2](PF6)4 (60) was prepared bythis method utilizing the previously reported analogous Ag4 complex,[Ag4(py(CH2HimCH2py)2)2](PF6)4 (55). Ligand precursor [py(CH2Himpy)2](PF6)2 (22), did notreact with Ag2O or Cu2O.All complexes were fully characterized by NMR, UV-vis, luminescence spectroscopiesand high-resolution mass spectrometry. Additionally, complexes 26, 28, 30 - 35, 40 - 42, 44 -54, 60, 61, 63, and 64 were characterized by single-crystal X-ray diffraction. The triangularmetal complexes contain a nearly equilateral triangle held together by three bridging NHCligands. In the Ag3 species, 26, 28, 30 - 35 the Ag - Ag distances range from 2.7164(3) -2.7832(3) Å while in Cu3 complexes 40 - 42, 44 - 54 the Cu - Cu distances range from2.4658(13) to 2.5634(9) Å. The rhomboidal metal complexes 60, 61, 63 and 64 contain aiiirhomboid metal center wrapped by two larger bridging NHC ligands. The Ag - Ag and Ag - Cudistances range from 2.7837(18) to 2.8490(7) Å and 2.6690(9) to 2.7830(9) Å, respectively.The Cu3-containing species are highly blue photoluminescent in both solution and in thesolid state. Upon UV excitation in CH3CN, complexes 40 - 54 emit at 429, 443, 441, 474, 438,429, 452, 454, 445, 453, 445, 449, 452, 455, and 459 nm, respectively while in the solid state thisemission shifts to 440, 430, 441, 475, 429, 432, 444, 450, 450, 452, 439, 479, 444, 460, and 453nm, respectively. The analogous Ag3-containing complexes are weakly photoemissive in thesolid state.In CH3CN solution, the Ag4-containing species 55 - 59 are strong blue emitters withemission maxima at 420, 395, 304, 367, and 383 nm, respectively, and are only weaklyphotoemissive in the solid state. In CH3CN, the analogous mixed-metal Ag2Cu2 species 60 - 64are highly orange photoluminescent with emission maxima at 623, 508, 598, 607, and 580 nm,respectively, while in the solid state these bands move to 671, 596, 603, 582, 584, respectively
