Waring type congruences involving factorials modulo a prime

In this paper we prove that any residue class λ modulo a large prime number p can be represented in the form

The sum-product estimate for large subsets of prime fields

Let be the field of prime order It is known that for any integer one can construct a subset with such that

One of the results of the present paper implies that if with then

     

Solid-phase extraction followed by dispersive liquid–liquid microextraction for the sensitive determination of selected fungicides in wine

A novel approach for the determination of seven fungicides (metalaxyl-M, penconazole, folpet, diniconazole, propiconazole, difenoconazole and azoxystrobin) in wine samples is presented. Analytes were extracted from the matrix and transferred to a small volume of a high density, water insoluble solvent using solid-phase extraction (SPE) followed by dispersive liquid–liquid microextraction (DLLME). Variables affecting the performance of both steps were thoroughly investigated (metalaxyl-M was not included in some optimisation studies) and their effects on the selectivity and efficiency of the whole sample preparation process are discussed. Under optimised conditions, 20 mL of wine were first concentrated using a reversed-phase sorbent and then target compounds were eluted with 1 mL of acetone. This extract was mixed with 0.1 mL of 1,1,1-trichloroethane (CH₃CCl₃) and the blend added to 10 mL of ultrapure water. After centrifugation, an aliquot (1–2 μL) of the settled organic phase was analyzed by gas chromatography (GC) with electron capture (ECD) and mass spectrometry (MS) detection. The method provided enrichment factors (EFs) around 200 times and an improved selectivity in comparison to use of SPE as single sample preparation technique. Moreover, the yield of the global process was similar for red and white wine samples and the achieved limits of quantification (LOQs) (from 30 to 120 ng L⁻¹ and from 40 to 250 ng L⁻¹, for GC–ECD and GC–MS, respectively) were low enough for the determination of target species in commercial wines. Among compounds considered in this work, metalaxyl-M and azoxystrobin were found in several wines at concentrations from 0.8 to 32 ng mL⁻¹.     

Tandem double-intramolecular [4+2]/[3+2] cycloadditions of nitroalkenes. Studies toward a total synthesis of daphnilactone B: piperidine ring construction

[reaction: see text] Two model studies in support of a total synthesis of the complex polycyclic alkaloid daphnilactone B have been completed. The objectives of the models studies were to demonstrate the use of a tandem double-intramolecular [4+2]/[3+2] nitroalkene cycloaddition for the stereocontrolled construction of four of the rings in the core of the natural product. The first model study established the ability to create a pyrrolidine ring corresponding to ring A of daphnilactone B through a modification of the dipolarophile and subsequent functional group manipulations. The second model study required the modification of the dienophile in the [4+2] cycloaddition to accommodate the formation of a piperidine ring (ring B of daphnilactone B). Nitroalkene 26 containing a diene as the dienophile served well in the tandem cycloaddition to afford the nitroso acetal 38a in 77% yield. Subsequent functional group manipulations allowed for the high-yielding conversion to the core of daphnilactone B.     

Simultaneous determination of benzotriazole and benzothiazole derivatives in aqueous matrices by mixed-mode solid-phase extraction followed by liquid chromatography–tandem mass spectrometry

An improved selectivity method for the simultaneous determination of four benzotriazoles (benzotriazole, 4-methylbenzotriazole, 5-methylbenzotriazole, and 5,6-dimethyl-1H-benzotriazole) and six benzothiazoles (benzothiazole, 2-hydroxybenzothiazole, 2-benzothiazolamine, mercaptobenzothiazole, 2-methylbenzothiazole, and 2-methylthiobenzothiazole) in aqueous matrices has been developed. Under optimal conditions, analytes are concentrated using a MAX solid-phase extraction (SPE) cartridge, based on divinylbenzene-N-vinylpyrrolidone functionalized with quaternary amine groups, which allows reversed-phase interactions in combination with ionic exchange. Selected compounds are recovered with methanol–acetone 7:3 (v/v) whereas acidic interferences remained attached to the sorbent, and as determined by liquid chromatography coupled to tandem mass spectrometry (LC-MS/MS), LOQs for surface, urban and industrial wastewater are in the range of 0.002–0.29 ng/mL. Figures of merit of the method revealed good precision (RSD% <12%), linearity (R ² > 0.99) and accuracy (%R = 80–100%) for surface waters and effluents allowing direct external standard quantification. For more complex samples, such as urban and industrial raw wastewater, either the standard addition method or pseudo-external standard calibration using matrix matched standards are recommended. Analysis of different real samples, surface, urban wastewater and, for the first time, metal industry wastewater, reflected concentrations up to 310 ng/mL. The methylbenzotriazole isomers ratio was also determined.     

Thermodynamic Efficiency of Interactions in Self-Organizing Systems

The emergence of global order in complex systems with locally interacting components is most striking at criticality, where small changes in control parameters result in a sudden global reorganization. We study the thermodynamic efficiency of interactions in self-organizing systems, which quantifies the change in the system’s order per unit of work carried out on (or extracted from) the system. We analytically derive the thermodynamic efficiency of interactions for the case of quasi-static variations of control parameters in the exactly solvable Curie–Weiss (fully connected) Ising model, and demonstrate that this quantity diverges at the critical point of a second-order phase transition. This divergence is shown for quasi-static perturbations in both control parameters—the external field and the coupling strength. Our analysis formalizes an intuitive understanding of thermodynamic efficiency across diverse self-organizing dynamics in physical, biological, and social domains.     

Degradation of cellulose nitrates in low-water media

The kinetics of degradation of cellulose nitrates in low-water organic solvents in the presence of various catalysts was examined.