The lower central and derived series of the braid groups of the projective plane

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups B(n)(RP(2)) of the projective plane RP(2) were originally studied by Van Buskirk during the 1960s. and are of particular interest due to the fact that they have torsion. The group B(1)(RP(2)) (resp. B(2)(RP(2))) is isomorphic to the cyclic group Z(2) of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n > 2, we first prove that the lower central series of B(n)(RP(2)) is constant from the commutator subgroup onwards. We observe that Gamma(2)(B(3)(RP(2))) is isomorphic to (F(3) X Q(8)) X Z(3), where F(k) denotes the free group of rank k, and Q(8) denotes the quaternion group of order 8, and that Gamma(2)(B(4)(RP(2))) is an extension of an index 2 subgroup K of P(4)(RP(2)) by Z(2) circle plus Z(2). As for the derived series of B(n)(RP(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group B(n)(RP(2)) being finite and soluble for n <= 2, the critical cases are n = 3, 4. We are able to determine completely the derived series of B(3)(RP(2)). The subgroups (B(3)(RP(2)))((1)), (B(3)(RP(2)))((2)) and (B(3)(RP(2)))((3)) are isomorphic respectively to (F(3) x Q(8)) x Z(3), F(3) X Q(8) and F(9) X Z(2), and we compute the derived series quotients of these groups. From (B(3)(RP(2)))((4)) onwards, the derived series of B(3)(RP(2)), as well as its successive derived series quotients, coincide with those of F(9). We analyse the derived series of B(4)(RP(2)) and its quotients up to (B(4)(RP(2)))((4)), and we show that (B(4)(RP(2)))((4)) is a semi-direct product of F(129) by F(17). Finally, we give a presentation of Gamma(2)(B(n)(RP(2))). (C) 2011 Elsevier Inc. All rights reserved.

Ion pair stabilization effects on a series of procaine structural analogs

In this work, a series of 10 structural procaine analogs have been synthesized in order to investigate the structural features affecting the stability of ion pair formation and its influence on the lipophilicity of ionizable compounds. The structural variation within this series was focused on the terminal nitrogen substituents and on the intermediate chain linkage nature. The hydrophobic parameters log P(n) and log P(i) (partition coefficient of the neutral and ionic species, respectively), as well as the ionization constants pK(a) and pK(a)(oct), were obtained from log D-pH profiles measured at pH values ranging from 2 to 12. The difference between log P(i) and log P(n) values (i.e. difflog P) of each prepared compound was considered a measure of the stability of ion pair formation. In this set, the difflog P values varied nearly over one log unit, ranging from -2.40 to -3.37. It has been observed that the presence of hydrogen bonding groups (especially donor) and low steric hindrance around the terminal amine ionizable group increases the relative lipophilicity of the ionic species as compared to the corresponding neutral species. These results were interpreted as due to the increased stability of ion pairs of the compounds bearing these structural features. (C) 2010 Elsevier B.V. All rights reserved.

Insights into the Molecular Requirements for the Anti-obesity Activity of a Series of CB1 Ligands

Two-dimensional and 3D quantitative structure-activity relationships studies were performed on a series of diarylpyridines that acts as cannabinoid receptor ligands by means of hologram quantitative structure-activity relationships and comparative molecular field analysis methods. The quantitative structure-activity relationships models were built using a data set of 52 CB1 ligands that can be used as anti-obesity agents. Significant correlation coefficients (hologram quantitative structure-activity relationships: r 2 = 0.91, q 2 = 0.78; comparative molecular field analysis: r 2 = 0.98, q 2 = 0.77) were obtained, indicating the potential of these 2D and 3D models for untested compounds. The models were then used to predict the potency of an external test set, and the predicted (calculated) values are in good agreement with the experimental results. The final quantitative structure-activity relationships models, along with the information obtained from 2D contribution maps and 3D contour maps, obtained in this study are useful tools for the design of novel CB1 ligands with improved anti-obesity potency.