Dinuclear nickel complexes with a Ni(2)O(2) core: a structural and magnetic study

The acetylacetonate complexes [Ni(2)L(1)(acac)(MeOH)] x H(2)O, 1 x H(2)O and [Ni(2)L(3)(acac)(MeOH)] x 1.5H(2)O, 2 x 1.5H(2)O (H(3)L(1) = (2-(2-hydroxyphenyl)-1,3-bis[4-(2-hydroxyphenyl)-3-azabut-3-enyl]-1,3-imidazolidine and H(3)L(3) = (2-(5-bromo-2-hydroxyphenyl)-1,3-bis[4-(5-bromo-2-hydroxyphenyl)-3-azabut-3-enyl]-1,3-imidazolidine) were prepared and fully characterised. Their crystal structures show that they are dinuclear complexes, extended into chains by hydrogen bond interactions. These compounds were used as starting materials for the isolation of the corresponding [Ni(2)HL(x)(o-O(2)CC(6)H(4)CO(2))(H(2)O)] x n MeOH and [Ni(2)HL(x)(O(2)CCH(2)CO(2))(H(2)O)]x nH(2)O dicarboxylate complexes (x = 1, 3; n = 1-3). The crystal structures of [Ni(2)HL(1)(o-O(2)CC(6)H(4)CO(2))(H(2)O)] x MeOH, 3 x MeOH, [Ni(2)HL(3)(o-O(2)CC(6)H(4)CO(2))(H(2)O)] x 3 MeOH, 4 x 3 MeOH and [Ni(2)HL(1)(O(2)CCH(2)CO(2))(H(2)O)] x 2.5H(2)O x 0.25 MeOH x MeCN, 5 x 2.5H(2)O x 0.25 MeOH x MeCN, were solved. Complexes 3-5 show dinuclear [Ni(2)HL(x)(dicarboxylate)(H(2)O)] units, expanded through hydrogen bonds that involve carboxylate and water ligands, as well as solvate molecules. The variable temperature magnetic susceptibilities of all the complexes show an intramolecular ferromagnetic coupling between the Ni(II) ions, which is attempted to be rationalized by comparison with previous results and in the light of molecular orbital treatment. Magnetisation measurements are in accord with a S = 2 ground state in all cases.     

Self-assembly of a tetranuclear Ni4 cluster with an S = 4 ground state: the first 3d metal cluster bearing a mu4-eta2:eta2-O,O carbonate ligand

Reaction of nickel(II) acetate with H(3)L (2-(5-bromo-2-hydroxyphenyl)-1,3-bis[4-(5-bromo-2-hydroxyphenyl)-3-azabut-3-enyl]-1,3-imidazolidine) yields [Ni(2)L(OAc)(H(2)O)(2)].3MeCN.2H(2)O (1.3MeCN.2H(2)O), crystallographically characterized. 1 is unstable in solution for a long time and hydrolyzes to give [Ni(2)L(o-OC(6)H(3)BrCHO)(H(2)O)].2.25MeCN.H(2)O (2.2.25MeCN.H(2)O). In addition, 1 uptakes CO(2) from air in a basic methanol/acetonitrile solution, yielding [[Ni(2)L(MeOH)](2)(CO(3))].1.5MeOH.MeCN.H(2)O (3.1.5MeOH.MeCN.H(2)O). The X-ray characterization of 3 reveals that it is a tetranuclear nickel cluster, which can be considered as the result of a self-assembly process from two dinuclear [Ni(2)L](+) blocks, joined by a mu(4)-eta(2):eta(2)-O,O carbonate ligand. The coordination mode of the carbonate anion is highly unusual and, to the best of our knowledge, it has not been described thus far for first-row transition metal complexes or magnetically studied until now. Magnetic characterization of 1 and 3 shows net intramolecular ferromagnetic coupling between the metal atoms in both cases, with S = 2 and S = 4 ground states for 1 and 3, respectively.     

Mahler measures and computations with regulators

In this work we apply the techniques that were developed in [M.N. Lalín, An algebraic integration for Mahler measure, Duke Math. J. 138 (2007), in press] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulators. Moreover, we apply the same techniques to the computation of generalized Mahler measures, in the sense of Gon and Oyanagi [Y. Gon, H. Oyanagi, Generalized Mahler measures and multiple sine functions, Internat. J. Math. 15 (5) (2004) 425-442].     

The Spectral Action and Cosmic Topology

The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincaré dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincaré homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces.     

Principal differential analysis of the Aneurisk65 data set

We explore the use of principal differential analysis as a tool for performing dimensional reduction of functional data sets. In particular, we compare the results provided by principal differential analysis and by functional principal component analysis in the dimensional reduction of three synthetic data sets, and of a real data set concerning 65 three-dimensional cerebral geometries, the AneuRisk65 data set. The analyses show that principal differential analysis can provide an alternative and effective representation of functional data, easily interpretable in terms of exponential, sinusoidal, or damped-sinusoidal functions and providing a different insight to the functional data set under investigation. Moreover, in the analysis of the AneuRisk65 data set, principal differential analysis is able to detect interesting features of the data, such as the rippling effect of the vessel surface, that functional principal component analysis is not able to detect.     

Higher Mahler measure for cyclotomic polynomials and Lehmer’s question

The k-higher Mahler measure of a non-zero polynomial P is the integral of log  ^k |P| on the unit circle. In this note, we consider Lehmer’s question (which is a long-standing open problem for k=1) for k>1 and find some interesting formulas for 2- and 3-higher Mahler measure of cyclotomic polynomials.     

Unimodularity of zeros of self-inversive polynomials

We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang as well as by Lalín and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lalín and Rogers on computational evidence.     

Middle points and inner products

Let X be a real normed space with unit sphere S. It is provedthat X is an inner product space if and only if there is a realnumber , 0 < < 1, and , such that every chord of S that supports S touches S at itsmiddle point.