Topological multiplicity of the maximum eigenvalue of graph 1-Laplacian

In this paper, we reveal the connection between the independent number of a graph and the topological multiplicity of the maximal eigenvalue of the corresponding graph 1-Laplacian. The pseudo independent number of a graph is introduced, which provides a better lower estimate of the topological multiplicity of the maximum eigenvalue. The technique of our proof is based on the localization property of the eigenvector for graph 1-Laplacian, the Krasnoselski genus, and its relation to the topological join.     

Homogeneous variable exponent Besov and Triebel–Lizorkin spaces

We introduce homogeneous Besov and Triebel–Lizorkin spaces with variable indexes. We show that their study reduces to the study of inhomogeneous variable exponent spaces and homogeneous constant exponent spaces. Corollaries include trace space characterizations and Sobolev embeddings.     

Influence of ionic liquid counterions on activity and selectivity of ethylene trimerization using chromium-based catalysts in biphasic media

In recent years ionic liquids (ILs) have attracted much interest because of their widespread use in various fields. Several trimerization and oligomerization catalysts have been evaluated in ILs with different organic–inorganic hybrid structures. High catalytic activity and selectivity, easy product separation, and recycling of the catalyst are the advantages of a biphasic catalyst system compared to the homogeneous catalysts. In this study, the influence of IL counter-anions on activity and selectivity of the ethylene trimerization catalysts based on Cr-SNS-R was investigated. All synthesized materials were characterized using Fourier-transform infrared spectroscopy, ¹H NMR, ¹³C NMR, UV–Vis. spectroscopy, thin-layer chromatography, and elemental analysis (CHNS). In ethylene trimerization reaction, the dodecyl substituent in the SNS ligand exhibited better activity and selectivity than the butyl substituent. The results revealed that the presence of BF₄⁻ as a counter-anion in the IL led to an increase in activity and selectivity compared to Br⁻ and I⁻ counter-anions. It was found that the BF₄⁻ counter-anion plays a conclusive role in the development of 1-hexene activity and selectivity to a maximum amount of 71,132 g_1-C6/(g_Cr × h) and more than 99%, respectively. Finally, the catalyst was reused thrice without losing its 1-hexene selectivity.     

Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness

We prove a sharp estimate for the k-modulus of smoothness, modelled upon a $L^p$-Lebesgue space, of a function f in $W^k{L}^{\frac{pn}{n+kp},p}(\mathrm{\Omega })$, where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, $k,n\in \mathbb{N}$, $k<n$ and $\frac{n}{n?k}<p<+\infty$. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.     

Facile preparation of Ag/Ag2WO4/g‐C3N4 ternary plasmonic photocatalyst and its visible‐light photocatalytic activity

Introducing plasmonic metals into semiconductor materials has been proven to be an attractive strategy for enhancing photocatalytic activity in the visible region. In this work, a novel and efficient Ag/Ag₂WO₄/g‐C₃N₄ (AACN) ternary plasmonic photocatalyst was successfully synthesized using a facile one‐step in situ hydrothermal method. The composition, structure, morphology and optical absorption properties of AACN were investigated using X‐ray diffraction, Fourier transform infrared spectroscopy, scanning electron microscopy, and UV–visible diffuse reflectance spectroscopy, respectively. Photocatalytic performance of AACN was evaluated via rhodamine B and tetracycline degradation. The results indicated that AACN had excellent photocatalytic performance for rhodamine B degradation with a rate constant of 0.0125 min^?1, which was higher than those of Ag₂WO₄ and Ag/Ag₂WO₄. Characterization and photocatalytic tests showed that the strong coupling effect between the Ag/Ag₂WO₄ nanoparticles and the exfoliated ultrathin g‐C₃N₄ nanosheets was superior for visible‐light responsivity and reduced the recombination rate of photogenerated electrons and holes. A proposed mechanism is also discussed according to the band energy structure and the experimental results.     

Biosynthesis of Pd/MnO2 nanocomposite using Solanum melongena plant extract and its application for the one‐pot synthesis of 5‐substituted 1H‐tetrazoles from aryl halides

In this work, for the first time, Solanum melongena plant extract was used for the green synthesis of Pd/MnO₂ nanocomposite via reduction osf Pd(II) ions to Pd(0) and their immobilization on the surface of manganese dioxide (MnO₂) nanoparticles (NPs) as an effective support. The synthesized nanocomposite were characterized by various analytical techniques such as Fourier transform infrared (FT‐IR), X‐ray diffraction (XRD), transmission electron microscopy (TEM), field emission scanning electron microscopy (FESEM), energy dispersive X‐ray spectroscopy (EDS) and UV–Vis spectroscopy. The catalytic activity of Pd/MnO₂ nanocomposite was used as a heterogeneous catalyst for the one‐pot synthesis of 5‐substituted 1H‐tetrazoles from aryl halides containing various electron‐donating or electron‐withdrawing groups in the presence of K ₄ [Fe (CN) ₆ ] as non‐toxic cyanide source and sodium azide. The products were obtained in good yields via a simple methodology and easy work‐up. The nanocatalyst can be recycled and reused several times with no remarkable loss of activity.     

Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function on the upper half‐spaces. Using the decomposition, we study the boundedness of . We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

Quantitative estimations of bivariate summation‐integral–type operators

In this paper, we study the approximation properties of bivariate summation‐integral–type operators with two parameters . The present work deals within the polynomial weight space. The rate of convergence is obtained while the function belonging to the set of all continuous and bounded function defined on ([0],∞)(×[0],∞) and function belonging to the polynomial weight space with two parameters, also convergence properties, are studied. To know the asymptotic behavior of the proposed bivariate operators, we prove the Voronovskaya type theorem and show the graphical representation for the convergence of the bivariate operators, which is illustrated by graphics using Mathematica. Also with the help of Mathematica, we discuss the comparison by means of the convergence of the proposed bivariate summation‐integral–type operators and Szász‐Mirakjan‐Kantorovich operators for function of two variables with two parameters to the function. In the same direction, we compute the absolute numerical error for the bivariate operators by using Mathematica and is illustrated by tables and also the comparison takes place of the proposed bivariate operators with the bivariate Szász‐Mirakjan operators in the sense of absolute error, which is represented by table. At last, we study the simultaneous approximation for the first‐order partial derivative of the function.     

Inhibition effect of butan-1-ol on the corrosion behavior of austenitic stainless steel (Type 304) in dilute sulfuric acid

The electrochemical behavior of austenitic stainless steel (Type 304) in 3 M sulfuric acid with 3.5% recrystallized sodium chloride at specific concentrations of butan-1-ol was investigated with the aid of potentiodynamic polarization, open circuit measurement and weight loss technique. Butan-1-ol effectively inhibited the steel corrosion with a maximum inhibition efficiency of 78.7% from weight-loss analysis and 80.9% from potentiodynamic polarization test at highest concentration studied. Adsorption of the compound obeyed the Freundlich isotherm. Thermodynamic calculations reveal physiochemical interactions and spontaneous adsorption mechanism. Surface characterizations showed the absence of corrosion products and topographic modifications of the steel. Statistical analysis depicts the overwhelming influence and statistical significance of inhibitor concentration on the inhibition performance.