Recovering an Homogeneous Polynomial from Moments of Its Level Set

Let $\mathbf{K }:=\left\{ \mathbf{x }: g(\mathbf{x })\le 1\right\} $ K : = x : g ( x ) ≤ 1 be the compact (and not necessarily convex) sub-level set of some homogeneous polynomial $g$ g . Assume that the only knowledge about $\mathbf{K }$ K is the degree of $g$ g as well as the moments of the Lebesgue measure on $\mathbf{K }$ K up to order $2d$ 2 d . Then the vector of coefficients of $g$ g is the solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order $2d$ 2 d of the Lebesgue measure on $\mathbf{K }$ K encode all information on the homogeneous polynomial $g$ g that defines $\mathbf{K }$ K (in fact, only moments of order $d$ d and $2d$ 2 d are needed).     

Synthesis and Structure of a Copper Complex of a Macrocyclic Ligand with Dissimilar HN+N(Amino)O2 and N2(Imino)O2 Inners

Abstract The copper complex [Cu(C₂₃H₂₇Br₂N₄O₂)(H₂O)(NO₃)]₂ was synthesized and characterized by X-ray diffraction. The complex crystallizes in the triclinic system with space group P-1, lattice parameters: a = 13.2293(18) (?), b = 14.184(20) (?), c = 16.522(23) (?), α = 91.923(2)°, β = 111.777(2)°, γ = 111.259(2)°, V = 2632.0(6) ?³, Z = 4, D _c = 0.88 Mg m⁻³. In the title compound [Cu(C₂₃H₂₇Br₂N₄O₂)(H₂O)(NO₃)]₂, the coordination polyhedron of the Cu atom is a square-planar environment. The structure consists of mononuclear units in which the copper(II) ions are linked in the plane by N, N′ iminato and O, O′ phenoxo atoms. In addition the crystal lattice contains one water molecule per unit. Graphical Abstract Dissymmetric ligand was efficiently synthesized and used to prepfsare a copper(II) complex which was characterized by X-ray diffraction      

A simple and versatile method to synthesize N-acyl-benzotriazoles

An efficient method for the synthesis of N-acyl-benzotriazoles from a wide variety of protected amino acids, as well as from compounds frequently used in drug discovery such as biotin and N-Fmoc polyethylene glycol, has been developed. The reaction of carboxylic acid derivatives with benzotriazole in the presence of T3P® yielded the corresponding N-acyl-benzotriazoles, which were obtained in high purity following a simple work-up procedure, in most cases via precipitation of the desired product in water.     

Synthesis of Bimetallic Copper‐Rich Nanoclusters Encapsulating a Linear Palladium Dihydride Unit

The structurally precise Cu‐rich hydride nanoclusters [PdCu₁₄H₂(dtc/dtp)₆(C≡CPh)₆] (dtc: di‐butyldithiocarbamate ( 1 ); dtp: di‐isopropyl dithiophosphate ( 2 )) were synthesized from the reaction of polyhydrido copper clusters [Cu₂₈H₁₅(S₂CNⁿBu₂)₁₂]⁺ or [Cu₂₀H₁₁{S₂P(OⁱPr)₂}₉] with phenyl acetylene in the presence of Pd(PPh₃)₂Cl₂. Their structures and compositions were determined by single‐crystal X‐ray diffraction and the results supported by ESI‐mass spectrometry. Hydride positions in 1 were confirmed by single‐crystal neutron diffraction. Each hydride is connected to one Pd⁰ and four Cu^I atoms in slightly distorted trigonalbipyramidal geometry. The anatomies of clusters 1 and 2 are very similar and DFT calculations allow rationalizing the interactions between the encapsulated [PdH₂]^2? unit and its Cu₁₄ bicapped icosahedral cage. As a result, Pd has the highest coordination number (14) so far recorded.     

Diamondoid Nanostructures as sp3‐Carbon‐Based Gas Sensors

Diamondoids, sp³‐hybridized nanometer‐sized diamond‐like hydrocarbons (nanodiamonds), difunctionalized with hydroxy and primary phosphine oxide groups, enable the assembly of the first sp³‐C‐based chemical sensors by vapor deposition. Both pristine nanodiamonds and palladium nanolayered composites can be used to detect toxic NO₂ and NH₃ gases. This carbon‐based gas sensor technology allows reversible NO₂ detection down to 50 ppb and NH₃ detection at 25–100 ppm concentration with fast response and recovery processes at 100 °C. Reversible gas adsorption and detection is compatible with 50 % humidity conditions. Semiconducting p‐type sensing properties are achieved from devices based on primary phosphine–diamantanol, in which high specific area (ca. 140 m² g^?1) and channel nanoporosity derive from H‐bonding.     

Carbonylation of C−N Bonds in Tertiary Amines Catalyzed by Low‐Valent Iron Catalysts

The first iron catalysts able to promote the formal insertion of CO into the C?N bond of amines are reported. Using low‐valent iron complexes, including K₂[Fe(CO)₄], amides are formed from aromatic and aliphatic amines, in the presence of an iodoalkane promoter. Inorganic Lewis acids, such as AlCl₃ and Nd(OTf)₃, have a positive influence on the catalytic activity of the iron salts, enabling the carbonylation at a low pressure of CO (6 to 8 bars).     

Bioinspired Oxidative Cyclization of the Geissoschizine Skeleton for Enantioselective Total Synthesis of Mavacuran Alkaloids

Reported is the enantioselective total syntheses of mavacuran alkaloids, (+)‐taberdivarine H, (+)‐16‐hydroxymethyl‐pleiocarpamine, and (+)‐16‐epi‐pleiocarpamine, and their postulated biosynthetic precursor 16‐formyl‐pleiocarpamine. This family of monoterpene indole alkaloids is a target of choice since some of its members are subunits of intricate bisindole alkaloids such as bipleiophylline. Inspired by the biosynthetic hypothesis, an oxidative coupling approach from the geissoschizine framework to form the N1?C16 bond was explored. Quaternization of the aliphatic nitrogen center was key to achieving the oxidative coupling induced by KHMDS/I₂ as it masks the nucleophilicity of the aliphatic nitrogen center and locks in the required cis conformation.