Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

Synthesis,structures, and electrochemistry of diiron toluene-3,4-dithiolate complexes containing phosphine ligands

Transition Metal Chemistry - In this paper, four diiron toluene-3,4-dithiolate complexes with phosphine ligands were synthesized and characterized. Treatment of complex...     

Solubility and preferential solvation of acetaminophen in methanol + water mixtures at 298.15 K

The equilibrium solubility of acetaminophen in methanol + water binary mixtures at 298.15 K was determined and correlated with the Jouyban–Acree model. Preferential solvation parameters by methanol (δx_1,3) were derived from their thermodynamic solution properties by means of the inverse Kirkwood–Buff integrals method. δx_1,3 values are negative in water-rich mixtures but positive in compositions from 0.32 in mole fraction of methanol to pure methanol. It is conjecturable that in the former case, the hydrophobic hydration around non-polar groups plays a relevant role in the solvation. The higher solvation by methanol in mixtures of similar cosolvent compositions and methanol-rich mixtures could be explained in terms of the higher basic behavior of this cosolvent.     

Directed Electron Delivery from a Pb-Free Halide Perovskite to a Co(II) Molecular Catalyst Boosts CO2 Photoreduction Coupled with Water Oxidation

The development of high-performance photocatalytic systems for CO₂ reduction is appealing to address energy and environmental issues, while it is challenging to avoid using toxic metals and organic sacrificial reagents. We here immobilize a family of cobalt phthalocyanine catalysts on Pb-free halide perovskite Cs₂AgBiBr₆ nanosheets with delicate control on the anchors of the cobalt catalysts. Among them, the molecular hybrid photocatalyst assembled by carboxyl anchors achieves the optimal performance with an electron consumption rate of 300±13 μmol g⁻¹ h⁻¹ for visible-light-driven CO₂-to-CO conversion coupled with water oxidation to O₂, over 8 times of the unmodified Cs₂AgBiBr₆ (36±8 μmol g⁻¹ h⁻¹), also far surpassing the documented systems (<150 μmol g⁻¹ h⁻¹). Besides the improved intrinsic activity, electrochemical, computational, ex-/in situ X-ray photoelectron and X-ray absorption spectroscopic results indicate that the electrons photogenerated at the Bi atoms of Cs₂AgBiBr₆ can be directionally transferred to the cobalt catalyst via the carboxyl anchors which strongly bind to the Bi atoms, substantially facilitating the interfacial electron transfer kinetics and thereby the photocatalysis.     

Semiconductor SERS on Colourful Substrates with Fabry-Pérot Cavities

Non-metallic materials have emerged as a new family of active substrates for surface-enhanced Raman scattering (SERS), with unique advantages over their metal counterparts. However, owing to their inefficient interaction with the incident wavelength, the Raman enhancement achieved with non-metallic materials is considerably lower with respect to the metallic ones. Herein, we propose colourful semiconductor-based SERS substrates for the first time by utilizing a Fabry-Pérot cavity, which realize a large freedom in manipulating light. Owing to the delicate adjustment of the absorption in terms of both frequency and intensity, resonant absorption can be achieved with a variety of non-metal SERS substrates, with the sensitivity further enhanced by ≈100 times. As a typical example, by introducing a Fabry-Pérot-type substrate fabricated with SiO₂/Si, a rather low detection limit of 10⁻¹⁶ M for the SARS-CoV-2S protein is achieved on SnS₂. This study provides a realistic strategy for increasing SERS sensitivity when semiconductors are employed as SERS substrates.     

J-aggregation of a water-soluble tetracationic porphyrin in mixed LB films with a calix[8]arene carboxylic acid derivative

The molecular organization of a mixed film, containing a water-soluble tetracationic porphyrin (TMPyP) and a p-tert-butyl calix[8]arene octacarboxylic acid derivative (C8A), at the air-water interface and on a solid support (LB film), has been investigated. Although the TMPyP aggregation was not detected at the air-water interface, TMPyP J-aggregates have been found in the LB films (Y-type). Unlike tetraanionic porphyrins, for example TSPP, the TMPyP J-aggregates are not induced by a zwitterion formation. The TMPyP J-aggregation is a result of a "double comb" configuration, where porphyrins from opposite layers are interwoven in a linear infinite J-aggregate. Our results confirm that TMPyP molecules tend to self-aggregate strongly, provided the electrostatic repulsions of their peripheral groups are cancelled by the anionic groups of the C8A matrix.     

Liquid chromatographic method development for steroids determination (corticoids and anabolics). Application to animal feed samples

A LC isocratic separation study of a complex mixture containing 18 steroids (corticoids and anabolics), used potentially as growth promoters, was carried out. For this purpose, using a Hypersil ODS column at controlled temperature, mobile phases (from binary to quaternary) prepared from water and MeOH, ACN or THF as organic modifiers and UV detection at 245 nm, were employed (dehydroepiandrosterone was detected at 200 nm). The optimum separation was achieved using water/acetonitrile (65:35, v/v) as mobile phase at 30 degrees C, allowing the separation of 16 out of 18 steroids in about 30 min. The retention scale using optimized binary mobile phases was related with steroids hydrophobicity and structure, allowing a classification into three groups for these compounds. To improve the separation several alkyl-silica packings were tested: Type A (Lichrospher C8) and Type B (Luna C18, Kromasil C18, Purospher C18 and Synergy C12). Taking into account resolution, number of separated compounds and run time analysis the Hypersil column was selected as the best choice for further applications. Calibration graphs were obtained using fluorocortisone, fluoxymesterone or methylprednisolone as internal standard. The optimized separation was applied to the analysis of piglet feed samples spiked with steroids. The sample preparation process included solvent extraction using diethyleter and solid phase extraction using silica cartridges. The recoveries were in the range 70-92%. Decision limits and detection capability were in the range 34-198 and 41-249 microg/kg, respectively. Repeatability was also evaluated.     

Synthesis, characterization, and catalytic properties of new half-sandwich zirconium(IV) complexes

A number of new half-sandwich zirconium(IV) complexes bearing N,N-dimethylaniline-amido ligands with the general formula Cp*ZrCl(2)[ortho-(RNCH(2))(Me(2)N)C(6)H(4)] [R = 2,6-Me(2)C(6)H(3) (1), 2,6-(i)Pr(2)C(6)H(3) (2), (i)Pr (3), (t)Bu (4)] were synthesized by the reaction of Cp*ZrCl(3) with the corresponding ortho-(Me(2)N)C(6)H(4)CH(2)NRLi. All new zirconium complexes were characterized by (1)H and (13)C NMR, elemental analyses and single crystal X-ray diffraction analysis. The molecular structural analysis reveals that the NMe(2) group does not coordinate to the zirconium atom in all cases. Complexes 1-4 all have a pseudo-tetrahedral coordination environment in their solid state structures and adopt a three-legged piano stool geometry for the zirconium atoms with the amide N atom and the two Cl atoms being the three legs and the Cp* ring being the seat. Variable-temperature (1)H NMR experiments for all complexes 1-4 were performed to investigate the possible intramolecular interaction between the N atom in the NMe(2) group and the central zirconium atom in solution. Upon activation with Al(i)Bu(3) and Ph(3)CB(C(6)F(5))(4), complexes 1-4 all exhibit moderate to good catalytic activity for ethylene polymerization and copolymerization with 1-hexene, producing linear polyethylene or poly(ethylene-co-1-hexene) with moderate molecular weight and reasonable 1-hexene incorporation.