Chebyshev type inequalities for Hilbert space operators

We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if $\mathscr{A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$, $\alpha: T\rightarrow [0, +\infty)$ is a measurable function and $(A_t)_{t\in T}, (B_t)_{t\in T}$ are suitable continuous fields of operators in ${\mathscr A}$ having the synchronous Hadamard property, then \begin{align*} \int_{T} \alpha(s) d\mu(s)\int_{T}\alpha(t)(A_t\circ B_t) d\mu(t)\geq\left(\int_{T}\alpha(t) A_t d\mu(t)\right)\circ\left(\int_{T}\alpha(s) B_s d\mu(s)\right). \end{align*} We apply states on $C^*$-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive $n\times n$ matrices. Several applications are given as well.Comment: 18 pages, to appear in J. Math. Anal. Appl. (JMAA

Caffeine as A Catalyst for A Four-Component Synthesis of Dihydropyrano [2, 3-C]Pyrazoles in Water

We describe a one-pot four component synthesis of pyrano[2,3-c]pyrazoles from hydrazine hydrate or phenyl hydrazine, ethyl acetoacetate, an aldehyde, and malononitrile in the presence of catalytic amounts of a caffeine as a green catalyst in water at 50 ºC. The present protocol offers the advantages of a clean reaction, a short reaction time, a high product yield, and an environmentally-friendly, easily-purified, and economically-available catalyst

A Highly Efficient and Green Catalytic Synthesis of 3,4-dihydro-pyrimidin-2-(1H)-ones (Thiones) Using 3-sulfonic Acid-1-imidazolopyridinium Hydrogen Sulfate under Solvent-free Conditions

The ionic liquid catalyst 3-sulfonic acid-1-imidazolopyridinium hydrogen sulfate, [Simp]HSO4 was found to be a highly active and green catalyst for the synthesis of 3,4-dihydropyrimidin-2(1H)-ones (thiones) under the solvent-free conditions. Avoiding organic solvents during the chemical reactions leading to an economic approach is effective. The reactions are characterized by high efficiency, short reaction time, high product yield, simple experimental procedure, availability of catalyst, and environmentally-friendly reaction conditions. This work is licensed under a Creative Commons Attribution 4.0 International License

Regioselective Syntheses of 3-Benzyl-Substituted 7H-Thiazolo[3,2-a]pyrimidine-7-ones through Palladium-Catalyzed Heteroannulation of Acetylenic Compounds

An efficient synthesis of 3-benzyl-substituted 7H-thiazolo[3,2-a]pyrimine-7-ones in acetonitrile is accomplished via Pd- and Cu-catalyzed reaction of 2-mercaptopropargylpyrimidone with various aryl iodides in the presence of triethylamine as the base

A new approach for one-pot, green synthesis of new polycyclic indoles in aqueous solution

Electro-oxidation of phenylamine derivatives (1a and 1b) have been studied in the presence of pyrazolidine-3,5-dione (3) as a nucleophile in phosphate buffer solution mixed with ethanol, using voltammetric and spectroscopic techniques. The obtained results indicated that the oxidized form of phenylamines (2a and 2b) participate in Michael addition type reactions with pyrazolidine-3,5-dione (3) and via ECECCCCC mechanisms convert to the corresponding new polycyclic indoles (12a and 12b). In the present study, new polycyclic indole derivatives were synthesized with good yields and high purity using a facile, one-pot and environmentally friendly electrochemical method, without any chemical catalysts, toxic solvents and hard conditions. (C) 2016 Chinese Chemical Society and Institute of Materia Medica, Chinese Academy Medical Sciences. Published by Elsevier B.V. All rights reserved

Clean and Green Synthesis of New Benzothiazole Derivatives via Electrochemical Oxidation of Catechol Derivatives

Electrochemical oxidation of the catechols 1a and 1b is studied in the presence of 6-methyl-2-thouracil (3b) and 6-propyl-2-thiouracil (3a) as nucleophiles in a phosphate buffer (0.15 mol L−1, pH = 6.8)/DMF (95:5) solution using cyclic voltammetry and controlled-potential coulometry. The results obtained indicate that the quinones derived from the catechols participate in 1,4-Michael-addition reactions with the nucleophiles to form the corresponding new benzothiazole compounds. In this work, we derive a variety of products with good yields using controlled potential at graphite electrodes in an undivided cell. <br><a rel="license" href="http://creativecommons.org/licenses/by/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by/4.0/80x15.png" /></a> This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</a>

New Hybrid Compounds from Imidazole and 1,2,3-Triazole: efficient Synthesis of Highly Substituted Imidazoles and Construction of Their Novel Hybrid Compounds by Copper-Catalyzed Click Reaction

New hybrid compounds were successfully prepared from highly substituted imidazoles and 1,2,3-triazoles using copper-catalyzed click reactions. Also, an efficient method described for the synthesis of tri- and tetrasubstituted imidazoles in the presence of ZnCl2/urea as an efficient catalyst by multi-component reaction under solvent-free conditions. The advantages of this one-pot methodology are experimental simplicity, easy work-up, and afford desired products with high to excellent yields. </p