On the best constant in a Wente‐type inequality for the fractional Laplace operator

In this paper, we consider the solution to Wente's problem with the fractional Laplace operator (?Δ)^α/2, where 0 < α < 2. We derive a Wente‐type inequality for this problem. Next, we compute the optimal constant in such inequality. Copyright © 2015 John Wiley & Sons, Ltd.     

Biodegradation of Tetracycline Under Various Conditions and Effects on Microbial Community

Five hundred tons of antibiotics are consumed yearly in the world. In this study, the biodegradation characteristics of tetracycline (TET) under nitrate-reducing, sulfate-reducing, and methanogenic conditions were determined by batch tests. Also, effects of TET on mixed microbial cultures were revealed by microbiological analysis. In this scope, gas generation and composition, dissolved organic carbon, and electron acceptor concentrations were monitored during 120 days. Additionally, changes on quantities of specific microbial groups were determined by Q-PCR. TET showed non-biodegradable behavior under nitrate- and sulfate-reducing conditions, whereas slightly biodegradable behavior under methanogenic conditions approximately 46 % degradation. The effects of TET on the abundance of mixed culture varied according to taxonomic units. Sulfate-reducing bacteria were inhibited by TET, while archaeal, bacterial, and methanogenic populations were not affected significantly.     

Synthesis, characterization and biological evaluation of a novel Cu(II) complex with the mixed ligands 2,6-pyridinedicarboxylic acid and 2-aminopyridine

A novel bridged binuclear Cu(II) complex with mixed ligands, di-μ-(2-aminopyridine(N,N′))-bis[(2,6-pyridinedicarboxylate)aquacopper(II)] tetrahydrate, formulated as [Cu(μ-ap)(dipic)(H₂O)]₂·4H₂O (1) (dipic = 2,6-pyridinedicarboxylate, ap = 2-aminopyridine), has been synthesized and characterized by elemental, spectral (IR and UV–Vis.), thermal analysis, magnetic measurements and single crystal X-ray diffraction analysis. The central Cu(II) ion resides on a centre of symmetry in a distorted square-pyramid coordination environment comprising of two N atoms, one from dipic and one from the ap ring, two carboxylate O atoms from dipic, and one O atom from water. Intermolecular N–HO and O–HO hydrogen bonds and π–π stacking interactions seem to be effective in the stabilization of the crystal structure. The free ligands and the complex were also evaluated for their antimicrobial and radical scavenging activities (DPPH = 1,1-diphenyl-2-picrylhydrazyl hydrate) using in vitro microdilution methods. Antimicrobial screening of the free ligands and their complex showed that the free ligands and the complex possess antifungal activity against Candida sp.     

Synthesis of heterocyclic systems with carbohydrate fragment

The reactions of levoglucosenone with amides of several -nitrocarboxylic acids and acetoacetic acid result in tetra hydropyridones fused with a carbohydrate fragment. In the case of acetoacetic acid amides, mixtures of keto and enol tautomers were obtained. The stereochemistry of cyclization is discussed in detail.For Communication 1, see Ref. 1.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 409–414, February, 1996.     

Synthesis and structure of azido-and amino-substituted dibenzoxazepinones

1,3-Dinitro[b, f][1,4]dibenzoxazepin-11(10H)-one enters nucleophilic substitution reactions with N-nucleophiles, azide ion preferably replacing the nitro group in position 3, whereas amines the one in position 1. Structures of the substitution products were confirmed by X-ray diffraction and ¹H NMR NOE spectroscopy. The selectivity observed in the reaction with amines was supposed to be caused by the stabilization of the intermediate σ-complex with the NH…O intramolecular hydrogen bond. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 10, pp. 2018–2022, October, 2007.     

Reaction of Polynitrotoluenes with Phthalic Anhydride

2,4-Dinitro- and 2,4,6-trinitrotoluenes react with phthalic anhydride in boiling N,N-dimethylaniline to give nitro group reduction products.     

Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces

Let X be a non-empty set and F:X×X→X be a given mapping. An element (x,y)∈X×X is said to be a coupled fixed point of the mapping F if F(x,y)=x and F(y,x)=y. In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393].     

On a new generalization of metric spaces

In this paper, we introduce the \({\mathcal {F}}\)-metric space concept, which generalizes the metric space notion. We define a natural topology \(\tau _{{\mathcal {F}}}\) in such spaces and we study their topological properties. Moreover, we establish a new version of the Banach contraction principle in the setting of \({\mathcal {F}}\)-metric spaces. Several examples are presented to illustrate our study.     

On Erdélyi–Kober Quadratic Functional-Integral Equation in Banach Algebra

Using the technique of a suitable measure of noncompactness in Banach algebra, we prove an existence theorem for some functional-integral equations in Banach algebra, namely Erdélyi–Kober quadratic functional-integral equation. This equation contains as particular cases a lot of integral equations that appear in many branches of nonlinear analysis and its applications. The obtained result is illustrated by an example.     

Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations

In this paper, we establish certain fixed point theorems in metric spaces with a partial ordering. Presented theorems extend and generalize several existing results in the literature. As application, we use the fixed point theorems obtained in this paper to study existence and uniqueness of solutions for fourth-order two-point boundary value problems for elastic beam equations.