Synthese arylsubstituierter 2-Pyrimidinamine,Dihydro-2-pyrimidinamine und Pyrimido[1,2-a]pyrimidine durch Reaktion von Guanidin mit Chalkonen

4- and 4-methoxy-, 4-hydroxy- and 2-hydroxychalcone4 e,k,f andg, respectively, are transformed by action of guanidine in benzene to yield 4-(4-methoxyphenyl)-6-phenyl-2-pyrimidinamine (6 e=6 k) and hydroxyphenylpyrimidinamines6 f andg, respectively. In contrast, 4- and 4-chlor-, 4-brom- and 4-phenylchalcone4 h,l–n resp. react with guanidine under analogous conditions to give 4,6-diaryl-1,4-dihydro-2-pyrimidinamines5 h,l–n. The bases5 h,l–n also tend to aromatize, but they can be stabilized by transformation into salts5 h,l–n · HCl with hydrochloric acid. Heating of5 n inDMF under atmospheric oxygen yields 4-(4-biphenyl)-6-phenyl-2-pyrimidinamine (6 n). Action of guanidine on 4-nitrochalcone4 o in chloroform affords 2-amino-4-nitrophenyl-6-phenyl-tetrahydro-4-pyrimidinol8 o, which is transformed by hydrochloric acid into 6-nitrophenyl-dihydropyrimidinamine-hydrochloride5 o · HCl. Treating of the latter with sodiummethylat in methanol yields the very stable dihydropyrimidinamine5 o. Action of guanidine on 4-chlor- and 4-bromchalcone4 h andm respectively (in addition to5 h andm) yields 2,4,6,8-tetraaryl-1,4-dihydro-2H-pyrimido[1,2-a]pyrimidines7 h andm, respectively.

Herrn Prof. Dr.Erwin Schauenstein mit den besten Wünschen zum 65. Geburtstag gewidmet.     

Optical metrics and birefringence of anisotropic media

The material tensor of linear response in electrodynamics is constructed out of products of two symmetric second rank tensor fields which in the approximation of geometrical optics and for uniaxial symmetry reduce to “optical” metrics, describing the phenomenon of birefringence. This representation is interpreted in the context of an underlying internal geometrical structure according to which the symmetric tensor fields are vectorial elements of an associated two-dimensional space.     

Structural properties, conformational stability and oxygen binding properties of Penaeus monodon hemocyanin

Hemocyanin sequences allineament shows the presence of highly invariant regions especially in the active site and in the tight intersubunits interaction sites. Comparing the aminoacids in contact regions between monomers is possible to interpret the stability of hexamers.     

A Short Remark on Weierstrass Points at Infinity on <Emphasis Type="Italic">X</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">N</Emphasis>)

We point out that the formalism of the trace map and reduction modulo p can be used to give a short proof for the fact first proved by Ogg that is not a Weierstrass point on X₀(pM) where p is a prime not dividing M and the genus of X₀(M) is zero.     

Letter: Statefinder Parameters for Interacting Dark Energy

We argue that the recently introduced statefinder parameters (Sahni et al., JETP Lett. 77, 201 (2003)), that include the third derivative of the cosmic scale factor, are useful tools to characterize interacting quintessence models. We specify the statefinder parameters for two classes of models that solve, or at least alleviate, the coincidence problem.     

Scaling Cosmology

We show that with the help of a suitable coupling between dark energy and cold dark matter it is possible to reproduce any scaling solution _{X M} a , where _X and _M are the densities of dark energy and dark matter, respectively. We demonstrate how the case = 1 alleviates the coincidence problem. Future observations of supernovae at high redshift as well as quasar pairs which are planned to discriminate between different cosmological models will also provide direct constraints on the coupling between dark matter and dark energy.     

Divisorial Linear Algebra of Normal Semigroup Rings

We investigate the minimal number of generators and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely many divisorial ideals I such that (I)C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover, we determine the minimal depth of all divisorial ideals and the behaviour of and depth in arithmetic progressions in the divisor class group.The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the nonnegative solutions of an inhomogeneous system of linear diophantine equations.We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.