On the Alexander-Hirschowitz theorem

The Alexander-Hirschowitz theorem says that a general collection of k double points in imposes independent conditions on homogeneous polynomials of degree d with a well-known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on the previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d=3, where our proof is shorter. We end with an account of the history of the work on this problem.     

An Integration by Parts Formula in Sobolev Spaces

We prove the following formula

QuEChERS method combined to liquid chromatography high-resolution mass spectrometry for the accurate and sensitive simultaneous determination of pyrrolizidine and tropane alkaloids in cereals and spices

Within the last decades, in the EU, there has been an increasing interest in toxic plant alkaloids as food contaminants, especially after the continuous and growing consumption of plant-based foods compared with food of animal origin. In this regard, the once neglected presence of these tropane alkaloids (TAs) and pyrrolizidine alkaloids (PAs) has recently been reconsidered by the European Food Safety Authority, highlighting the lack of data and the need to develop risk assessment strategies. For this reason, the emphasis has been placed on detecting their occurrence in food through the development of accurate and sensitive analytical methods to achieve the determination of these compounds. The present study aims to elaborate and validate an analytical method based on QuEChERS sample preparation approach, exploiting the UHPLC coupled to the HRMS to simultaneously identify and quantify 21 PAs and two TAs in cereals and spices. For TAs, the obtained limit of detection (LOD) is 0.1 μg·kg⁻¹ and the limit of quantification (LOQ) is 0.4 μg·kg⁻¹, while for PAs, the LODs values ranging between 0.2 to 0.3 μg·kg⁻¹ and the LOQ, between 0.4 and 0.8 μg·kg⁻¹, ensuring compliance with the recently established European Regulations. Several commercial samples were analysed to further verify the applicability of this comprehensive analytical approach.     

Nanostructure Explains the Behavior of Slippery Covalently Attached Liquid Surfaces

Slippery covalently-attached liquid surfaces (SCALS) with low contact angle hysteresis (CAH, <5°) and nanoscale thickness display impressive anti-adhesive properties, similar to lubricant-infused surfaces. Their efficacy is generally attributed to the liquid-like mobility of the constituent tethered chains. However, the precise physico-chemical properties that facilitate this mobility are unknown, hindering rational design. This work quantifies the chain length, grafting density, and microviscosity of a range of polydimethylsiloxane (PDMS) SCALS, elucidating the nanostructure responsible for their properties. Three prominent methods are used to produce SCALS, with characterization carried out via single-molecule force measurements, neutron reflectometry, and fluorescence correlation spectroscopy. CO₂ snow-jet cleaning was also shown to reduce the CAH of SCALS via a modification of their grafting density. SCALS behavior can be predicted by reduced grafting density, Σ, with the lowest water CAH achieved at Σ≈2. This study provides the first direct examination of SCALS grafting density, chain length, and microviscosity and supports the hypothesis that SCALS properties stem from a balance of layer uniformity and mobility.     

A biomimetic alternative to poly(ethylene glycol) as an antifouling coating: resistance to nonspecific protein adsorption of poly(L-lysine)-graft-dextran

Poly( L-lysine)- graft-dextran (PLL- g-dex), graft copolymers with dextran side chains grafted onto a poly( L-lysine) backbone, previously shown to be effective as stabilizers of DNA triple helices and as carriers of functional genes to target cells or tissues, were employed in this work to prevent nonspecific adsorption of proteins, as determined by means of optical waveguide lightmode spectroscopy. PLL- g-dex copolymers readily adsorb from aqueous solution onto negatively charged oxide surfaces and significantly reduce nonspecific protein adsorption onto bare silica-titania surfaces. While effective and equivalent surface adsorption and antifouling properties were observed for PLL- g-dex copolymers in a variety of architectures, nanotribological analysis by atomic force microscopy was able to distinguish between the different brush densities produced.     

Kinetic and thermochemical study of the antioxidant activity of sulfur-containing analogues of vitamin E

Sulfur-containing analogues of vitamin E (thiachromanols), either linked or not to a catechol moiety, were synthesized and their hydrogen-atom donating ability evaluated. The determination of the O--H bond dissociation enthalpy (BDE) of the alpha-tocopherol analogue 4 by the electron paramagnetic resonance (EPR) equilibration technique provided a value of 78.9 kcal mol(-1), that is, approximately 1.8 kcal mol(-1) higher than that of alpha-tocopherol. The kinetic rate constants for the reaction with peroxyl radicals (kinh), measured by inhibited autoxidation studies, showed that thiachromanols react 2.5 times slower than the corresponding tocopherols, in agreement with the higher BDE value. This behavior was explained, on the basis of crystallographic analyses and DFT calculations, in terms of a change in the molecular geometry caused by insertion of a sulfur atom into the framework of vitamin E. This behavior implies a greater deviation of the condensed ring from coplanarity with the aromatic ring, thus giving rise to a decrease in the conjugative stabilization of the phenoxyl radical and consequently to an increase in the O--H bond strength. Although less reactive than tocopherols, thiachromanols may, however, act as bimodal antioxidants as a result of the hydroperoxide decomposing ability of the sulfur atom.     

One-step synthesis of low molecular weight poly(p-phenyleneethynylenevinylene)s via polyaddition of aromatic diynes by catalysis of the [Ru(p-cymene)Cl2]2/AcOH system

pi-Conjugated low molecular weight polymers characterized by regio- and stereoregular alternation of phenylene and ( E)-1-en-3-yne moieties have been synthesized by polyaddition of 1,4-diethynylbenzene or of 2,5-diethynyl-1,4-alkoxybenzene monomers, employing the commercially available di-micro-chlorobis[( p-cymene)chlororuthenium(II)] complex as the metal catalyst source, under homogeneous, atom-economical, amine- and phosphine-free conditions. Bulk materials of poly( p-phenyleneethynylenevinylene) derivatives are obtained with yields larger than 80%, from which polymers readily soluble in chlorinated solvents and in tetrahydrofuran are extracted in 60-75% yields. The polymers with average degrees of polymerization in the range n AV = 4-8 display optical properties in solution similar to those of the higher molecular weights analogues.     

New copper(I) and silver(I) triazolato-complexes: Synthesis, reactivity and catalytic activity in olefin cyclopropanation

A new class of azolate ligands, deriving from the equimolar condensation of 3,5-diamino-1,2,4-triazole with salicylaldehyde (H₃L¹) and o-anisaldehyde (H₃L²) was prepared. In their anionic form, these species act as bridging moieties upon coordination to Cu(I) and Ag(I), giving rise to the formation of dinuclear complexes with the ligand in the typical N,N′-exobidentate conformation. The copper derivative [Cu(H₂L¹)(CH₃CN)]₂ (1) showed attractive reactivity in the replacement of the labile acetonitrile molecules. In particular, it was possible to isolate a dinuclear copper(I)-carbonyl complex [Cu(H₂L¹)(CO)]₂ (4), by substitution of the nitrile with carbon monoxide. Moreover, the reaction of 1 with ethyl diazoacetate (EDA) in CH₂Cl₂ afforded a mono-carbene product, as established by ¹³C NMR data. Finally, the copper derivative 1 proved to be a highly diastereoselective catalyst in olefin cyclopropanation in the presence of ethyl diazoacetate. In the case of internal alkenes a trans:cis ratio of up to 97:3 was reached. The X-ray structure of a dinuclear Ag(I) complex, namely [Ag(H₂L¹)(PPh₃)]₂ (3), obtained by reacting the polymeric [Ag(H₂L¹)]_n (2), with triphenylphosphine, is also reported.     

AgY zeolite as catalyst for the effective three-component synthesis of propargylamines

The preparation of substituted propargylamines via three-component reaction between aldehydes, terminal alkynes, and secondary amines following an environmentally acceptable methodology has been developed. The process has been carried out under solvent-free conditions in the presence of AgY zeolite. The catalyst can be easily recovered and reused for at least four cycles without significant decreases in yield and selectivity.