Generalized convex set-valued maps

The aim of this paper is to show that under a mild semicontinuity assumption (the so-called segmentary epi-closedness), the cone-convex (respectively, cone-quasiconvex) set-valued maps can be characterized in terms of weak cone-convexity (respectively, weak cone-quasiconvexity), i.e., the notions obtained by replacing in the classical definitions the conditions of type “for all x,y in the domain and for all t in ]0,1[…” by the corresponding conditions of type “for all x,y in the domain there exists t in ]0,1[….”     

Hilbert function of powers of ideals of low codimension

We study the Hilbert function of the powers of homogeneous ideals which are either Cohen-Macaulay of codimension 2 or Gorenstein of codimension 3. We show that that if I is an ideal in one of these classes and it is of linear type then for all k the Hilbert function of depends only on the Hilbert function of I. In other words, if I andJ are ideals in one of the above mentioned classes which are both of linear type and they have the same Hilbert function then also and have the same Hilbert function for all k. Received July 29, 1997; in final form January 23, 1998     

Base‐Induced Decarboxylation of Polyunsaturated α‐Cyano Acids Derived from Malonic Acid: Synthesis of Sesquiterpene Nitriles and Aldehydes with β‐, φ‐, and ψ‐End Groups

Catalytic base‐induced decarboxylation of polyunsaturated α‐cyano‐β‐methyl acids derived from malonic acid led to the corresponding nitriles 3 (Schemes 2 and 3), 6 (Scheme 5), and 9 (Scheme 6). This decarboxylation occurred with previous deconjugation of the α,β‐alkene moiety of the α‐cyano‐β‐methyl acid, leading to an α‐cyano‐β‐methylene propanoic acid which was easily decarboxylated (see Scheme 2). β‐Methylene intermediates, in some cases, could be isolated; mechanistic pathways are proposed. The nitriles 3, 6 , and 9 were reduced to the sesquiterpene aldehydes 4 (β‐end group), 7 (φ‐end group), and 10 (ψ‐end group), respectively.     

New advances in the synthesis of tripyridinophane macrocycles suitable to enhance the luminescence of Ln(III) ions in aqueous solution

A series of four new 18-membered hexaaza macrocyclic ligands bearing three endocyclic pyridine units and acetate or methylenephosphonate pendant arms has been prepared. The new synthetic procedure is based on the use of amine and diamine precursors incorporating masked carboxylate or phosphonate functions and on an efficient sodium template effect which controls the crucial macrocyclization step (yields of macrocyclization reactions: 62–88%). This procedure appears as a suitable alternative compared to the classical Richman-Atkins methodology generally used for the preparation of this class of macrocycles. As demonstrated with the Eu^III and Tb^III complexes derived from two ligands, these tripyridinophane chelators form luminescent and stable mononuclear Ln^III complexes in aqueous solution at physiological pH. In such a medium, Tb^III complexes exhibit a brightness of 1700 (λ_exc?=?279?nm) and 3000 (λ_exc?=?268?nm)?M^?1?cm^?1.     

Novel π2s+π2a Electrocyclization of Triethylenic‐Malonic Acids Exemplified for a One‐Pot Synthesis of New γ‐Dilactones cis‐Fused with a Cyclopentene

A new, easy and rapid synthesis of γ‐dilactones is cis‐fused with a cyclopentenic ring via cyclization of 7‐chlorotriethylenic‐malonic acids. The key step implicates an intramolecular cyclization to a cyclopentenyl cation, according to an electrocyclic π_2s + π_2a conrotatory process. This cyclopentenyl cation led to unstable γ‐lactones intermediates that are rearrange to more stable isomers. δ‐lactones (6Z and 6E‐(3‐chlorobut‐2‐en‐2‐yl)‐5‐methyl‐3,6‐dihydro‐2H‐pyran‐2‐one) were obtained as secondary products. Mechanistic pathways were considered. The structures of the newly synthesized compounds were established by elemental and spectral data.     

Detection of 28 neurotransmitters and related compounds in biological fluids by liquid chromatography/tandem mass spectrometry

This work presents two liquid chromatography/tandem mass spectrometry (LC/MS/MS) acquisition modes: multiple reaction monitoring (MRM) and neutral loss scan (NL), for the analysis of 28 compounds in a mixture. This mixture includes 21 compounds related to the metabolism of three amino acids: tyrosine, tryptophan and glutamic acid, two pterins and five deuterated compounds used as internal standards. The identification of compounds is achieved using the retention times (RT) and the characteristic fragmentations of ionized compounds. The acquisition modes used for the detection of characteristic ions turned out to be complementary: the identification of expected compounds only is feasible by MRM while expected and unexpected compounds are detected by NL. In the first part of this work, the fragmentations characterizing each molecule of interest are described. These fragmentations are used in the second part for the detection by MRM and NL of selected compounds in mixture with and without biological fluids. Any preliminary extraction precedes the analysis of compounds in biological fluids.     

The period-index problem for real surfaces

Publications mathématiques de l'IHÉS - We study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that...     

The size of the Dini subdifferential

Given a lower semicontinuous function , we prove that the points of , where the lower Dini subdifferential contains more than one element, lie in a countable union of sets which are isomorphic to graphs of some Lipschitzian functions defined on . Consequently, the set of all these points has a null Lebesgue measure.