Minimal Hilbert series for quadratic algebras and the Anick conjecture

We study the question on whether the famous Golod–Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper ‘Generic algebras and CW-complexes’, Princeton Univ. Press, where he proved that the estimate is attained for the number of quadratic relations $d\leq n^2/4$
and $d\geq n^2/2$, and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to $n(n-1)/2$ was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We announce here the result that over any infinite field, the Anick conjecture holds for $d \geq 4(n2+n)/9$ and an arbitrary number of generators. We also discuss the result that confirms the Vershik conjecture over any field of characteristic 0, and a series of related
asymptotic results.

ROOT PROBLEM FOR CONVENIENT MAPS

In this paper we study when the minimal number of roots of the so-called convenient maps horn two-dimensional CW complexes into closed surfaces is zero We present several necessary and sufficient conditions for such a map to be root free Among these conditions we have the existence of specific fittings for the homomorphism induced by the map on the fundamental groups, existence of the so-called mutation of a specific homomorphism also induced by the map, and existence of particular solutions of specific systems of equations on free groups over specific subgroups

Spherical space forms - Homotopy self-equivalences and homotopy types, the case of the groups Z/a x (Z/b x TL(2)(F(p)))

Let G = Z/a x(mu) (Z/b x TL(2)(F(p))) and X(n) be an n-dimensional CW-complex with the homotopy type of the n-sphere. We determine the automorphism group Aut(G) and then compute the number of distinct homotopy types of spherical space forms with respect to free and cellular G-actions on all CW-complexes X(2dn - 1), where 2d is a period of G. Next, the group E(X(2dn - 1)/alpha) of homotopy self-equivalences of spherical space forms X(2dn - 1)/alpha, associated with such G-actions alpha on X(2dn - 1) are studied. Similar results for the rest of finite periodic groups have been obtained recently and they are described in the introduction. (C) 2009 Elsevier B.V. All rights reserved.

Algumas considerações sobre Espaços de Eilenberg

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Modified polyoxometalates: Hydrothermal syntheses and crystal structures of three novel reduced and capped Keggin derivatives decorated by transition metal complexes

Three novel polyoxometalate derivatives decorated by transition metal complexes have been hydrothermally synthesized. Compound 1 consists of [(PMo6Mo2V8O44)-Mo-VI-V-V-O-IV{CO (2,2'-bipy)(2)(H2O)}(4)](3+) polyoxocations and [(PMo4Mo4V8O44)-Mo-IV-V-V-O-IV{Co(2,2'-bipy)(2)(H2O)}(2)](3-) polyoxoanions, which are both built on mixed-metal tetracapped [PMo8V8O44] subunits covalently bonded to four or two {Co(2,2'-bpy)(2)(H2O)}(2+), clusters via terminal oxo groups of the capping V atoms. Compound 2 is built on [(PMo8V6O42)-V-VI-O-IV{Cu-I(phen)}(2)](5-) clusters constructed from mixed-metal bicapped [(PMo8V6O42)-V-VI-O-IV](7-) subunits covalently bonded to two {Cu(phen)}(+) fragments in the similar way to 1. The structure of 3 is composed of [(PMo9Mo3O40)-Mo-VI-O-V](6-) units capped by two divalent Ni atoms via four bridging oxo groups.

A two-dimensional molybdenum (V) phosphate with covalently bonded transition metal coordination complexes: hydrothermal synthesis and structure characterization of Na-2[{Mn(phen)(2)(H2O)}{Mn(phen)(2)}(3){Mn (Mo12O24)-O-V (HPO4)(6)(PO4)(2)(OH)(6)}] center dot 4H(2)O (phen=1,10-phenanthroline)

An organic-inorganic hybrid molybdenum phosphate, Na-2[{Mn(phen)(2)(H2O)} {Mn(phen)(2)}(3){(MnMo12O24)-O-v (HPO4)(6)(PO4)(2) (OH)(6)}] . 4H(2)O (phen=1,10-phenanthroline), involving molybdenum present in V oxidation state and covalently bonded transition metal coordination complexes, has been hydrothermally synthesized and structurally characterized by single-crystal X-ray diffraction. Deep brown-red crystals are formed in the triclinic system, space group P (1) over bar, a=16.581(l)Angstrom, b=18.354(1)Angstrom, c=24.485(2)Angstrom, alpha=80.589(l)degrees, beta=71.279(1)degrees, gamma=67.084(1)degrees, V=6493.8(8)Angstrom(3), Z=2, lambda(MoKalpha)=0.71073Angstrom (R(F)=0.0686 for 29,053 reflections). Data were collected on a Bruker Smart Apex CCD diffractometer at 293 K in the range of 1.76 < theta < 28.06degrees using omega-2theta scans technique. The structure of the title compound may be considered to be based on {Mo6O12(HPO4)(3)(PO4)(OH)(3)} units bonded together with {Mn(phen)(2)} subunits into a two-dimensional network. Two types of tunnels are observed in the solid of the title compound.

The synthesis and crystal structure of vanadium complexes: [(VO2)-O-IV(phen)(2)]center dot 6H(2)O and [2,2 '-(bipy)(2)(VO2)-O-V](H2BO3)center dot 3H(2)O

The organic-inorganic hybrid materials vanadium oxide [(VO2)-O-IV(phen)(2)](.)6H(2)O (1) and [(2,2'-bipy)(2)(VO2)-O-V](H2BO3)(.)3H(2)O (2) have been conventional and hydrothermal synthesized and characterized by single crystal X-ray diffraction, elemental analyses, respectively. Although the method and the ligand had been used in the syntheses of the compounds (1) and (2) are different, they almost possess similar structure. They all exhibit the distorted octahedral [VO2N4] unit with organonitrogen donors of the phen and 2,2'-bipy ligands, respectively, which coordinated directly to the vanadium oxide framework. And they are both non-mixed-valence complexes. But the compound (1) is isolated, and the compound (2) consists of a cation of [(2,2'-bipy)(2)(VO2)-O-V](+) and an anion of (H2BO3)(-). So the valence of vanadium of (1) and (2) are tetravalence and pentavalence, respectively. Meanwhile it is noteworthy that pi-pi stacking interaction between adjacent phen and 2,2'-bipy groups in compounds I and 2 also play a significant role in stabilization of the structure. Thus, the structure Of [(VO2)-O-IV(phen)(2)](.)6H(2)O and [(2,2'-bipy)(2)(VO2)-O-V](H2BO3)(.)3H(2)O are both further extended into interesting three-dimensional supramolecular.

Hydrothermal syntheses and crystal structures of bimetallic cluster complexes [{Cd(phen)(2)}(2)V4O12]center dot 5H(2)O and [Ni(phen)(3)](2)[V4O12]center dot 17.5H(2)O

The hydrothermal reactions of vanadium oxide starting materials with divalent transition metal cations in the presence of nitrogen donor chelating ligands yield the bimetallic cluster complexes with the formulae [{Cd(phen(2))(2)V4O12].5H(2)O (1) and [Ni(phen)(3)](2)[V4O12] . 17.5H(2)O (2). Crystal data: C48H52Cd2N8O22V4 (1), triclinic. P (1) over bar, a = 10.3366(10), b = 11.320(3), c = 13.268(3) Angstrom, alpha = 103.888(17)degrees, beta = 92.256(15)degrees, gamma = 107.444(14)degrees, Z = 1; C72H131N12Ni2O29.5V4 (2), triclinic. P (1) over bar, a = 12.305(3), b = 13.172(6), c = 15.133(4), alpha = 79.05(3)degrees, beta = 76.09(2)degrees, gamma = 74.66(3)degrees, Z = 1. Data were collected on a Siemens P4 four-circle diffractometer at 293 K in the range 1.59degrees < theta < 26.02degrees and 2.01degrees < 0 < 25.01degrees using the omega-scan technique, respectively. The structure of 1 consists of a [V4O12](4-) cluster covalently attached to two {Cd(phen)(2)}(2+) fragments, in which the [V4O12](4-) cluster adopts a chair-like configuration. In the structure of 2, the [V4O12](4-) cluster is isolated. And the complex formed a layer structure via hydrogen bonds between the V4O12](4-) unit and crystallization water molecules.

Hydrothermal synthesis and crystal structure of [{Cu(2,2 '-bpy)(2)}(2)Mo8O26]: a beta-octamolybdate cluster covalently bonded to two {Cu(2,2 '-bpy)(2)}(2+) coordination complexes via bridging oxo groups

An organic-inorganic hybrid solid, (Cu(2,2'-bpy)(2))(2)Mo8O26, has been hydrothermally synthesized and structurally characterized by single-crystal X-ray diffraction. Dark green crystals crystallize in the orthorhombic system, space group Pna21, a = 24.164(5), b = 18.281(4), c = 11.877(2) Angstrom, alpha = 90 degrees, beta = 90 degrees, gamma = 90 degrees, V= 5247(2) Angstrom (3), Z = 4, lambda (MoK alpha) = 0.71073 Angstrom (R(F) = 0.0331 for 5353 reflections). Data were collected on a Siemens P4 four-circle diffractometer at 293 K in the range 1.69 degrees < theta < 25.04 degrees using the omega -scan technique. The structure was solved by the direct method and refined by full-matrix least squares on F-2 using SHELXL-93. The structure of this compound consists of discrete (Cu(2,2'-bpy)(2))(2)Mo8O26 clusters, constructed from beta -octamolybdate subunits ((Mo8O26)(4-)) covalently bonded to two (Cu(2,2'-bpy)(2))(2+) coordination complexes via bridging oxo groups that connect two adjacent molybdenum sites. (C) 2001 Academic Press.

Electrochemistry of Heteroleptic Tris(phthalocyaninato) Rare Earth(III) Complexes

The electrochemical characteristics of a series of heteroleptic tris(phthalocyaninato) complexes with identical rare earths or mixed rare earths (Pc)M(OOPc)M(OOPc) [M = Eu...Lu, Y; H2Pc = unsubstituted phthalocyanine, H2(OOPc) = 3,4,12,13,21,22,30,31-octakis(octyloxy)phthalocyanine] and (Pc)Eu(OOPc)Er(OOPc) have been recorded and studied comparatively by cyclic voltammetry (CV) and differential pulse voltammetry (DPV) in CH2Cl2 containing 0.1 M tetrabutylammonium perchlorate (TBAP). Up to five quasi-reversible one-electron oxidations and four one-electron reductions have been revealed. The half-wave potentials of the first, second and fifth oxidations depend on the size of the metal center, but the fifth changes in the opposite direction to that of the first two. Moreover, the difference in redox potentials of the first oxidation and first reduction for (Pc)M(OOPc)M(OOPc), 0.85−0.98 V, also decreases linearly along with decreasing rare earth ion radius, clearly showing the rare earth ion size effect and indicating enhanced π−π interactions in the triple-deckers connected by smaller lanthanides. This order follows the red-shift seen in the lowest energy band of triple-decker compounds. The electronic differences between the lanthanides and yttrium are more apparent for triple-decker sandwich complexes than for the analogous double-deckers. By comparing triple-decker, double-decker and mononuclear [ZnII] complexes containing the OOPc ligand, the HOMO−LUMO gap has been shown to contract approximately linearly with the number of stacked phthalocyanine ligands.

Sandwich complexes of naphthalocyanine with the rare earth metals

Over the past two decades and in particular the past five years, numerous sandwich-type rare earth complexes containing naphthalocyanine ligands have been synthesized. The more extended delocalized π-electron system of naphthalocyanine in comparison with phthalocyanine generates unique physical, spectroscopic, electrochemical and photoelectrochemical properties which have aroused significant research interest in these compounds. This review summarizes recent progress in research on this important class of molecular materials and overviews the current status of the field.

Comparative Electrochemical Study of Unsubstituted and Substituted Bis(phthalocyaninato) Rare Earth(III) Complexes

The electrochemistry of homoleptic substituted phthalocyaninato rare earth double-decker complexes M(TBPc)2 and M(OOPc)2 [M = Y, La...Lu except Pm; H2TBPc = 3(4),12(13),21(22),30(31)-tetra-tert-butylphthalocyanine, H2OOPc = 3,4,12,13,21,22,30,31-octakis(octyloxy)phthalocyanine] has been comparatively studied by cyclic voltammetry (CV) and differential pulse voltammetry (DPV) in CH2Cl2 containing 0.1 M tetra-n-butylammonium perchlorate (TBAP). Two quasi-reversible one-electron oxidations and three or four quasi-reversible one-electron reductions have been revealed for these neutral double-deckers of two series of substituted complexes, respectively. For comparison, unsubstituted bis(phthalocyaninato) rare earth analogues M(Pc)2 (M = Y, La...Lu except Pm; H2Pc = phthalocyanine) have also been electrochemically investigated. Two quasi-reversible one-electron oxidations and up to five quasi-reversible one-electron reductions have been revealed for these neutral double-decker compounds. The three bis(phthalocyaninato)cerium compounds display one cerium-centered redox wave between the first ligand-based oxidation and reduction. The half-wave potentials of the first and second oxidations and first reduction for double-deckers of the tervalent rare earths depend on the size of the metal center. The difference between the redox potentials of the second and third reductions for MIII(Pc)2, which represents the potential difference between the first oxidation and first reduction of [MIII(Pc)2]−, lies in the range 1.08−1.37 V and also gradually diminishes along with the lanthanide contraction, indicating enhanced π−π interactions in the double-deckers connected by the smaller, lanthanides. This corresponds well with the red-shift of the lowest energy band observed in the electronic absorption spectra of reduced double-decker [MIII(Pc′)2]− (Pc′ = Pc, TBPc, OOPc).

Vibrational spectroscopy of phthalocyanine and naphthalocyanine in sandwich-type (na)phthalocyaninato and porphyrinato rare earth complexes : (Part 10) The infrared and Raman characteristics of phthalocyanine in heteroleptic bis(phthalocyaninato) rare earth complexes with decreased molecular symmetry

The infrared (IR) spectroscopic data for a series of eleven heteroleptic bis(phthalocyaninato) rare earth complexes MIII(Pc)[Pc(α-OC5H11)4] (M = Sm–Lu, Y) [H2Pc = unsubstituted phthalocyanine, H2Pc(α-OC5H11)4 = 1,8,15,22-tetrakis(3-pentyloxy)phthalocyanine] have been collected with 2 cm−1 resolution. Raman spectroscopic properties in the range of 500–1800 cm−1 for these double-decker molecules have also been comparatively studied using laser excitation sources emitting at 632.8 and 785 nm. Both the IR and Raman spectra for M(Pc)[Pc(α-OC5H11)4] are more complicated than those of homoleptic bis(phthalocyaninato) rare earth analogues due to the decreased molecular symmetry of these double-decker compounds, namely C4. For this series, the IR Pc√− marker band appears as an intense absorption at 1309–1317 cm−1, attributed to the pyrrole stretching. With laser excitation at 632.8 nm, Raman vibrations derived from isoindole ring and aza stretchings in the range of 1300–1600 cm−1 are selectively intensified. In contrast, when excited with laser radiation of 785 nm, the ring radial vibrations of isoindole moieties and dihedral plane deformations between 500 and 1000 cm−1 for M(Pc)[Pc(α-OC5H11)4] intensify to become the strongest scatterings. Both techniques reveal that the frequencies of pyrrole stretching, isoindole breathing, isoindole stretchings, aza stretchings and coupling of pyrrole and aza stretchings depend on the rare earth ionic size, shifting to higher energy along with the lanthanide contraction due to the increased ring-ring interaction across the series. The assignments of the vibrational bands for these compounds have been made and discussed in relation to other unsubstituted and substituted bis(phthalocyaninato) rare earth analogues, such as M(Pc)2 and M(OOPc)2 [H2OOPc = 2,3,9,10,16,17,23,24-octakis(octyloxy)phthalocyanine].