the direct three component modified mannich reaction via condensation of aldehydes, 2-naphthol or 2,7-naphthalendiol and piperidine to generate betti bases has been carried out over l-proline (20 mol%) with high efficiency under solvent free conditions at 70 oc. also, the reaction of aromatic aldehyde and 4-hydroxycoumarin in the presence of l-proline (20 mol%) in the ethanol under reflux condi...

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of IG and its initial ideal (with respect to a natural term order) coinc...

In this thesis we give a complete description of the syzygies of irreducible, nonsingular, canoncial curves C of genus 9. This includes a collection of all possible Betti tables for C. Moreover a direct correspondence between these Betti tables and the number and types of special linear series on C is given. Especially for Cliff(C) = 3 the curve C is contained in determinantal surface Y on a 4d...

For a smooth variety X over a base field of characteristic 0 we have its algebraic de Rham cohomology H · dR(X) := H (XZar,Ω · X); for nonsmooth X, one defines H · dR(X) using cohomological descent as in Deligne [D]. If the base field is C, then one has the Betti cohomology H · B(X) := H (Xcl,Q) and a canonical period isomorphism (“integration of algebraic differential forms over topological cy...

Let ∆ be a stable simplicial complex on n vertexes. Over an arbitrary base field K, the symmetric algebraic shifted complex ∆s of ∆ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring K[x1, x2, . . . , xn] of the symmetric algebraic shifted, exterior algebraic shifted and combinatorial shifted complexes of ∆ are equal.

We define the reduced horseshoe resolution and the notion of conjoined pairs of ideals in order to study the minimal graded free resolution of a class of p-Borel ideals and recover Pardue’s regularity formula for them. It will follow from our technique that the graded betti numbers of these ideals do not depend on the characteristic of the base field k.