Restricted testing conditions for the multilinear maximal operator

Restricted testing conditions were considered recently. For the maximal operator, Hytönen, Li and Sawyer [8] first obtained parental testing condition. Later, they [9] showed that it suffices to restrict testing to doubling cubes. Chen and Lacey [3] gave a similar restricted testing condition. In our paper, we discuss a version of the latter in the multilinear setting.

     

Biosynthesis of zinc oxide nanoparticles using Euphorbia milii leaf constituents: Characterization and improved photocatalytic degradation of methylene blue dye under natural sunlight

A facile biosynthesis route was followed to prepare zinc oxide nanoparticles (ZnO NPs) using Euphorbia milii (E. milii) leaf constituents. The SEM images exhibited presence of spherical ZnO NPs and the corresponding TEM images disclosed monodisperse nature of the ZnO NPs with diameter ranges between 12 and 20 nm. The Brunauer–Emmett–Teller (BET) analysis revealed that the ZnO NPs have specific surface area of 20.46 m²/g with pore diameter of 2 nm–10 nm and pore volume of 0.908 cm³/g. The EDAX spectrum exemplified the existence of Zn and O elements and non-appearance of impurities that confirmed pristine nature of the ZnO NPs. The XRD pattern indicated crystalline peaks corresponding to hexagonal wurtzite structured ZnO with an average crystallite size of 16.11 nm. The FTIR spectrum displayed strong absorption bands at 512 and 534 cm^?1 related to ZnO. The photocatalytic action of ZnO NPs exhibited noteworthy degradation of methylene blue dye under natural sunlight illumination. The maximum degradation efficiency achieved was 98.17% at an illumination period of 50 min. The reusability study proved considerable photostability of the ZnO NPs during photocatalytic experiments. These findings suggest that the E. milii leaf constituents can be utilized as suitable biological source to synthesis ZnO NPs for photocatalytic applications.     

Nickel/Photoredox-Catalyzed Asymmetric Reductive Cross-Coupling of Racemic α-Chloro Esters with Aryl Iodides

A unique nickel/organic photoredox co-catalyzed asymmetric reductive cross-coupling between α-chloro esters and aryl iodides is developed. This cross-electrophile coupling reaction employs an organic reductant (Hantzsch ester), whereas most reductive cross-coupling reactions use stoichiometric metals. A diverse array of valuable α-aryl esters is formed under these conditions with high enantioselectivities (up to 94 %) and good yields (up to 88 %). α-Aryl esters represent an important family of nonsteroidal anti-inflammatory drugs. This novel synergistic strategy expands the scope of Ni-catalyzed reductive asymmetric cross-coupling reactions.     

Integral Representations of the Large and Little Schröder Numbers

In the paper, the authors establish several integral representations for the generating functions of the large and little Schröder numbers and for the large and little Schröder numbers.     

Vinylic MIDA Boronates: New Building Blocks for the Synthesis of Aza‐Heterocycles

A two‐step synthesis of structurally diverse pyrrole‐containing bicyclic systems is reported. ortho‐Nitro‐haloarenes coupled with vinylic N‐methyliminodiacetic acid (MIDA) boronates generate ortho‐vinyl‐nitroarenes, which undergo a “metal‐free” nitrene insertion, resulting in a new pyrrole ring. This novel synthetic approach has a wide substrate tolerance and it is applicable in the preparation of more complex “drug‐like” molecules. Interestingly, an ortho‐nitro‐allylarene derivative furnished a cyclic β‐aminophosphonate motif.