Comonotone approximation with interpolation at the ends on an interval

Let a function f ∈ C[-1, 1], changes its monotonisity at the finite collection Y := {y1,… ,ys} of s points yi ∈ (-1, 1). For each n ≥ N(Y), we construct an algebraic polynomial Pn, of degree ≤ n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and |f(x)-Pn(x)|≤c(s)ω2(f,(√1-x2)/n), x∈[-1,1],where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s and ω2 (f, t) is the second modulus of smoothness of f.     

Crystalline and spectroscopic characterization of poly(2-aminoethyl methacrylate hydrochloride) chains grafted onto poly[(R)-3-hydroxybutyric acid]

The present study is aimed to investigate the degree of crystallinity of poly(3-hydroxybutyrate) P(3HB) grafted with poly(2-aminoethyl methacrylate hydrochloride) (PAEMA) chains using WAXS, micro Raman, and FTIR spectroscopy. The samples were obtained by radiation induced graft polymerization of the monomer in the substrate using different solvents for comparison. The results of crystallinity are consistent with those obtained of lower crystallinity in grafting copolymer relative to the substrate P(3HB). The low crystallinity is directly related to the increase of the degree of grafting, meaning that although the P(3HB) amorphous region is grafted, the crystalline zone is also affected in some extent by the grafting process and the environment of the new molecule. Three different methods were surveyed to determine the variation of crystallinity degree with the grafting degree. It is shown that all methods provide linear relationships between these variables, but WAXS method was found more acceptable than the others (FTIR and Raman). A detailed characterization of the vibrational bands characteristic of amorphous and helical crystalline structure of the grafting copolymers are also highlighted.     

Recent Progress towards Transition‐Metal‐Catalyzed Synthesis of Fluorenes

Fluorenes are a commonly encountered structural motif in materials science, pharmaceutical chemistry, and organic synthesis. Among various strategies towards the synthesis of this unique structure, transition metal‐catalyzed functionalization has emerged as one of the most efficient methods. This Minireview presents an overview of the recent advances in this emerging area by highlighting the reactions’ specificity and applicability and, where possible, provides a mechanistic rationale.     

Ruthenium‐Catalyzed Oxidative Coupling of Primary Amines with Internal Alkynes through CH Bond Activation: Scope and Mechanistic Studies

The oxidative coupling of primary amines with internal alkynes catalyzed by Ru complexes is presented as a general atom‐economy methodology with a broad scope of applications in the synthesis of N‐heterocycles. Reactions proceed through regioselective C?H bond activation in 15 minutes under microwave irradiation or in 24 hours with conventional heating. The synthesis of 2,3,5‐substituted pyridines, benzo[h]isoquinolines, benzo[g]isoquinolines, 8,9‐dihydro‐benzo[de]quinoline, 5,6,7,8‐tetrahydroisoquinolines, pyrido[3,4g]isoquinolines, and pyrido[4,3g]isoquinolines is achievable depending on the starting primary amine used. DFT calculations on a benzylamine substrate support a reaction mechanism that consists of acetate‐assisted C?H bond activation, migratory‐insertion, and C?N bond formation steps that involve 28–30 kcal mol^?1. The computational study is extended to additional substrates, namely, 1‐naphthylmethyl‐, 2‐methylallyl‐, and 2‐thiophenemethylamines.     

Stoichiometric and Catalytic Inter‐ and Intramolecular Hydroamination of Terminal Alkynes by Frustrated Lewis Pairs

Frustrated Lewis pairs (FLPs) based on sterically encumbered anilines and the Lewis acid B(C₆F₅)₃ were found to react with terminal alkynes effecting intermolecular hydroamination affording iminium alkynylborate species of the form [RPhN?C(R′)Me][R′CCB(C₆F₅)₃]. In these cases, the reagent ratio of borane, aniline, and alkyne is 1:1:2. These reactions could also be performed in an intramolecular fashion by using anilines with alkynyl substituents effecting cyclization reactions. The use of 10 mol % B(C₆F₅)₃ under a H₂ atmosphere provides a one‐pot synthesis of the pyrrolidine 12 , the piperidines 13 – 15 , the azepane 16 , the isoindoline 17 , and the benzoxazine 18 .     

Iron‐Catalyzed C(sp2)H and C(sp3)H Methylations of Amides and Anilides

The so‐called magic methyl effect significantly boosts the bioactivities and physical properties of pharmacologically active drugs. Direct introduction of the methyl group by C?H activation was accomplished with a versatile iron catalyst, which enabled the C?H methylation of (hetero)benzamides, anilides, alkenes, and even alkanes by triazole assistance in a chemo‐, site‐ and diastereo‐selective fashion.     

Combined Experimental and Computational Investigations of Rhodium‐Catalysed CH Functionalisation of Pyrazoles with Alkenes

Detailed experimental and computational studies have been carried out on the oxidative coupling of the alkenes C₂H₃Y (Y=CO₂Me ( a ), Ph ( b ), C(O)Me ( c )) with 3‐aryl‐5‐R‐pyrazoles (R=Me ( 1 a ), Ph ( 1 b ), CF₃ ( 1 c )) using a [Rh(MeCN)₃Cp*][PF₆]₂/Cu(OAc)₂ ? H₂O catalyst system. In the reaction of methyl acrylate with 1 a , up to five products ( 2 aa – 6 aa ) were formed, including the trans monovinyl product, either complexed within a novel Cu^I dimer ( 2 aa ) or as the free species ( 3 aa ), and a divinyl species ( 6 aa ); both 3 aa and 6 aa underwent cyclisation by an aza‐Michael reaction to give fused heterocycles 4 aa and 5 aa , respectively. With styrene, only trans mono‐ and divinylation products were observed, whereas with methyl vinyl ketone, a stronger Michael acceptor, only cyclised oxidative coupling products were formed. Density functional theory calculations were performed to characterise the different migratory insertion and β‐H transfer steps implicated in the reactions of 1 a with methyl acrylate and styrene. The calculations showed a clear kinetic preference for 2,1‐insertion and the formation of trans vinyl products, consistent with the experimental results.     

Synthesis of 8‐Aminoquinolines by Using Carbamate Reagents: Facile Installation and Deprotection of Practical Amidating Groups

Described herein is the development of practical routes to 8‐aminoquinolines by using readily installable and easily deprotectable amidating reagents. Two scalable procedures were optimized under Rh^III‐catalyzed conditions: i) the use of pre‐generated chlorocarbamates and ii) a two‐step one‐pot process that directly employs carbamates. Both approaches are highly convenient for the gram‐scale synthesis of 8‐aminoquinolines under mild conditions. Facile deprotection of the synthetically versatile amidating groups was achieved under the Pd‐catalyzed transfer hydrogenation conditions with simultaneous deoxygenation of quinoline N‐oxides, thus yielding 8‐aminoquinolines in excellent overall efficiency.     

Synthesis of Extended π‐Systems through C–H Activation

Activation of aromatic C? H bonds by a transition metal catalyst has received significant attention in the synthetic chemistry community. In recent years, rapid and site‐selective extension of π‐electron systems by C–H activation has emerged as an ideal methodology for preparing organic materials with extended π‐systems. This Review focuses on recently reported π‐extending C–H activation reactions directed toward new optoelectronic conjugated materials.     

Cooperative Lewis Pairs Based on Late Transition Metals: Activation of Small Molecules by Platinum(0) and B(C6F5)3

A Lewis basic platinum(0)–CO complex supported by a diphosphine ligand and B(C₆F₅)₃ act cooperatively, in a manner reminiscent of a frustrated Lewis pair, to activate small molecules such as hydrogen, CO₂, and ethene. This cooperative Lewis pair facilitates the coupling of CO and ethene in a new way.