The Mellin Transform in Quantum Calculus

A q-analogue of the Mellin transform is introduced by using a standard method of q-calculus involving the q-Jackson integral. In this paper, we study some of its properties coinciding with the corresponding classical ones when q tends to 1. In addition to several examples given, we establish the q-inversion formula, the q-analogue of the convolution product, and the q-extension of the known Titchmarsh theorem. Finally, we prove the q-Mellin summation formula related to some q-zeta function.     

(C2N2H10)2Mg(HP2O7)2·2H2O synthesis and crystal structure

(C₂N₂H₁₀)₂Mg(HP₂O₇)₂·2H₂O, is a new inorganic organic hybrid structure. It has been synthetized using wet chemistry. Its crystal structure consists of cis- and trans-edge sharing [MgO₄(H₂O)₂] octahedra resulting in chains, which are linked via [HP₂O₇] units to form [Mg(HP₂O₇)₂(H₂O)₂]⁴⁻ layers. The Mg²⁺ cations and the ethylendiammonium cations are located on centers of inversion. The ethylendiammonium cations are alternately located in the interlayer space. The cohesion of the crystal is well ensured by coulombic interactions between anions and cations and by several hydrogen bonds. The diphosphate anion shows an eclipsed conformation.     

Triple scale analysis of periodic solutions and resonance of some asymmetric non linear vibrating systems

We consider small solutions of a vibrating mechanical system with smooth non-linearities for which we provide an approximate solution by using a triple scale analysis; a rigorous proof of convergence of the triple scale method is included; for the forced response, a stability result is needed in order to prove convergence in a neighbourhood of a primary resonance. The amplitude of the response with respect to the frequency forcing is described and it is related to the frequency of a free periodic vibration.     

Symbolic-numeric Gaussian cubature rules

It is well known that Gaussian cubature rules are related to multivariate orthogonal polynomials. The cubature rules found in the literature use common zeroes of some linearly independent set of products of basically univariate polynomials. We show how a new family of multivariate orthogonal polynomials, so-called spherical orthogonal polynomials, leads to symbolic-numeric Gaussian cubature rules in a very natural way. They can be used for the integration of multivariate functions that in addition may depend on a vector of parameters and they are exact for multivariate parameterized polynomials. Purely numeric Gaussian cubature rules for the exact integration of multivariate polynomials can also be obtained.We illustrate their use for the symbolic-numeric solution of the partial differential equations satisfied by the Appell function F₂, which arises frequently in various physical and chemical applications. The advantage of a symbolic-numeric formula over a purely numeric one is that one obtains a continuous extension, in terms of the parameters, of the numeric solution. The number of symbolic-numeric nodes in our Gaussian cubature rules is minimal, namely m for the exact integration of a polynomial of homogeneous degree 2m−1.In Section 1 we describe how the symbolic-numeric rules are constructed, in any dimension and for any order. In Sections 2, 3 and 4 we explicit them on different domains and for different weight functions. An illustration of the new formulas is given in Section 5 and we show in Section 6 how numeric cubature rules can be derived for the exact integration of multivariate polynomials. From Section 7 it is clear that there is a connection between our symbolic-numeric cubature rules and numeric cubature formulae with a minimal (or small) number of nodes.     

Convenient Synthesis of 1,2-Diamines from β-Chloro Amines: Precursors of New Substituted Piperazin-2-ones

A practical synthesis of 1,2-diamines from β-chloro amines is reported. The reaction proceeds through the intermediacy of an aziridinium ion opening by amines. The conversion of these diamines into polysubstituted piperazinones is also investigated.

Supplemental materials are available for this article. Go to the publisher's online edition of Synthetic Communications® to view the free supplemental file.     

A One-Pot Synthesis of trans-N-Alkylaziridine-2-carboxylates from Amino Alcohol Esters

A series of new trans-N-alkylaziridine-2-carboxylates 3 was conveniently synthesized in a one-pot reaction by treatment of 1,2-amino alcohol mono and diesters 1 with methanesulfonyl chloride in the presence of (iPr)₂NEt. The products were obtained in good to excellent yields with high trans-selectivity.

Angular dependence of the critical current density and the temperature scaling of the flux pinning force in YBCO thin film

In this paper, the critical current J_c(Θ) have been investigated as a function of magnetic-field angle Θ. Θ is the angle between the c-axis and the applied magnetic field direction. This investigation concerned three temperature values (60?K, 78?K and 81?K). The normalized pinning force f_p versus the normalized magnetic field h was also studied (f_p?=?F_p / F_pmax and h?=?H / H_max). The F_p expression was determined based on the Kramer model.The studied sample was a single crystal of YBaCuO thin film deposited by the ablation laser method on the surface (100) of a SrTiO₃ substrate.The results of this work show the existence of point core pinning of the normal centers in the low field regime and the occurrence of the flux creep in high field regime.     

Evaluation of the UFXC32k photon‐counting detector for pump–probe experiments using synchrotron radiation

This paper presents the performance of a single‐photon‐counting hybrid pixel X‐ray detector with synchrotron radiation. The camera was evaluated with respect to time‐resolved experiments, namely pump–probe–probe experiments held at SOLEIL. The UFXC camera shows very good energy resolution of around 1.5 keV and allows the minimum threshold setting to be as low as 3 keV keeping the high‐count‐rate capabilities. Measurements of a synchrotron characteristic filling mode prove the proper separation of an isolated bunch of photons and the usability of the detector in time‐resolved experiments.     

A General Synthetic Route to High-Quality Perovskite Oxide Nanoparticles and Their Enhanced Solar Photocatalytic Activity

The main limitations of current methods for synthesizing perovskite oxide (ABO₃) nanoparticles (NPs), e.g., the high reagent costs and sophisticated equipment, the long time and high-temperature processing, or multiple post-processing and thermal treatment steps, hamper their full study and potential application. Here, we use a facile low temperature (50 °C) chemical bath synthesis and only one annealing step to successfully produce high phase purity and crystalline quality nano-shaped rare-earth-based REMO₃ NPs (RE=La, Nd, Sm, Gd; M=Fe, Mn, Al). We also show the versatility of this approach by fabricating La_0.7Sr_0.3MnO₃ solid solution and non-RE-based BiFeO₃ perovskite. To assess the potential of the as-prepared REFeO₃ and REMnO₃ NPs, they are used for photocatalytic degradation of the norfloxacin antibiotic and show high efficiency. We believe this easy, robust, versatile, and general route for synthesizing ABO₃-based NPs can be further explored in the vast perovskite family and beyond.