A novel synthesis of linear polyenes via conjugate addition of cuprates to α,β-γ,δ-diunsaturated sulfones followed by SO2 extrusion

The two 1,3-butadienyl 2-propenyl sulfones $\text{5}$ and $\text{6}$, 1,3,5-heptatriene synthons, have been transformed into the tri- and tetraenes $\text{1}$ - $\text{4}$ by alkylcuprate addition and Ramberg-Bäcklund SO₂ extrusion. The reaction stereochemistry is discussed.     

Durch wasserstoff-transfer induzierte radikal-eliminierungen aus ortho-substituierten benzoesäuremethylestern

The investigation of some ²H-labelled compounds as well as the analysis of energetic properties (appearance potential measurement, kinetic energy release) reveal some unusual elimination of radicals (CH₃^., OH^. resp. CHO^.) from ortho substituted bencoic acid methyl esters. It can be shown that most of the reactions are induced by a novel hydrogen transfer to the carboxyl function.     

Double Proton Tautomerism via Intra- or Intermolecular Pathways? The Case of Tetramethyl Reductic Acid Studied by Dynamic NMR: Hydrogen Bond Association,Solvent and Kinetic H/D Isotope Effects

Using dynamic liquid-state NMR spectroscopy a degenerate double proton tautomerism was detected in tetramethyl reductic acid (TMRA) dissolved in toluene-d₈ and in CD₂Cl₂. Similar to vitamin C, TMRA belongs to the class of reductones of biologically important compounds. The tautomerism involves an intramolecular HH transfer that interconverts the peripheric and the central positions of the two OH groups. It is slow in the NMR time scale around 200 K and fast at room temperature. Pseudo-first-order rate constants of the HH transfer and of the HD transfer after suitable deuteration were obtained by line shape analyses. Interestingly, the chemical shifts were found to be temperature dependent carrying information about an equilibrium between a hydrogen bonded dimer and a monomer forming two weak intramolecular hydrogen bonds. The structures of the monomer and the dimer are discussed. The latter may consist of several rapidly interconverting hydrogen-bonded associates. A way was found to obtain the enthalpies and entropies of dissociation, which allowed us to convert the pseudo-first-order rate constants of the reaction mixture into first-order rate constants of the tautomerization of the monomer. Surprisingly, these intrinsic rate constants were the same for toluene-d₈ and CD₂Cl₂, but in the latter solvent more monomer is formed. This finding is attributed to the dipole moment of the TMRA monomer, compensated in the dimer, and to the larger dielectric constant of CD₂Cl₂. Within the margin of error, the kinetic HH/HD isotope effects were found to be of the order of 3 but independent of temperature. That finding indicates a stepwise HH transfer involving a tunnel mechanism along a double barrier pathway. The Arrhenius curves were described in terms of the Bell–Limbach tunneling model.     

The true prosoluble completion of a group: Examples and open problems

The true prosoluble completion of a group Γ is the inverse limit of the projective system of soluble quotients of Γ. Our purpose is to describe examples and to point out some natural open problems. We discuss a question of Grothendieck for profinite completions and its analogue for true prosoluble and true pronilpotent completions. Goulnara Arzhantseva and Zoran Šunić were the authors of the Appendix.     

Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”-and “ ”-norms, where ɛ ⩾ 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution, the errors are of order O(h ^p + τ ^q ). The estimates hold true even in the hyperbolic case when ɛ = 0.     

Colouring Cubic Graphs by Small Steiner Triple Systems

Given a Steiner triple system , we say that a cubic graph G is -colourable if its edges can be coloured by points of in such way that the colours of any three edges meeting at a vertex form a triple of . We prove that there is Steiner triple system of order 21 which is universal in the sense that every simple cubic graph is -colourable. This improves the result of Grannell et al. [J. Graph Theory 46 (2004), 15–24] who found a similar system of order 381. On the other hand, it is known that any universal Steiner triple system must have order at least 13, and it has been conjectured that this bound is sharp (Holroyd and Škoviera [J. Combin. Theory Ser. B 91 (2004), 57–66]). Research partially supported by APVT, project 51-027604. Research partially supported by VEGA, grant 1/3022/06.