An identity for the divisor generating function arising from sorting theory

In the present paper we prove

$\text{∑}\text{n=1}\text{∞}\text{nx}{}^{\mathrm{n}}\text{∏}\text{j=n+I}\text{∞}\text{(1?x}{}^{\mathrm{j}}\text{)}$

$\text{=}\text{∑}\text{n=1}\text{∞}\text{(?1)}{}^{\mathrm{n?1}}\text{x}{}^{\mathrm{<mtext>n(n+1)</mtext><mtext>2</mtext>}}\text{(1?x)(1?x}{}^2\text{)?(1?x}{}^{\mathrm{n?1}}\text{)(1?x}{}^{\mathrm{n}}\text{)}{}^2$

$\text{=}\text{∑}\text{n=1}\text{∞}\text{d(n)x}{}^{\mathrm{n}}$

     

Adsorption Characteristics of Pb(II) and Cr(III) onto C‐4‐Methoxyphenylcalix[4]Resorcinarene in Batch and Fixed Bed Column Systems

A study on the adsorption characteristics of Pb(II) and Cr(III) cations onto C‐4‐methoxyphenylcalix‐[4]resorcinarene (CMPCR) in batch and fixed bed column systems has been conducted. CMPCR was produced by one step synthesis from resorcinol, 4‐methoxybenzaldehyde, and HCl. The synthesis was carried out at 78 °C for 24 hours and afforded the adsorbent in 85.7% as a 3:2 mixture of C_4ν:C_2ν isomer. Most parameters in batch and fixed bed column systems confirm that CMPCR is a good adsorbent for Pb(II) and Cr(III), though Pb(II) adsorption was more favorable than that of Cr(III). The adsorption kinetic of Pb(II) and Cr(III) adsorptions in batch and fixed bed column systems followed a pseudo 2ⁿ order kinetics model. The rate constant of Pb(II) was higher than that of Cr(III) in the batch system, but this result was contrary to the result obtained in a fixed bed column system. Desorption studies to recover the adsorbed Pb(II) and Cr(III) were performed sequentially with distilled water and HCl, and the results showed that the adsorption was dominated by chemisorption.     

Synthesis of ZrO<Subscript>2</Subscript> nanoparticles for liquid scintillators used in the detection of neutrinoless double beta decay

We synthesized liquid scintillators incorporating ZrO₂ nanoparticles for application in neutrinoless double beta decay experiments. ZrO₂ nanoparticles of less than 10 nm in size were synthesized with sub- and supercritical hydrothermal methods. The Zr concentrations in the liquid scintillators were determined to be up to 1.4 wt% with inductively coupled plasma analysis, and the liquid scintillators were transparent to scintillation. These results indicate that these methods are applicable for the preparation of liquid scintillators for neutrinoless double beta decay experiments.     

Water inclusion as a trigger for modulation of anthracene arrangement and fluorescence emission of organic salt

An organic salt composed 9,10-bis(4-aminophenyl)anthracene and hypophosphorous acid emits yellow-green fluorescence in the crystalline state. The salt also gives water inclusion crystal which provides blue-green fluorescence. Single X-ray crystal studies revealed that water molecules induced the fluorescence change resulting from reaggregation of molecular packing.     

5,11-Diazadibenzo[hi,qr]tetracene: Synthesis,Properties, and Reactivity toward Nucleophilic Reagents

5,11-Diazadibenzo[hi,qr]tetracene was synthesized as a new nitrogen-substituted polycyclic heteroaromatic compound by Pd-catalyzed cycloisomerization of an alkyne precursor followed by oxidative cyclization with bis(trifluoroacetoxy)iodobenzene. The substitution of imine-type nitrogen atoms significantly enhanced its electron-accepting character and facilitated the direct nucleophilic addition of arylamines under strongly basic conditions to afford the desired amino-substituted products. The introduction of amino groups induced a remarkable red-shift in their absorption spectra; the tetrasubstituted product exhibited intense near-infrared absorbing property. Furthermore, the π-electronic system, which includes a redox-active 1,4-diazabutadiene moiety, underwent reversible interconversion to its corresponding reduced form upon reduction with NaBH₄ and aerobic oxidation.     

Photoredox‐Catalyzed Alkenylation of Benzylsulfonium Salts

Visible light‐mediated radical alkenylation of benzylsulfonium salts was achieved by means of fac‐Ir(ppy)₃ as a photocatalyst, giving allylbenzenes as products. A variety of functional groups, such as halogen, ester, and cyano, were well tolerated in this transformation. Starting benzylsulfonium salts could be readily prepared from benzyl alcohols by an acid‐mediated substitution, increasing the synthetic utility of this transformation.     

Homo‐double helix formation of an optically active conjugated polymer bearing carboxy groups and amplification of the helicity upon complexation with achiral and chiral amines

An optically active, m‐terphenyl‐based π‐conjugated polymer bearing carboxy groups was synthesized by the copolymerization of the diethynyl monomer bearing a carboxy group with (S,S)‐2,5‐bis(2‐methylbutoxy)‐1,4‐dibromobenzene using Sonogashira reaction. The copolymer showed a weak circular dichroism (CD) in the main‐chain chromophore region due to a homo‐double helix formation with an excess helical handedness biased by the chiral alkoxy substituents through self‐association. However, upon complexation with achiral amines, such as piperidine, the CD intensity of the polymer significantly increased resulting in the formation of a greater excess one‐handed homo‐double helix via hydrogen‐bonded inclusion complexation with the achiral amines between each strand, leading to the amplification of the helicity. A preferred‐handed homo‐double helix was also induced in the polymer in the presence of nonracemic amines. The effect of the achiral and chiral amines on the homo‐double helix formation was investigated by comparing the CD spectra of the polymer to those of its model dimer. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 990–999     

Stereoselective analysis of flecainide enantiomers using reversed‐phase liquid chromatography for assessing CYP2D6 activity

Stereoselective analyses of flecainide enantiomers were performed using reversed‐phase high‐performance liquid chromatography (HPLC) equipped with a polysaccharide‐based chiral column (Chiralpak AS‐RH) and fluorescence detector. Excitation and emission wavelengths were set at 300 and 370 nm, respectively. Flecainide enantiomers in serum and urine were extracted using diethyl ether. The mobile phase solution, comprising 0.1 m potassium hexafluorophosphate and acetonitrile (65:35, v/v), was pumped at a flow rate of 0.5 mL/min. The recoveries of flecainide enantiomers were greater than 94%, with the coefficients of variation (CVs) <6%. The calibration curves of flecainide enantiomers in serum and urine were linear in the concentration range 5–500 ng/mL and 0.75–15 µg/mL (r > 0.999), respectively. CVs in intra‐day and inter‐day assays were 1.8–5.8 and 3.4–7.5%, respectively. In a pharmacokinetic study, the ratios of (S)‐ to (R)‐flecainide (S/R ratio) in the area under the curve and the amount of flecainide enantiomers excreted in urine were lower in a subject carrying CYP2D6*10/*10 than in subjects carrying CYP2D6*1/*2. The S/R ratio of trough serum flecainide concentration ranged from 0.79 to 1.16 in patients receiving oral flecainide. The present HPLC method can be used to assess hepatic flecainide metabolism in a pharmacokinetic study and therapeutic drug monitoring. Copyright © 2014 John Wiley & Sons, Ltd.     

Encapsulation and Enhanced Luminescence Properties of IrIII Complexes within a Hexameric Self‐Assembled Capsule

Encapsulation and luminescence studies of [Ir(ppy)₂(bpy)]Cl (ppy=2‐phenylpyridinate, bpy=2,2′‐bipyridine) within a hexameric resorcinarene capsule are reported. One Ir^III complex cation was encapsulated within the capsule, as demonstrated by NMR and dynamic light scattering (DLS) studies. The emission color of the Ir^III complex was drastically changed from orange to yellow by encapsulation, in contrast with the lack of significant changes in the absorption spectrum. The hexameric capsule effectively hampers the non‐radiative pathway to increase both the luminescence quantum yield and the exited state lifetime. The luminescent properties of the encapsulated Ir^III complex depend on the ratio of Ir^III complex to the resorcinarene monomer as well as the concentration of resorcinarene monomer owing to the reversible process of self‐assembly of the hexameric capsule. Quenching experiments revealed that the Ir^III complex in the capsule was effectively separated from quenchers.