Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(\mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}$ and tensors of the form ${T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${\bar{g}}$ , conformal to g, so that ${A_{\bar g}=T}$ , where ${A_{\bar g}}$ is the Schouten Tensor of the metric ${\bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${\bar g}$ is complete on ${\mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${\bar g}$ , conformal to g, such that ${\sigma_2 ({\bar g })}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${\bar{g} = g/{\varphi^2}}$ , such that ${\sigma_2 ({\bar g }) = f_1}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}$ .     

A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modeling sedimentation-consolidation processes

We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definition of entropy solution of the model in the sense of Kru kov and prove convergence of the scheme to the unique entropy solution of the problem, up to satisfaction of one of the boundary conditions.

     

Beijing Lectures on the Grade Restriction Rule

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model.The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rφdland.A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.     

Lipid-Based Nanostructures for the Delivery of Natural Antimicrobials

Encapsulation can be a suitable strategy to protect natural antimicrobial substances against some harsh conditions of processing and storage and to provide efficient formulations for antimicrobial delivery. Lipid-based nanostructures, including liposomes, solid lipid nanoparticles (SLNs), and nanostructured lipid nanocarriers (NLCs), are valuable systems for the delivery and controlled release of natural antimicrobial substances. These nanostructures have been used as carriers for bacteriocins and other antimicrobial peptides, antimicrobial enzymes, essential oils, and antimicrobial phytochemicals. Most studies are conducted with liposomes, although the potential of SLNs and NLCs as antimicrobial nanocarriers is not yet fully established. Some studies reveal that lipid-based formulations can be used for co-encapsulation of natural antimicrobials, improving their potential to control microbial pathogens.     

Green Extraction Processes for Complex Samples from Vegetable Matrices Coupled with On-Line Detection System: A Critical Review

The detection of analytes in complex organic matrices requires a series of analytical steps to obtain a reliable analysis. Sample preparation can be the most time-consuming, prolonged, and error-prone step, reducing the reliability of the investigation. This review aims to discuss the advantages and limitations of extracting bioactive compounds, sample preparation techniques, automation, and coupling with on-line detection. This review also evaluates all publications on this topic through a longitudinal bibliometric analysis, applying statistical and mathematical methods to analyze the trends, perspectives, and hot topics of this research area. Furthermore, state-of-the-art green extraction techniques for complex samples from vegetable matrices coupled with analysis systems are presented. Among the extraction techniques for liquid samples, solid-phase extraction was the most common for combined systems in the scientific literature. In contrast, for on-line extraction systems applied for solid samples, supercritical fluid extraction, ultrasound-assisted extraction, microwave-assisted extraction, and pressurized liquid extraction were the most frequent green extraction techniques.     

Theoretical and electrochemical studies on organometallic symmetrical schiff base complexes of Zn(II), Cu(II), Ni(II) and Co(II)

The electronic communication between two redox centres through a Schiff base complex has been investigated in a series of ethylenediimine-bis(1-ferrocenyl-1,3-butanedionate) complexes of Zn(II) 1, Cu(II) 2, Ni(II) 3 and Co(II) 4. Cyclic voltammetry experiments of 1 and 2 exhibit a unique two-electron reversible oxidation wave, whereas in the case of 3 and 4 two and three one-electron oxidation processes are, respectively, observed. These results suggest some electronic interaction between the iron atoms of the ferrocenyl groups. DFT calculations carried out on model complexes show that for all the studied compounds the removal of the first two electrons corresponds to the oxidation processes of the iron centres in the weakly coupled ferrocenyl termini. The electronic communication between the two iron centres increases on going from 1 to 4. Finally, a re-indexation of the bands observed in the UV-Visible spectra has been carried out using TDDFT calculations.     

Crystalline polysilicate magadiite with intercalated n-alkylmonoamine and some correlations involving thermochemical data

Synthesized hydrated lamellar acidic crystalline magadiite (H₂Si₁₄O₂₉·2H₂O) nanocompound was used as host for intercalation of polar n-alkylmonoamine molecules of the general formula H₃C(CH₂)_nNH₂ (n = 1–6) in aqueous solution. The original interlayer distance (d) of 1500 pm, determined by X-ray powder diffraction patterns, increases after intercalation. The values correlated with the number of aliphatic amine carbon (n_c) atoms: d = [(1312 ± 11) + (21 ± 2)]n_c. The amount of intercalated amines (N_s), decreased as n_c increased: N_s = [(5.82 ± 0.04) − (0.45 ± 0.01)]n_c. The acidic layered nanocompound was calorimetrically titrated with the amines and the thermodynamic data gave exothermic values for all guest molecules, as shown by the correlation: Δ_intH = −[(24.45 ± 0.49) − (1.91 ± 0.10)]n_c and d = [(1576 ± 16) − (10.8 ± 1.0)]Δ_intH. The negative values of the Gibbs energies and the positive entropies also presented the correlations: Δ_intG = −[(22.8 ± 0.2) − (0.2 ± 0.1)]n_c and Δ_intS = [(6 ± 1) + (5 ± 1)]n_c, respectively.     

Studies on quinones. VIII. The application of michael adducts from 2-hydroxy-1,4-naphthoquinones for the preparation of dihydronaphthopyrandiones

Dihydronaphthopyran-5,10-diones 7,13, 16 and their corresponding dihydronaphthopyran-5,6-diones 8, 14,17 have been obtained starting from Michael adducts of 2-hydroxy- and 2-hydroxy-7-methoxy-1,4-naph-thoquinone ( 1, 9 ). An efficient synthesis of hydroxyhydrolapachol ( 4 ) employing 2-hydroxy-3-(3-oxobutyl)-1,4-naphthoquinone ( 5 ) is described.     

Spirolactone syntheses through a rhodium-catalyzed intramolecular C-H insertion reaction: model studies towards the synthesis of syringolides

Model studies towards the total synthesis of syringolides using a rhodium-catalyzed intramolecular C-H insertion reaction as the key step are described. A highly stereospecific synthesis of spirolactones is achieved employing this methodology.