Sign-changing solutions to a partially periodic nonlinear Schrödinger equation in domains with unbounded boundary

We consider the problem

$$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$

where \(\Omega \) is either \(\mathbb {R}^{N}\) or a smooth domain in \(\mathbb {R} ^{N}\) with unbounded boundary, \(N\ge 3,\) \(V_{\infty }>0,\) \(V\in \mathcal {C} ^{0}(\mathbb {R}^{N}),\) \(\inf _{\mathbb {R}^{N}}V>-V_{\infty }\) and \(2<p<\frac{2N}{N-2}\). We assume V is periodic in the first m variables, and decays exponentially to zero in the remaining ones. We also assume that \(\Omega \) is periodic in the first m variables and has bounded complement in the other ones. Then, assuming that \(\Omega \) and V are invariant under some suitable group of symmetries on the last \(N-m\) coordinates of \(\mathbb {R}^{N}\), we establish existence and multiplicity of sign-changing solutions to this problem. We show that, under suitable assumptions, there is a combined effect of the number of periodic variables and the symmetries of the domain on the number of sign-changing solutions to this problem. This number is at least \(m+1\)

     

1,3-Dipolar cycloaddition of 1,2-o-isopropylidene-α-d-xylopentadialdo-1,4-furanosenitrile oxide to some alkenes

The synthesis of five 3-(1,2-O-isopropylidene-α-D-xylofuranosyl)-5-substituted-2-isoxazolines obtained by 1,3-dipolar cycloaddition are described, as well as the intermediate products from the synthetic route. The physical and spectroscopic characterization of these compounds are reported.     

Alternative catalysts for MTBE production

MTBE synthesis has been studied over catalysts having different acidities: zeolites, fluorine-promoted SiO₂−Al₂O₃ and sulfur-promoted ZrO₂. Some catalysts perform similarly to acidic resins and a clear relationship between acidity and activity is not apparent.     

Photoassisted vapor generation in the presence of organic acids for ultrasensitive determination of Se by electrothermal-atomic absorption spectrometry following headspace single-drop microextraction

A method is described for the determination of selenium at the pg/mL level by electrothermal-atomic absorption spectrometry using in situ photogeneration of Se vapors, headspace sequestration onto an aqueous microdrop containing Pd(II) and subsequent injection in a graphite tube. Several organic acids (formic, oxalic, acetic, citric and ethylenediaminetetraacetic) have been tried for photoreduction of Se(IV) into volatile Se compounds under UV irradiation. Experimental variables such as UV irradiation time, organic acid concentration, Pd(II) concentration in the drop, sample and drop volumes, extraction time and pH were fully optimized. Low-molecular weight acids such as formic and acetic provided optimal photogeneration of volatile Se species at a 0.6 mol/L concentration. Citric and ethylenediaminetetraacetic acid allowed to use a concentration as low as 1 mmol/L, but extraction times were longer than for formic and acetic acids. Photogeneration of (CH₃)₂Se from Se(IV) in the presence of acetic acid provided a detection limit of 20 pg/mL, a preconcentration factor of nearly 285 and a precision, expressed as relative standard deviation, of 4%. Analytical performance seemed to depend not only on the photogeneration efficiency obtained with each acid but also on the stability of the vapors in the headspace. The method showed a high freedom from interferences caused by saline matrices, but interferences were observed for transition metals at a relatively low concentration.     

Capillary electrophoresis for the analysis of short-chain organic acids in coffee

A simple and rapid capillary electrophoresis method for low-molecular mass carboxylic acids measurement in coffee has been optimised and validated. Regarding separation conditions, phosphate concentration in the background electrolyte, surfactant type [cetyltrimethylammonium bromide (CTAB), tetradecyltri methylammonium bromide (TTAB) and hexadimethrine bromide (HDB)], percentages of organic modifier and pH were assayed. The best conditions were: 500 mM phosphate buffer at pH 6.25 with CTAB 0.5 mM. The separation was carried out with an uncoated fused-silica capillary (57 cm x 50 microm i.d.) which was operated at -10 kV potential. Detection was performed at 200 nm. In such conditions 17 short-chain organic acids: oxalic, formic, fumaric, mesaconic, succinic, maleic, malic, isocitric, citric, acetic, citraconic, glycolic, propionic, lactic, furanoic, pyroglutamic, quinic acids plus nitrate were separated, identified and measured. Validation parameters of the method allow us to consider it lineal, accurate and precise and, therefore, reliable for its employment in food composition studies or for quality control. Results in coffees with different industrial treatment allow the detection of important differences in the organic acid profile.     

Iridium solutes effect C-H bond activation and C-C bond forming reactions of C6H6-MeOCH2CH2OMe solvent mixtures

The in situ generated [Tp(Me2)Ir(C(6)H(5))(2)] fragment induces both aromatic and aliphatic C-H bond activation reactions, along with C-C bond formation, when heated with benzene and 1,2-dimethoxyethane.     

Diastereoselective synthesis of 1,2-O-isopropylidene-1,6-dioxaspiro[4,4]nonane applying the methodology of generation of radical cations under non-oxidizing conditions

We report the stereoselective synthesis of an optically pure spiroketal via an intramolecular tandem hydrogen abstraction reaction promoted by an alkoxy radical. Expanding the use of alkene radical cation under non-oxidizing conditions in the synthetic scenario.     

Highly efficient biocatalytic resolution of cis- and trans-3-aminoindan-1-ol: syntheses of enantiopure orthogonally protected cis- and trans-indane-1,3-diamine

The efficient chemoenzymatic synthesis of enantiopure 1,3-difunctionalized indane derivatives has been achieved. The corresponding cis and trans N-protected amino alcohols were successfully resolved by acetylation using lipase B, which is a biocatalyst isolated from Candida antarctica. All the possible isomers were obtained in very good chemical yields and ee values (>99 %). The utility of these compounds was subsequently shown by the preparation of orthogonally protected cis- and trans-indane-1,3-diamine using a Mitsunobu reaction. Both enantiomers of the trans isomer and a desymmetrized cis diastereomer were prepared in enantiopure form. Complete inversion of configuration during the Mitsunobu reaction was demonstrated by a combination of NMR techniques and molecular modeling. The utility and versatility of the strategy was also demonstrated by the selective deprotection of each nitrogen atom under mild reaction conditions.     

Recent efforts to increase greenness in chromatography

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Direct oxidation of a heterocyclic nitroso dimer

A cyclic cis-nitroso dimer, 3,4-dibromo-3,5,5-trimethylpyrazoline 1,2-dioxide was oxidized directly to an open chain 1,3-dinitro compound by a peroxyacid but nitrosyl chloride displaced bromine, added, and oxidized the same compound to 3-chloro-4-nitro-3,5,5-trimethylpyrazoline 1,2-dioxide.