Isomorphism and elementary equivalence of multilinear maps

In [14], we proved that two finitely generated finite-by-nilpotent groups G,H are elementarily equivalent if and only if Z×G and Z×H are isomorphic. In the present paper, we obtain similar characterizations of elementary equivalence for the following classes of structures:

1. the (n+2)-tuples (A ₁…,A _n+1,f),where n≥2 is an integerA ₁…,A _n+1 are disjoint finitely generated abelian groups and f A ₁×…×A _n →A _n+1: is a n-linear map;

2. the triples (A,B f), where n≥2 is an integerA,B are disjoint finitely generated abelian groups and f : A ⁿ →B is a n-linear map;

3. the couples (A,f), where n≥2 is an integerA is a finitely generated abelian group and f:A ⁿ →A is a n-linear map.

For each class, we show that elementary equivalence does not imply isomorphism. In particular, we give an example of two nonisomorphic finitely generated torsion-free Lie rings which are elementarily equivalent.     

An application of ramsey’s theorem to groups with homogeneous theory

For any integers n >m ≥ 2, we say that a complete theory T is (m, n)-homogeneous if, for each model M of T, two n-tuples [abar],[bbar] in M have the same type if the corresponding m-tuples from [abar] and [bbar] have the same type. It was conjectured by H. Kikyo that, if M is an infinite group, with possibly additional structure, then the theory of M is not (m, n)-homogeneous. We prove a general result on structures with (m, n)-homogeneous theory which implies that, if M is a counterexample to this conjecture, then there exists an integer h such that each abelian subgroup of M has at most h elements. It follows that there exist an integer k such that M ^k = 1, and an integer l such that each finite subgroup of M has at most l elements.     

First 2-hydroxy-3-methylbut-3-enyl substituted xanthones isolated from plants: structure elucidation,synthesis and antifungal activity

Two new 2-hydroxy-3-methylbut-3-enyl substituted xanthones, (+/-)-caledol 1 and (+/-)-dicaledol 2 were isolated from a dichloromethane extract of the leaves of Calophyllum caledonicum (Clusiaceae). Compounds 1 and 2 are the first 2-hydroxy-3-methylbut-3-enyl substituted xanthones isolated from natural source. Their structures were elucidated by means of combined analytical methods including HRFABMS, 1D and 2D NMR spectroscopies and also confirmed by total synthesis using biomimetic ortho-prenylphenols photooxygenation (1O2) as a key step. The antifungal activity against Aspergillus fumigatus is reported.     

Synthesis of 2-hydroxy-3-methylbut-3-enyl substituted coumarins and xanthones as natural products. Application of the Schenck ene reaction of singlet oxygen with ortho-prenylphenol precursors

Application of our original photooxidation-reduction methodology to prenylated dihydroxycoumarin and trihydroxyxanthone compounds led to the corresponding ortho-(2-hydroxy-3-methylbut-3-enyl)phenol derivatives with yields ranging from 8 to 65%. In most of the reported experiments, the oxidation products distribution, after the photooxygenation step, was controlled by the competition between the large group effect and the stabilising phenolic assistance effect. We also showed that ortho-(3-hydroxy-3-methylbut-1-enyl)phenol derivatives could be considered as biogenetic precursors of 2,2-dimethylbenzopyranic structures.