Pd(0)‐ S‐propyl‐2‐aminobenzothioate immobilized onto functionalized magnetic nanoporous MCM‐41 as efficient and recyclable nanocatalyst for the Suzuki,Stille and Heck cross coupling reactions

The present work describes the use of Pd(0)‐ S‐propyl‐2‐aminobenzothioate Complex immobilized onto functionalized magnetic nanoporous MCM‐41(Fe₃O₄@MCM‐41@Pd‐SPATB) as efficient and recyclable nano‐organometallic catalyst for C–C bond formation between various aryl halides with phenylboronic acid (Suzuki reaction), aryl halides with triphenyltin chloride (Stille reaction), and aryl halides with n‐butyl acrylate (Heck reaction). All the reactions were carried out in PEG‐400 as green solvent with short reaction time and good to excellent yields. This catalyst was characterized by FT‐IR spectroscopy, XRD, TGA, VSM, ICP‐OES, TEM, EDX and SEM techniques. Ease of operation, high efficiency, recovery and reusability for five continuous cycles without significant loss of its catalytic activities or metal leaching are the noteworthy features of the currently employed heterogeneous catalytic system.     

Anchoring of Cu(II)–vanillin Schiff base complex on MCM‐41: A highly efficient and recyclable catalyst for synthesis of sulfides and 5‐substituted 1H–tetrazoles and oxidation of sulfides to sulfoxides

A copper(II)–vanillin complex was immobilized onto MCM‐41 nanostructure and was used as an inexpensive, non‐toxic and heterogeneous catalyst in the synthesis of symmetric aryl sulfides by the cross‐coupling of aromatic halides with S₈ as an effective sulfur source, in the oxidation of sulfides to sulfoxides using 30% H₂O₂ as a green oxidant and in the synthesis of 5‐substituted 1H –tetrazoles from a smooth (3 + 2) cycloaddition of organic nitriles with sodium azide (NaN₃). The products were obtained in good to excellent yields. This catalyst could be reused several times without loss of activity. Characterization of the catalyst was performed using Fourier transform infrared, energy‐dispersive X‐ray and atomic absorption spectroscopies, X‐ray diffraction, thermogravimetric analysis, and scanning and transmission electron microscopies.     

Strichartz estimates for non-degenerate Schrödinger equations

We consider the Schrödinger equation with a non-degenerate metric on the Euclidean space. We study local in time Strichartz estimates for the Schrödinger equation without loss of derivatives including the endpoint case. In contrast to the Riemannian metric case, we need the additional assumptions for the well-posedness of our Schrödinger equation and for proving Strichartz estimates without loss.     

Zirconium@guanine@MCM-41 nanoparticles: An efficient heterogeneous mesoporous nanocatalyst for one-pot,multi-component tandem Knoevenagel condensation–Michael addition–cyclization Reactions

MCM-41-supported nanoscale guanine bonded with Zr (IV) was prepared using sol–gel method and characterized by FT-IR, Raman, XRD, BET, TGA, EDX, ICP, AAS, X-Ray mapping, SEM and TEM techniques. This compound was employed as an efficient, chemoselectivity and green heterogeneous catalyst in order to prepare a series of benzo[a]benzo[6,7]chromeno[2,3-c]phenazine and spiro[benzo[a]benzo[6,7]chromeno[2,3-c] phenazine] derivatives by one-pot, four-component domino reaction from the 2-hydroxy-1,4-naphthoquinone, benzene-1,2-diamines, 2-hydroxy-1,4-naphthoquinone and carbonyl compounds in PEG, at 100 °C and, also, Bis (pyrazolyl) methane derivatives using aldehydes and 3-Methyl-1-phenyl-5-pyrazolone in ethanol under reflux condition. Results indicated that all products were synthesized in short reaction times and high yields in the ranges 78–99%. The Zr-guanine-MCM-41 can be recycled four runs without any significant loss of its catalytic activity. In addition, the stability of the catalyst was confirmed by metal leaching, heterogeneity tests, SEM and FT-IR techniques.     

Primes dividing both 2 n –3 and 3 n –2 are rare

Let S(x) denote the number of primes p < x which divide both 2ⁿ–3 and 3ⁿ–2 for some We prove that

One-Dimensional Diffusions and Approximation

Consider the Voronovskaja operator A of a sequence of positive linear operators and let u(t, x) be the solution of the Cauchy problem for A. In the spirit of Altomare’s theory this solution can be studied by using the semigroup (T(t))_{t ≥ 0} generated by A and represented in terms of the operators L_n.One associates to A a stochastic equation; its solution can be also used in order to represent u(t, x).The relations between all these objects are described in the case of the operator A associated with some Meyer-König and Zeller type operators.     

On best error bounds for approximation by piecewise polynomial functions

Summary An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.The work presented in this paper was supported by the ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University, under Contract E(11-1)-3077 with the Energy Research and Development Administration.     

Synthesis,characterization of Nafion-functionalized MCM-41 and its catalytic application in preparation of CL-20 via HNO3 electrolyte involved nitration of TAIW

Hybrid organic–inorganic MCM-41 (Mobil Composition of Matter No. 41) silicas functionalized with Nafion (perfluoroalkylsulfonic acid analogous) were prepared and characterized by Fourier Transform infrared spectroscopy (FTIR), X-ray diffraction (XRD), and N₂ adsorption analysis. The Brønsted acidity was determined by UV spectrophotometer. The prepared catalyst (SA/MCM-41) exhibited high catalytic activity in the nitration of TAIW (tetraacetylhexaazaisowurtzitan) aiming at synthesizing CL-20 (hexanitrohexaazaisowurtzitane), with the yield up to 93%. The leaching problem was not observed after several runs, demonstrating that the catalyst could be recycled and reused without losing activity.     

Construction of conjugate quadrature filters with specified zeros

Let ℂ denote the complex numbers and denote the ring of complex-valued Laurent polynomial functions on ℂ\{0}. Furthermore, we denote by the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials , which have no common zeros in ℂ\{0} there exists a pair of Laurent polynomials satisfying the equation Q ₁ P ₁ + Q ₂ P ₂ = 1. We provide some information about the minimal length Laurent polynomials Q ₁ and Q ₂ with these properties and describe an algorithm to compute them. We apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset Λ⊂ℂ\{0} with the property that for every implies and . This revised version was published online in June 2006 with corrections to the Cover Date.