A quantitative discrete H2-regularity estimate for the Shortley-Weller scheme in convex domains

Summary We investigate the discreteH ²-regularity properties of the Shortley-Weller discretization of Poisson's equation (with homogeneous Dirichlet boundary condition) in bounded convex domains ². It is shown that the regularity constant is 1 independent of the mesh sizeh if theH ²-seminorm is defined in a way assigning less weight to the (unsymmetric) differences near the boundary.     

Über die Darstellung von substituierten 3,4-Dihydro-2(1H)-pyrimidin-iminen und 2-Aminopyrimidinen aus Guanidin und α,β-ungesättigten Ketonen

Guanidine reacts with the 2-alken-1-ones4 a-f and5 to give the unstable dihydropyrimidinimines (or-amines respectively)8 a-f (I, II or III respectively) and the hexahydroquinazolinimine (-amine)9 (I, II or III);8 a-f lose H₂, partly in the course of the reaction, partly during recrystallization to yield the 2-pyrimidinamines10 a-c, e, f, and the 4-methyl-5,6,7,8-tetrahydro-2-quinazolinamine11. With picric acid the unstable compounds8 d, f and9, resp. are converted into the stable 2-amino-3,4-dihydro-1H-2-pyrimidinylium picrates12 d and into the 2-amino-4-methyl-3,5,6,7,8,8 a-hexahydro-1H-2-quinazolinium picrate (13 resp.14, 15)¹⁸, whereas8 e reacts with HCl to give the chloride12 e. The structures of12 d-f follow from their NMR-spectra, and of10 a-c, e, f and11 by alternative syntheses by known methods^12–17 (e.g. from -diketones and guanidine). The reaction of8 d-f and9 to10 d-f and11 respectively is compared with previously described dehydrogenation and disproportionation reactions, especially of 3,4-dihydro-2(1H)-pyrimidinones (6 d ⁴,23 ²⁰) and-thiones (30 ¹) of similar structure and formulated as a base-catalysed elimination of H₂.

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HerrnA. Fuchsgruber danken wir für die Aufnahme und Hilfe bei der Interpretation der NMR-Spektren.     

Über 4,4,6-trisubstituierte 3,4-Dihydro-2(1H)-pyrimidinimine,Hexahydro-2(1H)-chinazolinimine bzw.-one bzw.-thione und Spiro[cycloalkan-1,4′(1′H)-cycloalkeno-pyrimidin]-2′ (3′H)-imine

The 2-alken-1-ones2 b, 2 c and3–8, which have very different structures, were reacted with guanidine to give cyclic compounds:2 b and2 c resp. are transformed by guanidine to the trisubstituted 3,4-dihydro-2(1H)-pyrimidinimines1 b and1 c resp., action of guanidine on the cyclo-hexylidenaceton (3) yields 4-methyl-1,3-diaza-4-spiro[5,5]-undecen-2-imine (11); isopropylidencyclohexanone4 and the isomeric (methylcyclohexenyl)ethanone5 condense with guanidine resp. urea resp. NH₄CNS to give 4,4-dimethyl-3,4,5,6,7,8-hexahydro-2(1H)-quinazolinimine (15), resp.-one (16), resp.-thione (17) and 4,8a-dimethyl-3,4,6,7,8,8a-hexahydro-2(1H)-quinazolinthione (19). With the cycloalkyliden-alkanones6–8 guanidine reacts to yield the spiro[cycloalkan-1,4(1H)-cycloalkenopyrimidin]-2(3H)-imines24–26. The structure of the compounds cited is derived from their NMR-and (partially) mass spectra; from most of the bases picrates were also synthesized.

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Herrn Prof. Dr.H. Grubitsch zum 70. Geburtstag gewidmet.     

On the computation of a-invariants

Thea-invariant of a graded Cohen-Macaulay ring is the least degree of a generator of its graded canonical module. We compute thea-invariants of (i) graded algebras with straightening laws on upper semi-modular lattices and (ii) the Stanley-Reisner rings of shellable weighted simplicial complexes. The formulas obtained are applied to rings defined by determinantal and pfaffian ideals.