Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations

Synergistic palladium-phosphoric acid catalysis in (3 + 2) cycloaddition reactions between vinylcyclopropanes and imines

The palladium-catalyzed (3 + 2) cycloaddition reaction between vinylcyclopropanes (VCPs) bearing geminal EWG\u2019s and imines represents a straightforward and flexible entry to polysubstituted pyrrolidine derivatives. In this paper, we demonstrate that using a synergistic catalysis approach, based on the combination of phosphoric acid and palladium catalysts, it is possible to engage for the first time N-aryl and N-benzyl imines in this cycloaddition reaction. A range of polysubstituted pyrrolidines is obtained with moderate to good yields and diastereoselectivities, using a simple palladium species (Pd(PPh3)4) and an archetypical phosphoric acid as catalyst combination. A two-step scheme which exploits the same palladium catalyst for two consecutive and mechanistically distinct reactions (the cycloaddition and a Suzuki\u2013Miyaura cross-coupling) is also presented. This synergistic catalysis approach is well posited for the development of the enantioselective version of this reaction. A screening of common BINOL-derived chiral phosphoric acids as catalyst component identified a species giving the product with moderate, yet promising, enantioselectivity (64% ee)

Metabolomic adjustments in the orchid mycorrhizal fungus <em>Tulasnella calospora </em>during symbiosis with <em>Serapias vomeracea</em>.

All orchids rely on mycorrhizal fungi for organic carbon, at least during early development. In fact, orchid seed germination leads to the formation of a protocorm, a heterotrophic postembryonic structure colonized by intracellular fungal coils, thought to be the site of nutrient transfer. The molecular mechanisms underlying mycorrhizal interactions and metabolic changes induced by this symbiosis in both partners remain mostly unknown. We studied plant-fungus interactions in the mycorrhizal association between the Mediterranean orchidSerapias vomeraceaand the basidiomyceteTulasnella calosporausing nontargeted metabolomics. Plant and fungal metabolomes obtained from symbiotic structures were compared with those obtained under asymbiotic conditions. Symbiosis induced substantial metabolomic alterations in both partners. In particular, structural and signaling lipid compounds increased markedly in the external fungal mycelium growing near the symbiotic protocorms, whereas chito-oligosaccharides were identified uniquely in symbiotic protocorms. This work represents the first description of metabolic changes occurring in orchid mycorrhiza. These results - combined with previous transcriptomic data - provide novel insights on the mechanisms underlying the orchid mycorrhizal association and open intriguing questions on the role of fungal lipids in this symbiosis