Some new proofs of modular relations for the Göllnitz-Gordon functions

In this paper, we find new proofs of modular relations for the Göllnitz-Gordon functions established earlier by S.-S. Huang and S.-L. Chen. We use Schröter’s formulas and some simple theta-function identities of Ramanujan to establish the relations. We also find some new modular relations of the same nature.     

Banana pulp extract mediated synthesis of Cu2O nanoparticles: An efficient heterogeneous catalyst for the ipso-hydroxylation of arylboronic acids

A green and facile novel procedure has been developed for the synthesis of Cu₂O nanoparticles within a very short reaction time using banana pulp extract as a reducing agent. The synthesized nanoparticles are well characterized by SEM (Scanning Electron Microscope), TEM (Transmission Electron Microscope) and powder XRD (X-ray Diffraction) methods. An environmental benign and highly efficient protocol for the ipso-hydroxylation of aryl and hetero arylboronic acids using bio-fabricated Cu₂O nanoparticles as a catalyst and aqueous H₂O₂ as an oxidant has also been developed. The main advantages of this protocol are the base free reaction condition, reusable and heterogeneous catalytic system, and short reaction time with excellent yields.     

Optical and dielectric properties of isothermally crystallized nano-KNbO3 in Er3+-doped K2O–Nb2O5–SiO2 glasses

Precursor glass of composition 25K₂O–25Nb₂O₅–50SiO₂ (mol%) doped with Er₂O₃ (0.5 wt% in excess) was isothermally crystallized at 800 °C for 0–100 h to obtain transparent KNbO₃ nanostructured glass–ceramics. XRD, FESEM, TEM, FTIRRS, dielectric constant, refractive index, absorption and fluorescence measurements were carried out to analyze the morphology, dielectric, structure and optical properties of the glass–ceramics. The crystallite size of KNbO₃ estimated from XRD and TEM is found to vary in the range 7–23 nm. A steep rise in the dielectric constant of glass–ceramics with heat-treatment time reveals the formation of ferroelectric nanocrystalline KNbO₃ phase. The measured visible photoluminescence spectra have exhibited green emission transitions of ²H_11/2, ⁴S_3/2 → ⁴I_15/2 upon excitation at 377 nm (⁴I_15/2 → ⁴G_11/2) absorption band of Er³⁺ ions. The near infrared (NIR) emission transition ⁴I_13/2 → ⁴I_15/2 is detected around 1550 nm on excitation at 980 nm (⁴I_15/2 → ⁴I_11/2) of absorption bands of Er³⁺ ions. It is observed that photoluminescent intensity at 526 nm (²H_11/2 → ⁴I_15/2), 550 nm (⁴S_3/2 → ⁴I_15/2) and 1550 nm (⁴I_13/2 → ⁴I_15/2) initially decrease and then gradually increase with increase in heat-treatment time. The measured lifetime (τ_f) of the ⁴I_13/2 → ⁴I_15/2 transition also possesses a similar trend. The measured absorption and fluorescence spectra reveal that the Er³⁺ ions gradually enter into the KNbO₃ nanocrystals.     

Structural, optical and electrical properties of tin oxide thin film deposited by APCVD method

Tin oxide (SnO₂) thin films have been grown on glass substrates using atmospheric pressure chemical vapour deposition (APCVD) method. During the deposition, the substrate temperature was kept at 400°C–500°C. The structural properties, surface morphology and chemical composition of the deposited film were studied by X-ray diffraction (XRD), scanning electron microscope (SEM) and Rutherford back scattering (RBS) spectrum. XRD pattern showed that the preferred orientation was (110) having tetragonal structure. The optical properties of the films were studied by measuring the transmittance, absorbance and reflectance spectra between λ = 254 nm to 1400 nm and the optical constants were calculated. Typical SnO₂ film transmits ∼ 94% of visible light. The electrical properties of the films were studied using four-probe method and Hall-voltage measurement experiment. The films showed room temperature conductivity in the range 1.08 × 10² to 1.69 × 10² Ω⁻¹cm⁻¹.     

Exact <Emphasis Type="Italic">S</Emphasis>-wave solution of the Schrödinger equation for three new potentials using the transformation method

We apply the extended transformation method to a non-power-law potential to generate a set of exactly solvable quantum systems in spaces of any dimensions. We derive exact analytic solutions of the Schrödinger equations with an exactly solvable non-power-law potential. For the transformed potentials obtained as a result, we calculate the quantized bound-state energy spectra and the corresponding wave functions.     

Studies on the reactivity of cis-RuCl2 fragment in Ru(PPh3)2(TaiMe)Cl2 with N,N-chelators (TaiMe = 1-methyl-2-(p-tolylazo)imidazole). Spectral and electrochemical characterisation of the products

Dechlorination of Ru(PPh₃)₂(TaiMe)Cl₂ (TaiMe = p-Me-C₆H₄-N=N-C₃H₂NN(1)-Me (1), 1-methyl-2-(p-tolylazo)imidazole) has been carried out in acetone solution by Ag⁺ and reacted with N,N’-chelators to synthesise [Ru(PPh₃)₂ (TaiMe)(N,N’)]²⁺. The complexes have been isolated as their perchlorate salts. The N,N’ chelators are 1-alkyl-2-(phenylazo)imidazoles (PaiX, X = Me, Et, CH₂Ph); 2-(arylazo)pyridines, (Raap,p-R-C₆H₄-N=N-C₅H₄N; R = H, Me, Cl); 2-(arylazo)pyrimidines (Raapm,p-R-C₆H₄-N=N-C₃N₂H₂; R = H, Me, Cl); 2,2’-bipyridine (bpy) and 1,10-phenanthroline (o-phen). Unsymmetrical N,N’ chelators may give two isomers and this is indeed observed. The¹H NMR spectral data refer to the presence of two isomers in the mixture in different proportions. With consideration of coordination pairs in the order of PPh₃, PPh₃; N,N (N refers to N(immidazole)) and N’,N (N’ refers to N(azo)), the complexes have been characterised astrans-cis-cis andtrans-trans-trans configuration; the former predominates in the mixture. Electrochemical studies exhibit high potential Ru(III)/Ru(II) couple and quasireversible N=N reduction. Electronic spectra show high intensity (ε ∼ 10⁴) MLCT transition in the visible region (520 ±10) nm along with a shoulder (ε ∼ 10³) in the longer wavelength region.     

Similarity relations, invertibility and eigenvalues of intuitoinistic fuzzy matrix

In this article, the similarity relations are studied, together with invertibility conditions and eigenvalues of intuitionistic fuzzy matrices (IFMs). Besides, idempotent, regularity, permutation matrix and spectral radius of IFMs are considered here with some properties and results for IFMs investigated.     

An analysis of quantum chromodynamic structure function beyond leading order

We obtain approximate solutions of Altarelli-Parisi equations beyond leading logarithmic approximation. Our results suggest quantitative utility of higher order terms in the structure function analysis of deep inelastic scattering.     

On the derived length of units in group algebra

Let G be a finite group G, K a field of characteristic p ≥ 17 and let U be the group of units in KG. We show that if the derived length of U does not exceed 4, then G must be abelian.     

Synthesis of silver nanoparticles in an aqueous suspension of graphene oxide sheets and its antimicrobial activity

A solution-based approach to the synthesis of silver (Ag) nanoparticles by chemical reduction of AgNO(3) in a graphene oxide (GrO) suspension is demonstrated. X-ray diffraction and transmission electron microscopy indicate that the Ag nanoparticles, of size range 5-25nm, were decorated on the GrO sheets. The size and shape of the Ag nanoparticles are dependent on the concentration of the AgNO(3) solution. Antimicrobial activity of such hybrids materials is investigated against the Gram negative bacteria Escherichia coli and Pseudomonous aeruginosa. The bacterial growth kinetics was monitored in nutrient broth supplemented with the Ag nanoparticle-GrO suspension at different conditions. It was observed that P. aeruginosa is comparatively more sensitive to the Ag nanoparticle-GrO suspension.